The effective exponent γ ( Q ) and the slope of the β function

P.M Stevenson

doi:10.1016/j.physletb.2016.08.061

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*.

Highlights

Of a renormalizable quantum field theory is renormalizationscheme (RS) dependent
It is related to the slope of the β function but has an extra term that is crucial for its RS invariance
In the infrared limit, the critical exponent γ∗ is the derivative of the EC β function at the fixed point

Summary

Introduction

The slope of the β function at a fixed point, is commonly believed to be scheme invariant. Consider two RS’s, primed and unprimed, whose renormalized coupling constants (couplants) are related by a general scheme transformation a′ = a(1 + v1a + v2a2 + . Since β(a) vanishes at the fixed point, it would seem that dβ′ da′ We define the “effective exponent” γ(Q), a Q-dependent “scaling dimension” associated with a specific physical quantity R.

Results

Conclusion