$\{\beta\}$-expansion in QCD, its conformal symmetry limit: theory + applications

TL;DR

This paper discusses the $eta$-expansion in QCD, its relation to conformal symmetry, and its applications, emphasizing the completeness of the expansion and its connection to the Crewther relation for key QCD observables.

Contribution

It introduces a complete form of the $eta$-expansion in QCD, compares it with PMC methods, and applies it to important QCD functions at three-loop order.

Findings

Complete $eta$-expansion terms at $O( ext{alpha}_s^3)$ are derived.

The $eta$-expansion relates to the Crewther relation for $D^{NS}$ and $C^{Bjp}_{NS}$.

The results are expressed in a PMC-type form for phenomenological use.

Abstract

The basis of the $\{\beta\}$-expansion for the perturbative series evaluated in the $\overline{MS}$ scheme for the renormalization group invariant quantities is summarized.Comparison with a similar representation,used within the BLM-motivated Principle of Maximal Conformality,is discussed.We stress that the original $\{\beta\}$-expansion contains a completed list of terms rather than its PMC analog. The arguments in favour of the complete $\{\beta\}$-expansion are presented. They are based on the relations which follow from the power $\beta$-function generalization of the Crewther relation for the nonsinglet $\overline{MS}$ contributions to the Adler $D^{NS}$-functionand to the Bjorken sum rule $C^{Bjp}_{NS}$ of the polarized lepton-nucleon scattering. The terms of the complete $\{\beta\}$-expansionat the $O(\alpha_s^3)$ level for $D^{NS}$ and $C^{Bjp}_{NS}$ are presented. These perturbative results are expressed in the PMC-type form. The problem of applications of these expressions for phenomenological applications is summarized.