Representation of the RG-invariant quantities in perturbative QCD through powers of the conformal anomaly

TL;DR

This paper explores representing RG-invariant quantities in perturbative QCD as expansions in powers of the conformal anomaly, analyzing several key observables and their relations within this framework.

Contribution

It introduces a unified approach to decompose various QCD quantities into conformal anomaly powers, extending previous results and applying it to new observables.

Findings

Successful decomposition of the Adler function and Bjorken sum rule coefficients.

Relation established between the Coulomb potential and cusp anomalous dimension.

Formal results for the heavy quark mass ratio in this framework.

Abstract

In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly $\beta(\alpha_s)/\alpha_s$ in the ${\rm{\overline{MS}}}$-scheme. We remind that such expansion is possible for the Adler function of the process of $e^+e^-$ annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the Crewther-Broadhurst-Kataev relation. In addition, we study the discussed decomposition for the static quark-antiquark Coulomb-like potential, its relation with the quantity defined by the cusp anomalous dimension and the coefficient function of the Bjorken unpolarized sum rule of neutrino-nucleon scattering. In conclusion we also present the formal results of applying this approach to the non-renormalization invariant ratio between the pole and ${\rm{\overline{MS}}}$-scheme running mass of heavy quark in QCD and compare them with those already known in the literature. The arguments in favor of the validity of the considered representation in powers of $\beta(\alpha_s)/\alpha_s$ for all mentioned perturbative quantities are discussed.