Remarks on Exact RG Equations

TL;DR

This paper discusses the role of the anomalous dimension in exact renormalization group equations, emphasizing the conditions for fixed points and extending the analysis to supersymmetric theories.

Contribution

It provides a detailed analysis of the anomalous dimension in exact RG equations, introduces a method to determine it via fixed points, and applies the framework to supersymmetric theories.

Findings

The anomalous dimension is a free parameter reflecting RG equation freedom.

The exact value of $ta$ is fixed by IR fixed point existence.

In supersymmetric theories, $ta$ may be large at fixed points.

Abstract

Exact RG equations are discussed with emphasis on the role of the anomalous dimension $\eta$. For the Polchinski equation this may be introduced as a free parameter reflecting the freedom of such equations up to contributions which vanish in the functional integral. The exact value of $\eta$ is only determined by the requirement that there should exist a well defined non trivial limit at a IR fixed point. The determination of $\eta$ is related to the existence of an exact marginal operator, for which an explicit form is given. The results are extended to the exact Wetterich RG equation for the one particle irreducible action $\Gamma$ by a Legendre transformation. An alternative derivation of the derivative expansion is described. An application to $\N=2$ supersymmetric theories in three dimensions is described where if an IR fixed point exists then $\eta$ is not small.