text:
\section{Introduction}

Chiral perturbation theory (ChPT) is a frequently used tool in the analysis of Lattice Quantum Chromodynamics (QCD) data. Well-known examples are the calculation of finite-volume (FV) effects or the light quark mass dependence of physical observables. ChPT results for the quark mass dependence are commonly used in the chiral extrapolation to relate unphysical lattice results obtained at heavy quark masses to the physical point with quark masses as light as in Nature. However, the need for a chiral extrapolation has been slowly fading away. Constant progress in computer power as well as advances in simulation algorithms have made lattice simulations possible with the light quark masses set to their physical values. Such {\em physical point  simulations} eliminate altogether the systematic uncertainties associated with a chiral extrapolation.

In this talk I briefly report on a different application of ChPT to Lattice QCD, namely the study of excited-state contaminations. In physical point simulations one can expect multi-particle states with additional light pions to become a non-negligible excited-state contamination in many correlation functions measured on the lattice. The calculation of nucleon structure observables is a familiar example where the excited-state contamination is known to be a source of significant systematic uncertainty. The multi-particle states expected to be most relevant in this case are two-particle nucleon-pion ($N\pi$) states, and their contribution is accessible by ChPT.

\section{$N\pi$-state contribution in nucleon correlation function}

Let us start with a simple but illustrative example, the nucleon 2pt function
\begin{equation} \label{Def2ptfunc}
G_{\rm 2pt}(t) =\sum_{\vec{x}} \, \langle  N(\vec{x},t) \overline{N}(0,0)\rangle\,,
\end{equation}
which is measured in Lattice QCD to compute the nucleon mass. $N,\overline{N}$ denote interpolating fields with the quantum numbers of the nucleon. The sum over the finite spatial volume $V=L^3$ projects to vanishing spatial momentum. Performing the standard spectral decomposition the 2pt function is found to be given by a sum of exponentials,
\begin{equation}
\label{ExpAnsatz}
G_{\rm 2pt}(t) = c_0 e^{-M_N t} + c_1 e^{-E_1 t} + c_2 e^{-E_2 t} + \ldots\,,
\end{equation}
where $t > 0$ and the ordering $M_N < E_1 < E_2 < \ldots $ for the energies is assumed. By construction, the first exponential contains the nucleon mass. The coefficient $c_0$ associated with it is the squared matrix element of the interpolating field between the vacuum and the nucleon at rest. The other terms stem from excited states with the same quantum numbers as the nucleon, and the coefficients $c_j$  involve the matrix elements with the excited states instead of the nucleon state.

In lattice simulations one usually computes the effective nucleon mass, defined as the negative time derivative of $\ln G_{\rm 2pt}(t)$. With \pref{ExpAnsatz} we obtain ($\Delta E_k = E_k-M_N$)
\begin{equation}\label{Meff}
M_{N,{\rm eff}}  = M_N + \frac{c_1}{c_0} \Delta E_1 e^{-\Delta E_{1} t} + \frac{c_2}{c_0} \Delta E_2 e^{-\Delta E_{2} t}+\ldots \,.
\end{equation}
In the limit $t\rightarrow \infty$ the effective mass converges to the nucleon mass. For large but finite $t$ there is an excited-state contamination present.  Although exponentially suppressed the euclidean time separation needs to be sufficiently large for the excited-state contamination to be small. In practice the signal-to-noise problem \cite{Parisi:1983ae,Lepage:1989hd} prevents us from going to very large euclidean time separations to make the excited-state contribution arbitrarily small. The smaller the pion mass the shorter the time separation for which the effective mass can be measured with small statistical errors. Typical time separations at present are between 1 to 1.5 fm.

From experiment we can get some idea about the states that contribute to the 2pt function in a finite spatial volume. There is a contribution from resonance states that are associated to the nucleon resonances in infinite volume, the most prominent one being the Roper resonance $N^*(1440)$.\footnote{The precise relation between infinite volume resonance properties and the finite volume energy spectrum has been worked out first in Ref.\ \cite{Luscher:1991cf}.} In addition there is the contribution from multi-particle states, e.g.\ 2-particle $N\pi$ states, 3-particle $N\pi\pi$ states etc. The multi-particle-state contribution is expected to become rapidly relevant  for pion masses approaching the physical value. This is easily seen by ignoring the interaction energy of the particles, which is expected to be rather small since the pions interact only weakly with the nucleon and with themselves. Within this approximation the energy of the $N\pi\pi$ state with all three particles at rest is equal to $M_N+2M_{\pi}\approx 1.3 M_N$. For a two-particle $N\pi$ state to contribute both nucleon and pion need non-vanishing and opposite spatial momenta. If we assume periodic boundary conditions for the finite spatial volume the spatial momenta are discrete, $\vec{p}_k=2\pi\vec{k}/L$, with $\vec{k}$ having integer-valued components. The larger the lattice extent $L$ the smaller the discrete energies of the $N\pi$ states allowed by the boundary conditions. If we assume the typical value $M_{\pi}L=4$ we find three $N\pi$ states with an energy less than $M_{N^*}\approx 1.5M_N$ (see fig.\ \ref{fig:sketch}). This number increases to seven for $M_{\pi}L=6$, as realized in the simulations of the PACS collaboration \cite{Ishikawa:2015rho}.  At this conference Y.~Kuramashi reported on simulations on even larger lattices with $M_{\pi}L\approx 7.7$  \cite{Kuramashi:Lattice17}, and such volumes imply ten $N\pi$ states with energy below $1.5M_N$. The main conclusion is that quite a few multi-particle states are expected to contribute to the sum in eq.\ (\ref{ExpAnsatz}) before a contribution of the first resonance state appears. Thus, in physical point simulations the multi-particle states dominate the excited-state contamination in the asymptotic regime of large but still finite $t$.

\begin{figure}[tb]
\centerline{\includegraphics[scale=0.4]{figures/sketch}}
\caption{Sketch of the energies of the lowest $N\pi$ states (diamonds) that contribute to the nucleon 2pt function. Discrete momenta corresponding to  $M_{\pi}L=4$ (orange) and $M_{\pi}L=6$ (purple) are considered and interaction energies are ignored. The blue square represents the energy of the $N\pi\pi$ state with all three particles at rest. }
\label{fig:sketch}
\end{figure}

The impact of the multi-particle states to the 2pt function depends also on the size of the coefficients $c_j, j\ge1$. Not much is known  a priori about these coefficients, except for the fact that a coefficient associated with an $n$-particle state is, in a finite spatial volume, suppressed by $(1/L^3)^{n-1}$. Note that this volume suppression cannot be taken as an argument for a small multi-particle-state contribution if $L$ is large. Although the contribution of each individual multi-particle state decreases, more and more states contribute the larger the volume is. In particular, the multi-particle-state contribution is not a FV effect and will not disappear in the infinite-volume limit.

Other observables than the nucleon mass are expected to be even more afflicted by an excited state contamination. Still rather straightforward to compute numerically are nucleon charges and Mellin moments of parton distribution functions. All these observables refer to nucleon matrix elements $\langle N(\vec{p})|O_X| N(\vec{p})\rangle$ of a local quark bilinear $O_X$ between nucleon states and with zero momentum transfer. In case of the standard axial vector current as well as the tensor and scalar density for $O_X$ ($X=A,T,S$) one obtains the axial, tensor and scalar charges $g_A, g_T$ and $g_S$.\footnote{Throughout I will exclusively refer to the flavor nonsinglet charges.} The axial charge is known at the per mille level form neutron beta decay \cite{Olive:2016xmw} and considered to be a benchmark observable for lattice QCD. The scalar and tensor charge are only poorly known, but enjoy a revived interest. New nuclear beta-decay experiments aim at an order of magnitude more precise upper bounds for these charges. In order to constrain beyond-standard-model physics lattice QCD estimates for these charges with 10-15\% uncertainties will be needed \cite{Bhattacharya:2011qm}.

Parton distribution functions (pdfs) are difficult to access directly in Euclidean space-time. Instead, one can define and calculate Mellin moments that can be related to matrix elements of local one-derivative operators. For example, the average quark momentum fraction $\langle x\rangle_{u-d}$ (also called the first unpolarized moment) involves the operator $ V_{\mu\nu} \,=\, \overline{u}\gamma_{{\{\mu}}D^-_{\nu\}} u - \overline{d}\gamma_{{\{\mu}}D^-_{\nu\}} d$ with the (color covariant) derivative  given by $D^-_{\mu} = ( {\overrightarrow D}_{\mu} -  \overleftarrow{D}_{\mu})/2$, and the curly brackets refer to symmetrization with respect to the indices $\mu,\nu$ and subtraction of the trace. The helicity moment $\langle x \rangle_{\Delta u - \Delta d}$ and the transversity moment $\langle x \rangle_{\delta u - \delta d}$ are defined analogously and differ in the gamma matrices involved. Pdf moments are accessible phenomenologically and some of them have been determined with percent accuracy. For example, in Ref.\ \cite{Alekhin:2012ig} the average quark momentum fraction is found as $\langle x\rangle_{u-d,{\rm phen}}=0.1655(39)$ (in the $\overline{\rm MS}$ scheme at 2 GeV).

The standard methods to compute the charges and moments require the computation of the 3pt function.\footnote{The recently proposed method \cite{Bouchard:2016heu} inspired by the Feynman-Hellman theorem avoids the explicit calculation of the 3pt function.} To be specific, we need the 3pt function
\begin{equation}
G_{{\rm 3pt},A}(t,t') = \sum_{\vec{x},\vec{y}} \,\Gamma'_{k,\alpha\beta} \langle N_{\beta}(\vec{x},t) A^3_k(\vec{y},t') \overline{N}_{\alpha}(\vec{0},0)\rangle\,
\end{equation}
in case of the axial charge. The nucleon interpolating fields are the same as in the 2pt function. A spatial component of the non-singlet axial vector current is placed at some operator insertion time $t'$ between source and sink, and the projector $\Gamma'_{k}$ is chosen such that the leading part of the 3pt function is proportional to $g_A$. Forming the ratio with the 2pt function,
\begin{equation}\label{Defratiogeneric}
R_A(t,t')=\frac{G_{{\rm 3pt},A}(t,t')}{G_{\rm 2pt}(t)}\,,
\end{equation}
and performing the spectral decomposition of both numerator and denominator one finds the representation
\begin{equation}
\label{DefRatio}
R_A(t,t')= g_A + b_{A,1} e^{-\Delta E_1 (t-t')} + \tilde{b}_{A,1} e^{-\Delta E_1 t'} + \tilde{c}_{A,1} e^{-\Delta E_1 t } +\ldots\,
\end{equation}
for the ratio. In the limit that all times $t,t'$ and $t-t'$ go to infinity the ratio tends to a constant, the axial charge. For finite times there are again excited-state contributions present stemming from resonance and multi-hadron states. The coefficients $b_{A,1},\tilde{b}_{A,1},\tilde{c}_{A,1}$ in \pref{DefRatio} are ratios of  matrix elements involving the axial vector current as well as the interpolating fields and the states for the vacuum, the nucleon at rest and the first excited state.

The excited-state contributions in eq.\ \pref{DefRatio} involves the three time separations $t-t', t'$ and $t$. As already mentioned, the source-sink separations $t$ accessible in numerical simulations are limited due to the signal-to-noise problem. Since $t-t'$ and $t'$ are even smaller than $t$ we anticipate a significantly larger excited-state contamination in calculations of the axial charge (as well as the other charges and the pdf moments). For large time separations, however, we expect the multi-hadron states to be the dominant source for the excited-state contribution, since these have the smallest gap to the energy of the single nucleon state.

Topic of this talk is that the correlation function ratios can be computed in ChPT. Such calculations provide the coefficients  $b_{A,j},   \tilde{b}_{A,j},\tilde{c}_{A,j} $ in \pref{DefRatio} associated with multi-particle-state contributions. Results are available for the two-particle $N\pi$ contribution in the ratios for the three nucleon charges and the three pdf moments introduced before \cite{Bar:2015zwa,Bar:2016uoj,Bar:2016jof}. For a recent review see Ref.\ \cite{Bar:2017kxh}. Calculations for contributions  of the  $N\pi\pi$ and $\Delta\pi$ states will be analogous but have not been performed yet.

That BChPT can be employed to compute the multi-particle state contribution in correlation functions is neither surprising nor a new idea \cite{Tiburzi:2009zp,Bar:2012ce}. In fact,
Brian Tiburzi \cite{Tiburzi:2009zp} was probably the first to compute the $N\pi$-state contribution to $R_A$ in an attempt to explain, at least qualitatively, why most lattice results at the time underestimated the experimental value of the nucleon axial charge. However,  the $N\pi$-state contribution is found to lead to an overestimation of the axial charge and the idea did not receive much attention.

In the following I will present results for the nucleon axial, tensor and scalar charges and the three pdf moments introduced before. The calculation differs in some details from the early one in Ref.\ \cite{Tiburzi:2009zp}. The discreteness of the nucleon and pion momenta due to a finite spatial volume is taken into account. In addition, the mapping of smeared nucleon interpolating fields, commonly used in lattice calculations, to ChPT has been put on firmer grounds \cite{Luscher:2013vga,Bar:2013ora}. More importantly, having results for six instead of only one observable we find the $N\pi$-state contribution in all six observables to be related. This is nothing but chiral symmetry at work.

\section{The correlation functions in ChPT}

Chiral perturbation theory refers to the low-energy effective theory of QCD based on spontaneous chiral symmetry breaking \cite{Weinberg:1978kz,Gasser:1983yg,Gasser:1984gg}. It has been around for many years and there seems no need to introduce or review it here.\footnote{There exist numerous reviews and lecture notes on ChPT that differ in scope and length, e.g.\ Refs.\ \cite{Colangelo:2000zw,Gasser:2003cg,Kubis:2007iy,Scherer:2009bt,Golterman:2009kw,Scherer:2012xha,Ecker:2013xja}. A particularly useful review of Baryon ChPT that also covers applications to lattice QCD is given in Ref.\ \cite{Bernard:2007zu}.}

The two most prominent applications in Lattice QCD are the calculation of the quark mass dependence of observables and FV effects due to the pions. Starting point for the application we have in mind here, the excited-state contribution caused by two-particle $N\pi$ states, is the same well-established formulation of Baryon ChPT (BChPT) \cite{Gasser:1987rb,Becher:1999he} that is basis for the other applications as well. We work to leading order only, thus we do not encounter any of the subtleties associated with the loop expansion in BChPT. We consider SU(2) BChPT and assume isospin symmetry. In this case the chiral effective theory contains the three mass degenerate pions $\pi^a$ and the nucleon doublet $\Psi=(p,n)^T$ involving the mass degenerate proton and neutron fields. The interaction term relevant here is given by\footnote{We work in Euclidean space time.}
\begin{equation}
{\cal L}_{{\rm int},1\pi} =\frac{ig_A}{2f}\overline{\Psi}\gamma_{\mu}\gamma_5\sigma^a\Psi \partial_{\mu}\pi^a\,,
\end{equation}
which implies the well-known one-pion-exchange-potential (OPEP) between a nucleon pair.  It is proportional to the ratio $g_A/f$ of two LO low-energy coefficients (LECs), the chiral limit values of the axial charge and the pion decay constant.

The effective operators for the axial vector current and the scalar density  have been known for a long time \cite{Gasser:1987rb,Fettes:2000gb}. The operators needed for the average quark momentum fraction and the helicity moment too can be taken from the literature \cite{Dorati:2007bk,Wein:2014wma}. The expressions associated to the tensor operators, needed for the tensor charge and the transversity moment, have been derived following the general procedure described in Ref.\ \cite{Fettes:2000gb}. The explicit expressions are not displayed here but can be found in Refs.\ \cite{Bar:2016uoj,Bar:2016jof}. Still, it is worth pointing out that the chiral limit values of all charges and pdf moments we are interested in appear as LECs in these effective operators. These are input parameters of ChPT.

The chiral expressions for the nucleon interpolating fields are also needed. These too are derived based on the transformation properties under chiral symmetry and parity \cite{Nagata:2008zzc}. The standard local 3-quark baryon operators \cite{Ioffe:1981kw,Espriu:1983hu} have been mapped to ChPT in Ref.\ \cite{Wein:2011ix}. Smeared interpolating fields are mapped to the same pointlike expressions but with different LECs provided the smeared quark fields transform as their local analogues under parity and chiral symmetry. This, for instance, is the case in Gaussian and exponential smearing \cite{Gusken:1989ad,Gusken:1989qx,Alexandrou:1990dq} or the gradient flow \cite{Luscher:2013cpa}. However, the smearing radius must be small compared to the Compton wavelength of the pion, i.e.\ $R_{\rm smear} \ll 1/M_{\pi}$ \cite{Luscher:2013vga,Bar:2013ora}. For physical pion masses this inequality seems reasonably well satisfied for smearing radii up to a few tenths of a fermi.

\begin{figure}[tb]
\centering
\includegraphics[scale=0.45]{figures/figs2pta}\hspace{1cm}\includegraphics[scale=0.45]{figures/figs2ptb}\\
a)\hspace{4.4cm} b)\\[3ex]
\includegraphics[scale=0.45]{figures/figs2ptc}\hspace{1cm}\includegraphics[scale=0.45]{figures/figs2ptd}\\
c)\hspace{4.4cm} d)
\caption{Feynman diagrams for the nucleon 2pt function. The squares represent the nucleon interpolating fields at times $t$ and $0$. The circles represent a vertex insertion at an intermediate space-time point; and an integration over this point is implicitly assumed. The solid and dashed lines represent nucleon and pion propagators, respectively. }
\label{fig:Npidiagrams2pt}
\end{figure}

With these elements the calculation of the 2pt and 3pt functions is a standard perturbative calculation in ChPT. It seems convenient to use the time-momentum representation for the pion and nucleon propagators since it directly yields the time dependence of the correlation functions.
Figure \ref{fig:Npidiagrams2pt} shows the leading diagrams with an $N\pi$-state contribution to the 2pt function. These diagrams contain a contribution that drops off exponentially with the total energy $E_{N\pi,n} = E_{N,n}+E_{\pi,n}$ of a nucleon-pion state, where the discrete spatial momenta are back-to-back, $\vec{p}_{N,n}=-\vec{p}_{\pi,n}$. We are interested in the prefactor of this exponential. Note that even though the diagrams in fig.\ \ref{fig:Npidiagrams2pt} are 1-loop diagrams the calculation of the $N\pi$-state contribution does not involve a summation over some undetermined momentum, i.e.\ it is a tree-level calculation.\footnote{Diagrams b) - d) also contain single-nucleon contributions dropping off with $\exp(-M_N t)$, and this contribution does involve a sum over all momenta of the intermediate $N\pi$ pair. Diagram b), for instance, is essentially the nucleon self-energy diagram that leads to the well-known $M_{\pi}^3$ term in the 1-loop ChPT result for the nucleon mass \cite{Gasser:1987rb}.}
Figure \ref{fig:Npidiagrams3pt} shows the leading diagrams with an $N\pi$-state contribution to the 3pt functions.

The calculation of the Feynman diagrams is straightforward. The result for the ratio of the 3pt and 2pt functions is of the general form
\begin{equation}
\label{DefRatio2}
R_X(t,t')= g_X \Big[1+ \sum_{\vec{p}_n} \Big(b_{X,n} e^{-\Delta E_n (t-t')} + \tilde{b}_{X,n} e^{-\Delta E_n t'} + \tilde{c}_{X,n} e^{-\Delta E_n t }\Big)\Big]
\end{equation}
for all three nucleon charges ($X=A,T,S$), with $\Delta E_n = E_{N\pi,n}-M_N$. An analogous result is found for the moments. The non-trivial results of the calculation are the coefficients $b_{X,n}, \tilde{b}_{X,n}, \tilde{c}_{X,n}$ in all six ratios. The exact expressions for these coefficients can be found in Refs.\ \cite{Bar:2016uoj,Bar:2016jof} and are not displayed here. Still, it is noteworthy that all coefficients involve a product of two universal factors, for example
\begin{equation}
 \tilde{c}_{{X},n} = \frac{1}{16(fL)^2E_{\pi,n}L} \left(1-\frac{M_N}{E_{N,n}}\right)\Bigg(\ldots\Bigg)\,.
\end{equation}
The first factor on the right hand side shows the anticipated volume suppression factor $1/L^3$ of a two-particle state in a finite spatial volume. It combines with the pion decay constant and the pion energy to the dimensionless combination $1/(fL)^2E_{\pi,n}L$. The second factor $(1-M_N/E_{N,n})$ vanishes if the momentum of the nucleon (and the pion) is zero. This has to be the case since the nucleon-pion state with both particles at rest does not contribute to the correlation function because it is parity odd. The remaining factor is non-universal and not very illuminating.

\begin{figure}[t]
\centering
\includegraphics[scale=0.45]{figures/VCa}\hspace{0.3cm}\includegraphics[scale=0.45]{figures/VCm}\hspace{0.3cm}\includegraphics[scale=0.45]{figures/VCn}\\[4ex]
\includegraphics[scale=0.45]{figures/VCo}\hspace{0.3cm} \includegraphics[scale=0.45]{figures/VCe}\hspace{0.3cm}\includegraphics[scale=0.45]{figures/VCf}\\[4ex]
\includegraphics[scale=0.45]{figures/VCi}\hspace{0.3cm}\includegraphics[scale=0.45]{figures/VCj}\hspace{0.3cm}
\includegraphics[scale=0.45]{figures/VCc}\\[4ex]
\includegraphics[scale=0.45]{figures/VCd}\hspace{0.3cm}\includegraphics[scale=0.45]{figures/VCh}\hspace{0.3cm}\includegraphics[scale=0.45]{figures/VCl}\\[4ex]
\caption{Feynman diagrams for the LO nucleon-pion contribution in the 3pt functions. The diamond stands for the operator insertion at time $t'$.}
\label{fig:Npidiagrams3pt}
\end{figure}

More interesting than the exact expressions is the fact that the coefficients do not depend on the LECs associated with the nucleon interpolating fields. To LO these cancel in the ratio $R_X$. Consequently, the LO $N\pi$-state contribution to the ratios is the same for pointlike and for smeared interpolating fields, since the difference between these interpolators is encoded in their different values for the LECs.
However, this universality property will be lost at higher orders in the chiral expansion.

The coefficients depend on a few parameters only: The pion mass and pion decay constant, the finite volume via $M_{\pi}L$ and four more LECs (the chiral limit values of the nucleon mass, the axial charge, the average quark momentum fraction and the helicity moment). All these parameters can be estimated using their experimental or phenomenological values. With this input we can estimate the $N\pi$-state contribution to the ratios calculated in lattice simulations.

\section{Impact on lattice calculations}
The simplest estimator for the charges and pdf moments is the so-called midpoint estimate. It is based on a simple observation: For a given source-sink separation $t$ the excited-state contribution to $R_X$ is minimized if the operator insertion time $t'$ is in the middle between source and sink. Therefore, the best estimate for the charges and moments is the midpoint value $R_X(t,t/2)$. It is essentially equivalent to what is called plateau estimate, and we will use this terminology in the following.

The experimental/phenomenological values for the masses and LECs can be found in the literature \cite{Olive:2016xmw,Alekhin:2012ig,Blumlein:2010rn}.\footnote{For the results presented here the following simplified values have been used: $M_{\pi}=140$ MeV, $M_N=940$ MeV, $g_A=1.27$, $f_{\pi}= 93$ MeV, $\langle x\rangle_{u-d}=0.165$ and $\langle x \rangle_{\Delta u - \Delta d}=0.19$. Errors in these estimates are ignored since they are too small to be significant for the LO ChPT results.} With these input parameters being fixed the LO ChPT result for  $N\pi$-state contribution is a function of the spatial volume only.

\begin{figure}[t]
\begin{center}
$R_A(t,t/2)/g_A$\\
\includegraphics[scale=0.85]{figures/figaxialcharge}\\
$t$ [fm]
\caption{The plateau estimate $R_A(t,t/2)$ normalized by $g_A$ as a function of the source-sink separation $t$. Results are shown for $M_{\pi}L=4$ (solid lines),  $M_{\pi}L=5$ (dotted lines) and $M_{\pi}L=6$ (dashed lines). Different colors distinguish between three different upper energy bounds for the $N\pi$ states taken into account (see main text).}
\label{fig:axialcharge}
\end{center}
\end{figure}

Fig.\ \ref{fig:axialcharge} shows the plateau estimate $g_{A,{\rm plat}}(t)\equiv R_A(t,t/2)$ for the axial charge divided by $g_A$
 as a function of the source-sink separation $t$.
Without the excited-state contribution this ratio would be equal to 1. Any deviation from this constant value shows directly the $N\pi$ contribution to the plateau estimate in percent. Results are shown for three different lattice sizes with $M_{\pi}=4,5$ and 6 (solid, dotted and dashed lines).  Different colors refer to different numbers of $N\pi$ states taken into account in the sum over the discrete spatial momenta $\vec{p}_n$, cf.\ equation \pref{DefRatio2}. For the black lines $N\pi$ states with a total energy up about 1.35 GeV are included. The blue and red lines correspond to total energies up to about 1.6 and 1.9 GeV.
Note that the number of states corresponding to these energies depends on the spatial lattice extent $L$. For example, the black lines in fig.\ \ref{fig:axialcharge}
correspond to 2, 3 and 5 states for $M_{\pi}L=4,5$ and 6, respectively.

Apparently, the differences between lines of the same color are very small, i.e.\ the $N\pi$ contribution is essentially the same for the three different $M_{\pi}L$ values displayed in the figure. As already mentioned in the introduction, the $1/L^3$ suppression of the two-particle-state matrix elements is expected to be largely compensated by a growing number of $N\pi$ states in a given energy interval. Still, it is perhaps somewhat surprising that the differences between the three different volumes are so small.

Fig.\ \ref{fig:axialcharge} also shows that the $N\pi$ contribution leads to an overestimation of the axial charge by the plateau estimate. Before reading off a number from the plot recall that ChPT is an expansion in small pion momenta. We may expect reasonably reliable ChPT results for the low-momentum $N\pi$ states, depicted by the black lines in the plot. The chiral expansion of the high-momentum $N\pi$ states, included in the blue and red curves, is expected to work less well if it works at all.

This restriction implies that the source-sink separation $t$ needs to be large enough for the contribution of the high-momentum states to be sufficiently suppressed such that the low-momentum states provide the dominate part of the total $N\pi$ contribution. According to fig.\ \ref{fig:axialcharge} this seems satisfied for $t\approx 2.5$ fm, but certainly not for $t\approx 1.5$ fm where the low-momentum states contribute less than half of the contribution given by the red curves. It is at about $t\approx 2$ fm that the black lines start to capture the dominant $N\pi$ contribution, thus we may conclude that source-sink separations of about 2 fm and larger are required for ChPT to make reliable estimates. How large the higher order corrections at these source-sink separations are is difficult to predict. Naive error estimates suggest a 30-50\% uncertainty for $t\gtrsim2$ fm. A more solid error estimate requires the calculation at NLO.

\begin{figure}[t]
\centering$R_X(t,t/2)/g_X$ and $R_X(t,t/2)/\Pi_X$ \\
\includegraphics[scale=0.6]{figures/figall}\\
$t$ [fm]
\caption{The plateau estimates $R_X(t,t/2)$ normalized by the asymptotic values for all charges and moments. Results are shown for $M_{\pi}L=4$ with $N\pi$ states included with energies up to 1.6 GeV. The result for the axial charge (blue solid line) is the same as the blue solid line in fig.\ \ref{fig:axialcharge}.}
\label{fig:resultsall}
\end{figure}

Qualitatively the same results are found for the tensor and scalar charges and the three moments. Fig.\ \ref{fig:resultsall} shows the plateau estimates normalized by either the charge or the moment. As for the axial charge we only find a small dependence on $M_{\pi}L$ in all cases,
so fig.\ \ref{fig:resultsall}  shows the $M_{\pi}L=4$ results only. In all cases we find the low-momentum $N\pi$ states start to dominate for $t\gtrsim 2$ fm, thus the results displayed in fig.\ \ref{fig:resultsall} are expected to be reliable only for $t\gtrsim2$ fm.

According to fig.\ \ref{fig:resultsall} the $N\pi$-state contribution leads to an overestimation for all three charges and the three moments by their plateau estimates. The largest overestimation is found for the scalar charge, which is about twice as large as for the axial charge. The $N\pi$ contribution to the average momentum fraction is close to the one for the scalar charge. In summary we find an overestimation of about 5-10\% for the charges and moments at $t\approx 2$ fm, and it slowly decreases to 3-6\% at $t\approx 2.5$ fm.

We stress once again that these numbers are results at LO, i.e.\ at O($p$).  The higher order corrections are hard to quantify without having done the calculation at NLO. However, naive arguments can be made for the O($p^2$) corrections and suggest error estimates up to about 50\%.

\section{Comparison with lattice data}

Most of the existing lattice results for the charges and moments have been obtained for pion masses larger than the physical value. There exist many reviews summarizing these results, for example Refs.\  \cite{Syritsyn:2014saa,Green:2014vxa,Constantinou:2015agp,Alexandrou:2016hiy}. Moreover, Sara Collins presented a review talk at Lattice 2016 in Southampton \cite{Collins:Lattice16}, and I refer to it for details concerning the present status of the lattice calculations.

Some collaborations have carried out lattice calculations with the pion mass at or near the physical value \cite{Bali:2014nma,Abdel-Rehim:2015owa,Bhattacharya:2016zcn,vonHippel:2016wid,Green:2012ud,Bali:2014gha}. The main obstacle for directly applying the ChPT results to these lattice calculations are the rather small source-sink separations in these simulations. In most cases the maximal source-sink separation  used to extract the charges and moments with the plateau method is below 1.5 fm. As discussed in the previous section, ChPT is not expected to provide solid results unless the source-sink separations are about 2 fm or even larger. Still, it is useful to compare the existing lattice results with our ChPT predictions, since the latter need to be reproduced eventually.

In the following we restrict ourselves to data obtained with the plateau method with pion masses not larger than 165 MeV. Lattice results obtained with the summation method \cite{Green:2012ud}, two-state fits \cite{Bhattacharya:2016zcn} or the method based on the Feynman-Hellman theorem \cite{Bouchard:2016heu,Berkowitz:2017gql} cannot be used in our comparison.

\begin{figure}[t]
\centering
$g_{A,{\rm plat}}(t)/g_{A,{\rm exp}}$ \\
\includegraphics[scale=0.85]{figures/compgA_new}\\
$t$ [fm]
\caption{Lattice plateau estimates for the axial charge normalized by the experimental value. Lattice data from ETMC (blue), NME (magenta) and RQCD (red). The ChPT result of fig.\ \ref{fig:resultsall} is also shown for $t\ge 2$ fm by the blue solid line together with a 50\% error band as a naive estimate for higher order corrections.}
\label{fig:compgA}
\end{figure}

Figure \ref{fig:compgA} shows the renormalized plateau estimates obtained by  ETMC \cite{Abdel-Rehim:2015owa,Alexandrou:PC}, NME \cite{Yoon:2016jzj,Gupta:Lattice17} and RQCD \cite{Bali:2014nma}, divided by the experimental value $g_{A,{\rm exp}}$ \cite{Olive:2016xmw}.\footnote{The ETMC results shown in fig.\ \ref{fig:compgA} are based on larger statistics compared to the ones given in Ref.\ \cite{Abdel-Rehim:2015owa}. They have smaller error bars but are still preliminary \cite{Alexandrou:PC}. I thank C.\ Alexandrou and C.\ Kallidonis for sharing these results. I also thank R.\ Gupta, Y.-C.\ Jang and B.\ Yoon for sending me numerical results published in Ref.\ \cite{Yoon:2016jzj}. } The ETMC results (blue symbols) were obtained with $N_f=2$ twisted mass fermions with $M_{\pi}\approx 130$ MeV, $M_{\pi}L\approx 3$ and $a\approx 0.093$. The NME results (magenta symbols) were generated with $N_f=2+1$ Wilson-clover fermions  with $M_{\pi}\approx 165$ MeV, $M_{\pi}L\approx 3.7$ and $a\approx 0.09$. The remaining data point (red symbol) from RQCD was obtained with $N_f=2$ improved Wilson fermions with $M_{\pi} \approx 150$ MeV, $M_{\pi}L\approx 3.5$ and $ a\approx 0.071$ fm. More details about the simulation setup are given in the original papers.

The plateau estimates shown in fig.\ \ref{fig:compgA} were obtained for source-sink separations between 0.9 and 1.5 fm. They are all below the experimental value although the NME result at $t\approx1.5$ fm agrees with $g_{A,{\rm exp}}$ within the statistical error.

The ChPT prediction for the overestimation due to the $N\pi$ states is also shown for $t\ge 2$ fm. A naive guess for the higher order corrections is also included in form of a 50\% error band. Apparently, the lattice data is not in conflict with the ChPT result. It seems very plausible that the lattice data will  connect smoothly to the ChPT prediction when $t$ is increased. Whether this indeed happens needs to be checked. This requires lattice calculations of $g_A$ at $t$ larger than 1.5 fm with a statistical error of a few percent.\footnote{In addition one should also check that the underestimation of the axial charge at small source-sink separations persists in the continuum limit.}

If indeed realized, such a scenario can be quite misleading in practice. Since the plateau estimate approaches the experimental value at some $t$ well before the asymptotic region is reached, one might be tempted to stop simulating at larger source-sink separations. In that case one reproduces the correct experimental value $g_{A,{\rm exp}}$, but for the wrong reason. The excited-state contributions are not small because $t$ is sufficiently large to be in the asymptotic regime. Instead, various excited-state contributions are still sizeable but accidentally cancel each other.

A concrete model for such a scenario was suggested by M.\ Hansen and H.\ Meyer in Ref.\ \cite{Hansen:2016qoz}. Based on plausible assumptions concerning the higher order corrections to the LO ChPT results, the high-momentum $N\pi$ states with energies larger than about 1.5 $M_N$ contribute {\em negatively} to the plateau estimate. Summing up the total $N\pi$ contamination the positive contribution from the low-momentum states is overcompensated by the contribution from the high-momentum states, leading to an underestimation of the axial charge for source-sink separations below about $1.5$ fm, in agreement with lattice results.

Whether the high-momentum $N\pi$ states are indeed responsible for the compensation of the low-momentum ones needs to be corroborated. Other excited states (e.g.\ $N\pi\pi$ and $\Delta\pi$ states) are expected to play a non-negligible role too for source-sink separations below 2 fm. In any case, the model in Ref.\ \cite{Hansen:2016qoz} supports the expectation that source-sink separations $t\gtrsim2$ fm are needed for the LO ChPT results to be applicable.

\begin{figure}[p]
\centering
$\langle x\rangle_{u-d,{\rm plat}}(t)/\langle x\rangle_{u-d,{\rm phen}}$ \\
\includegraphics[scale=0.85]{figures/compavxUpdate}\\
$t$ [fm] \\[5ex]
$\langle x\rangle_{\Delta u-\Delta d,{\rm plat}}(t)/\langle x\rangle_{\Delta u-\Delta d,{\rm phen}}$  \\
\includegraphics[scale=0.85]{figures/comphelmomUpdate}\\
$t$ [fm]
\caption{Lattice plateau estimates for the average momentum fraction (upper panel) and the helicity moment (lower panel) normalized by their phenomenological values.  Lattice data from ETMC (blue) and RQCD (red). The ChPT result of fig.\ \ref{fig:resultsall} is also shown for $t\ge 2$ fm together with a 50\% error band as a naive estimate for higher order corrections.}
\label{fig:compavmom}
\end{figure}

Fig.\ \ref{fig:compavmom}  shows the plateau estimates of ETMC \cite{Abdel-Rehim:2015owa,Alexandrou:PC} and RQCD \cite{Bali:2014gha} for the average quark momentum fraction $\langle x\rangle_{u-d}$ and the helicity moment $\langle x \rangle_{\Delta u - \Delta d}$, normalized by their phenomenological values given in \cite{Alekhin:2012ig,Blumlein:2010rn}. For both observables the plateau estimates overestimate the phenomenological value. The absolute discrepancy is significantly larger than for the axial charge (note the different scales in fig.\ \ref{fig:compavmom}).  The statistical errors are rather large, but the plots suggest that the plateau estimates decrease as the source-sink separation increases. Compared with the results for the axial charge it is here much simpler to imagine a simple monotonic decrease until contact with ChPT can be made at source sink-separations between 2 and 2.5 fm.

In case of the scalar and tensor charges  as well as the transversity moment we do not have experimental values at our disposal. Thus, comparisons between the lattice plateau estimates and the ChPT predictions cannot be done. However, there is no reason to believe that the asymptotic region is reached at significantly smaller source-sink separations for these observables. In fact, the absence of experimental results
elevates the role ChPT can play as a guide for the systematic uncertainties due the $N\pi$ states, provided one is in the asymptotic regime where ChPT can be applied.

The main conclusion we can draw from our comparison is that present lattice data at source-sink separations $t\lesssim 1.5$ fm is still far away from the asymptotic regime where the lowest lying $N\pi$ states are responsible for the dominant excited-state contribution in the plateau estimates. Lattice data at significantly larger source-sink separations (and with sufficiently small statistical errors) are needed to make contact with the asymptotic region where ChPT can be applied.

\section{Miscellaneous comments}

An alternative way to compute the charges and pdf moments is provided by the summation method \cite{Maiani:1987by,Capitani:2012gj}.
It starts from the ratio $R_X(t,t')$ and computes the sum over all operator insertion times between source and sink. In case of the axial charge the result of the sum is of the form
\begin{equation}\label{sumestimate}
\sum_{t'=0}^t R_A(t,t') = t[g_A + {\rm O}(e^{-\Delta E_n t})] + \dots\,.
\end{equation}
The slope $[\ldots]$ is called the summation estimate of the axial charge. For finite $t$ it differs by exponentially suppressed excited state contributions from $g_A$.

Using the ChPT results for the ratios in eq.\ \pref{sumestimate} the $N\pi$-state contribution to the summation estimate is easily computed. However, the sum on the left hand side involves the 3pt function at short distances when $t'$ approaches either source or sink. In these cases the ChPT results do not properly capture the full 3pt function and the ratio.

A possible remedy for this limitation would be to keep all time separations large by summing over a central subinterval only. This, however, requires even larger source-sink separations than those needed for the plateau method. Based on our results we would expect to need a minimal separation of about 1 fm for $t-t'$ and $t'$. In addition we need a non-zero time interval to sum over. Both requirements imply minimal source-sink separations of about 2.5 fm if not larger. Note that this is not a requirement for the summation method per se, but for ChPT to have something to say about it.

For the computation of the various ratios we had to compute the 2pt function. It is interesting to also look at the $N\pi$ contribution to the effective nucleon mass, c.f.\ eg.\ \pref{Meff}. Fig.\ \ref{fig:effmass} shows the ratio $M_{N,{\rm eff}}(t)/M_N$ as a function of the source-sink separation $t$.
Without the excited state contribution this ratio would be equal to 1, any deviation from this value stems from the $N\pi$ contribution. As in figure \ref{fig:axialcharge}, results are shown for the three different lattice sizes (solid, dashed and dotted lines)  and the three different energies $E_{N\pi,{n_{\rm max}}}$ (black, blue and red lines) considered before.

Qualitatively figs.\ \ref{fig:axialcharge} and \ref{fig:effmass} look very similar. Only a very small FV dependence is visible, provided the number of states are adjusted according to the energy interval for the $N\pi$ states taken into account for the $N\pi$ contribution. The larger $t$ the more dominates the contribution of the low-momentum $N\pi$ states (black curves). The actual time $t$ where this happens, however, is much smaller compared to the result for the axial charge. Even more importantly, the $N\pi$ state contamination in the effective mass is significantly smaller. Even allowing for an error as big as 100\% for the $N\pi$ contribution the overestimation of the nucleon mass is smaller than 2\% at source-sink separations of about  1.2 fm and larger. This contamination can be ignored unless the lattice data have statistical errors at the sub-percent level.

\begin{figure}[t]
\centering
$M_{\rm eff}(t)/M_N$\\
\includegraphics[scale=0.85]{figures/figeffmass}\\
$t$ [fm]
\caption{The ratio $M_{N,{\rm eff}}(t)/M_N$ as a function of the source sink separation $t$. Results are shown for $M_{\pi}L=4$ (solid lines),  $M_{\pi}L=5$ (dotted lines) and $M_{\pi}L=6$ (dashed lines). Different colors distinguish between three different upper energy bounds for the $N\pi$ states taken into account (see main text).}
\label{fig:effmass}
\end{figure}

Note that the significantly larger excited-state contamination in the charges and moments stems from the contributions associated with the time separations $t-t'$ and $t'$. For this reason the $N\pi$ contribution at a given time $t$ in fig.\ \ref{fig:effmass} should be compared the one at $2t$ in fig.\ \ref{fig:axialcharge}. In that case the $N\pi$ contribution in both observables is roughly of the same size.\footnote{The factors $\Delta E_n/M_N$ in $M_{\rm eff}/M_N$, c.f.\ eq.\ \pref{Meff}, causes an additional suppression of the excited-state contribution in the effective mass.}

The conclusions we have drawn are based on the LO results for the $N\pi$-state contribution. This contribution dominates for large time separations in physical point simulations. A calculation of the NLO corrections is certainly desirable, mainly to obtain firmer error estimates for the LO results.
In addition, it needs to be checked that contributions of other excited states are small and do not change qualitatively the results we have found. Other multi-hadron-state contributions have their origin in three-particle $N\pi\pi$ states and two-particle $\Delta \pi$ states. ChPT calculations of these contributions are essentially analogous to the calculation of the $N\pi$ contribution and in principle straightforward to perform.

\section{Conclusions}

Physical point simulations eliminate the systematic uncertainties associated with a chiral extrapolation. Excited-state effects due to multi-hadron-states involving one or more pions, on the other hand, become more pronounced for pion masses as light as in Nature. Using Baryon ChPT we have computed the two-particle $N\pi$ state contamination in the plateau estimates of various nucleon charges and pdf moments. This particular excited-state contamination leads to an overestimation at the 5-10\% level for source-sink separations of about 2 fm. This is uncomfortably large and cannot be not ignored if results with percent precision are the goal of lattice calculations.

Even more troublesome is that lattice calculations of the charges and moments with source-sink separations of 2 fm and larger are out of reach with present simulation techniques. So far most lattice results have been obtained with source-sink separations of 1.5 fm and smaller. At time separations that small one expects excited states other than $N\pi$ states to contribute a sizeable excited-state contamination too.

In case of the axial charge the lattice data together with the ChPT result strongly suggests a non-monotonic $t$ dependence of the plateau estimate. The lattice data underestimate the experimental value for $t\lesssim 1.5$ fm while ChPT predicts an overestimation for larger time separations. This peculiar behavior, neither seen for the average quark momentum fraction nor the helicity moment, can be attributed to other excited states that contribute a negative contamination to the plateau estimate. In this case we expect a cancellation of the various excited-state contributions for some source-sink separation and an accidental agreement between the lattice result and the experimental value for the axial charge. Given the benchmark character of $g_A$ it thus seems vital to carefully check the excited-state contamination in this quantity with lattice simulations at source-sink separations larger than 1.5 fm.

The main conclusion we can draw from our ChPT calculation is that physical point simulations require significantly larger source-sink separations than those with heavier pion masses. Simulations with  larger time separations require new simulation techniques to overcome the notorious signal-to-noise problem. Some ideas in that respect have been recently proposed \cite{Ce:2016idq,Ce:2016ajy,Wagman:2017xfh} and presented at this conference \cite{Giusti:Lat17,Wagman:Lat17}, but remain to be tested in actual lattice calculations of the nucleon correlation functions considered here.

Alternatively one may resort to the variational method \cite{Luscher:1990ck,Blossier:2009kd} to get a handle on the nucleon matrix elements at smaller time separations. The variational method can be used provided interpolating fields for the $N\pi$ states are taken into account \cite{Lang:2012db,Kiratidis:2015vpa,Lang:2016hnn,Kiratidis:2016hda}. In this approach too the numerical cost grows significantly since the number of Wick contractions involved in computing the correlation functions grows rapidly once interpolating fields for multi-particle states are used \cite{Lang:2016hnn}.

\section*{\small Acknowledgments}
Communication with C.~Alexandrou, R.~Gupta, Y.-C.~Jang, M.~Kallidonis and R.~Sommer is gratefully acknowledged.
This work was supported by the Japan Society for the Promotion of Science (JSPS) with an Invitation Fellowship for Research in Japan (ID No. L16520), and by the German Research Foundation (DFG), Grant ID BA 3494/2-1.

redpajama_set_name: RedPajamaArXiv


BOSTON, MA - For the second year in a row, the Red Sox Foundation, in partnership with KIND Healthy Snacks, will hold yoga sessions at Fenway Park on Sunday, June 8. "FenwaYoga" will raise funds for two of the foundation's cornerstone programs, the Red Sox Scholars Program and the RBI Youth Baseball and Softball Programs.
Two classes, each accommodating 222 participants along the warning track, will be taught by celebrity yoga instructor Mandy Ingber, a New York Times bestselling author and the creator of Yogalosophy. For over 18 years Mandy has motivated people with her light-hearted approach and unique teaching style. Her public appeal grew as a celebrity yoga instructor when she did a magazine spread in 2006 with client Jennifer Aniston. Other clients include Kate Beckinsale, Ricki Lake, Brooke Shields, and Helen Hunt to name a few.
The two morning classes will take place at 9 a.m. and 10:30 a.m. and are geared towards all levels of experience. Each participant must commit to raising a minimum of $250 for the foundation. There will be prizes awarded for the top three fundraisers. KIND will provide snacks made from nutritionally-rich ingredients following each session.
Those interested in signing up can register at fenwayoga.com.


Christian boarding schools in Saint Marys, Georgia. Free help finding the best Christian boarding school in Saint Marys.
Is your teenager spinning out of control? Looking for a therapeutic boarding school near Saint Marys, Georgia?
Let us assist you in researching the best program whether or not it is near Saint Marys, Georgia. We've made it our life goal to know all we can about every school, so we can help parents (without cost) find the best possible choice for their troubled teen. We want to help you avoid the same mistake we did.
Now, every day we hear from many frustrated parents in towns like Saint Marys, Georgia, who share how their once normal and happy child has become dangerously out of control — just like our son was. They talk about how it has disrupted their family and how they fear for their child's future and very life. "My child is no longer who she used to be," is an often repeated cry from these parents.
If your child is out of control, he or she needs you to intervene. That's why you've landed on this site, so allow us to be the help you need to help you find just the right program, hopefully near Saint Marys, Georgia. The downward spiral your teen is on can have tremendous destructive potential with lifelong consequences, or even bring a teen's life to a quick end. Don't wait. Act today based on what you know is true — your faith, your own beliefs, and what you know is best for your child.
Give us the opportunity to help you find a Christian boarding school closest to Saint Marys, Georgia that will provide your teen with the help and mentoring he or she needs, and within your budget. We've visited most of these programs and we know the staff there. We know what they are good at, and what they are not so good at. We poll parents who have placed their teen in these programs, so we know how well they have done to help the teen.
You may not want to put your child in a boarding school nearby Saint Marys merely out of convenience (like we did). Rather, find the program that is most likely to help your child, even if it is outside of your state. Give us the chance to tell you about a few of them. Fill in the inquiry form now to begin that process. There is NO CHARGE for this service.
More about Saint Marys, Georgia:St. Marys is a city in Camden County, Georgia, United States. The city is the gateway to Cumberland Island National Seashore, the largest of the Georgia Coast's barrier islands. The Islands visitor center and boat access are both located at the St. Marys waterfront. The city is also home to the annual St. Marys Rock Shrimp Festival, the St. Marys Submarine Museum, the Kings Bay Naval Submarine Base, and to eight Ohio-class submarines. The population of St. Marys was 29,986 at the 2010 census.


text:
Christie's Unveils Stellar Selection For London's The Art of The Surreal Auction
11 January 2015 / Art Categories Art Market, News / Art Tags 2015, Auction, Christie's, The Art of The Surreal / / / / /
Christie's stellar selection of 36 lots in the much anticipated annual London evening sale of The Art of The Surreal on 4 February 2015. The auction comprises works from many important collections, led by Joan Miró's masterful L'oiseau au plumage déployé vole vers l'arbre argenté, 1953, from a Distinguished European Collection (estimate: £7-9 million, illustrated above left). Tailored to meet the market's ever growing appetite for surrealist works, further highlights include notable examples by Magritte, Chagall, Picabia, Arp, Ernst, Tanguy and Dominguez. Christie's overall evening of Impressionist, Modern and Surrealist Art on 4 February has a total pre-sale estimate of £92.8 million to £133.8 million (please see separate Impressionist and Modern Art Evening Sale release).
Olivier Camu, Deputy Chairman, Impressionist and Modern Art, Christie's: "This February Christie's is offering the strongest and most valuable grouping of Surrealist art to be offered at auction, worldwide, to date. At the core of the group are two anonymous private collections which are each the result of over 50 years of passionate collecting by their respective owners. We are proud to present the most important and impressive group of works by Magritte and Miró to come to the market. Many works are at auction for the very first time and are thus totally fresh to the market and should appeal to today's broad group of discerning international collectors, who increasingly recognise and appreciate the influence that the Surrealists had on subsequent artists of the 20th and 21st centuries."
This auction comprises a total of six works by Joan Miró, providing the market with what is arguably the best group of works by Miró to be offered in a single sale. The top lot of the group is Joan Miró's masterful L'oiseau au plumage déployé vole vers l'arbre argenté, 1953, which is a remarkable example of his artistic developments in the early 1950s (estimate: £7-9 million, illustrated above centre). Covering a large canvas, the painting combines a wealth of effects, pairing precise, graphic details with expressive effusions of paint.
Works from A Distinguished European Collection, the same collection from which Francis Bacon's major portrait Seated Figure (Red Cardinal), which was recently sold in New York on 12 November for $45 million/ £28 million. The full collection of 31 twentieth century works of art is expected to realise a combined final total in excess of £64 million ($100 million), for the press release on this collection please click here.
A total of 9 works by René Magritte are presented in the sale, 8 of which are from the Distinguished European Collection which comprises 4 oils on canvas and 4 gouaches on paper. This season's offering by the artist spans Magritte's oeuvre and includes much of his most iconic imagery, together constituting the most extraordinary and extensive group of works by Magritte to come to the market since the landmark Harry Torczyner sale in 1998, which took place at Christie's New York and established new market price levels for the artist. Leading the group is Les compagnons de la peur, a triumphantly eerie work from 1942 which centres on the subject of transformation, challenging the realms of possibility and turning the traditions of landscape and portraiture painting on their head, to present – perched atop a mountain – four owls transforming from bird to plant world, or vice-versa
(estimate: £2.7-3.5 million, illustrated left). Magritte's highly poetic gouache Le thérapeute, 1962, subverts the nature of pictures and the concept of the portrait with the captivating absence at the heart of the composition: the torso of the subject is a void through which the blue sky and clouds of a summer's day entices the viewer beyond the twinkling night sky (estimate: £600,000-900,000, illustrated page 1 and right).
A beautifully composed painting that dates to a period of great happiness and stability for Marc Chagall, Jeune fille au cheval, 1927-1929, is offered at auction for the first time, having been acquired by A Private European Family almost 60 years ago (estimate: £2.2-2.8 million, illustrated right). A mirage of surreal and magical lyricism and blissful romance which encompasses Chagall's favoured themes of love, memory, music and fantasy, Jeune fille au cheval exemplifies the artist's unique and deeply personal artistic vision. Executed in iridescent, delicate
Chagall's colours and soft brushstrokes, it depicts a woman adorned in flowers sitting atop a horse emerging from a misty blue haze, while an airborne violinist plays across from her, with a rural street scene from Chagall's beloved hometown, Vitebsk, beyond. Chagall's works from this period have a soothing, gentle atmosphere. Filled with flowers, lovers or fiddlers, they all exude a poetic harmony. In Jeune fille au cheval the components of the composition are unified through the delicate light, rich colours, and tender, romantic mood. The rich blue, dreamlike haze from which the image emerges could have developed from Chagall's fascination with the French landscape, particularly the Côte d'Azur, which the artist had visited for the first time in 1926, three years before Jeune fille au cheval was painted. The gentle glow of light lends the painting a pictorial cohesion and compositional unity while evoking the fantastical, imaginary context.
The sale also presents surrealist works by a total of 15 artists, including an exquisite painting by Yves Tanguy from 1935, Composition surréaliste, depicting strange rock-like forms or personages stretching out across the painting like a strip of calligraphy (estimate: £450,000-550,000, illustrated left); and Madamme, 1937, by Oscar Domínguez, which dates from the artist's partisan years within Surrealism, when he contributed extensively to the movement and produced some of his most significant works (estimate: £400,000-600,000, illustrated right). The other artists featured are: Salvador Dalí, Paul Delvaux, Alberto Giacometti, André Masson, Matta, Odilon Redon and Antoni Tàpies.

redpajama_set_name: RedPajamaCommonCrawl


text:
package in.biconsoft.controller;

import javax.inject.Singleton;
import javax.ws.rs.GET;
import javax.ws.rs.Path;
import javax.ws.rs.Produces;
import javax.ws.rs.core.MediaType;
import javax.ws.rs.core.Response;
import javax.ws.rs.core.Response.Status;

import org.springframework.beans.factory.annotation.Autowired;

/*
 * Scope discussion here
 * http://jersey.576304.n2.nabble.com/Spring-framework-support-for-Jersey-2-td7580673.html
 */
@Path("/user")
@Singleton
public class Hello {
	@Autowired
	public String abc;

	@GET
	@Path("/hey")
	@Produces(MediaType.TEXT_HTML)
	public Response topQueriesXml() {
			return Response.status(Status.ACCEPTED).entity("jims says hello to you !!")
				.build();
	}
}

redpajama_set_name: RedPajamaGithub


The Hungarian government is in "continuous communication" with Ukraine's ethnic Hungarian KMKSZ party over recent developments in Ukraine, Foreign Minister Peter Szijjarto said on Monday.
Speaking in Bonn, Szijjarto called it "obviously bad news"that "martial law has been declared in a neighboring country". It is in Hungary's interest that peace should prevail "in countries in which lots of Hungarians live", he said, adding that security for Ukraine's 150,000-strong Hungarian community was "crucially important". Szijjarto said hopefully the situation in Ukraine would be resolved soon.
Foreign Minister Peter Szijjarto, in a lecture at the University of Bonn on Monday, argued for a strong Europe composed of strong nation states. "If there is any country that needs a strong Europe, it is definitely Hungary," he said, noting that Hungary, located in the middle of Europe with an open economy, has a history of losing out in conflicts between east and west. Europe, he added, faces historic challenges, so it is natural that there are debates about its future and on how to secure a strong Europe.
Whenever "someone strays from the mainstream" in the debate on the European Union's future, it is wrong that they immediately get labelled "un-European", he said. Szijjarto said some wanted weak EU member states within a kind of united states of Europe. Yet strong integration between weak countries is hard to achieve, he argued, adding that Hungary believes strong member states are needed.
Szijjarto named migration among the EU's ongoing challenges. He said migration had heightened the threat of terrorism in Europe. When hundreds of thousands are allowed to enter the continent unchecked, certain terrorists take advantage of such a lax policy, he said. This is why migration is also a security issue, he said. Meanwhile, he said every country should have the opportunity to provide its own solutions to its demographic challenges, and this did not necessarily have to be in the form of immigration. Border protection is also a question of sovereignty and even the Schengen Agreement mandates the protection of Europe's external borders, Szijjarto said. The minister also said Europe should return to its Christian roots. He called it "unacceptable" that Christian symbols were being removed in certain places in western Europe.
He also touched on the issue of the EU's next budget, saying that its implementation should be preceded by a fair debate. The distribution of EU funds does not depend on the generosity of western European countries, Szijjarto said. "We have a right to these funds," he said.
In response to a question, Szijjarto noted that a few years ago several thousand people crossed Hungary's border illegally "and it is not clear how they would have had the right to do that". He said the migrants had rejected any sort of cooperation with the authorities and "took over public spaces". Hungary does not judge another country if it wants to build a multicultural society, but other countries should not judge Hungary, either, for following a different path, Szijjarto said.


Visual Technologies recently fabricated and supplied over 500m of premium high power LED strip lighting for these 4.5 star luxury apartments.
Visual Technologies are firm believers in quality education. As they say, knowledge is power, and this is so true. BUT unfortunately when it comes to LEDs and lighting a sign, knowledge gets left behind and price means everything.


There is a new CCNA subcommittee, the Emergency Preparedness Committee. Even if you were unable to attend our last general meeting on emergency preparedness, we urge you to join in with the subcommittee. There will be a meeting of this subcommittee at Suzanne DeBenedittis' house, either on March 22 or March 27th, please RSVP.
Christine Parra, CC Fire Department & I (Suzanne DeBenedittis) will facilitate this hour long meeting. (If need be, this meeting will be 90 minutes max).
Provide each family in the Crest the opportunity to adequately prepare so that your family is not imposed on in an emergency.
Work as a team for cost-savings bulk buying.
Invite neighbors on your Street to get actively involved so that you can rely on each other.
Assure the Golden State water generators will pump immediately if there is need.
Anything else you deem needed to insure our resilience.
Meeting will close with Sign-ups for sub-committees/teams to carry each aspect of our Emergency Prep Project to success.
Yummy refreshments will be served! Bring neighbors & family too, as "many hands make light work" Let's make it fun too!!


Gooch & Housego manufactures precision optical components for industrial, medical and research applications; providing support to a global customer base to achieve optimized system performance. Unique skills and processes allow reliable production for the most demanding applications. Products include: precision optics (UV to IR), thin-film coatings, glass engineering and accoust-optics.
Experience in manufacturing precision optical components has enabled G&H to become the worlds leading manufacturer of AO Q-switches. G&H can apply precision and optical expertise to the manufacture of components in a wide variety of optical materials, offering precision optical finishes with extremely high accuracy and tolerances on engineered parts. As well as precision drilling, shaping, polishing, and honing of optical materials, G&H have an in-house coating capability, providing high quality AR, HR, broadband and narrowband coatings. Our comprehensive knowledge of both optical and mechanical properties of optical materials combined with the ability to handle all aspects of component manufacture on one site ensures the highest quality product, from a lens to a complex, multi-faceted CNC machined glass part.


text:
[Blu Review] Shout Factory's 'My Bloody Valentine' Is A Love Letter To Horror Fans
February 14, 2020 by Joe Lipsett
Thanks to Scream Factory, a Canadian cult classic slasher gets the deluxe Blu treatment that it deserves.
Spoilers follow…
My Bloody Valentine is one of the most greatest slasher films of all time. Shot on location in a real (out of commission) mine in Sydney, Nova Scotia, Canada, there's an unmistakable air of authenticity to the entire production.
Director George Mihalka and screenwriter John Beaird dedicate nearly as much time to their core cast as they do to the elaborate death sequences and the result is a film that is as emotionally compelling as it is gory. These aren't simply forgettable characters to be put through the meat grinder for the audience's delight; they are fleshed out people with legitimate class issues. The realities of living in a small town driven by a single economic industry is starkly presented and serves to underline most of the film's conflicts, particularly the romantic triangle between TJ (Paul Kelman), Axel (Neil Affleck) and Sarah (Lori Hallier), as well as the tragic history of the fictitious town of Valentine Bluffs.
Even if the drama between the film's three leads doesn't entice viewers, the film is filled with memorable supporting characters, including kindly Ms. Mabel (Patricia Hamilton), bemusing Howard (Alf Humphreys) and my ride-or-die slasher fave Hollis (Keith Knight), arguably the most lovable character in any 80s slasher.
Throw in some truly epic kills, including Mabel's laundromat death, the sexy pick-axe through the heart opening and, of course, the incomparable head through the shower faucet, and a great, authentic costume for killer "Harry Warden" (dancer/stuntman Peter Cowper) and My Bloody Valentine is a quintessential classic of the subgenre.
As with nearly every Scream Factory release, the true reason to pick up the latest release is twofold: it looks great and it has a bounty of special features.
The film looks exceptional – the gory scenes that were omitted by MPAA request from the film's theatrical release (expanded upon in a featurette on the first disc) have been expertly cleaned up and incorporated into the unrated edition seamlessly. This is particularly evident in the scene where the Sheriff discovers Mabel's body in the laundromat – there is quite a bit more footage of her corpse spinning around to gloriously disgusting effect. There is a little bit of grain in some of the transfer, but these moments are exceedingly rare and, overall, this is the best that the film has ever looked.
Of particular note in the Special Features camp is director George Mihalka's audio commentary on the second disc's unrated cut. It's informative and candid in all of the ways that you hope for when someone is reflecting on one of their favourite films. Mihalka is occasionally prompted with questions to help ensure a near continuous flow of facts and reminiscing, including fascinating details about the aggressive shooting schedule required due to the lengthy time it took to move equipment up and down the old mine, which despite being out of commission was still dangerous due to methane gas build-up.
For genre enthusiasts, his reservations about including one of the more aggravating tropes of the slasher genre ("let's not tell anyone there's a murderer on the loose in order to protect them"), as well as his interest in reflecting the actual lived experience of miners – right down to the clothes, dining and drinking habits – confirms that the director put the utmost care into what could have been a generic horror film during the glut of Cycle 1 slashers back in the early 80s.
What is also evident in both the interviews (on the first disc) and the cast reunion for the film's 35th anniversary at the Bay of Blood convention in Tampa (on the second disc) is just how much they love and respect the film. It makes sense given the film's longevity in the horror community, but hearing everyone involved in the production discuss their experiences with such reverence and enthusiasm is a dream for horror fans.
Full List of Bonus Features
DISC 1 – Theatrical Version:
NEW 4K scan of the original camera negative
NEW An Anemic Valentine – an interview with director George Mihalka
NEW From the Heart – an interview with actor Paul Kelman
NEW Friends of Mine – an interview with actress Lori Hallier
NEW Axel, Be My Valentine – an interview with actor Neil Affleck
NEW Becoming Sylvia – an interview with actress Helene Udy
NEW The Secret Keeper – an interview with actor Rob Stein
NEW Broken Hearts and Broken Bones – an interview with special makeup effects designer Tom Burman
NEW Holes in the Heart – a look at the difference between the theatrical version and the uncut version
Still Gallery
DISC 2 – Uncut Version:
NEW 4K scan of the uncut original camera negative
NEW Audio Commentary with director George Mihalka
MY BLOODY VALENTINE 35th Anniversary Cast Reunion panel at the Bay of Blood Convention in Florida featuring director George Mihalka, cast members Lori Hallier, Helene Udy, Rob Stein, Peter Cowper, Thomas Kovacs, Jim Murchison, Alf Humphreys and hosted by Brian Singleton
Thomas Kovacs performs "The Ballad of Harry Warden" at the Bay of Blood Convention with Peter Cowper and Jim Murchison
The deluxe two disc edition of My Bloody Valentine is now available for purchase everywhere.
Filed Under: Horror, Horror Film Reviews Tagged With: Canadian film, canadian horror, my bloody valentine, scream factory, shout factory, slasher


text:
\subsection{Quantum Annealing and Ising Models}
Quantum annealing is a physical implementation of adiabatic quantum computation~\cite{quant-ph/0001106} with a focus on obtaining the ground state of an objective function (i.e., a Hamiltonian) by minimizing the energy of Ising models:
\begin{equation}\label{eq:1}
    E(s) = \sum_{i} h_i s_i + \sum_{i,j}J_{ij}s_is_j,\quad  s_i\in\{-1,1\},
\end{equation}
where $s$ is the classical spin variable, $h$ is the external longitudinal magnetic field strength vector and the matrix $J$ represents the coupler interactions.   This problem is   NP-hard~\cite{barahona1982computational} for arbitrary $h$ and $J$.

A Quadratic Unconstrained Optimization Problem (QUBO) of the  form
\begin{equation}\label{eq:2}
    Q(x) = \sum_{i} h_i x_i + \sum_{i,j}J_{ij}x_ix_j,\quad  x_i\in\{0,1\}
\end{equation}
can be converted to an Ising problem by a one-to-one mapping of the variables: $x_i = \frac{1+s_i}{2}$. We will use QUBO for the rest of the paper but it is worth noting that quantum annealers from D-Wave require the QUBOs to be transformed into Ising models before running on the processor.

Given a standard binary optimization problem with a linear or quadratic objective function,
\begin{equation}\label{eq:3}
    \begin{aligned}
        \min_{x} \quad & f(x)\\
        \textrm{s.t.} \quad & Ax = b \\
                            & x\in\{0,1\}^n,
    \end{aligned}
\end{equation}
we can rewrite it as a QUBO
\begin{equation}\label{eq:4}
    Q(s) = f(x)  + \lambda(Ax-b)^T(Ax-b)
\end{equation}
to be minimized by quantum annealers. We will discuss how to convert the Markowitz optimization problem to have continuous variables in Section~\ref{sec:markowitz}.

\subsection{D-Wave Quantum Annealer}\label{sec:quantum-annealer}
Here we start by discussing the latest quantum annealing technologies offered by D-Wave as we will see, the solubility of the problem is deeply dependent on the offered architecture. D-Wave quantum annealers are specifically designed to solve Ising problems natively.  Currently two types of quantum annealers are offered by D-Wave: `2000Q' processor with Chimera topology and `Advantage' processor with Pegasus topology. The later one was made publicly available  in 2020 and it has more qubits (5760 vs. 2048) and better connectivity. The qubits in the Chimera topology have 5 couplers per qubit while in the Pegasus topology they have 15 couplers per qubit~\cite{advantage}.
It is not always possible to formulate an optimization problem to match the Chimera or Pegasus topologies exactly. Therefore minor embedding are necessary to map the problems to D-Wave processors. Such embeddings usually require the users to map multiple physical qubits to one logical variable which significantly reduces the total size of the problems that can be solved in the quantum annealers.

Furthermore, it is advisable to have uniform chain lengths - numbers of qubits representing a single variable - and uniform number of physical couplers for each logical coupler for more predictable chain dynamics during the anneal~\cite{Venturelli_2015}. Algorithms in~\cite{cliqueembedding} detail such procedures for cliques which is the underlying logical graph for the portfolio optimization problem. A full-yield 2000Q processor can map up to 64 logical variables and an Advantage processor can map around 180 logical variables. A comparison between the embedding of the two topologies is shown in Figure~\ref{fig:chimera_pegasus}. In the experiments we use the \emph{find\_clique\_embedding} function from \emph{dwave-system} to map fully-connected graphs to either Chimera or Pegasus topology.

\begin{figure*}[t!]
\includegraphics[width=0.32\linewidth]{Plots/chimera_8_3.eps}
\includegraphics[width=0.32\linewidth]{Plots/chimera_16_5.eps}
\includegraphics[width=0.32\linewidth]{Plots/chimera_24_7.eps}\\
\includegraphics[width=0.31\linewidth]{Plots/pegasus_8_2.eps}
\includegraphics[width=0.31\linewidth]{Plots/pegasus_16_2-3.eps}
\includegraphics[width=0.31\linewidth]{Plots/pegasus_16_2-3.eps}
\begin{center}
\hspace{2em}(a) $K=8$\hspace{12em} (b) $K=16$\hspace{12em} (c) $K=24$\hfill
\end{center}

\caption{A comparison between minor embeddings on Chimera and Pegasus lattices for cliques (fully connected graphs) of different sizes. The vertices with the same color or label represent the same logical variable in Eq.~(\ref{eq:1}). Each Chimera cell is a 4 by 4 complete bipartite graph ($K_{4,4}$) with 4 additional edges connecting neighboring cells. Each Pegasus cell has 24 qubits which include 3 $K_{4,4}$ graphs as in the Chimera cell and $K_{2,4}$ edges connecting them. To minor embed cliques of 8 vertices ($K=8$), the chain length on the Chimera lattice is 3 while on the Pegasus lattice it is 2. With $K=16$, the chain lengths are 5 and 2-3, respectively, and with $K=24$, they are 7 and 3-4,  respectively. This shows that Pegasus processor scales better for larger clique problems, which may lead to better performance.\label{fig:chimera_pegasus}}
\end{figure*}

\subsection{Previous Work}
\citet{port_1} solved multi-period portfolio optimization problem using D-Wave's quantum annealer:
\begin{equation}\label{eq:5}
    \begin{aligned}
        \max_{w} \quad & \sum_{t=1}^T (\mu_t^T w_t - \frac{\gamma}{2}w_t^T\Sigma_tw_t-\Delta w_t^T \Lambda_t \Delta w_t + \Delta w_t^T \Lambda_t^{'} \Delta w_t)\\
        \textrm{s.t.} \quad   &\sum_{n-1}^N w_{nt} = K,\forall t, \\
                           &   w_{nt} \leq K', \forall t , \forall n.
    \end{aligned}
\end{equation}
Eq.~\eqref{eq:5} seeks to maximize returns considering the portfolio variance and transaction costs. The sum of asset holdings is constrained by $K$ and the maximum allowed holdings of each asset is $K'$. For small problems ranging from 12 to 584 variables,
D-Wave's 512 and 1152-qubit systems were able to find optimal solutions with high probability.

\citet{port_2} focused on the following QUBO where the task is to select $M$ assets from a pool of $N$ assets:
\begin{equation}\label{eq:6}
   \sum_{i=1}^N a_i q_i + \sum_{i=1}^{N}\sum_{j=i+1}^N b_{ij}q_iq_j + P\left(M-\sum_{i=1}^N q_i\right)
\end{equation}
Coefficients $a_i$ indicates attractiveness of the $i$-th asset and $b_{ij}$ is the pairwise diversification penalties (positive) or rewards (negative). $ q_i$  is set to 1 if the $i$-th asset is selected and $ P$ scales the constraint to make sure it is satisfied during the annealing process. The authors also explore the benefits of reverse annealing on D-Wave systems and report one to three orders of magnitude speed-up in time-to-solution.

The problem considered by   \citet{phillipson2020portfolio}  is similar to the Markowitz Optimization problem but with binary variables indicating asset selections instead of real weights. The authors report comparable results from D-Wave's hybrid solver to other state of the art classical algorithms and solvers including simulated annealing, genetic algorithm, linear programs (Gurobi with Branch and Bound Algorithm)  and local search.

\citet{port_3} benchmark the Markowitz Optimization problem on D-Wave 2000Q processor with real weight variables and uniform price data and explore how embeddings, spin reversal and reverse annealing affect the success probability. The novel contribution in our paper is to follow the general QUBO formulation in~\cite{port_3}, and benchmark with real-world ETF and currency data. Additionally, we present the results on the newly-available Advantage processor and experiment on problems with $30$  and up to $115$ logical spins, up from $20$
in~\cite{port_3}.
\subsection{Markowitz Model in Mean-Shortfall Optimization and its QUBO Formulation}\label{sec:markowitz}
The expected shortfall is defined as the expected return on the portfolio in the $\alpha\%$ worse cases. It is possible to write the mean-shortfall optimization problem as a linear program~\cite{uryasev_rockafellar_2001} but it requires adding $T+1$ variables and $2T$ constraints where $T$ is the number of assets. However, under the assumption that the assets' historical returns follow a Gaussian distribution we can approximate the expected shortfall by:
\begin{equation}\label{eq:7}
    \textrm{ES}_{\alpha}(P) = \mu + \sigma \frac{\phi(\Phi^{-1}(\alpha))}{1-\alpha},
\end{equation}
where $\mu$ is the mean of the return data, $\sigma$ is the volatility of the portfolio, and $\phi(x)$ and $\Phi(x)$ are the Gaussian probability distribution and cumulative distribution function, respectively~\cite{norton2019calculating}.  It is clear that the expected shortfall has positive correlation with the volatility and in turn, the variance of the portfolio.

Thus instead of minimizing the expected shortfall directly in the optimization program, we use the Markowitz Model~\cite{Markowitz} to find a portfolio with minimum volatility given the target return is met:
\begin{equation}\label{eq:8}
    \begin{aligned}
        \min_{w} \quad & w^T C_t w\\
        \textrm{s.t.} \quad & \mu_t^Tw = p, \\
                            &\sum{w_i} = 1, w_i\geq 0 \enspace \forall i.
    \end{aligned}
\end{equation}

Here $w$ is the weight vector, $C_t$ is the co-variance matrix of the assets' returns during time window $t$, $R$ is the historical return data, $\mu_t = \text{mean}(R)$ is the mean return vector of time window $t$, and $p$ is the target return.
The constraint $\mu_t^Tw = p$ ensures that the target return is met,  $\sum{w}=1$ indicates we want to invest all of the resources, and $w \geq 0$ means short selling is not allowed. With all the constraints satisfied, we minimize $w^T C_t w$, i.e., the variance of the  portfolio.

To convert Eq.~(\ref{eq:8}) to a QUBO, first we write it as an unconstrained optimization program with Lagrangian mutlipliers $\lambda_1, \lambda_2$ and $\lambda_3$ :
\begin{multline}\label{eq:9}
    \mathcal{L} = \left(\sum_i^n \sum_j^n C_{i,j}w_iw_j\right) +\lambda_1 \left(\sum_i^n \mu_{t,i} w_i - p\right)^2 + \\ \lambda_2  \left(\sum_i^n w_i -1 \right)^2
\end{multline}
where $\lambda_1$ and $\lambda_2$  scale  the constraint penalties. It is also equivalent to
\begin{multline}\label{eq:10}
   \mathcal{L} = \left(\sum_i^n \sum_j^n C_{i,j}w_iw_j\right) + \\\lambda_1 \left[\left(\sum_i^n \mu_{t,i} w_i\right)^2 -2p\sum_i^n \mu_{t,i} w_i\right]\\
    + \lambda_2 \left[\left(\sum_i^n w_i\right)^2 -2\sum_i^n w_i \right]\\
\end{multline}
after expanding the squared terms and eliminating the constants. When the constraints are satisfied exactly, we have
\begin{equation}\label{eq:11}
    \lambda_1 \left[\left(\sum_i^n \mu_{t,i} w_i\right)^2 -2p\sum_i^n \mu_{t,i} w_i\right] = -\lambda_1 p^2,
\end{equation}
and
\begin{equation}\label{eq:12}
    \lambda_2 \left[\left(\sum_i^n w_i\right)^2 -2\sum_i^n w_i \right] = -\lambda_2.
\end{equation}

We use $k$ binary variables $x_1\dotsc x_k \in \{0,1\}$ to approximate the continuous variable $w$ in Eq.~(\ref{eq:8})
\begin{equation}\label{eq:13}
    w_i = \sum_{a=1}^k 2^{-a} x_i.
\end{equation}

The larger $k$ is, the more precision $w_i$ has. However, smaller $k$ also widens the differences between the coupler strengths - $J$ terms in Eq.~(\ref{eq:2}). Although the coupler strengths for D-Wave annealers can be set at any double-precsion floating point number  between $-1$ and $1$,  errors in precision pose a challenge to D-Wave annealers due to integrated control errors (ICE)~\cite{dwave-doc}. In reality D-Wave processors solve a slightly modified version of Eq.~(\ref{eq:2}),
\begin{equation}\label{eq:14}
    Q^{\delta}(s) = \sum_{i} (h_i+\delta h_i(s)) x_i + \sum_{i,j}(J_{ij} + \delta J_{ij}(s))x_ix_j.
\end{equation}
Here $\delta h$ and $\delta J$ are time-dependent and~\cite{2000Q_5_specs} shows an example of their distribution during the anneal for a  2000Q processor.

In our experiments, we set $k=5$ which achieves a good enough approximation on the weights while keeping all coupler strengths close. For larger $k$ we will risk the errors dominating the coupler coefficients, rendering those additional qubits unreliable. We also set $\lambda_1$ to $p^{-2}$ and $\lambda_2$ to 1 such that the QUBO will have a theoretical minimum objective value of $-2 + \lambda_3 \left(\sum_i \sum_j C_{i,j}w_iw_j\right)$ when the constraints are satisfied exactly by Eq.~(\ref{eq:11}) and Eq.~(\ref{eq:12}).

Substituting Eq.~(\ref{eq:13}) in to  Eq.~(\ref{eq:10}), we have the final binary optimization formalism
\begin{multline}\label{eq:15}
    f(x) = \left[\left(\sum_i^n \sum_a^k \mu_{t,i} 2^{-a}x_i\right)^2 -2p\sum_i^n\sum_a^k \mu_{t,i} 2^{-a}x_i\right] \\+ p^{-2} \left[\left(\sum_i^n \sum_a^k 2^{-a}x_i\right)^2 -2\sum_i^n\sum_a^k 2^{-a}x_i \right]\\
    +\lambda_3 \left(\sum_i^n \sum_j^n \sum_a^k \sum_b^k C_{i,j}2^{-a-b}x_ix_j\right),
\end{multline}
which is quantum-annealable as it only has linear and quadratic interactions.

\subsection{Quantum-Classical Algorithm for Portfolio Optimization}
Since expected shortfall cannot be expressed as a quadratic formulation exactly, we opt to use it as a convergence criterion instead of  including it in the optimization problem directly. We use a hybrid approach where we optimize the portfolio weights and the constrains on the target return, $p$, in two different steps – the optimization of the weights are performed in a quantum annealer, while the adjustments of $p$ were evaluated classically based on the quantum annealing results.

\begin{algorithm*}
\SetAlgoLined
\KwIn{$\mu_t$, $R_t$, $\sigma_{ref}$,$\sigma_{ref\_t}$, $ES_{ref}$}
\KwOut{$w$}
    Let $p=\text{mean}(\mu_t)$  \Comment*[r]{Initialize the target return}
    Let $EST_t = \frac{\sigma_{ref}}{\sigma_{ref\_t}} ES_{ref}$ \Comment*[r]{Initialize the target expected shortfall}
    Let $C_t = \text{cov}(R_t)$ \Comment*[r]{Compute the co-variance matrix from the returns}
    \While{True} {
        \uIf{ $p > \max{(\mu_t)}$} {
            Return $\mathbf{0}$ \Comment*[r]{Return constraint cannot be satisfied}
        }
        Solve Eq.~(\ref{eq:10}) for $w$ \Comment*[r]{with Quantum Annealing}
        Let $ESt$ be the mean of the lowest $\alpha$\% values of $w^TR_t$ \Comment*[r]{Compute the expected shortfall from the asset weights}
        \uIf{ $\frac{|ESt|}{|EST_t|} > 1 + \epsilon$ } {
            Let $p = p  ( 1 - \delta)$ \Comment*[r]{Decrease target return for lower risks}
        } \uElseIf{$\frac{|ESt|}{|EST_t|} < 1 - \epsilon$} {
            Let $p = p  ( 1 + \delta)$ \Comment*[r]{Increase target return as there is room for more risks}
        } \uElse{
            Return $w$\;
        }
    }
 \caption{Expected Shortfall based Dynamic Asset Allocation during $t$.\label{alg:asset}}
\end{algorithm*}

Here $\sigma$ is the volatility of the reference asset's returns during time window $t$ or 2008, $\alpha$ is the risk level parameter, $EST_t$ is
the target expected shortfall at time window $t$, $ESt$ is the expected shortfall for the computed portfolio during the optimization process at time window $t$, $\epsilon$ is the error tolerance parameter, and $\delta$ is the momentum parameter which should be adjusted dynamically.

In the previous section we explain the usage of the Markowitz Model in a mean-shortfall problem. Fig.~\ref{fig:ratio} is one example from Algorithm~\ref{alg:asset} solved with an exact solver.
\begin{figure}[!t]
\includegraphics[width=8.6cm]{Plots/method_1.eps}
\caption{Y-axis tracks the ratio between the variance and expected shortfall in later iterations against their respective values in the first iteration. The expected shortfall decrease at different rates from the variance but each iteration in the algorithm is guaranteed to make  progress towards the target expected shortfall, which ensures convergence. \label{fig:ratio}}
\end{figure}

\subsection{Test Input and Annealer Parameters}
We pick the top-six ETFs by trading volumes, EEM, QQQ, SPY, SLV, SQQQ and XLF, and six major currencies' USD exchange rates, AUD, EUR, GBP, CNY, INR and JPY,  for most of the tests below and extracted their return data from Yahoo Finance. The reference assets for ETF and currency tests are SPY and EUR, respectively. For the final tests which use 12 and 23 assets respectively, we choose the top ETFs by trading volumes again.

We can control a range of annealer parameters that may impact the solution quality in various degrees. Specifically, we set the number of spin reversal transforms~\cite{spin_reversal_2} to 100 and readout thermalization to $\SI{100}{\micro\second}$ as suggested in~\cite{thermalization,spin_reversal}. We set the anneal time at $\SI{1}{\micro\second}$ as longer anneal time sees no statistically significant improvements to the solutions similarly reported in~\cite{port_3}. Results from 2000Q and Advantage processors are both included in the following sections. Additionally, we report the results from D-Wave's post processing utility on 2000Q processors, which decomposes the underlying graph induced by the QUBO into several low tree-width subgraphs~\cite{markowitz_1957} then solve them exactly using belief propagation on junction trees~\cite{Jensen1990BayesianUI}.

We sample all QUBOs 30,000 times with both D-Wave backends and report the samples with lowest energy each time. Fig.~\ref{fig:dist} shows an example distribution of the samples.
\begin{figure}[!ht]
\centering
\includegraphics[width=8.6cm]{Plots/solution_dist.eps}
\caption{The distribution of the samples for 4 different QUBOs with both D-Wave backends. Each QUBO is sampled 30,000 times and the energy of the  samples is scaled to be between (-1,1). We divide the energy range into 50 equally-spaced bins and count the number of samples in each bin. All four samples exhibit the Poisson distribution, and  thus we only report the samples with the lowest energy for the experiments in this section.\label{fig:dist}}
\end{figure}

\subsection{Embedding Comparison on D-Wave Annealers}
As discussed in Section~\ref{sec:quantum-annealer}, D-Wave quantum annealers require the problems to be minor embedded to the Chimera or Pegasus topology. For small problems this means there may be multiple valid embeddings and in this section we will measure how different embeddings can make an impact on the solution quality.

We compute 4 different embeddings that use different subgraphs from both Chimera and Pegasus topology and sample the same QUBO - first iteration of Algorithm~\ref{alg:asset} on the ETFs from December, 19 to May, 20 - 10 times with 10,000 samples each. We then pick the best solutions in terms of QUBO energy from all 10 sample sets for each embedding and obtain their average and minimum values. Table~\ref{tab:embed-comp} represents the results as ratios against the ground energy computed by simulated annealing for better readability.

\begin{table}[!ht]
\centering
\caption{Embedding comparison on the 2000Q processor with 30 logical qubits or 270 physical qubits after minor embedding. The energy is defined as the objective value from Eq.~(\ref{eq:15}) and is normalized against the simulated annealer. The second embedding out of these four is able to find the solution with better average and best energy. \label{tab:embed-comp}}
\begin{tabular}{c|c|c}
\hline
\hline
Embedding    & Average Energy & Best Energy\\
\hline
1 & $95.66\%$ & $98.78\%$ \\
\hline
2 & $96.83\%$ & $99.66\%$ \\
\hline
3 & $96.53\%$ & $98.37\%$ \\
\hline
4 & $96.21\%$ & $98.49\%$ \\
\hline
\hline
\end{tabular}

\end{table}

\begin{table}[!ht]
\centering
\caption{Embedding comparison on the Advantage processor with 30 logical qubits or 134 physical qubits after minor embedding. Different embeddings on the Advantage processor show no statistically significant differences.}
\begin{tabular}{c|c|c}
\hline
\hline
Embedding    & Average Energy & Best Energy\\
\hline
1 & 99.25\% & 99.89\% \\
\hline
2 & 99.52\% & 99.94\% \\
\hline
3 & 99.22\% & 99.95\% \\
\hline
4 & 99.48\% & 99.89\% \\
\hline
\hline
\end{tabular}
\end{table}

We can see that the impact that different embeddings make is statistically insignificant. However, it is clear that Pegasus processors have higher energy ratios than Chimera ones, which we will address next.

\subsection{Annealing Results Comparison}
We benchmark our algorithm on both simulated and quantum annealers against a  classical optimization solver, namely, \texttt{cvxpy}~\cite{diamond2016cvxpy}.
The results are normalized against the optimal classical solution. The algorithm fails to converge on the first two currency tests on the 2000Q processor and they are reported as $0$ in the chart.

\begin{figure*}
\includegraphics[width=0.9\linewidth]{Plots/nj/return.eps}
\caption{\label{fig:wide}The comparison of the final returns between all 4 backends. Each test uses 100 days of return data with different starting dates from 2010 to 2020. The results from 2000Q with post processing yields identical results from simulated annealing. Both 2000Q and Advantage processors are able to compute returns that are consistently more than $80\%$ of the optimal except the two currency test cases where the algorithm fails to converge on the 2000Q.}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{Plots/nj/shape.eps}
\caption{\label{fig:wide2}The comparison of the final Sharpe ratios between all 4 backends. Sharpe ratio is calculated as $\mbox{return}/\mbox{std}$. The results confirm that the portfolio variance returned by the quantum processors are close to optimal results obtained from classical optimization methods,  and it is effective to solve standard constrained optimization problems as a QUBO.}
\end{figure*}

With $5$ qubits to represent each asset weight variable, the simulated annealing results follow the optimal solutions closely in most tests. We note that the quantum annealers produce results better than their classical counterparts in terms of returns and sharpe ratios in some cases. We also observe the currency tests generally perform better than ETF tests on both quantum annealer backends. Figure 6 shows how quantum annelears perform with respect to the average of absolute correlation coefficients over all pairs of assets in each test.

\begin{figure}[!ht]
\centering
\includegraphics[width=8.6cm]{Plots/corr_vs_return.eps}
\caption{Final returns obtained from both quantum annealers against the average of the absolute correlation coefficients. All currency samples in our tests have higher average correlation values than those of ETF samples.\label{fig:corr_vs_return}}
\end{figure}

Although we acknowledge there may be other factors contributing to our observations that currency tests do better than ETF tests on quantum annealers, Fig.~\ref{fig:corr_vs_return} implies that more correlated assets tend to perform better. Detailed analysis on which attributes of the assets have an impact on the quantum annealing performance and how much are the impacts requires more research in the future. A reference study from~\cite{barbosa_pelofske_hahn_djidjev_2021} used machine learning models such as decision tree and regression to predict the accuracy of D-Wave's quantum annealer on maximum clique problems.

\subsection{Alternate QUBO Formulation For the Markowitz Model}
Instead of having to optimize both $\lambda_1$ and $\lambda_2$ in Eq.~(\ref{eq:10}), we can reformulate Eq.~(\ref{eq:8}) to have one fewer Lagrangian multiplier:
\begin{equation}\label{eq:16}
    \begin{aligned}
        \min_{w} \quad & w^T C_t w - q\mu_t^Tw\\
        \textrm{s.t.} \quad &\sum{w} = 1, \ w\geq0.
    \end{aligned}
\end{equation}

To prove this, note that the Lagrangians for Eq.~(\ref{eq:8}) and Eq.~(\ref{eq:16}) are
\begin{equation}\label{eq:17}
    \mathcal{L}_1(w,\lambda_1, \lambda_2)= w^TC_tw + \lambda_1(\mu_t^Tw-p) + \lambda_2(1^Tw - 1),
\end{equation}
and
\begin{equation}\label{eq:18}
    \mathcal{L}_2(w,\lambda_2)= w^TC_tw - q\mu_t^Tw + \lambda_2(1^Tw - 1).
\end{equation}
Subtracting Eq.~(\ref{eq:18}) from Eq.~(\ref{eq:17}), we have
\begin{equation}\label{eq:19}
    \mathcal{L}_1(w,\lambda_1, \lambda_2) - \mathcal{L}_2(w,\lambda_2)= (\lambda_1 - q) \mu_t^Tw - \lambda_1 p,
\end{equation}
which is a constant. Therefore Eq.~(\ref{eq:8}) and Eq.~(\ref{eq:16}) are equivalent and are both quantum-annealable.

However, the quantum annealing results from the alternating QUBO formulation is significantly worse than that of the original formulation. One possible explanation for the poor results is that the variance ($w^TC_tw$) and the expected return ($q\mu_t^Tw$) have different scales and without the additional Lagrangian multiplier in the original QUBO formulation to regulate them, it is harder for the quantum annealers with limited accuracy to find the ground state.

\subsection{State of The Art on D-Wave Annealers}
The embeddings of the $6$ asset tests on both 2000Q and Advantage processors leave plenty of unused qubits. D-Wave's clique embedding algorithm~\cite{BoothbyTomas2015Fcmg} suggests that we can embed a fully-connected graphs of vertex counts of 64 and 180 to full-yield 2000Q and Advantage processors, respectively. Due to the defective qubits and connectors in the currently available Advantage processor, experimentally we can embed up to 119 qubits. This means we can solve portfolio optimization problems with $12$ and $23$ assets natively on the D-Wave processors.

\begin{figure}[!ht]
\centering
\includegraphics[width=8.6cm]{Plots/12_asset_comp.eps}
\caption{The energy comparison of the 12 asset test between all 4 backends. The solutions from 2000Q deviate from the ground states by a large margin while the Advantage processor is able to keep up closely. Post processing is able to improve the 2000Q results to match simulated annealing, again.}
\end{figure}

On the $12$ asset test, the 2000Q processor struggles to find the ground state as its embedding chain length reaches $16$ while the Advantage processor stays closely with the simulated annealing and post-processed results. However, neither quantum annealer succeeds in converging as the classical counterparts do.

\begin{table}[!ht]
\centering
\caption{Energy of Last 5 Iterations from simulated annealing and Advantage. Even though the differences in the energy are small, the results in terms of the portfolio weights vary greatly.}
\begin{tabular}{c|c|c}
\hline
\hline
SA    & Pegasus & difference\\
\hline
-1.026824759 & -1.016152942 &1.0393\%\\
\hline
-0.951012554 & -0.950129253 & 0.0928\%\\
\hline
-0.879067674 & -0.878089411 & 0.1112\%  \\
\hline
-0.811026432 & -0.809642008 & 0.1707\%  \\
\hline
-0.74698612 & -0.746420758 & 0.0757\%  \\
\hline
\hline
\end{tabular}
\end{table}

Comparing the last $5$ iterations in terms of energy from simulated annealing and the Advantage processor shows that even the energy is hardly different, the solution quality is increasingly sensitive to the QUBO objective value that for a problem this size, a $0.1\%$ change in the objective value leads to $30\%$ difference in the portfolio variance. One potential reason is that solving larger problems increases the size of co-variance matrix and also the condition number. And as shown in the previous section larger condition number generally equates to worse quantum annleaer performance. In this case, either the quantum annealers need to be more accurate to find the ground state or our QUBO setup needs to change to account for higher asset counts.

\begin{figure}[!ht]
\centering
\includegraphics[width=8.6cm]{Plots/23_asset_comp.eps}
\caption{The energy comparison of the 23 asset test between simulated annealing and the Advantage processor. Due to the high chain lengths of the embedding, the Advantage processor fails to either reach the ground state or get close to it in all iterations, rendering the processor incapable of solving  problems of such sizes. }
\end{figure}
With the embedding chain lengths going up to 17, the Advantage processor fails to find the ground state by a large margin. Even though we can physically map a problem of this size, the results reflect the limitations of current-generation quantum annealers.

\section{Introduction}
\input{1.introduction}

\section{Background}
\label{sec:back}
\input{2.background}

\section{Methods}
\label{sec:method}
\input{3.methods}

\section{Results}
\label{sec:result}
\input{4.results}

\section{Conclusion and  Further Research}
\label{sec:conclude}
\input{5.conclusion}

\begin{acknowledgments}
We like to thank Dr. Erica Grant, Dr. Andrew King, Dr. Isil Ozfidan and Dr. Travis Humble for helpful discussions. This research performed using D-Wave used resources of the Oak Ridge Leadership Computing Facility (OLCF) at the Oak Ridge National Laboratory (ORNL), which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. AB and SD likes to thank the ORNL Quantum Computing Institute and the Purdue Quantum Science and Engineering Institute for support.
\end{acknowledgments}