The ${\rm{\bar{MS}}}$-scheme $\alpha_s^5$ QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold $\{\beta\}$-expanded representation

TL;DR

This paper develops a two-fold expansion method in powers of the conformal anomaly and strong coupling for QCD contributions to the Adler function and Bjorken sum rule, providing new five-loop order corrections and relations between different loop terms.

Contribution

It introduces a novel two-fold $eta$-expanded representation for non-singlet QCD contributions, enabling the estimation of five-loop corrections and fixed $eta$-expansion terms.

Findings

Derived five-loop order corrections for Adler function and Bjorken sum rule.

Established relations between different loop order terms within the $eta$-expansion.

Fixed analytical $ zeta_4$-contributions for generic gauge groups.

Abstract

We consider the two-fold expansion in powers of the conformal anomaly and of the strong coupling $\alpha_s$ for the non-singlet contributions to Adler $D$-function and Bjorken polarized sum rule calculated previously in the $\MSbar$-scheme at the four-loop level. This representation provides relations between definite terms of different loop orders appearing within the $\{\beta\}$-expansion of these quantities. Supposing the validity of this two-fold representation at the five-loop order and using these relations, we obtain some $\mathcal{O}(\alpha_s^5)$ corrections to the $D$-function, to the $R$-ratio of $e^+e^-$-annihilation into hadrons and to Bjorken polarized sum rule. These corrections are presented both analytically in the case of the generic simple gauge group and numerically for the $SU(3)$ color group. The arguments in the favor of validity of the two-fold representation are given at least at the four-loop level. Within the $\{\beta\}$-expansion procedure the analytical Riemann $\zeta_4$-contributions to the five-loop expressions for the Adler function and Bjorken polarized sum rule are also fixed for the case of the generic simple gauge group.