Background field method in the gradient flow

TL;DR

This paper introduces a modified background field method for the Yang--Mills gradient flow that simplifies perturbative calculations, enabling faster one-loop computations relevant to lattice gauge theory observables.

Contribution

A new gauge-covariant background field formulation of the gradient flow is proposed, streamlining perturbative calculations in gauge theories.

Findings

Enables quick one-loop calculations of small flow time expansions.

Preserves gauge covariance under background gauge transformations.

Provides detailed calculation methods not previously published.

Abstract

The Yang--Mills gradient flow and its extension to the fermion field provide a very general method to obtain renormalized observables in gauge theory. The method is applicable also with non-perturbative regularization such as lattice. The gradient flow thus offers useful probes to study non-perturbative dynamics of gauge theory. In this work, aiming at possible simplification in perturbative calculations associated with the gradient flow, a modification of the gauge-fixed version of the flow equation, which preserves gauge covariance under the background gauge transformation, is proposed. This formulation allows for example a very quick one-loop calculation of the small flow time expansion of a composite operator that is relevant to the construction of a lattice energy--momentum tensor. Some details of the calculation, which have not been given elsewhere, are presented.