text:
\section{Introduction}

Sentiment Analysis \citep[SA,][]{Pang08opinionmining} deals with the automatic processing of subjective information in natural language texts. Early work on SA focused on conceptually simpler tasks, such as polarity classification at the sentence or document level. With the advances in natural language processing (NLP), more fine-grained and complex tasks have been proposed, such as detecting the entity that expresses an opinionated chunk of text, or the entity that was targeted.
More particularly, \citet{barnes-etal-2021-structured} consider sentiment analysis as a (graph) structured task, and discuss up to five subtasks: (i) sentiment expression extraction, (ii) sentiment target extraction, (iii) sentiment holder extraction, (iv) defining the relationship between these elements, and (v) assigning a polarity label. They discuss that although these tasks have been extensively studied by different authors \citep[\emph{inter alia}]{turney-2002-thumbs,pontiki-etal-2015-semeval,zhang2019end}, they are not addressed all together. They also discuss that such subdivision into subtasks might have a negative impact in the general analysis of the sentence, and that a joint analysis could translate into a holistic approach. To do so, they propose to encapsulate all these tasks in the form of a sentiment graph. Formally, the goal is to find the set of opinion tuples $\{O_1,\ldots,O_i,\ldots,O_n\}$ in a given text, where each opinion $O_i$ is a tuple of the form $(h, t, e, p)$ where $h$ is a holder who expresses a polarity $p$ towards a target $t$ through a sentiment expression $e$, implicitly defining pairwise relationships between elements of the same tuple. We illustrate an example in Figure \ref{fig:sentiment-graph-example}.

\begin{figure}[thbp!]
    \centering
    \small
    \includegraphics[width=\columnwidth]{charts/head-final_labels.png}
    \caption{An example of sentiment graph as defined by \citet{barnes-etal-2021-structured}. The sentence has a holder (`I'), two sentiment expressions (`got' and `at no cost') and one target (`an upgrade to Executive suite')}
    \label{fig:sentiment-graph-example}
\end{figure}

More particularly, for the SemEval-2022 Task 10 \cite{barnes-etal-2022-semeval}, the organizers proposed both a monolingual\footnote{We use the term monolingual as it was the term used by the organizers, but this setup allowed the use of any resource, including resources in different languages.} and a cross-lingual (zero-shot) setup. They considered 5 languages (and 7 treebanks): English, Spanish, Catalan, Basque, and Norwegian. For the zero-shot setup Basque, Catalan, and Spanish were the target languages.

\paragraph{Our approach} is based on the idea of viewing this task as semantic dependency parsing \cite{oepen-etal-2015-semeval}, since both tasks are structurally similar even if the graphs have different meaning. More specifically, we rely on a bi-affine graph-based parser \cite{dozat-manning-2018-simpler} and different large pre-trained language models (LM), such as BERT \cite{devlin-etal-2019-bert}, RoBERTa \cite{liu2019roberta} or XLM-R \cite{conneau-etal-2020-unsupervised}. For the monolingual setup we train a semantic parsing model on single and merged treebanks, and compare the performance using different LMs. For the cross-lingual setup, we first do a word-level translation of the datasets in a different language than the target treebank, and then proceed similarly to the monolingual setup. Overall, the approach relies on off-the-shelf tools already available, but traditionally used for other purposes. We here re-purpose them for their use for sentiment analysis as graph-based parsing.

\section{The role of parsing in SA}

Parsing has been used in the past for SA, with different motivations, such as integrating syntactic knowledge as a component of the model's architecture, or producing structured sentiment outputs.

\paragraph{Polarity classification.} Since early times, authors have studied the importance of language structure to deal with relevant linguistic phenomena for polarity classification, first focusing on simpler strategies such as the use of n-grams or lexical rules \cite{pang-etal-2002-thumbs,taboada2011lexicon}. Later on, more complex syntactic structures were incorporated as well, both for rule-based and machine learning approaches.

For instance, for the rule-based paradigm, \citet{poria2014sentic} used dependency relations for concept-level sentiment analysis, so sentiment could flow from one concept to another to better contextualize polarity. \citet{VilAloGomNLE2015,vilares2017universal} proposed a model to compute the sentiment of sentences that was driven by syntax-based rules to deal with specific relevant phenomena in SA, and that could be easily re-purposed for any language for which a dependency parser was available. \citet{kanayama-iwamoto-2020-universal} built on top of \citeauthor{vilares2017universal}'s idea, and proposed a multilingual syntax-based system that achieved a high precision for 17 languages.

From the machine learning perspective, \citet{joshi2009generalizing,vilares2015usefulness} used dependency triplets to train data-driven (pre-neural) models and obtain slight improvements over purely lexical approaches. \citet{socher-etal-2013-recursive} collected sentiment labels for phrases and sentences that were previously automatically represented as constituent (sub)trees, to then train a compositional model that used a recursive neural network. This work has some relevant resemblances with \citet{barnes-etal-2021-structured}'s proposal for structured sentiment analysis. \citeauthor{socher-etal-2013-recursive} were among the first to provide tree-shaped annotated sentiment data (in this case just for polarity classification), while most of previous work had focused on using tree knowledge as external information to the models, but with sentiment annotations only associated with plain texts. This publicly available data later encouraged many authors to design models that could exploit tree-shaped annotated data to obtain better performing models \cite[\emph{inter alia}]{tai-etal-2015-improved,zhang-zhang-2019-tree}.

\paragraph{Aspect-based sentiment analysis (ABSA).} ABSA is a task that is particularly suitable for the integration of syntactic information, since its main goal is to associate sentiment with specific entities and aspects that occur in the sentence \cite{pontiki-etal-2015-semeval}. Related to this, \citet{popescu-etzioni-2005-extracting} already used dependency trees to constrain an unsupervised sentiment analysis system that extracted a set of product features and their sentiment, given a particular item.
More recently, with the wide adoption of neural networks in NLP, different authors have integrated syntactic knowledge and syntactic structures in different network architectures, such as long short-term memory networks \cite[LSTMs,][]{tang-etal-2016-effective}, recursive neural networks \cite{nguyen-shirai-2015-phrasernn}, convolutional networks \cite{xue-li-2018-aspect}, and graph attention networks  \cite{huang-etal-2020-syntax,sun-etal-2019-aspect}.\\

\noindent Overall, it is clear that parsing has had a high relevance in SA. Yet, the novelty of the shared task is in using graphs to represent richer annotations. This makes it possible to use parsing algorithms as sentiment models, i.e. not just to use them as a component of the model architecture, but as the model responsible of producing the whole sentiment structure of the chunk of text. Also, this is especially relevant in the era of large neural models, where the utility of parsers for downstream tasks is sometimes questioned, with some studies questioning its need in the presence of pretrained models that implicitly learn syntax \cite{tenney-etal-2019-bert,glavas-vulic-2021-supervised,dai-etal-2021-syntax} while others still achieve extra accuracy from their use in conjuntion with such models \cite{sachan-etal-2021-syntax,xu-etal-2021-syntax,li-etal-2021-improving-bert,Zhang2020syntax}. In any case, tasks like this one show that graph structures can be also useful to re-purpose traditional tasks such as SA, while taking advantage of research that the NLP community has done on parsing algorithms for decades.

\section{Brief overview of the shared task}

The goal of the task is to produce graph structures that reflect the sentiment of a sentence, as we showed in Figure \ref{fig:sentiment-graph-example}. More particularly, the organizers released 7 treebanks in 5 different languages: OpeNER \citep[][English and Spanish]{Agerri2013}, MPQA \cite[][English]{Wiebe2005b}, Darmstadt\_unis \citep[][English]{toprak-etal-2010-sentence}, MultiBooked \cite[][Basque and Catalan]{barnes-etal-2018-multibooked}, and NoReC\_fine \cite[][Norwegian]{ovrelid-etal-2020-fine}.\footnote{For more detailed information see \url{https://github.com/jerbarnes/semeval22_structured_sentiment}} Table \ref{tab:detalles-datasets} details the main statistics for the datasets.

\begin{table}[thbp!]
    \centering
    \tiny
    \begin{tabular}{l|ccccc}
        \textbf{Dataset} & \textbf{Language} & \textbf{\# sents} & \textbf{\# holders} & \textbf{\# targets} & \textbf{\# expr.} \\
        \hline
        NoReC\_fine & Norwegian & 11437 & 1128 & 8923 & 11115 \\
        MultiBooked & Basque & 1521 & 296 & 1775 & 2328 \\
        MultiBooked & Catalan & 1678 & 235 & 2336 & 2756 \\
        OpeNER & Spanish & 2057 & 255 & 3980 & 4388 \\
        OpeNER & English & 2494 & 413 & 3850 & 4150 \\
        MPQA & English & 10048 & 2279 & 2452 & 2814 \\
        Darmstadt\_unis & English & 2803 & 86 & 1119 & 1119 \\
        \hline
    \end{tabular}
    \caption{General statistics of the treebanks used in the shared task.}
    \label{tab:detalles-datasets}
\end{table}

\noindent{The sentiment of a sentence is composed of all the opinions, \textit{O\textsubscript{i}}, that make it up. Each opinion can have up to four elements: a holder (\textit{h}) who expresses a polarity (\textit{p}) towards a target (\textit{t}) through a sentiment expression (\textit{e}). These four elements implicitly define the pairwise relationships between the elements of a tuple.} The previous example, Figure \ref{fig:sentiment-graph-example}, shows a sentence with two sentiment expressions (\textit{got} and \textit{at no cost}) that express the polarity (\textit{Positive}) of the sentiment that a holder (\textit{I}) has towards one target (\textit{an upgrade to Executive suite}).

\paragraph{Preprocessing} The organizers of the shared task proposed two possible ways to address the task: as a sequence labeling or as graph-based parsing problem. As mentioned above, we opted for the latter. We use the scripts available in the official repository to transform the JSON files to the CoNLL-U based format and \emph{vice versa}, and we applied the needed changes to make it compatible with \texttt{supar} (see \S \ref{section-model}).\footnote{\url{https://github.com/MinionAttack/conllu-conll-tool}}
Under the graph-based paradigm, the problem is approached as a bilexical dependency graph prediction task, with some assumptions. To convert the data, the organizers suggest two possible conversions, namely \textit{head-first} and \textit{head-final}. In \textit{head-first}, it is assumed that the first token of the sentiment expression is a root node, and that the first token of each holder or target spans is the head node of such span, while the other ones are dependents. Meanwhile, in \textit{head-final}, the final token of the holder and target spans is set as the head of the span, and the final token of the sentiment expression as a root node (Figure \ref{fig:sentiment-graph-example} is a head-final example). In this work, we have chosen \textit{head-final}, which is the default option for the shared task and also delivered better results than \textit{head-first} in the experiments carried out by \citet{barnes-etal-2021-structured} (see Table 3 in that paper).

\paragraph{Subtasks} More in detail, the challenge is divided into two subtasks:
\begin{enumerate}
    \item Monolingual setup: When training and development data is available for the same treebank/language, i.e. the goal is to train one model per treebank. It was allowed to use extra resources or tools that could boost performance, even from different languages.
    \item Cross-lingual, zero-shot setup: It is assumed that there is no gold training data in the language of the target treebank. The organizers specified that it is possible to use treebanks in other languages, translation tools, and any other resources that do not include sentiment annotations in the target language.
\end{enumerate}

\paragraph{Metrics} Each subtask is evaluated independently, and the ranking metric was \emph{sentiment graph F1} \cite{barnes-etal-2021-structured}, where true positives are exact matches at the graph level, weighting the overlap between the predicted and gold spans for each element, and averaged across all three spans. To compute precision, it
weights the number of correctly predicted tokens divided by the total number of predicted tokens, while for recall it weights the number of correctly predicted tokens divided by the total number of gold tokens. Also, as mentioned earlier, it is possible to have tuples with empty holders and targets.

\section{Our model}\label{section-model}

We rely on the \citet{dozat-manning-2018-simpler} parser, a widely used state-of-the-art model both for syntactic and semantic dependency parsing. Inspired in previous graph-based parsers \cite{mcdonald-etal-2005-non,kiperwasser-goldberg-2016-simple}, the parser first computes contextualized representations for each word using bidirectional LSTMs \citep[biLSTMs;][]{hochreiter1997long,schuster1997bidirectional}. After that, the model computes a \emph{head} and a \emph{dependent} representation for each term, to establish through a bi-affine attention whether an edge exists between each pair of tokens, and if so, what is the semantic relationship between them.
In particular, in this paper we follow the implementation used in the \texttt{supar}\footnote{\url{https://github.com/yzhangcs/parser}} package, as it has been widely adopted by the community and it is available for other flavors of parsing as well, such as constituent or dependency parsing. We preferred this implementation over the graph-based baseline provided in the SemEval repository, since early experiments showed a superior performance, and it also offered a simpler integration of large language models. We left the parser hyperparameters, except the learning rate, at their default value.

\paragraph{Pre-trained language models} For each language we looked for avaliable monolingual and multilingual pre-trained LMs at \url{https://huggingface.co/}. Specifically, for each language, we included:

\begin{itemize}
    \item \texttt{Basque}: berteus-base-cased, RoBasquERTa.
    \item \texttt{Catalan}: julibert, roberta-base-ca, calbert-base-uncased.
    \item \texttt{English}: bert-base-cased, bert-base-uncased, bert-large-cased, bert-large-uncased, roberta-base, roberta-large, albert-base-v2, albert-large-v2, xlnet-base-cased, xlnet-large-cased, electra-base-discriminator, electra-large-discriminator, electra-base-generator, electra-large-generator.
    \item \texttt{Norwegian}: norbert, nb-bert-base, nb-bert-large, electra-base-norwegian-uncased-discriminator.
    \item \texttt{Spanish}: bio-bert-base-spanish-wwm-uncased, bert-base-spanish-wwm-cased, roberta-base-bne, roberta-large-bne, selectra\_medium, zeroshot\_selectra\_medium.
\end{itemize}

\noindent With respect to the cross-lingual LMs, we considered: xlm-roberta-base and xlm-roberta-large.

\subsection{Monolingual models}

For this task, we use: (i) pre-trained language models, (ii) \texttt{supar}, and (iii) the official training  and development files to build our models. Also note that we train end-to-end models, using words as the only input (later tokenized into subword pieces by the language models), but ignoring the part-of-speech tags and syntactic information provided in the sentiment treebanks. We did not use part-of-speech tags (or other morphosyntactic annotations) since these are not used in \texttt{supar} together with BERT encoders, and using them would require to adapt the code, which was exactly what we tried to avoid in this work.

\paragraph{Training procedure} We fine-tuned parsing models considering for each treebank the proposed LMs, and combining them with \texttt{supar}. Since training is time-consuming, many model configurations are proposed, and the performance of \texttt{supar} is stable independently of the seed, we decided to train a single model per LM. Specifically, all models have been trained with the default seed used by \texttt{supar}, which is 1. The only parameter that was modified was the learning rate (\textit{l\textsubscript{r}}), as we observed that for some models (specially the larger language models) the fine-tuning process did not converge. We started with \begin{math}5\cdot 10^{-5}\end{math}, and did a small grid search down to $1\cdot 10^{-6}$, where if a model still did not converge it was discarded.\footnote{For both monolingual and cross-lingual subtasks, all the selected models used a \textit{l\textsubscript{r}} of \begin{math}5\cdot 10^{-5}\end{math}. The only exception is Norwegian in the monolingual subtask, for which we used \begin{math}5\cdot 10^{-6}\end{math}.}
Additionally, to train the parsing models, we considered three strategies:
\begin{enumerate}
    \item \label{mono-approach-1} Single monolingual training and development files: We train each model on a single treebank and validate its performance in the corresponding dev set, i.e., the standard monolingual training and development methodology.
\end{enumerate}

For the best model obtained for each treebank according to strategy \ref{mono-approach-1}, we explored a couple of harmonized training strategies (harmonized in the sense that different treebanks follow the same annotation guidelines):

\begin{enumerate}
\setcounter{enumi}{1}
    \item \label{mono-approach-2} Merged training and development files from different treebanks: We considered to merge all the available training files and all the available development files, treating them as a single dataset. Thus, we trained a single model that could predict all test files, but with the disadvantage that model selection is based on multilingual performance, which could hurt the performance in this setup.
    \item \label{mono-approach-3} Merged training files, single development file: Similar to \ref{mono-approach-2}, but merging only the training files. For the development phase, we proceeded as in \ref{mono-approach-1} and used each dataset's dev file for model selection. The idea was to have training data that can benefit from multilingual information, but that still considers a monolingual file for a language-dependent model selection, i.e., given $n$ treebanks, we still need to train $n$ models, one per treebank.
\end{enumerate}

We detail the experimental results for the training/development phase in \S \ref{section-results}.

\subsection{Cross-lingual (zero-shot) models}

In this setup, we rely on two main components: (i) available translation systems to perform word-level translations from source language to target language treebanks, and (ii) both monolingual and cross-lingual language models. Our goal with (i) is to obtain noisy, unlikely-grammatical data, but that still can provide sentiment annotations for a given target language, exploring the viability of this approach. Regarding the learning rate, we used $5\cdot 10^{-5}$ in all cases.

\paragraph{Auxiliary translation models} From the CoNLLU converted files\footnote{\url{https://github.com/MinionAttack/corpus-translator}}, we translated the sentences at the word level using the Helsinki-NLP translation models\footnote{\url{https://huggingface.co/Helsinki-NLP}} \cite{tiedemann-thottingal-2020-opus} available at \texttt{huggingface}. Table \ref{tab:crosslingual-translation-languages} lists the language pairs for which we could obtain translated versions for the cross-lingual setup.

\begin{table}[thbp!]
    \centering
    \scriptsize
    \begin{tabular}{l|cccc}
        \textbf{Dataset} & \textbf{Language} & \textbf{Basque} & \textbf{Catalan} & \textbf{Spanish} \\
        \hline
        NoReC\_fine & Norwegian &  &  & \checkmark \\
        MultiBooked & Basque &  &  & \checkmark \\
        MultiBooked & Catalan &  &  & \checkmark \\
        OpeNER & Spanish & \checkmark & \checkmark &  \\
        OpeNER & English & \checkmark & \checkmark & \checkmark \\
        MPQA & English & \checkmark & \checkmark & \checkmark \\
        Darmstadt\_unis & English & \checkmark & \checkmark & \checkmark \\
        \hline
    \end{tabular}
    \caption{Treebanks and the languages to which they were translated for the cross-lingual experiments.}
    \label{tab:crosslingual-translation-languages}
\end{table}

Then, to train the models we proceeded similarly to strategy \ref{mono-approach-2} used in the monolingual setup: we combined the translated training and validation files coming from treebanks in other languages, and used the micro-averaged F1-score on the translated development set for model selection.

\paragraph{Post-evaluation (and better) baseline} After the deadline to submit proposals, we also tested a baseline consisting on training, using an XLM-RoBERTa LM as the base component, a cross-lingual model that uses all the English datasets (without any kind of translation) as the source data. We discuss these results as well in \S \ref{section-crosslingual-task}.

\section{Results}\label{section-results}

Here, we detail and discuss the results that we got for both subtasks (see \S \ref{section-monolingual-task} and \ref{section-crosslingual-task}) on: (i) the official development sets, and (ii) the official test sets of the shared tasks.

\subsection{Monolingual setup}
\label{section-monolingual-task}

Tables \ref{tab:monolingual-dev-scores-english-before} and \ref{tab:monolingual-dev-scores-non-english-before} show the results for the development phase on the English and non-English datasets, respectively, including different LMs and training setups.

\begin{table}[hbtp!]
    \centering
    \scriptsize
    \begin{tabular}{c|c|c|c}
        \textbf{Corpus} & \textbf{Model} & \textbf{Strategy} & \textbf{F1} \\
        \hline
        OpeNER\_en & \makecell{xlm-roberta-large\\electra-base-discriminator\\xlm-roberta-large\\xlm-roberta-base\\roberta-large\\xlnet-base-cased\\xlm-roberta-base\\xlm-roberta-base\\roberta-base\\electra-large-discriminator\\bert-large-uncased\\electra-large-generator\\xlm-roberta-large\\bert-base-uncased\\bert-large-cased\\xlnet-large-cased\\bert-base-cased\\electra-base-generator\\albert-base-v2\\albert-large-v2} & \makecell{3\\1\\1\\2\\1\\1\\3\\1\\1\\1\\1\\1\\2\\1\\1\\1\\1\\1\\1\\1} & \makecell{0.714\\0.710\\0.707\\0.686\\0.683\\0.681\\0.679\\0.673\\0.663\\0.662\\0.660\\0.652\\0.643\\0.640\\0.640\\0.639\\0.612\\0.612\\0.590\\0.297} \\
        \hline
        MPQA & \makecell{roberta-base\\roberta-large\\electra-base-discriminator\\xlm-roberta-large\\xlnet-base-cased\\xlm-roberta-large\\bert-base-cased\\xlm-roberta-base\\electra-large-generator\\xlm-roberta-large\\bert-large-uncased\\bert-base-uncased\\xlm-roberta-base\\xlm-roberta-base\\bert-large-cased\\electra-base-generator\\albert-base-v2\\xlnet-large-cased} & \makecell{1\\1\\1\\1\\1\\3\\1\\2\\1\\2\\1\\1\\3\\1\\1\\1\\1\\1} & \makecell{0.374\\0.365\\0.351\\0.346\\0.338\\0.327\\0.306\\0.303\\0.301\\0.298\\0.297\\0.294\\0.285\\0.277\\0.269\\0.253\\0.236\\0.209} \\
        \hline
        Darmstadt\_unis & \makecell{xlm-roberta-large\\xlm-roberta-large\\electra-base-discriminator\\xlm-roberta-base\\xlm-roberta-large\\roberta-base\\xlm-roberta-base\\xlnet-base-cased\\xlnet-large-cased\\roberta-large\\electra-large-generator\\electra-large-discriminator\\xlm-roberta-base\\bert-large-uncased\\bert-base-uncased\\bert-large-cased\\electra-base-generator\\albert-base-v2\\bert-base-cased} & \makecell{3\\1\\1\\3\\2\\1\\1\\1\\1\\1\\1\\1\\2\\1\\1\\1\\1\\1\\1} & \makecell{0.329\\0.309\\0.306\\0.306\\0.301\\0.301\\0.276\\0.276\\0.269\\0.268\\0.267\\0.264\\0.262\\0.257\\0.251\\0.237\\0.229\\0.217\\0.212} \\
        \hline
    \end{tabular}
    \caption{Scores on the development set for the English treebanks and the monolingual setup. Models trained on the training data before its updated version.}
    \label{tab:monolingual-dev-scores-english-before}
\end{table}

\begin{table}[hbpt!]
    \centering
    \tiny
    \begin{tabular}{c|c|c|c}
        \textbf{Corpus} & \textbf{Model} & \textbf{Strategy} & \textbf{F1} \\
        \hline
        NoReC\_fine & \makecell{nb-bert-large\\nb-bert-base\\xlm-roberta-large\\xlm-roberta-large\\xlm-roberta-large\\xlm-roberta-base\\xlm-roberta-base\\xlm-roberta-base\\electra-base-norwegian\\-uncased-discriminator\\norbert} & \makecell{1\\1\\1\\3\\2\\3\\2\\1\\1\\\\1} & \makecell{0.479\\0.459\\0.450\\0.439\\0.427\\0.414\\0.411\\0.401\\0.382\\\\0.298} \\
        \hline
        MultiBooked\_eu & \makecell{xlm-roberta-large\\xlm-roberta-large\\xlm-roberta-base\\berteus-base-cased\\xlm-roberta-base\\xlm-roberta-base\\xlm-roberta-large\\RoBasquERTa} & \makecell{3\\2\\3\\1\\2\\1\\1\\1} & \makecell{0.662\\0.623\\0.613\\0.602\\0.597\\0.571\\0.569\\0.496} \\
        \hline
        MultiBooked\_ca & \makecell{xlm-roberta-base\\xlm-roberta-large\\xlm-roberta-large\\xlm-roberta-large\\xlm-roberta-base\\roberta-base-ca\\xlm-roberta-base\\julibert\\calbert-base-uncased} & \makecell{1\\1\\2\\3\\2\\1\\3\\1\\1} & \makecell{0.694\\0.683\\0.679\\0.679\\0.674\\0.672\\0.653\\0.590\\0.579} \\
        \hline
        OpeNER\_es & \makecell{xlm-roberta-large\\xlm-roberta-base\\xlm-roberta-base\\xlm-roberta-large\\xlm-roberta-base\\xlm-roberta-large\\bert-base-spanish-wwm-cased\\selectra\_medium\\roberta-base-bne\\zeroshot\_selectra\_medium\\roberta-large-bne\\bio-bert-base-spanish-wwm-uncased} & \makecell{3\\2\\3\\2\\1\\1\\1\\1\\1\\1\\1\\1} & \makecell{0.666\\0.662\\0.657\\0.639\\0.635\\0.635\\0.630\\0.622\\0.616\\0.610\\0.605\\0,457} \\
        \hline
    \end{tabular}
    \caption{Scores on the development set for the non-English treebanks and the monolingual setup. Models trained on the training data before its updated version.}
    \label{tab:monolingual-dev-scores-non-english-before}
\end{table}

With respect to the results on the English treebanks, an interesting trend is that despite being the monolingual setup, using cross-lingual language models, and in particular XLM-RoBERTa, performed surprisingly well. Combined with the training strategy \ref{mono-approach-3} (merged training sets, single development set), such models obtained the best results for 2 out of 3 English corpora (OpeNER and Darmstadt), while they still ranked well in the other dataset (MPQA). Across monolingual LMs, we also observe trends: electra-base-discriminator and (both base and large) RoBERTa models obtain overall the best results. On the other hand, we did not obtain equally robust results with ALBERT, and to a lesser extent, with BERT architectures. This is not totally surprising, since among the tested LMs, BERT is among the oldest ones, and ALBERT is a lite BERT, so some computational power is lost and it is understandable that this translates into some performance loss too, compared to larger LMs.

With respect to the experiments on the non-English datasets, we observe certain similarities, although the number of available models is much smaller than in the English cases. Again, XLM-RoBERTa overall obtains the best results. The only exception is the Norwegian dataset, where we obtained the best results with a BERT architecture.

Yet, a more thoughtful discussion would be needed to determine if some architectures truly behave better than others. Note that all these LMs are usually pre-trained using different and heterogeneous text sources, and specially for the less-resourced languages, some constraints are usually imposed during training. For instance, it is hard to conclude that berteus-base-cased (BERT) \cite{agerri-etal-2020-give} is worse than XLM-RoBERTa \cite{conneau-etal-2020-unsupervised}, since the amount of resources to train the former was more constrained.

Finally, a few days before the submission deadline, the training files of some treebanks were slightly updated by the organizers, due to minor bugs in the segmentation process that corrupted some sentences.  As we did not have time to rerun all models and update the results, we chose to re-train only the model that obtained the best performance on the previous version of the treebanks. Therefore, all the outputs submitted for the test sets correspond to models trained on the updated, uncorrupted files. In Table \ref{tab:monolingual-dev-scores-after} we compare the performance of the models trained on the updated and deprecated versions of the training files. Overall, we observed relatively small, but non-negligible differences, usually obtaining a better performance with the updated version of the treebank.

\begin{table}[thbp!]
    \centering
    \scriptsize
    \begin{tabular}{c|c|c|c|c}
        \textbf{Corpus} & \textbf{Model} & \textbf{Strategy} & \textbf{Old F1} & \textbf{New F1} \\
        \hline
        NoReC\_fine & nb-bert-large & 1 & 0.479 & 0.492 \\
        MultiBooked\_eu & xlm-roberta-large & 3 & 0.662 & 0.648 \\
        MultiBooked\_ca & xlm-roberta-base & 1 & 0.694 & 0.699 \\
        OpeNER\_es & xlm-roberta-large & 3 & 0.666 & 0.709 \\
        OpeNER\_en & xlm-roberta-large & 3 & 0.714 & 0.716 \\
        MPQA & roberta-base & 1 & 0.374 & 0.374 \\
        Darmstadt\_unis & xlm-roberta-large & 3 & 0.329 & 0.357 \\
        \hline
    \end{tabular}
    \caption{Scores on the development set for the models trained on the corrupted and uncorrupted versions of the training files, on the monolingual setup. For each treebank, we only did the comparison for the best performing model, based on the performance on the corrupted version.}
    \label{tab:monolingual-dev-scores-after}
\end{table}

\paragraph{Official results on the test sets} In Table \ref{tab:model-test-scores-monolingual} we show the performance on the test sets of our submitted models, i.e. those that achieved the highest score in the corresponding development phase. The performance is stable across different test sets, obtaining slightly better results for Iberian languages. For a detailed comparison against the rest of participants, we refer the users to Appendix \ref{tab:monolingual-leaderboard} and the official shared task paper \cite{barnes-etal-2022-semeval}.

\begin{table}[t]
    \centering
    \scriptsize
    \begin{tabular}{c|c|c|c}
        \textbf{Dataset} & \textbf{Model} & \textbf{Strategy} & \textbf{Score} \\
    \hline
        NoReC\_fine & nb-bert-large & 1 & 0.462\textsubscript{(10)} \\
        MultiBooked\_eu & xlm-roberta-large & 2 & 0.680\textsubscript{(7)} \\
        MultiBooked\_ca & xlm-roberta-base & 1 & 0.653\textsubscript{(8)} \\
        OpeNER\_es & xlm-roberta-large & 3 & 0.692\textsubscript{(6)} \\
        OpeNER\_en & xlm-roberta-large & 3 & 0.698\textsubscript{(9)} \\
        MPQA & roberta-base & 1 & 0.349\textsubscript{(10)} \\
        Darmstadt\_unis & xlm-roberta-large & 3 & 0.414\textsubscript{(8)} \\
        \hline
    \end{tabular}
    \caption{Scores of our models, for the monolingual subtask, on each test set. Our ranking on the shared task for each test set is indicated as a subscript.}
    \label{tab:model-test-scores-monolingual}
\end{table}

The datasets of the shared task belong to different domains: OpeNER and MultiBooked deal with hotel reviews, NoReC with professional reviews in multiple domains, Darmstadt\_unis (the dataset for which we obtain the second lowest scores) contains English online university reviews, and MPQA (the dataset for which we obtain the lowest scores) is about news articles annotated with opinions and other private states
(i.e., beliefs, emotions, sentiments, speculations, \dots).
For the two lowest-scoring datasets, they have in common that they mostly contain single-opinion sentences, whereas the other datasets tend to have more variety in the number of opinions and their distribution. For instance, $\sim$85\% and $\sim$74\%  of the training sentences of the Darmstadt\_unis and MPQA datasets have only one opinion, while the next most `single-opinion' dataset is multibooked\_eu with only $\sim$53\% of the sentences. However, we need to perform more detailed analysis as future work to extract more robust conclusions.

\subsection{Cross-lingual setup}\label{section-crosslingual-task}

Table \ref{tab:crosslingual-dev-scores-before} shows the results for the development phase for the three target languages and their datasets. Again, XLM-RoBERTa models obtain overall the best performance, although in this case it is less surprising since cross-lingual LMs are expected to suit well this kind of challenges.
Similar to the case of the monolingual setup, we decided to retrain the best-performing model with the updated versions of the training files. In Table \ref{tab:crosslingual-dev-scores-after}, we show the comparison between the corrupted and uncorrupted versions of the datasets, which contrarily to the monolingual setup, often turned out into worse performing models.

\begin{table}[t]
    \centering
    \small
    \begin{tabular}{c|c|c}
        \textbf{Corpus} & \textbf{Model} & \textbf{F1} \\
        \hline
        Basque & \makecell{xlm-roberta-base\\berteus-base-cased\\RoBasquERTa} & \makecell{0.434\\0.416\\0.323} \\
        \hline
        Catalan & \makecell{roberta-base-ca\\xlm-roberta-base\\julibert\\calbert-base-uncased} & \makecell{0.564\\0.519\\0.486\\0.385} \\
        \hline
        Spanish & \makecell{xlm-roberta-base\\xlm-roberta-large\\bert-base-spanish-wwm-cased\\zeroshot\_selectra\_medium\\selectra\_medium\\roberta-base-bne\\roberta-large-bne\\bio-bert-base-spanish-wwm-uncased} & \makecell{0.605\\0.593\\0.583\\0.555\\0.536\\0.515\\0.438\\0.386} \\
        \hline
    \end{tabular}
    \caption{Scores on the development set for the \emph{translated} English treebanks and the cross-lingual setup. Models trained on the training data before its updated version.}
    \label{tab:crosslingual-dev-scores-before}
\end{table}

\begin{table}[t]
    \centering
    \small
    \begin{tabular}{l|c|c|c}
        \textbf{Language} & \textbf{Model} & \textbf{Old F1} & \textbf{New F1} \\
        \hline
        Basque & \makecell{berteus-base-cased\\xlm-roberta-base} & \makecell{0.416\\0.434} & \makecell{0.424\\0.416} \\
        \hline
        Catalan & roberta-base-ca & 0.564 & 0.572 \\
        \hline
        Spanish & \makecell{xlm-roberta-large\\xlm-roberta-base} & \makecell{0.593\\0.605} & \makecell{0.570\\0.569} \\
        \hline
    \end{tabular}
    \caption{Scores on the development set for the models trained on the corrupted and uncorrupted versions of the \emph{translated} training files, on the cross-lingual setup. For each treebank, we only did the comparison for the best performing model, based on the performance on the corrupted version.}
    \label{tab:crosslingual-dev-scores-after}
\end{table}

Finally, Table \ref{tab:crosslingual-dev-scores-from-english} shows the scores for the post-evaluation baseline (model trained on the English datasets with XLM-RoBERTa) on the dev set.
Very interestingly, the results show that this baseline outperformed our word-level translation approaches. We need more analysis to understand why this happens, but we hypothesize that the larger amount of English texts XLM-RoBERTa was pre-trained on could be playing an important role.

\begin{table}[thbp!]
    \centering
    \small
    \begin{tabular}{l|c|c}
        \textbf{Language} & \textbf{Model} & \textbf{Score} \\
        \hline
         Basque & \makecell{xlm-roberta-base\\xlm-roberta-large} & \makecell{0.678\\0.677} \\
        \hline
         Catalan & \makecell{xlm-roberta-base\\xlm-roberta-large} & \makecell{0.598\\0.625} \\
        \hline
         Spanish & \makecell{xlm-roberta-base\\xlm-roberta-large} & \makecell{0.663\\0.638} \\
        \hline
    \end{tabular}
    \caption{Scores on the development set of the trained English models (trained on MPQA, OpeNER\_en and Darmstadt\_unis corpora, \emph{without} word-level translation) for the cross-lingual subtask.}
    \label{tab:crosslingual-dev-scores-from-english}
\end{table}

\paragraph{Official results on the test sets} Finally, in Table \ref{tab:model-test-scores-crosslingual} we show our results on test sets of the cross-lingual, zero-shot setup, for which we obtain again stable results. Appendix \ref{tab:crosslingual-leaderboard} contains the results for all participants.

\begin{table}[thbp!]
    \centering
    \small
    \begin{tabular}{c|c|c}
        \textbf{Language} & \textbf{Model} & \textbf{Score} \\
        \hline
        \multicolumn{3}{l}{\textbf{\color{gray}\scriptsize Target language: Basque}} \\
        \hline
        Basque & berteus-base-cased & 0.509\textsubscript{(8)} \\
        \hline
        Catalan & roberta-base-ca & 0.554\textsubscript{(8)} \\
        \hline
        Spanish & xlm-roberta-large & 0.570\textsubscript{(7)} \\
        \hline
        \multicolumn{3}{l}{\textbf{\color{gray}\scriptsize Combined English corpora without word-level translation}} \\
        \hline
        Basque & \makecell{xlm-roberta-base\\xlm-roberta-large} & \makecell{0.649\textsubscript{(2)*}\\0.641\textsubscript{(2)*}} \\
        \hline
        Catalan & \makecell{xlm-roberta-base\\xlm-roberta-large} & \makecell{0.647\textsubscript{(2)*}\\0.655\textsubscript{(2)*}} \\
        \hline
        Spanish & \makecell{xlm-roberta-base\\xlm-roberta-large} & \makecell{0.670\textsubscript{(1)*}\\0.638\textsubscript{(2)*}} \\
        \hline
    \end{tabular}
    \caption{Scores of our models, for the cross-lingual subtask, on each test set. Our ranking on the shared task for each test set is indicated as a subscript. * indicates the ranking that we would obtain in the shared task using the post-evaluation baseline models.}
    \label{tab:model-test-scores-crosslingual}
\end{table}

\section{Conclusion}

This paper describes our participation at the Sem\-Eval Shared Task 10 on structured sentiment analysis. We participated both in the monolingual and cross-lingual (zero-shot) setups. We applied a simple, but effective approach, relying on off-the-shelf tools, traditionally used for other purposes, and used them to predict sentiment graphs instead. More particularly, for the monolingual setup, we linked pre-trained language models with bi-affine graph parsing and training over single and multiple treebanks. In the zero-shot setup, we followed a similar approach, but relied on publicly available translation models to obtain training data, by applying a word-level translation of treebanks, to then train models similarly to the monolingual setup.

\section*{Acknowledgements}

This work is supported by a 2020 Leonardo Grant for Researchers and Cultural Creators from the FBBVA,\footnote{FBBVA accepts no responsibility for the opinions, statements and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors.} as well as by the European Research Council (ERC), under the European Union's Horizon 2020 research and innovation programme (FASTPARSE, grant agreement No 714150). The work is also supported by ERDF/MICINN-AEI (SCANNER-UDC, PID2020-113230RB-C21), by Xunta de Galicia (ED431C 2020/11), and by Centro de Investigación de Galicia ''CITIC'' which is funded by Xunta de Galicia, Spain and the European Union (ERDF - Galicia 2014–2020 Program), by grant ED431G 2019/01.

redpajama_set_name: RedPajamaArXiv


text:
Youngsters steal show at Boot Harvest Home
A HOLME VALLEY village local has raked in thousands of pounds at a Harvest Home auction.
The landlords of the Scholes pub, The Boot and Shoe, said their 10-year-old daughter made one of the biggest profits of the day – selling a few flowers for more than £100.
In total, nearly £4,000 was raised and landlady, Debbie Whiteley said she was impressed with the massive sum, to be split between the West Yorkshire Air Ambulance, Scholes junior school and the play school.
She said: "I just want to say thank you to everybody. Times are hard and everybody knows they are, but they are still willing to come out and spend so much money which was raised in just four hours, which I think is phenomenal.
"People paid quite a lot of money for silly little things.
"One little girl called Rebecca Fowler bought a box of biscuits with her own pocket money then went round selling them for 50p and made £27.50.
"And our daughter, Toni-Marie went to Barnsley Market and bought 10 roses for £3 and then got on the microphone and auctioned them off for £105!"

redpajama_set_name: RedPajamaCommonCrawl


Finnish Functional Genomics Centre maintains a panel of instruments, which can be accessed and used independently after user training.
For targeted or small genome Next-Generation Sequencing.
For automated nucleic acid extraction.
For high-throughput nucleic acid QC and other applications.
For nucleic acid quantitation and QC.
For Real-Time PCR in 96-well or 384-well format.


text:
Trainwreck – Film Review
Nick Levine Aug 14, 2015 5:30 pm BST
Subversive comedian Amy Schumer makes a fresh and fearless big screen debut
Subversive stand-up Amy Schumer is about to become a seriously big deal. Highlights from her sketch show keep going viral – if you haven't seen her One Direction parody 'Girl, You Don't Need Makeup', YouTube it immediately – and this September, she's been booked to open for Madonna at Madison Square Garden. But it's Trainwreck, a fresh and fearless comedy film that she wrote for herself to star in, that's set to make the New Yorker a household name.
Schumer plays Amy Townsend, a Manhattan-based magazine journalist who's basically an exaggerated version of herself: frank, funny and fiercely unapologetic about sex. By day, Amy works for an awful fictional lads' mag called S'NUFF where meetings yield pitches for articles like Are You Really Gay – Or Is She Just Boring?. By night, she drinks heavily, smokes weed and frequently cheats on her boring musclebound boyfriend (professional wrestler/actor John Cena). However, when she's sent to interview unexpectedly charming sports surgeon Dr Aaron Conners (Saturday Night Live's Bill Hader), Amy finds herself questioning the dubious message that her adulterous father drummed into her as a child: "Monogamy is unrealistic."
At this point, Trainwreck morphs into a more traditional romantic comedy. But with "gross out" specialist Judd Apatow (Knocked Up) directing, it's still a long way from Sleepless In Seattle. Amy's journey towards monogamy takes a wrong turn when she goes out drinking with S'NUFF's underage intern Donald (We Need To Talk About Kevin's Ezra Miller), and she has trouble reining in her drink and drug intake. Even when Schumer's script gets sentimental about family, it's made abundantly clear that Amy could never settle down in the suburbs like her pregnant younger sister Kim (Brie Larson, 21 Jump Street).
Trainwreck has a few gags that don't land, paticularly an unfunny send-up of a pretentious indie film which squanders a cameo from Daniel Radcliffe. But for most of its two-hour running time, it's a total blast. Schumer, fantastically watchable in her first major film role, is well supported by the likeable Hader and a scene-stealing Tilda Swinton, who disguises herself with a tan and glossy haircut as Amy's awesomely vulgar editor. Professional basketball player LeBron James is also impressive playing Aaron's best friend, a fictionalised and surprisingly stingy version of himself.
Schumer's skyrocketing popularity means that Trainwreck is bound to be analysed to death in the coming weeks: her feminist credentials are already being called into question, and one British broadsheet recently ran a column accusing her – though not the film specifically – of having "a shockingly large blind spot around race". But at its core, this is a successful big screen debut from an undeniable comic talent.
BTS Credit: Getty Images
BTS to perform their 'Love Yourself: Speak Yourself' tour in Saudi Arabia later this year
Florence + The Machine at BST Hyde Park 2019 Credit: Ben Bentley / NME
Florence + The Machine celebrate a decade as a band at emotional BST Hyde Park show
John Carpenter Credit: Michael Tullberg/Getty Images
Horror legend John Carpenter set to write new DC Joker comic


text:
Pascal Stenzel (born 20 March 1996) is a German professional footballer who plays as a right-back for Bundesliga club VfB Stuttgart.

Club career

Borussia Dortmund
Stenzel is a Borussia Dortmund youth exponent, who played for the reserve team. He made his professional debut in the 3. Liga at 2 August 2014 against Holstein Kiel.

SC Freiburg
On 29 January 2016, Stenzel joined SC Freiburg on a loan until 30 June 2017. In April, he signed permanently with Freiburg, agreeing to a contract reportedly running until 2021.

VfB Stuttgart
For the 2019–20 season Stenzel was loaned out to VfB Stuttgart. On 1 July 2020, Stenzel moved permanently to Stuttgart and signed a contract until June 2024.

Career statistics

References

External links

1996 births
Living people
German footballers
Borussia Dortmund players
Borussia Dortmund II players
SC Freiburg players
VfB Stuttgart players
Association football midfielders
Bundesliga players
2. Bundesliga players
3. Liga players
Regionalliga players
Germany youth international footballers
Germany under-21 international footballers
Footballers from North Rhine-Westphalia
People from Herford (district)
Sportspeople from Detmold (region)

redpajama_set_name: RedPajamaWikipedia


text:
\section{INTRODUCTION}
\label{intro}

   One of the instrumental operators of nonperturbative
field-theoretic investigations of hadron structure is the hadron
interpolating field. This operator is used to excite a hadron of
specified quantum numbers from the QCD vacuum.  It has long been
established that there are two independent interpolating fields with
no derivatives having the quantum numbers of spin 1/2 and isospin 1/2.
Both are expected to excite the ground state nucleon from the vacuum.
Various linear combinations of these interpolators are used in
nonperturbative approaches to QCD.  What distinguishes the different
approaches is the manner in which the propagation of quarks in the
QCD vacuum is determined.

   Numerical simulations of the theory via lattice regulation is the
only method for probing deep into the nonperturbative regime of QCD.
Exploitation of the Operator Product Expansion (OPE) in QCD Sum Rules
(QCD-SR) allows the near perturbative regime of QCD to be explored.
Modeling of the QCD vacuum via instanton fluctuations in the Random
Instanton Liquid Model (RILM) has also produced some new insights into
QCD.  While there are formal field theoretic arguments indicating
nucleon properties are independent of the interpolating field, a
demonstration of this in practice is an important test of these
approaches to nonperturbative field theory.

   Some attention has been given previously to alternate nucleon
interpolators on the lattice \cite{leinweber91,bowler84}.  In these
analyses conclusions were limited as the correlation functions
deteriorated too quickly for ground state properties to be determined.
In a previous paper \cite{leinweber94a} it was established that the
properties of the lowest lying state may be extracted from the first
few points of two-point correlation functions with the use of a pole
plus QCD continuum model inspired by QCD Sum Rule analyses.  In this
paper, these techniques are used to investigate nucleon properties
obtained from correlation functions of unconventional nucleon
interpolating fields.  Some of these correlation functions suffer a
loss of signal prior to a clear ground state domination and the QCD
continuum model becomes a central element in fitting the correlation
functions.

   In the QCD-SR field there is a history of argument over the optimum
nucleon interpolating field to be used in analyses
\cite{chung82,chung84,dosch89a,ioffe81,ioffe83,leinweber90,furnstahl92,jin94}.
This issue is recognized to be of paramount importance and remains
unsettled \cite{shifman92}.  While some advocate an interpolating
field for which the leading terms of the OPE are stationary with
respect to the interpolating field mixing parameter
\cite{chung84,dosch89a}, others argue that a balance between
OPE convergence\footnote{Here and in the following, ``convergence'' of
the OPE simply means that the highest dimension terms considered in
the OPE, with their Wilson coefficients calculated to leading order in
perturbation theory, are small relative to the leading terms of the
OPE.} and QCD continuum contributions must be maintained
\cite{ioffe83,leinweber90,furnstahl92,jin94}.

   Ideally, one would like to simply calculate with alternate
interpolating fields and confirm that the nucleon properties remain
unchanged.  However, the limitations of the QCD-SR approach have
prevented one from doing this in practice.  Limitations include
uncertainties in the values of lower dimension condensates,
factorization of higher dimension operators, OPE truncation and
convergence issues, uncertainties surrounding the role of direct
instanton contributions to the sum rules and uncertainties in the
reliability of the continuum model for excited states.  Fortunately,
the lattice approach is not plagued with the same limitations and the
following analysis resolves this long standing debate.

   The format of this paper is as follows.  Section \ref{lattcorrfun}
introduces the interpolating fields explored in this analysis, the
lattice techniques used, and issues encountered in relating lattice
and continuum (lattice spacing $a \to 0$) formalisms.  Section
\ref{contmodel} highlights the QCD continuum model derivation.  The
analysis of the correlation functions is presented in Section
\ref{lattcorrfits} for each interpolating field combination.  The
results are compared with other approaches to QCD in Section
\ref{compare}.  Finally the conclusions regarding interpolating field
invariance and the optimal nucleon interpolator for QCD Sum Rules are
summarized in Section
\ref{conclusions}.

\section{LATTICE CORRELATION FUNCTIONS}
\label{lattcorrfun}

\subsection{Interpolating Fields}

   The commonly used interpolating field for the proton in lattice
calculations has the form
\begin{equation}
\chi_1(x) = \epsilon^{abc}
                 \left ( u^{Ta}(x) C \gamma_5 d^b(x) \right ) u^c(x)
\, .
\label{chiN1}
\end{equation}
Here, we follow the notation of Sakurai \cite{sakurai67}.  $C =
\gamma_4 \gamma_2$ is the charge conjugation matrix, $a,\ b,\ c$ are
color indices, $u(x)$ is a $u$-quark field, and the superscript $T$
denotes transpose.  Dirac indices have been suppressed.

   In the sum rule approach, it is common to find linear combinations
of this interpolating field and
\begin{equation}
\chi_2(x) = \epsilon^{abc}
                 \left ( u^{Ta}(x) C d^b(x) \right ) \gamma_5 u^c(x)
\, ,
\label{chiN2}
\end{equation}
which vanishes in the nonrelativistic limit.  {\it A priori}, there is
no reason to exclude such an interpolating field \cite{chung82}, as
the quark field operators of (\ref{chiN2}) annihilate the light
current quarks of QCD.  Of course, these quarks are highly
relativistic when bound in the nucleon.  With the use of the Fierz
relations, the combination of the above two interpolating fields with
a relative minus sign may be written
\begin{eqnarray}
\chi_{\rm SR}(x) &=& \epsilon^{abc}
                 \left ( u^{Ta}(x) C \gamma_\mu u^b(x) \right )
                 \gamma_5 \gamma^\mu d^c(x) \nonumber \\
           &=& 2 \left ( \chi_2 - \chi_1 \right ) \, ,
\label{chiSR}
\end{eqnarray}
giving the proton interpolating field often found in sum rule
calculations \cite{ioffe83,leinweber90,furnstahl92,jin94}.  The
alternate QCD Sum Rule interpolating field is
\begin{eqnarray}
\chi_{\rm A}(x) &=& {1 \over 2} \epsilon^{abc}
                 \left ( u^{Ta}(x) C \sigma_{\mu \nu} u^b(x) \right )
                 \sigma^{\mu \nu} \gamma_5 d^c(x) \nonumber \\
           &=& 2 \left ( \chi_2 + \chi_1 \right ) \, .
\label{chiA}
\end{eqnarray}

   In this analysis we will consider both interpolating fields
introduced in (\ref{chiN1}) and (\ref{chiN2}) and their interference
terms such that any linear combination of these interpolating fields
may be investigated.

\subsection{Correlation Functions at the Quark Level}

   Hadron masses are determined through the consideration of two-point
correlation functions.  Here we consider the nucleon correlator
\begin{equation}
G_2(t,\vec p) = \sum_{\vec x}\, e^{-i \vec p \cdot \vec x}
{\rm tr} \Bigl [ \Gamma_4
\bigm < 0 \bigm | T \{ \chi_N (x)\, \overline \chi_N (0) \} \bigm |
0 \bigm > \Bigr ]
\label{twopoint}
\end{equation}
where $\chi_N(x)$ may be either (\ref{chiN1}) or (\ref{chiN2}),
$\Gamma_4 = (1 + \gamma_4)/4$ projects positive parity states for
$\vec p = 0$, and tr indicates the trace over Dirac indices.

   Correlation functions at the quark level are obtained through the
standard procedure of contracting out time-ordered pairs of quark
field operators.  For the octet baryons it is convenient to define the
correlation function
\begin{eqnarray}
\lefteqn{
{\cal F}(S_{f_1},S_{f_2},S_{f_3}) = } \hspace{2.2cm} \nonumber \\
\epsilon^{abc} \epsilon^{a'b'c'}
\biggl \{
&& S_{f_1}^{a a'}(x,0) \, tr \left [ S_{f_2}^{b b'}(x,0) \,
   S_{f_3}^{c c' \, T}(x,0)
   \right ] \nonumber \\
&& + S_{f_1}^{a a'}(x,0) \, S_{f_3}^{c c' \, T}(x,0) \,
   S_{f_2}^{b b'}(x,0) \biggr \} \, ,
\label{calFdef}
\end{eqnarray}
where $S^{a a'}(x,0) = T \left \{ q^a(x), \, \overline q^{a'}(0)
\right \}$ and $f_1$, $f_2$, $f_3$ are flavor labels.  For the proton
interpolating field $\chi_1$ of (\ref{chiN1}), the two-point function
may be written
\begin{equation}
G_2(t,\vec p) = \sum_{\vec x} e^{-i \vec p \cdot \vec x}
   tr \left [ \Gamma_4 \, \,
   {\cal F} \left ( S_u, \, S_u, \, \widetilde C S_d
   \widetilde C^{-1} \right ) \right ] \, ,
\label{latticeCorrel1}
\end{equation}
where $\widetilde C = C \gamma_5$. Similarly the two-point function
corresponding to $\chi_2$ of (\ref{chiN2}) may be written in the form
\widetext
\begin{equation}
G_2(t,\vec p) = \sum_{\vec x} e^{-i \vec p \cdot \vec x}
   tr \left [ \Gamma_4 \, \,
   {\cal F} \left ( \gamma_5 S_u \gamma_5, \,
   \gamma_5 S_u \gamma_5, \, \widetilde C S_d
   \widetilde C^{-1} \right ) \right ] \, .
\label{latticeCorrel2}
\end{equation}
The interference contributions of these two interpolating fields are
\begin{equation}
G_2(t,\vec p) =  \sum_{\vec x} e^{-i \vec p \cdot \vec x}
   tr \biggl [ - \Gamma_4 \, \, \biggl \{
   {\cal F} \left ( S_u \gamma_5,\,
   S_u \gamma_5, \, \widetilde C S_d \widetilde C^{-1} \right )
   + {\cal F} \left ( \gamma_5 S_u,\, \gamma_5 S_u, \,
   \widetilde C S_d \widetilde C^{-1}
   \right ) \biggr \} \biggr ] \, ,
\label{latticeCorrelI}
\end{equation}

\narrowtext

\subsection{Lattice Techniques}

   Here we briefly summarize the lattice techniques used in the
following calculations.  Additional details may be found in Ref.\
\cite{leinweber91}.  Wilson's formulation is used for both the gauge
and fermionic action $(r=1)$.  $SU(2)$-isospin symmetry is enforced by
equating the Wilson hopping parameters $\kappa_u = \kappa_d = \kappa$.
Three values of $\kappa$ are selected and are denoted $\kappa_1 =
0.152$, $\kappa_2 = 0.154$ and $\kappa_3 = 0.156$.  To make contact
with the physical world, the mass and interpolating field coupling
strengths calculated at the three values of $\kappa$ are linearly
extrapolated to $\kappa_{\rm cr}=0.159\,8(2)$ where an extrapolation
of the squared pion mass vanishes.  Differences between linear
extrapolations to $m_\pi=0$ as opposed to the physical pion mass are
small and are neglected in the following.

  Twenty-eight quenched gauge configurations were generated by the
Cabibbo-Marinari \cite{cabibbo82} pseudo-heat-bath method on a $24
\times 12 \times 12 \times 24$ periodic lattice at $\beta=5.9$.
Configurations were selected after 5000 thermalization
sweeps from a cold start, and every 1000 sweeps thereafter
\cite{correlations}.

   Dirichlet boundary conditions are used for fermions in the time
direction.  Time slices are labeled from 1 to 24, with the
$\delta$-function source at $t=4$.  To minimize noise in the Green
functions, the parity symmetry of the correlation functions, and the
equal weighting of $\{U\}$ and $\{U^*\}$ gauge configurations in the
lattice action are exploited.  The nucleon mass determined from
$\chi_1$ of (\ref{chiN1}) is used to set the lattice spacing.  This
estimate lies between other estimates based on the string tension or
the $\rho$-meson mass.  The lattice spacing is determined to be
$a=0.132(4)$ fm and $a^{-1} = 1.49(5)$ GeV.

   Statistical uncertainties in the lattice correlation functions are
estimated by a single elimination jackknife \cite{efron79}.  A
covariance matrix fit of the pole plus QCD continuum model over a
range of 7 or more time slices is likely to be unreliable for 28 gauge
configurations \cite{michael94}.  Instead we use standard statistical
error analysis in which correlations among the fit parameters are
accounted for.  The Gauss-Newton method is used to minimize $\chi^2$.
Uncertainties are taken from the standard error ellipse \cite{rpp92}
at $\chi^2 = \chi^2_{\rm min} + 1$.

\subsection{Operator Mixing}

   The implementation of Wilson fermions on the lattice induces mixing
between the composite nucleon interpolating fields \cite{richards87}
of (\ref{chiN1}) and (\ref{chiN2}), reflecting the breaking of chiral
symmetry.  In Ref. \cite{richards87} the mixing is argued to occur
between
\begin{mathletters}
\begin{eqnarray}
O_\alpha &\equiv& \epsilon^{abc}
                  \left ( u^{Ta}_R(x) C d^b_R(x) \right ) u^c_L(x)
\, , \\
&=& {1 \over 4} \left ( 1 - \gamma_5 \right )
                \left ( \chi_1 - \chi_2 \right ) \, , \nonumber \\
O_\beta &\equiv& \epsilon^{abc}
                  \left ( u^{Ta}_L(x) C d^b_L(x) \right ) u^c_L(x)
\, , \\
&=& -{1 \over 4} \left ( 1 - \gamma_5 \right )
                 \left ( \chi_1 + \chi_2 \right ) \, , \nonumber
\end{eqnarray}
and a third operator
\begin{equation}
O_\gamma  \equiv \epsilon^{abc}
          \left ( u^{Ta}(x) C \gamma_\rho \gamma_5 d^b(x) \right )
          \gamma_L \gamma^\rho u^c(x) \, .
\end{equation}
\end{mathletters}
Here
\begin{mathletters}
\begin{equation}
u_R = {1 \over 2} \left ( 1 + \gamma_5 \right ) u \, , \qquad
u_L = {1 \over 2} \left ( 1 - \gamma_5 \right ) u \, ,
\end{equation}
\begin{equation}
\gamma_R = {1 \over 2} \left ( 1 + \gamma_5 \right ) \, ,
\quad {\rm and} \quad
\gamma_L = {1 \over 2} \left ( 1 - \gamma_5 \right ) \, .
\end{equation}
\end{mathletters}
However, there are only two operators having isospin-1/2 and spin-1/2
and it is possible to demonstrate
\begin{equation}
O_\gamma = - 2 O_\alpha \, ,
\end{equation}
via Fierz transformations.  For $\chi_1$ and $\chi_2$ the expressions
up to one-loop order in perturbation theory relating the operator
matrix elements in the Pauli-Villars (PV) and Lattice (L) schemes are
\cite{richards87}
\mediumtext
\begin{eqnarray}
\chi_1^{\rm PV} &=& \chi_1^{\rm L}
      - {\alpha_s \over 4 \pi} \,
        \left [ -2 \, \ln Q^2 a^2 +
        \left ( C_1^{\rm L} - C_2^{\rm L} - 2 C_3^{\rm L} \right )
        \right ] \, \chi_1^{\rm L}
      - {\alpha_s \over 4 \pi} \, 2 C_3^{\rm L} \, \chi_2^{\rm L}
\, , \\
\chi_2^{\rm PV} &=& \chi_2^{\rm L}
                  - {\alpha_s \over 4 \pi} \,
                    \left [ -2 \, \ln Q^2 a^2 +
                    \left ( C_1^{\rm L} + C_2^{\rm L} \right )
                    \right ] \, \chi_2^{\rm L} \, ,
\end{eqnarray}
\narrowtext
where
\begin{equation}
C_1^{\rm L} = 37.91, \quad C_2^{\rm L} = -3.21, \quad {\rm and} \quad
C_3^{\rm L} = -0.80,
\end{equation}
for the Wilson parameter $r=1$.  The important point is that the
interpolating field $\chi_2$ does not mix with $\chi_1$ to one-loop
order.  Moreover, the mixing of $\chi_1$ with $\chi_2$ is negligible.
Hence, it is possible to identify the properties of these
interpolating fields determined on the lattice with those of their
continuum $(a \to 0)$ counterparts to a good approximation.

   The dominant contribution to the coefficient $C_1^{\rm L}$ in the
above expressions is from the self-energy corrections to the quark
external lines.  These corrections are accounted for in the mean-field
improved approach, and the remaining renormalization $Z_\chi$
associated with composite operators is relatively small.  The
principle renormalization constant $C_1^{\rm L}$ has been determined
in the mean-field approach \cite{lepage93} and is used in the
following.  The nucleon coupling strength, $\lambda_N$, is determined
in absolute terms, without resorting to a ratio of the QCD continuum
contributions as done in \cite{chu93a,chu93b}.  In particular, the
renormalization at the scale of $1/a$ is
\begin{equation}
\chi^{\rm Continuum} = {Z_{\chi_N} \over a^{9/2}} \,
\chi^{\rm Lattice} \,
\left ( 1 - { 3 \kappa \over 4 \kappa_{\rm cr}} \right )^{3/2} \, ,
\end{equation}
and $Z_{\chi_N} = (1 - 0.73 \, \alpha_V) \simeq 0.80$ at $\beta =
5.9$.
The $\kappa$ dependence of this wave function renormalization is very
different from the na\"{\i}ve normalization
\begin{equation}
\chi^{\rm Continuum} = {1 \over a^{9/2}} \, \chi^{\rm Lattice} \,
\left ( 2 \kappa \right )^{3/2} \, ,
\end{equation}
and is crucial to recovering the correct mass independence of the
Wilson coefficient of the identity operator.

\section{THE QCD CONTINUUM MODEL}
\label{contmodel}

   Here we briefly review the QCD continuum model implementation in
Euclidean space as examined in detail in Ref.\ \cite{leinweber94a}.
We start with the two-point correlation function of (\ref{twopoint}).
At the phenomenological level, one inserts a complete set of states
$N^i$ and defines
\begin{equation}
\bigm < 0 \bigm | \chi_N (0) \bigm | N^i,p,s \bigm > \, =
\lambda_N^i\, u(p,s) \, ,
\end{equation}
where the coupling strength, $\lambda_N^i$, measures the ability of
the interpolating field $\chi_N$ to annihilate the i'th excitation
with nucleon quantum numbers.  For $\vec p = 0$ and Euclidean time $t
\to \infty$, the ground state dominates and $G_2(t) \to \lambda_N^2
e^{-M_N t}$.  The spectral representation is defined by
\begin{equation}
G_2(t) = \int_0^\infty \, \rho(s) \, e^{-st} \, ds \, ,
\label{SpectralRepr}
\end{equation}
and the spectral density is,
\begin{equation}
\rho(s) = \lambda_N^2 \, \delta(s-M_N) + \zeta(s)
\end{equation}
where $\zeta(s)$ provides the excited state contributions.

   The form of the spectral density used in the QCD continuum model is
determined by the leading terms of the OPE surviving in the limit $t
\to 0$.  Here, the combination of interpolators $\chi_1 \overline
\chi_1$ is considered.  The derivation of the QCD continuum model
contributions to other correlation functions proceeds in an analogous
fashion.  In Euclidean space, $G_2(t)$ has the following OPE
\begin{eqnarray}
G_2(t) &\simeq& {3 \cdot 5^2 \over 2^8 \pi^4} \, \biggl ( \,
{1 \over t^6} + {28 \over 25} \, {m_q a \over t^5}
              + {14 \over 25} \, {m_q^2 a^2 \over t^4} \nonumber \\
            &&- {56 \pi^2 \over 75} \, {\bigm < : \overline q q :
                                        \bigm > a^3
                \over t^3}
              + \cdots \biggr ) \, .
\label{OPE11}
\end{eqnarray}

   The spectral density used in the QCD continuum model is defined by
equating (\ref{SpectralRepr}) and (\ref{OPE11}).  The QCD continuum
model is defined through the introduction of a threshold which marks
the effective onset of excited states in the spectral density.
Keeping the first two terms of (\ref{OPE11}), the phenomenology of
$G_2(t)$ is \widetext
\begin{mathletters}
\label{phenom}
\begin{eqnarray}
G_2(t) &=& \lambda_1^2 \, e^{-M_N t} + \xi \,
\int_{s_0}^\infty \, \rho(s) \, e^{-st} \, ds  \, ,
\label{phenomCrypt} \\
&=& \lambda_1^2 \, e^{-M_N t} + \xi \,
{3 \cdot 5^2 \over (2^8 \pi^4)} \,
e^{-s_0 t} \times \label{phenomExpl} \\
&& \quad \Biggl ( \left \{ {1 \over t^6} + {s_0 \over t^5}
+ {1 \over 2}\,{s_0^2 \over t^4} + {1 \over 6}\,{s_0^3 \over t^3}
+ {1 \over 24}\,{s_0^4 \over t^2} + {1 \over 120}\,{s_0^5 \over t}
\right \}
+ {28 \, m_q a \over 25} \,  \left \{
  {1 \over t^5} + {s_0 \over t^4}
+ {1 \over 2}\,{s_0^2 \over t^3} + {1 \over 6}\,{s_0^3 \over t^2}
+ {1 \over 24}\,{s_0^4 \over t}
\right \} \Biggr ) \, .  \nonumber
\end{eqnarray}
\end{mathletters}
\narrowtext
The parameter $\xi$ governs the strength of the QCD continuum model.
In the continuum limit $(a \to 0)$ $\xi = 1$ but here is optimized
with $\lambda_N$, $M_N$, and $s_0$ to account for enhancement of the
correlator in the short time regime reflecting lattice anisotropy.
$\xi$ is an overall QCD continuum model strength and is expected to be
independent of the quark mass.  With this approach, the effects of
lattice anisotropy may be absorbed through a combination of a larger
QCD continuum model strength $(\xi > 1)$ and marginally larger
threshold $(s_0)$.

   Infrared lattice artifacts are not a significant problem for this
approach as the Fourier transform weight $\exp(-i \vec p \cdot \vec
x)$ is correct for all propagator paths including those which wrap
around the lattice spatial dimensions.  The ultraviolet lattice cutoff
may be modeled in a manner similar to that for the QCD continuum
model.  However the modeling becomes insignificant by the second time
slice following the source.  Instead we simply discard the source and
first time slice when fitting the correlation functions.

\section{LATTICE CORRELATOR FITS}
\label{lattcorrfits}

\subsection{\pmb{$\bigm < \chi_1 \overline \chi_1 \bigm >$}
            Correlation Function }

   $\chi_1$ is the standard nucleon interpolating field used in
lattice analyses.  Its overlap with the nucleon ground state is
excellent.  This lattice correlation function is fit with
(\ref{phenom}) in a four parameter search of $\lambda_N$, $M_N$, $s_0$
and $\xi$ in analysis intervals from $t=6 \to t_f$ where $t_f$ ranges
{}from 11 through 23.  Figure \ref{NuclCorrFn} illustrates the lattice
data and these 13 fits at our intermediate value of quark mass.
Similar results are seen for $\kappa = 0.152$ and 0.156.

   The similarity of the 13 pole plus QCD continuum fits establishes
that the QCD continuum model effectively accounts for excited state
contaminations in the correlation functions and allows the extraction
of the ground state properties from a regime as small as $t=6 \to 11$.
For an in-depth examination of this correlator see Ref.\
\cite{leinweber94a}.  The quark mass dependence of $\lambda_N$ is
illustrated in Figure \ref{lambda1}.  Table \ref{Tab:chi11C5ope}
summarizes the fit parameters for the regime $t=6 \to 20$.

\subsection{\pmb{$\bigm < \chi_2 \overline \chi_2 \bigm >$}
            Correlation Function }

   Unlike QCD Sum Rule analyses, correlators may be determined on the
lattice for any interpolating field without regard to the restrictions
of OPE convergence issues or operator factorization assumptions.
In constructing the QCD continuum model, only the first few terms of
the OPE are required.  The OPE for the interpolating fields $\chi_2
\overline \chi_2$ is
\begin{eqnarray}
G_2(t) &=& {3 \cdot 5^2 \over 2^8 \pi^4} \,
\biggl ( \, {1 \over t^6} -
{4 \over 5} \, {m_q a \over t^5} - {2 \over 5} \, {m_q^2 a^2 \over
t^4} \nonumber \\ &&+ {8 \pi^2 \over 15} \, {\bigm < \overline q q
\bigm > a^3 \over t^3} + \cdots \biggr ) \, ,
\label{OPE22}
\end{eqnarray}
and the QCD continuum model is derived in an analogous manner to that
outlined in Section \ref{contmodel}.  The phenomenological side of
$G_2(t)$ is
\widetext
\begin{eqnarray}
G_2(t) &=& \lambda_N^2 \, e^{-M_N t} +
           \xi \, {3 \cdot 5^2 \over (2^8 \pi^4)} \,
           e^{-s_0 t} \, \times \nonumber \\
&& \Biggl (
\left \{ {1 \over t^6} + {s_0 \over t^5}
+ {1 \over 2}\,{s_0^2 \over t^4} + {1 \over 6}\,{s_0^3 \over t^3}
+ {1 \over 24}\,{s_0^4 \over t^2} + {1 \over 120}\,{s_0^5 \over t}
\right \}
- {4 \, m_q a \over 5} \,  \left \{
  {1 \over t^5} + {s_0 \over t^4}
+ {1 \over 2}\,{s_0^2 \over t^3} + {1 \over 6}\,{s_0^3 \over t^2}
+ {1 \over 24}\,{s_0^4 \over t}
\right \} \Biggr ) \, .  \label{phenomExpl2}
\end{eqnarray}
\narrowtext

   Figure \ref{chi22} illustrates the lattice correlation function and
the final fit.  The choice of $\Gamma_4$ in (\ref{twopoint}) projects
out positive parity nucleon states when $\vec p = 0$ and therefore the
correlation function must remain positive.  At $t=13$ the lattice
correlation function data changes sign, and indicates a loss of
signal.

   The fit from $t = 6 \to 12$ using a pole plus QCD continuum model
leads to fit parameters where the pole lies above the continuum
threshold.  The position of the pole is insignificant as its removal
has little effect on the $\chi^2$/dof.  Fixing the pole at the
previously determined nucleon masses returns an optimum value for
$\lambda_2^2$ which is negative, and once again unphysical.  The fit
illustrated in Figure \ref{chi22} employs the QCD continuum model
alone.  Hence there is no evidence of any overlap of $\chi_2$ with the
ground state nucleon in this correlation function.  While the results
illustrated here are for our intermediate value of quark mass
considered on the lattice, similar results are seen for the lighter
and heavier quark masses.  The fit parameters are summarized in Table
\ref{Tab:chi22}.  The QCD continuum threshold is not too different
{}from that for $\chi_1 \overline \chi_1$.

   Figure \ref{chi22c6} illustrates the quark mass dependence of
$\xi$.  As anticipated, $\xi$ is independent of $m_q$.  This quark
mass independence confirms the negative sign of the $m_q$ correction
appearing in the OPE for $\chi_2 \overline \chi_2$ and the use of
mean-field improved operators.  It also confirms the perturbative role
of the quark mass operator.  While the heaviest current quark mass
used in this investigation is similar to that of the strange quark, it
is still light on the scale set by the nucleon mass.

   It is also interesting to note that $\xi$ is much closer to 1 than
for the correlators of $\chi_1 \overline \chi_1$.  That this might be
the case is eluded to by the opposite signs of the leading terms of
the OPE in (\ref{OPE22}), providing the possibility of cancelations
in the short-time perturbative regime of the correlation function.

\subsection{\pmb{$\bigm < \chi_1 \overline \chi_2 +
                          \chi_2 \overline \chi_1 \bigm >$}
            Correlation Function }

   Since the square of $\lambda_2$ is small, one might be able to
recover a signal for the overlap of the nucleon ground state and
$\chi_2$ by considering the correlation function for
$\chi_1 \overline \chi_2$.  Figure \ref{chi12} illustrates a three
parameter fit from $t = 6 \to 12$ using the pole plus QCD continuum
model derived from the OPE for
${1 \over 2} ( \chi_1 \overline \chi_2 + \chi_2 \overline \chi_1)$
\begin{eqnarray}
G_2(t) &=& {3 \cdot 5 \over 2^8 \pi^4} \, \biggl ( \,
{1 \over t^6} - {4 \over 5} \, {m_q a \over t^5}
              - {2 \over 5} \, {m_q^2 a^2 \over t^4} \nonumber \\
     &&+ {8 \pi^2 \over 15} \, {\bigm < \overline q q \bigm > a^3
                \over t^3}
              + \cdots \biggr ) \, .
\label{OPE12}
\end{eqnarray}
In this fit, the nucleon ground state pole position has been fixed at
the previously determined nucleon masses.  The fit parameters are
summarized in Table \ref{Tab:chi12}.  While there is sufficient
information in the correlation function to determine a value for the
nucleon mass, the corresponding uncertainties are large.

   The leading terms of the OPEs for $\chi_2 \overline \chi_2$ and
${1 \over 2} ( \chi_1 \overline \chi_2 + \chi_2 \overline \chi_1)$
given in (\ref{OPE22}) and (\ref{OPE12}) are equivalent up to a
normalization factor of five.  Since the continuum model is
constructed to accommodate these leading terms, one expects the QCD
continuum model parameters, $s_0$ and $\xi$, for these two correlators
to be similar.  A comparison of Tables \ref{Tab:chi22} and
\ref{Tab:chi12} indicates that this is indeed the case.

   Figure \ref{chi12c6} illustrates the expected quark mass
independence of $\xi$, again confirming the negative sign of the $m_q$
correction appearing in the OPE for
${1 \over 2} ( \chi_1 \overline \chi_2 + \chi_2 \overline \chi_1)$
and the use of mean-field improved operators.  Similarly $\xi \sim 1$
as anticipated by the opposite signs of the leading terms of the OPE
of (\ref{OPE12}).

   The linear extrapolation of $\left ( \lambda_1 \lambda_2
\right)^{1/2}$ to $\kappa_{\rm cr}$ is illustrated in Figure
\ref{chi12lambda}.  The combination $\left ( \lambda_1 \lambda_2
\right)^{1/2}$ shows little sensitivity to the quark mass.  This
contrasts the dependence of $\lambda_1$ illustrated in Figure
\ref{lambda1}, where $\lambda_1$ decreases as the quarks become
lighter.  Thus, $\lambda_2$ increases for decreasing quark mass.  This
reflects the fact that $\chi_2$ vanishes in a nonrelativistic
reduction.  At the chiral limit $\left ( \lambda_1 \lambda_2
\right)^{1/2} = 0.0014(10)\ {\rm GeV}^3$.  Systematic uncertainty in
the extrapolated value of $\left ( \lambda_1 \lambda_2 \right)^{1/2}$
may be estimated using the quark mass dependence suggested by Chiral
Perturbation Theory \cite{lee94} as in \cite{leinweber94a}.  Here the
systematic uncertainty in extrapolating is negligible relative to the
statistical uncertainties.

   With the previous result $\lambda_1 = 0.013(2)\ {\rm GeV}^3$, the
nucleon coupling strength for $\chi_2$ is found to be $\lambda_2 =
0.00016(22)\ {\rm GeV}^3$, approximately 100 times smaller than
$\lambda_1$.  In short, there is only one nucleon interpolating field
that has significant overlap with the nucleon ground state, namely
\begin{equation}
\chi_1(x) = \epsilon^{abc}
                 \left ( u^{Ta}(x) C \gamma_5 d^b(x) \right ) u^c(x)
\, ,
\end{equation}
and
\begin{eqnarray}
&\lambda_{\rm SR} &\simeq \lambda_{\rm A} \simeq 2 \lambda_1 =
                          0.026(4)\ {\rm GeV}^3 \, , \\
&\lambda_2 &\simeq 0 \, .
\end{eqnarray}
The results for all the considered interpolating fields are summarized
in Table \ref{InterSum} \cite{DiffRenorm}.

\section{COMPARISON WITH OTHER CALCULATIONS}
\label{compare}

   It is worth noting that the nonperturbative QCD Sum Rule
predictions for $\lambda_{\rm SR}$ have remained quite stable over the
years, despite the fact that the early calculations have a number of
numerical errors in Wilson coefficients and anomalous dimensions
\cite{review}.  This cannot be said for the more model dependent
predictions. It is ironic that, in some cases, the model calculations
were pursued due to reservations about the reliability or validity of
the QCD Sum Rule approach.  Table \ref{CompareSR} summarizes a
collection of predictions taken and updated from Ref.\
\cite{brodsky84} and \cite{gavela89}.

   QCD Sum Rule predictions of $\lambda_2$ or $\lambda_{\rm A}$ are
more uncertain.  This is largely due to a lack of rigor in the
analysis of the sum rules.  Often, the region of validity in Borel
space is simply postulated with little regard to OPE convergence or
the size of continuum model contributions.  Many authors have fixed
the continuum threshold to a preferred value or excitation energy
rather than leaving it as a search parameter to be optimized.  The
upper limit of the Borel region must be monitored as it is a function
of the three required fit parameters, $M_N$, $\lambda_N$ and $s_0$ and
varies for different interpolating fields.  The failure to monitor
these issues in existing analyses, is largely responsible for the
apparent inconsistencies between sum rules derived from different
interpolating fields.

   Ref.\ \cite{leinweber90} is one of the few sum rule analyses
where these issues are rigorously implemented.  However, the
interpolating field $\chi_2$ was not considered there.  The sign of
the quark condensate term in (\ref{OPE22}) indicates the two sum rules
will be saturated by both positive and negative parity states.  A
careful analysis of these sum rules has not yet been attempted.

   In the QCD-SR discussion of Ref.\ \cite{ioffe81} it was concluded
that the overlap of $\chi_{\rm A}$ with the nucleon, $\lambda_{\rm
A}$, must be negligible, due to the vanishing of most of the Wilson
coefficients to dimension 8.  However, this conclusion need not be the
case.  Higher order terms of the OPE starting at dimension 9 are not
zero and could easily give rise to large overlap with the nucleon as
discovered here.  The only conclusion that may be drawn from these sum
rules is that the pole contribution is small {\it relative} to the QCD
continuum model contribution.  The pole contribution is not
necessarily small in {\it absolute} terms.

   The first five entries of Table \ref{CompareSR} summarize results
for $\lambda_{\rm SR}$ obtained from the consideration of two-point
correlation functions, and these compare favorably.  The same cannot
be said for $\lambda_{\rm A}$.  The RILM prediction \cite{schafer94a}
is $\lambda_{\rm A} = 0.040(2)$ GeV${}^3$ and is large compared to the
lattice prediction of $\lambda_{\rm A} = 0.027(5)$ GeV${}^3$.  Figure
3 of Ref.\ \cite{schafer94a} displays significant discrepancies
between a global fit of the six nucleon correlators considered and two
of the correlators.  These two correlators are both dependent on
$\lambda_{\rm A}$ and these discrepancies are not reflected in their
quoted uncertainty of $\pm 0.002$ GeV${}^3$.  Their conclusion ``What
is even more important, the simple `nucleon pole plus continuum' model
gives a very good simultaneous description for the complete set of
correlation functions'' is difficult to justify in the RILM,
particularly in light of these new results.

\section{CONCLUSIONS}
\label{conclusions}

\subsection{Interpolating Field Invariance}

   Ground state nucleon properties {\it are} independent of the
interpolating field used to excite the baryon from the vacuum.  This
invariance is satisfied in a trivial manner.  The interpolating field
which vanishes in the nonrelativistic limit,
\begin{equation}
\chi_2(x) = \epsilon^{abc}
                 \left ( u^{Ta}(x) C d^b(x) \right ) \gamma_5 u^c(x)
\, ,
\end{equation}
has negligible overlap with the nucleon ground state.  Inclusion of
$\chi_2$ components in interpolating fields only increases the
statistical uncertainties of lattice QCD correlation functions.

\subsection{Optimal Interpolator for QCD Sum Rules}

   This analysis indicates that, to a good approximation, $\chi_2$
excites pure QCD continuum.  Since $\chi_2$ has negligible overlap
with the ground state nucleon, it is tempting to simply conclude that
the optimum interpolating field is $\chi_1$.  While this is certainly
the case for lattice QCD investigations, it is not obviously the case
for QCD Sum Rule analyses.

   The optimal nucleon interpolator must involve $\chi_1$ as this
interpolator is required to maintain overlap with the ground state.
The task is to determine the optimal mixing of $\chi_2$.  The Borel
improved QCD Sum Rules for the generalized interpolator
\begin{equation}
\chi_{\cal O} = \chi_1 + \beta \chi_2 \, ,
\end{equation}
are
\begin{mathletters}
\begin{eqnarray}
&&{5 + 2 \beta + 5 \beta^2 \over 64} \, M^6 \, L^{-4/9}
\left [ 1 - e^{-s_0^2/M^2}
\left ( {s_0^4 \over 2 M^4} + {s_0^2 \over M^2} + 1 \right ) \right ]
\nonumber \\
&&\quad +{5 + 2 \beta + 5 \beta^2 \over 256} \, b \, M^2 \, L^{-4/9}
\left [ 1 - e^{-s_0^2/M^2} \right ]
\nonumber \\
&&\quad +{7 - 2 \beta - 5 \beta^2 \over 24} \, a^2 \, L^{4/9}
        -{13 - 2 \beta - 11 \beta^2 \over 96} \,
         { m_0^2 \, a^2 \over M^2}
\nonumber \\
&&\qquad = \widetilde \lambda_{\cal O}^2 \, e^{-M_N^2/M^2}
+ \widetilde \lambda_{\cal O^*}^2 \, e^{-M_{N^*}^2/M^2} \, ,
\label{QCDSR1}
\end{eqnarray}
and
\begin{eqnarray}
&&{7 - 2 \beta  - 5 \beta^2 \over 16} \, a \, M^4
\left [ 1 - e^{-s_0^2/M^2}
\left ( {s_0^2 \over M^2} + 1 \right ) \right ] \nonumber \\
&&\quad -{3 (1 - \beta^2) \over 16} \, m_0^2 \, a \, M^2 \, L^{-4/9}
\left [ 1 - e^{-s_0^2/M^2} \right ] \nonumber \\
&&\quad +{3 + 2 \beta - 5 \beta^2 \over 2^7} \, a \, b
\nonumber \\
&&\qquad = \widetilde \lambda_{\cal O}^2 \, M_N \, e^{-M_N^2/M^2}
-\widetilde \lambda_{\cal O^*}^2 \, M_{N^*} \, e^{-M_{N^*}^2/M^2} \, ,
\label{QCDSR2}
\end{eqnarray}
\label{QCDSR}
\end{mathletters}
where
\begin{mathletters}
\begin{eqnarray}
a &=& - (2 \pi)^2 \bigm < \overline q q \bigm >
   = 0.450\ {\rm GeV}^3  \, , \\
b &=&   (2 \pi)^2 \bigm < {\alpha_s \over \pi} \,
        G_{\mu \nu}^a G^{a \mu \nu} \bigm >
   = 0.474\ {\rm GeV}^4 \, , \\
m_0^2 &=& - {\bigm < \overline q \, g \, \sigma \cdot G \, q \bigm >
           \over \bigm < \overline q q \bigm >}
       = 0.65 \, , \\
L &=& {\log(M/\Lambda_{\rm QCD}) \over
       \log(\mu/\Lambda_{\rm QCD})} \, , \\
\widetilde \lambda_{\cal O} &=& (2 \pi)^2 \lambda_{\cal O} \, .
\end{eqnarray}
\end{mathletters}
Here $M$ is the Borel parameter and plays a role similar to the
inverse Euclidean time of the Lattice approach.  The condensate values
are taken from Ref.\ \cite{leinweber90} where $\mu = 0.5$ GeV and
$\Lambda_{\rm QCD} = 0.1$ GeV.  The continuum model contributions are
indicated on the left hand side of the sum rules where they appear in
brackets as subtractions from the terms of the OPE surviving in the
limit $M \to \infty$.  To aid the following discussion, both a
positive parity ground state and a negative parity excited state are
included on the right-hand-side of the sum rules.

   The first QCD-SR of (\ref{QCDSR1}) is known to have uncontrollably
large perturbative corrections to the Wilson coefficient of the
identity operator \cite{jamin88}.  In leading order, these corrections
are independent of $\beta$ and are approximately 50\%.  As a result
this sum rule must be discarded \cite{fourquark}.

   In the QCD-SR approach, approximations are made at both the quark
level and the phenomenological level.  At the quark level, the OPE is
truncated and the Wilson coefficients are calculated perturbatively.
This sets a lower limit for the Borel mass.  At the phenomenological
level, the spectral density is approximated by a pole plus the QCD
continuum model.  Maintaining ground state dominance on the
phenomenological side of the sum rule sets an upper limit on the Borel
mass.  By including $\chi_2$ components in an interpolating field, one
can reduce the continuum contributions excited by $\chi_1$ and allow a
broader Borel analysis window.

   One of the most difficult things to monitor in the QCD-SR approach
is whether the OPE is sufficiently convergent for a particular value
of Borel mass.  The lattice results presented here indicate the
$\chi_2 \overline \chi_2$ correlator has the fastest converging OPE,
as its overlap with the nucleon ground state is negligible.
Similarly, the combination $\chi_1 \overline \chi_1$ produces an OPE
with the slowest convergence, as this correlation function is
dominated by the ground state nucleon for small Borel masses.

   Hence, errors made in truncating the OPE are dominated by errors in
the $\chi_1 \overline \chi_1$ component of the general correlator.
The relative error in the OPE truncation can be reduced by adding
$\chi_2$ components to the correlator.  However, the $\chi_2$
components in the OPE are simply subtracted off again by the continuum
model terms.  Hence the relevant error is the absolute error.  For
$|\beta|
\mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 1$,
this error is dominated by $\chi_1 \overline \chi_1$ components of the
correlator.  As a result, OPE truncation errors are approximately
independent of $\beta$.  This crucial point has been neglected in
previous arguments regarding the optimal nucleon interpolating field.

   Since $\chi_2$ has negligible overlap with the nucleon, the ground
state contribution is also independent of $\beta$.  Hence, the size of
the continuum model contributions is the predominant factor in
determining the optimal interpolator.  Figure \ref{optcon} illustrates
the contributions of the continuum model terms in (\ref{QCDSR2}) for
$M=0.938$ GeV and $s_0 = 1.4$ GeV.  The following discussion is not
dependent on the precise values of these parameters.  The first point
to be made is that contributions from the continuum model are largest
for $\beta \sim -0.2$.  This selection of mixing is the worst possible
choice for extracting information on the ground state nucleon.

   Figure \ref{optcon} also indicates it is possible to have vanishing
continuum model contributions at $\beta \simeq -1.5$ or $\beta = 1$.
However, we are relying on the continuum model to account for strength
in the correlator that does not have its origin in the ground state.
Without a continuum model, one would need to include additional poles
on the right-hand-side of (\ref{QCDSR2}) to account for positive
and negative parity excitation strength.  For $\beta < -1.5$ or $\beta
> 1.0$ the correlator is negative indicating the sum rule is saturated
by a negative parity state.

   Thus the optimal interpolator is $\beta \sim -1.2$ or $\beta \sim
0.8$.  To discriminate between these two regimes, we turn to the
higher dimension operators (HDO) which do not contribute to the
continuum model.  It is these terms that provide crucial information
on whether the strength in the correlator lies in the ground state or
the excited states.  If these terms are absent, the optimal fit of the
correlator is obtained when $\widetilde\lambda_{\cal O} \to 0$ and
$s_0 \to 0$. In this case the continuum model becomes the Laplace
transform and the fit is perfect.  Hence the HDO terms should be large
in magnitude.  A change in sign from the leading terms of the OPE will
also assist in distinguishing ground state strength from excited state
strength as the change in the curvature of the correlator will be more
prominent.

   The last term of (\ref{QCDSR2}) is a HDO term, and its value is
plotted as a function of beta in Figure \ref{optcon}.  The HDO
contributions are larger for $\beta \sim -1.2$ than for $\beta \sim
0.8$.  In addition, the sign of the HDO contribution is opposite that
of the continuum model contributions.  Hence the preferred regime is
$\beta \sim -1.2$.  In fact, optimization of the three fit parameters,
$M_N$, $\widetilde\lambda_{\cal O}$ and $s_0$
$(\widetilde\lambda_{\cal O^*} = 0)$ of (\ref{QCDSR2}) for $\beta =
0.6 \to 0.8$ results in fit parameters describing pure continuum with
$s_0 \sim M_N$.  Information to separate the ground state pole from
the continuum is insufficient for $\beta = 0.6 \to 0.8$.

   In summary, the lattice results indicate OPE convergence, and
ground state pole contributions are approximately independent of
$\beta$.  Considerations of the size of continuum model contributions
and the sign and magnitude of HDO operators leads to the preferred
value of
\begin{equation}
\beta = -1.2 \pm 0.1 \, .
\end{equation}
A more precise determination of $\beta$ will depend on the details
of limits for continuum model contributions, HDO values, condensate
values, and other parameters of the sum rules.

   Hence, this analysis supports the selection of $\beta = -1$
\cite{ioffe81,ioffe83,leinweber90,furnstahl92,jin94}, over $\beta =
-0.2$ \cite{chung82,chung84,dosch89a}.  At $\beta = -0.2$, where the
leading terms of the OPE are stationary with respect to $\beta$
\cite{chung84,dosch89a}, the continuum contributions are maximal.
The positive value and small magnitude of the HDO indicates the
stability of the leading terms of the OPE will not be realized as
stability in the ground state mass, coupling, nor in the continuum
threshold.

\subsection{Future Investigations}

   These techniques may be used to determine the optimal interpolating
field for any sum rule involving $\chi_1$ and $\chi_2$ components.
Each sum rule will have an optimal selection for $\beta$.  The overlap
of spin-1/2 and spin-3/2 interpolating fields are known to yield
nucleon sum rules which offer stability in the fit parameters
\cite{leinweber90} that cannot be obtained from the more common sum
rules considered here.  It will be interesting to discover if the
historical selection of $\beta = -1$ is indeed optimal.

   While it is important to establish the optimal mixing of
interpolating fields for QCD-SR analyses, one should not overlook the
fact that there is a range of values for $\beta$ where the sum rules
are expected to work.  Moreover, the ground state contribution to all
these sum rules is equivalent to the 1\% level.  In other words, the
right-hand-side of (\ref{QCDSR2}) for a single pole plus continuum
model is independent of $\beta$.  After the first sum rule is written
down, additional sum rules may be introduced with merely one new fit
parameter $(s_0)$ per sum rule.  Since direct instanton contributions
to the sum rules are not independent of $\beta$ \cite{forkel93}, one
has an excellent opportunity to see if direct instanton contributions
really are necessary to maintain sum-rule consistency
\cite{leinweber94c}.

   Future lattice QCD investigations should aim to make a direct
comparison of the OPE and lattice correlation functions.  The Wilson
coefficients and vacuum expectation values of normal ordered operators
could be determined directly from OPE fits to the lattice data.  Such
a comparison would test the validity of the OPE in the nonperturbative
sector and our understanding of quantum field theory \cite{wilczek93}.

   A direct comparison of lattice and continuum formalisms requires
the use of an improved \cite{gabrielli91} or perfect
\cite{hasenfratz93} lattice action to reduce or eliminate lattice
anisotropy in the short time regime of lattice correlation
functions.  Alternatively, the Wilson coefficients of the
Euclidean-space correlation function may be derived via lattice
perturbation theory.

   An extremely fine lattice spacing is required to provide a
sufficient number of lattice sites within the radius of convergence of
the OPE.  In the most optimistic case, the two invariant nucleon sum
rules of a given interpolator could be isolated such that, to
dimension eight, each correlation function would have up to four
parameters to be determined when extracting OPE coefficients.
Ultraviolet cutoff considerations, OPE convergence issues and the need
for error estimates in the fit parameters place the lattice spacing at
less than 0.05~fm.

   Essential information on the importance of direct instanton
contributions to the OPE can be obtained from such an investigation.
The OPE coefficients and vacuum expectation values are determined
first by matching OPE and lattice correlation functions in which there
are no direct instanton contributions.  This approach determines the
OPE in a self consistent manner.  Then, other correlators in which
direct instanton contributions are argued to be important
\cite{forkel93} can be examined.  Discrepancies between the OPE
and lattice correlation functions would signal the possible importance
of direct instanton contributions.  If existing predictions for direct
instanton contributions to the OPE resolve the differences in the
correlators, then one has compelling evidence of a non-trivial role
for direct instanton contributions to the OPE.  The importance of such
investigations warrants further effort in this direction.

\acknowledgements

   The correlation functions used in this analysis were obtained in
collaboration with Terry Draper and Richard Woloshyn in Ref.\
\cite{leinweber91}.  I wish to thank Dick Furnstahl and Xuemin Jin for
a number of interesting and helpful discussions.  Thanks also to
Marina Nielsen for her contributions and discussions surrounding the
generalized sum rules of (\ref{QCDSR}).  This research is supported by
the National Science Foundation under grants PHY-9203145, PHY-9258270,
PHY-9207889 and PHY-9102922.