Scheme Dependence of the Wilsonian Effective Action and Sharp Cutoff Limit of the Flow Equation

TL;DR

This paper explores how the Wilsonian effective action's dependence on cutoff schemes can be viewed as a coordinate transformation in theory space, revealing strong scheme dependence in Wilsonian actions but not in effective average actions, and connects the sharp cutoff limit to the Wegner-Houghton equation.

Contribution

It demonstrates the scheme dependence as a coordinate transformation and derives the sharp cutoff limit of the Polchinski RG equation, linking it to the Wegner-Houghton equation.

Findings

Wilsonian effective actions depend strongly on cutoff schemes.

Effective average actions are scheme-independent in the IR region.

The sharp cutoff limit of the Polchinski RG equation matches the Wegner-Houghton equation.

Abstract

The cutoff scheme dependence in the several formulations of the Exact Renormalization Group (ERG) is investigated. It is shown that the cutoff scheme dependence of the Wilsonian effective action is regarded as a certain coordinate transformation on the theory space. From this observation the Wilsonian effective actions are found to suffer from strong dependence on the schemes even in the infra-red asymptotic region for massive theories. However there is no such scheme dependence in the one particle irreducible parts of them, which is called the effective average actions. We also derive the explicit form of the Polchinski RG equation in the sharp cutoff limit. Finally this equation is shown to be identical with the Wegner-Houghton RG equation.