Derivation of a gradient flow from the exact renormalization group

Hidenori Sonoda,Hiroshi Suzuki

doi:10.1093/ptep/ptz020

Abstract

We establish a concrete correspondence between a gradient flow and the renormalization group flow for a generic scalar field theory. We use the exact renormalization group formalism with a particular choice of the cutoff function.

Highlights

The gradient flow, introduced in [1, 2], has been attracting much attention lately
The flow is much reminiscent of the renormalization group transformation [3], and it is especially important for lattice theories which had only discrete renormalization group transformations available
We introduce Wilson actions with a finite momentum cutoff using the formalism of the exact renormalization group (ERG) [3]

Summary

Introduction

The gradient flow, introduced in [1, 2], has been attracting much attention lately It is a continuous diffusion of local fields, well defined in continuum space and on discrete lattices. In the gradient flow we introduce a diffusion time t > 0, and extend the local field φ(x) in D-dimensional space by the solution of the diffusion equation (with no non-linear terms; see [16] for the motivation for this simple choice):. This is to prepare for the discussion of the gradient flow at small diffusion times in Sec. 5, where we derive the small time expansions of local products of the diffused field.

Overview of the ERG formalism

Derivation of a gradient flow

Gradient flow for a renormalizable theory

The small time expansions

Conclusions