A basis of the gradient flow exact renormalization group for gauge theory

TL;DR

This paper formulates the gradient flow exact renormalization group (GFERG) for gauge theories using the Reuter equation, clarifying its structure, divergences, and relation to flowed fields, thus enabling gauge-invariant computations.

Contribution

It introduces a new formulation of GFERG via the Reuter equation, clarifying its structure, divergence issues, and connection to flowed fields, advancing gauge-invariant RG analysis.

Findings

Unique ordering of functional derivatives removes ambiguity.

Unconventional UV divergences arise without gauge fixing.

Modified correlation functions match flowed field correlators.

Abstract

The gradient flow exact renormalization group (GFERG) is a variant of the exact renormalization group (ERG) for gauge theory that is aimed at preserve gauge invariance as manifestly as possible. It achieves this goal by utilizing the Yang--Mills gradient flow or diffusion for the block-spin process. In this paper, we formulate GFERG by the Reuter equation in which the block spinning is done by Gaussian integration. This formulation provides a simple understanding of various points of GFERG, unresolved thus far. First, there exists a unique ordering of functional derivatives in the GFERG equation that remove ambiguity of contact terms. Second, perturbation theory of GFERG suffers from unconventional ultraviolet (UV) divergences if no gauge fixing is introduced. This explains the origin of some UV divergences we have encountered in perturbative solutions to GFERG. Third, the modified correlation functions calculated with the Wilson action in GFERG coincide with the correlation functions of diffused or flowed fields calculated with the bare action. This shows the existence of a Wilson action that reproduces precisely the physical quantities computed by the gradient flow formalism (up to contact terms). We obtain a definite ERG interpretation of the gradient flow. The formulation given in this paper provides a basis for further perturbative/nonperturbative computations in GFERG, preserving gauge invariance maximally.