The Generalized Scheme-Independent Crewther Relation in QCD

TL;DR

This paper derives a scheme-independent generalized Crewther relation in QCD using the Principle of Maximal Conformality, enabling precise, scheme-independent connections between observables and improving fundamental tests of QCD.

Contribution

It introduces a new generalized Crewther relation in QCD that is independent of renormalization scheme and initial scale, utilizing PMC scale-setting for unambiguous predictions.

Findings

Derived a scheme-independent generalized Crewther relation in QCD.

Connected effective charges at their physical scales with negligible initial scale dependence.

Provided a framework for precision, scheme-independent tests of QCD.

Abstract

The Principle of Maximal Conformality (PMC) provides a systematic way to set the renormalization scales order-by-order for any perturbative QCD process. The resulting predictions are independent of the choice of renormalization scheme, a requirement of renormalization group invariance. The Crewther relation, which was originally derived for conformal theory, provides a remarkable connection between two observables when the $\beta$ function vanishes. The "Generalized Crewther Relation" relates these two observables for physical QCD with nonzero $\beta$ function; specifically, it connects the non-singlet Adler function ($D^{\rm ns}$) to the Bjorken sum rule coefficient for polarized deep-inelastic electron scattering ($C_{\rm Bjp}$) at leading twist. A scheme-dependent $\Delta_{\rm CSB}$-term appears in the analysis in order to compensate for the conformal symmetry breaking (CSB) terms from perturbative QCD. In conventional analyses, this normally leads to unphysical dependence in both the choice of the renormalization scheme and the choice of the initial scale at any finite order. However, by applying PMC scale-setting, we can fix the scales of the QCD coupling unambiguously at every order of pQCD. ...... Thus one obtains a new generalized Crewther relation for QCD which connects two effective charges, $\widehat{\alpha}_d(Q)= \sum_{i\geq1} \widehat{\alpha}^{i}_{g_1}(Q_i)$, at their respective physical scales. This identity is independent of the choice of the renormalization scheme at any finite order, and the dependence on the choice of the initial scale is negligible. Similar scale-fixed commensurate scale relations also connect other physical observables at their physical momentum scales, thus providing convention-independent, fundamental precision tests of QCD.