Finite-volume effects due to spatially nonlocal operators

Abstract

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 −mπ L form is recovered, but with a polynomial prefactor of the form L m /|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.

Parton distribution functions from Ioffe time pseudo-distributions

“…This periodicity drives the significant finite volume effects observed in [74], particularly for distances such as z L/2. It is also possible that C L should be augmented by powers of (L − z) as was found in [74]. For simplicity, in the following, these unknown powers will be neglected.…”

“…The case of a two current operator has been studied for a model of scalar "pions" and "nucleons" in [74]. This periodicity drives the significant finite volume effects observed in [74], particularly for distances such as z L/2. It is also possible that C L should be augmented by powers of (L − z) as was found in [74].…”

Lattice QCD and neutrino-nucleus scattering

“…First, large momentum translates into large (aP ) n , and therefore, ensembles with increasingly smaller lattice spacings a are needed. Given the need to keep the spatial size of the box sufficiently large to avoid significant finite-volume effects, which may be enhanced for some nonlocal matrix elements [166], this increases the computational cost. Second, as the momentum becomes larger, the signal-to-noise ratio degrades, even when using methods such as "momentum smearing" [167], designed to enhance the contribution of the lowest-lying state in correlation functions at nonzero three-momentum, thereby increasing the number of measurements that need to be made.…”

Parton distributions from lattice data: the nonsinglet case

“…5%. In addition to these "standard" FVE of lattice computations, it has been recently pointed out that the usage of a spatially extended operator, including a Wilson line, may lead to additional FVE [71]. The intuition behind this is that further FVE may appear when the Wilson line has non-negligible size with respect to the lattice length in the boost direction.…”

“…The analysis of Ref. [71] pertains to a toy scalar theory and predicts a FVE of the form exp(−M (L − z)) (possibly with a polynomial amplifying prefactor), with M being the analogue of the mass of the investigated hadron in the quasi-PDF approach. Given that the nucleon mass is at the physical point around 7 times larger than the pion mass, that would lead to totally irrelevant effects, since the maximum considered z is more than 3 times smaller than L. However, it can not be excluded that in QCD, the form of this FVE can be more severe, e.g.…”