Regularization Prescription for the Mixing Between Nonlocal Gluon and Quark Operators

TL;DR

The paper addresses an ambiguity in relating coordinate-space and momentum-space results for the mixing of nonlocal gluon and flavor-singlet quark operators, rooted in a singular $1/{\bf z_{12}}$ pole. It shows that dimensional regularization provides a simple, physically meaningful prescription that yields consistent factorization in both spaces and aligns with lattice-based extractions within the LaMET framework. Through forward and nonforward correlators, comparisons between coordinate-space OPE and momentum-space calculations, and analyses of Mellin-moment matching, the authors demonstrate that DR reproduces the expected local operator mixing and cures the ambiguities that plague other approaches. This unifies the coordinate- and momentum-space pictures, enabling reliable lattice determinations of gluon PDFs/GPDs in practical applications.

Abstract

It is well-known that in the study of mixing between nonlocal gluon and quark bilinear operators there exists an ambiguity when relating coordinate space and momentum space results, which can be conveniently resolved through Mellin moments matching in both spaces. In this work, we show that this ambiguity is due to the lack of a proper regularization prescription of the singularity that arises when the separation between the gluon/quark fields approaches zero. We then demonstrate that dimensional regularization resolves this issue and yields consistent results in both coordinate and momentum space. This prescription is also compatible with lattice extractions of parton distributions from nonlocal operators.