The effective exponent gamma(Q) and the slope of the beta function

TL;DR

This paper clarifies the relationship between the slope of the beta function at a fixed point and the critical exponent gamma*, introducing an effective exponent gamma(Q) that is RG invariant and converges to the true gamma*.

Contribution

The paper defines a proper RG invariant effective exponent gamma(Q) that accurately captures the critical behavior near fixed points, correcting previous misconceptions.

Findings

Gamma(Q) is RG invariant and converges to gamma*

The slope of the beta function is not strictly RG invariant

The effective exponent provides a more precise measure of critical behavior

Abstract

The slope of the beta function at a fixed point is commonly thought to be RG invariant and to be the critical exponent gamma* that governs the approach of any physical quantity R to its fixed-point limit: R*-R proportional to Q^gamma*. Chyla has shown that this is not quite true. Here we define a proper RG invariant, the "effective exponent" gamma(Q), whose fixed-point limit is the true gamma*.