Gradient flow and the renormalization group

TL;DR

This paper explores the connection between gradient flow and the renormalization group by proposing an effective action-based flow equation, analyzing its properties, and demonstrating its consistency with known RG fixed points through an epsilon expansion.

Contribution

It introduces a novel RG perspective on gradient flow using the effective action and derives results consistent with established fixed points.

Findings

The flow equation can be interpreted as an RG equation with a field-variable transformation.

The local potential approximation reproduces eigenvalues of linearized RG transformations.

The epsilon expansion aligns with known fixed point properties.

Abstract

We investigate the renormalization group (RG) structure of the gradient flow. Instead of using the original bare action to generate the flow, we propose to use the effective action at each flow time. We write down the basic equation for scalar field theory that determines the evolution of the action, and argue that the equation can be regarded as a RG equation if one makes a field-variable transformation at every step such that the kinetic term is kept to take the canonical form. We consider a local potential approximation (LPA) to our equation, and show that the result has a natural interpretation with Feynman diagrams. We make an $\varepsilon$ expansion of the LPA and show that it reproduces the eigenvalues of the linearized RG transformation around both the Gaussian and the Wilson-Fisher fixed points to the order of $\varepsilon$.