We perform a systematic study of the correlation functions of two quark currents in a pion using lattice QCD. We obtain good signals for all but one of the relevant Wick contractions of quark fields. We investigate the quark mass dependence of our results and test the importance of correlations between the quark and the antiquark in the pion. Our lattice data are compared with predictions from chiral perturbation theory.

Lattice QCD Chiral Lagrangians

Article funded by SCOAP3

Introduction

Correlation functions of two currents

Isospin decomposition and constraints

Predictions from chiral perturbation theory

Lattice techniques

Lattice contractions and their symmetry properties

Physical matrix elements

Simulation details

Renormalisation

Data quality and lattice artefacts

Plateaux and excited state contributions

Anisotropy effects

Finite volume effects

Finite momentum and boost invariance

Results for individual lattice contractions

Light quarks

Quark mass dependence

Test of factorisation

L/2$to the sum over all$\mvec{y}$amounts to 39\% for light and to 16\% for strange quarks. At such distances, the effect of periodic images discussed in section~\ref{sec:aniso} is considerable. Moreover, the slow decrease of$y^2\, C_{1}^{\ms VV}(\mvec{y})$with$y$implies that the numerator of~\eqref{naive-rms} misses important contributions from distances$y/a > \sqrt{3}\ms L /2$that are not included in the elementary cell. In the following we present a method to deal with this situation. Let us assume that the correlation function computed on the lattice has the form \begin{equation} \label{mirror-img} C_1^{\text{lat}}(\mvec{y}) = \sum_{\mvec{n} \in \mathbb{Z}_3} C_1^{}(\mvec{y} + \mvec{n} L a) \,, \end{equation} where$C_1(\mvec{y})$is the correlation function in the physical limit and the terms with$\mvec{n} \neq \mvec{0}$are due to the periodic boundary conditions on the lattice~\cite{Burkardt-ml-1994pw}. The rms radii can be computed from the Fourier transform \begin{equation} \label{cont-FT} \widetilde{C}_1(\mvec{q}) = \int d^3 y\; e^{i \mvec{q} \mvec{y}}\, C_1(\mvec{y}) \end{equation} as \begin{equation} \label{rms-deriv} r^2 = {}- 6 \ms \bigl[ \widetilde{C}_{1}(\mvec{0}) \bigr]^{-1} \, \frac{d\ms \widetilde{C}_1(\mvec{q})}{d\ms q^2} \biggl|_{q^2 = 0} \,, \end{equation} where we used that$\widetilde{C}_1$depends on$\mvec{q}$only via$q = |\mvec{q}|. On the lattice we can evaluate the discrete Fourier transform \begin{align} \label{discr-FT} \widetilde{C}_1^{\text{lat}}(\mvec{q}) &= a^3 \!\! \sum_{\mvec{y} \in E(\mvec{0})} e^{i \mvec{q} \mvec{y}}\, C_1^{\text{lat}}(\mvec{y}) & \text{ for } \mvec{q} = \mvec{k} p_0 \end{align} with\mvec{k} \in \mathbb{Z}_3$and$p_0 = 2\pi / (L a). We have \begin{align} \widetilde{C}_1^{\text{lat}}(\mvec{q}) &= a^3 \sum_{\mvec{n} \in \mathbb{Z}_3^{}} \; \sum_{\mvec{y} \in E(\smash{\mvec{0}})} e^{i \mvec{q} \mvec{y}}\, C_1^{}(\mvec{y} + \mvec{n} L a) = a^3 \sum_{\mvec{n} \in \mathbb{Z}_3^{}} \; \sum_{\mvec{y} \in E(\mvec{n})} e^{i \mvec{q} \mvec{y} - i \mvec{q} \mvec{n} L a}\, C_1(\mvec{y}) \nonumber \\[0.4em] &= a^3 \!\! \sum_{\mvec{y}/a \in \mathbb{Z}_3} e^{i \mvec{q} \mvec{y}}\, C_1(\mvec{y}) \,, \end{align} whereE(\mvec{n})$denotes the shifted lattice cell defined by$-L/2 < y^i/a + n^i \le L/2$for$i=1,2,3$, and in the last step we have used the condition$\mvec{q} = \mvec{k} p_0$. We thus find that for$a\to 0$the discrete Fourier transform~\eqref{discr-FT} becomes equal to the infinite-volume expression~\eqref{cont-FT}. The periodic images included in$\widetilde{C}_1^{\text{lat}}(\mvec{q})$provide the contribution of the infinite-volume correlator$C_1(\mvec{y})$at distances outside the elementary cell$E(\mvec{0})$. To evaluate the rms radii from~\eqref{rms-deriv}, we need to construct a smooth function of$\mvec{q}$out of$\widetilde{C}_1(\mvec{q})$at the points$\mvec{q} = \mvec{k} p_0$. Whilst direct computation of~\eqref{rms-def} would require us to \emph{extrapolate}$C_1(\mvec{y})$to large values of$y$and to remove the contributions from periodic images, we now need to \emph{interpolate} between discrete values of$\widetilde{C}_1(\mvec{q})$in the vicinity of$\mvec{q} = \mvec{0}$. The reliability of this interpolation is of course higher for a higher density of points$\mvec{k} p_0 = 2\pi \mvec{k} /(L a)$and thus for larger physical lattice size$L a$. In figure~\ref{fig:FFT-all-q} we show$\widetilde{C}_{1}^{\ms VV}(\mvec{q})$obtained from our lattice data for light and charm quarks. The results for strange quarks and those for$\widetilde{C}_{1}^{\ms AA}(\mvec{q})$look qualitatively similar. At values$q \sim \pi/a$we see a clear anisotropy in the Fourier transform (which in the continuum limit depends on the length but not on the direction of$\mvec{q}$). This is to be expected and reflects discretisation effects in$C_1(\mvec{y})$at distances$y$of a few lattice units$a$. With decreasing$q$, the anisotropy gradually disappears and we have a very clear and smooth signal. \begin{figure} \centering \includegraphics[width=0.5\textwidth]{plots/rms_a.pdf} ]]> Subtraction term for the annihilation graph Results for isospin amplitudes Isospin amplitudes Comparison with chiral perturbation theory 0.65 \fm$ and not shown here. \begin{figure} \centering \subfloat[$F_1, \ov, p=0, L=40$.]{\includegraphics[width=0.45\textwidth,trim=0 0 0 17pt,clip]{plots/static_chpt-F1-V4V4-p000-vol40.pdf}} \hfill \subfloat[$F_1, \oa, p=0, L=40$.]{\includegraphics[width=0.45\textwidth,trim=0 0 0 17pt,clip]{plots/static_chpt-F1-A4A4-p000-vol40.pdf}} \\ \subfloat[$F_2, \ov, p=0, L=40$.]{\includegraphics[width=0.45\textwidth,trim=0 0 0 17pt,clip]{plots/static_chpt-F2-V4V4-p000-vol40.pdf}} \hfill \subfloat[$F_2, \oa, p=0, L=40$.]{\includegraphics[width=0.45\textwidth,trim=0 0 0 17pt,clip]{plots/static_chpt-F2-A4A4-p000-vol40.pdf}} ]]>