Crewther's relation in different schemes

TL;DR

This paper investigates Crewther's relation within various renormalization schemes in QCD, emphasizing the importance of gauge parameter dependence and proposing modifications to preserve conformal properties at critical points.

Contribution

It demonstrates the need for an additional term in Crewther's relation in gauge-parameter dependent schemes to maintain conformal invariance at all QCD critical points.

Findings

Additional term is necessary in mMOM scheme for gauge invariance

Conformal properties are preserved with the modified relation

Gauge-parameter dependence affects Crewther's relation at high loop order

Abstract

We examine Crewther's relation at high loop order in perturbative QCD and demonstrate how the relation is accommodated in gauge-parameter dependent schemes where the running of the gauge parameter has to be explicitly considered. Motivated by ensuring that the conformal properties of the relation are preserved at all the critical points of QCD, including the Banks-Zaks and its infra-red stable twin, we demonstrate the necessity of an additional term in the relation for describing gauge running in the minimal momentum subtraction scheme (mMOM) and argue for its inclusion for all gauge-parameter dependent schemes.