Gauge-invariant renormalization of fermion bilinears and energy-momentum tensor on the lattice

TL;DR

This paper introduces a gauge-invariant renormalization scheme (GIRS) for lattice QCD, enabling nonperturbative renormalization of fermion bilinears and energy-momentum tensors without gauge fixing, and provides conversion factors to the MS scheme.

Contribution

The paper extends the coordinate space scheme to a gauge-invariant context, proposing variants that reduce noise and applying it to fermion bilinears and energy-momentum tensors with one-loop conversion factors.

Findings

GIRS is gauge-invariant and does not require gauge fixing.

Variants of GIRS can reduce statistical noise in lattice simulations.

One-loop conversion factors to MS scheme are computed.

Abstract

We study a gauge-invariant renormalization scheme (GIRS) for composite operators, regularized on the lattice, by extending the coordinate space (X-space) scheme proposed some years ago. In this scheme, Green's functions of products of gauge-invariant operators located at different spacetime points are considered. Due to the gauge-invariant nature of GIRS, gauge fixing is not needed in the lattice simulations. Also, when operator mixing occurs, the gauge-variant operators (BRST variations and operators which vanish by the equations of motion) can be safely excluded from the renormalization process. We propose a number of variants of GIRS, including integration over time slices of the operator insertion point in a Green's function, which may lead to reduced statistical noise in lattice simulations. We employ these variants in the renormalization of fermion bilinear operators and the study of mixing between the gluon and quark energy-momentum tensor operators. We extract the one-loop conversion factors relating the nonperturbative renormalization factors in different versions of GIRS to the reference scheme of $\overline{\rm MS}$.