The Yang-Mills gradient flow and lattice effective action

TL;DR

This paper explores the Yang-Mills gradient flow's connection to the Wilsonian renormalization group, defining an effective action that evolves with flow time and proposing a flow-dependent gradient akin to RG flow.

Contribution

It introduces a formalism linking the Yang-Mills gradient flow to the Wilsonian RG through an effective action and differential equations describing their flow dynamics.

Findings

Derived the exact differential equation for the effective action's flow time dependence.

Proposed a flow time dependent gradient similar to the renormalization group equation.

Discussed the interpretation of the flow as a Wilsonian RG flow.

Abstract

Recently, the Yang-Mills gradient flow is found to be a useful concept not only in lattice simulations but also in continuous field theories. Since its smearing property is similar to the Wilsoninan "block spin transformation", there might be deeper connection between them. In this work, we define the "effective action" which generates configurations at a finite flow time and derive the exact differential equation to investigate the flow time dependence of the action. Then Yang-Mills gradient flow can be regarded as the flow of the effective action. We also propose the flow time dependent gradient, where the differential equation becomes similar to the renormalization group equation. We discuss a possibility to regard the time evolution of the effective action as the Wilsonian renormalization group flow.