Finite-volume effects due to spatially nonlocal operators

Spatially nonlocal 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 nonlocal operators, composed of two currents displaced in a spatial direction by a distance ξ. 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−mπ(L−ξ), where mπ is the mass of the light state. For heavier external states, the usual e−mπL form is recovered, but with a polynomial prefactor of the form Lm/|L−ξ|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.