Abstract

Using symmetry properties, we determine the mixing pattern of a class of nonlocal quark bilinear operators containing a straight Wilson line along a spatial direction. We confirm the previous study that mixing among the lowest dimensional operators, which have a mass dimension equal to three, can occur if chiral symmetry is broken in the lattice action. For higher dimensional operators, we find that the dimension-three operators will always mix with dimension-four operators, even if chiral symmetry is preserved. Also, the number of dimension-four operators involved in the mixing is large, and hence it is impractical to remove the mixing by the improvement procedure. Our result is important for determining the Bjorken-x dependence of the parton distribution functions using the quasi-distribution method on a Euclidean lattice. The requirement of using large hadron momentum in this approach makes the control of errors from dimension-four operators even more important.

Highlights

  • Controlling the systematic uncertainties is critical for obtaining meaningful results in lattice QCD

  • We use the symmetries of lattice QCD to analyze the mixing pattern of a class of nonlocal quark bilinear operators defined in Eq (27)

  • We extend the analysis to nonlocal quark bilinear operators

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Summary

Introduction

Controlling the systematic uncertainties is critical for obtaining meaningful results in lattice QCD. The nonperturbative renormalization method of the Rome-Southampton collaboration [1] has been widely used to convert from the lattice scheme to continuum schemes, avoiding the introduction of errors from the slowly converging lattice perturbation theory. We use the symmetries of lattice QCD to analyze the mixing pattern of a class of nonlocal quark bilinear operators defined in Eq (27). The renormalization of the nonlocal quark bilinears in the continuum was studied in Refs. Instead of performing explicit computations, we use symmetries to systematically study the mixing patterns among nonlocal quark bilinears (part of this work was reported in [15, 58]). We first review the symmetry analysis of local quark bilinear operators, and move to the nonlocal ones

Review of local quark bilinear operators
Dimension-three local operators
Dimension-four local operators
Dimension-three nonlocal operators
Dimension-four nonlocal operators
A mixing example in perturbative theory
Summary

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