The Wilson-Polchinski exact renormalization group equation

TL;DR

This paper revisits the Wilson-Polchinski exact renormalization group equation, clarifies the role of the critical exponent η, and proposes a criterion for its optimal value to improve the systematic use of the RG in theoretical studies.

Contribution

It establishes an explicit relation between Wilson's and Polchinski's RG formulations, analyzes the equation up to next-to-leading order, and introduces a criterion for selecting the best η value.

Findings

Clarified the relation between Wilson and Polchinski RG equations.

Analyzed the RG equation up to next-to-leading order in derivative expansion.

Proposed a criterion for choosing the optimal η value.

Abstract

The critical exponent $\eta $ is not well accounted for in the Polchinski exact formulation of the renormalization group (RG). With a particular emphasis laid on the introduction of the critical exponent $\eta $, I re-establish (after Golner, hep-th/9801124) the explicit relation between the early Wilson exact RG equation, constructed with the incomplete integration as cutoff procedure, and the formulation with an arbitrary cutoff function proposed later on by Polchinski. I (re)-do the analysis of the Wilson-Polchinski equation expanded up to the next to leading order of the derivative expansion. I finally specify a criterion for choosing the ``best'' value of $\eta $ to this order. This paper will help in using more systematically the exact RG equation in various studies.