Background field method in the gradient flow

TL;DR

This paper introduces a background gauge covariant gauge-fixing method for the Yang--Mills gradient flow, enabling efficient perturbative calculations while preserving gauge invariance.

Contribution

It proposes a modified gauge-fixing term that maintains background gauge covariance and does not affect gauge-invariant quantities, improving computational methods.

Findings

Allows background gauge covariant perturbative expansion

Provides efficient computation of small flow time expansion coefficients

Can be extended to systems with fermions

Abstract

In perturbative consideration of the Yang--Mills gradient flow, it is useful to introduce a gauge non-covariant term ("gauge-fixing term") to the flow equation that gives rise to a Gaussian damping factor also for gauge degrees of freedom. In the present paper, we consider a modified form of the gauge-fixing term that manifestly preserves covariance under the background gauge transformation. It is shown that our gauge-fixing term does not affect gauge-invariant quantities as the conventional gauge-fixing term. The formulation thus allows a background gauge covariant perturbative expansion of the flow equation that provides, in particular, a very efficient computational method of expansion coefficients in the small flow time expansion. The formulation can be generalized to systems containing fermions.