Gradient flow and the renormalization group

Yoshihiko Abe,Masafumi Fukuma

doi:10.1093/ptep/pty081

Yoshihiko Abe, Masafumi Fukuma

Open Access

https://doi.org/10.1093/ptep/pty081

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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 $\varepsilon$.

Highlights

In recent years the gradient flow has attracted much attention for practical and conceptual reasons [1,2,3,4,5,6,7]
We make an ε expansion of the local potential approximation (LPA) and show that it reproduces the eigenvalues of the linearized renormalization group (RG) transformation around both the Gaussian and the Wilson–Fisher fixed points to the order of ε
We have investigated the RG structure of the gradient flow

Summary

Introduction

In recent years the gradient flow has attracted much attention for practical and conceptual reasons [1,2,3,4,5,6,7]. 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 an RG equation if one makes a fieldvariable transformation at every step such that the kinetic term is kept in 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.

Results

Conclusion