Finite-volume effects due to spatially non-local operators

TL;DR

This paper estimates finite-volume effects in lattice QCD calculations of non-local matrix elements, revealing dependence on operator separation and external states, which impacts systematic uncertainty assessments.

Contribution

The work provides a detailed analysis of how finite-volume effects vary with operator separation and external states in lattice QCD, offering formulas to estimate these effects.

Findings

Finite-volume corrections depend on the operator separation and the external state.

For light external states, corrections scale as e^{-m_pi (L - xi)}.

Heavier states exhibit corrections with polynomial prefactors, potentially enhancing volume effects.

Abstract

Spatially non-local matrix elements are useful lattice-QCD observables in a variety of contexts, for example in determining hadron structure. To quote credible estimates of the systematic uncertainties in these calculations, one must understand, among other things, the size of the finite-volume effects when such matrix elements are extracted from numerical lattice calculations. In this work, we estimate finite-volume effects for matrix elements of non-local operators, composed of two currents displaced in a spatial direction by a distance $\xi$. We find that the finite-volume corrections depend on the details of the matrix element. If the external state is the lightest degree of freedom in the theory, e.g.~the pion in QCD, then the volume corrections scale as $ e^{-m_\pi (L- \xi)} $, where $m_\pi$ is the mass of the light state. For heavier external states the usual $e^{- m_\pi L}$ form is recovered, but with a polynomial prefactor of the form $L^m/|L - \xi|^n$ that can lead to enhanced volume effects. These observations are potentially relevant to a wide variety of observables being studied using lattice QCD, including parton distribution functions, double-beta-decay and Compton-scattering matrix elements, and long-range weak matrix elements.