The $\{\beta\}$-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order $O(\alpha_s^4)$

TL;DR

This paper derives explicit expressions for the $eta$-expansion of the nonsinglet Adler function and Bjorken sum rule at order $O(\alpha_s^4)$, revealing properties linked to the Crewther and Broadhurst-Kataev relations in extended QCD.

Contribution

It provides the first explicit expressions for the $eta$-expansion at N$^4$LO for these quantities, incorporating recent high-order calculations in extended QCD.

Findings

Explicit $eta$-expansion expressions at N$^4$LO for Adler function and Bjorken sum rule.

Analysis of properties of the $eta$-expansion related to Crewther and Broadhurst-Kataev relations.

Insights into the structure of perturbative QCD at high orders.

Abstract

We derive explicit expressions for the elements of the $\{ \beta \}$-expansion for the nonsinglet Adler $D_A$-function and Bjorken polarized sum rules $S^{Bjp}$ in the N$^4$LO using recent results by Chetyrkin for these quantities computed within extended QCD including any number of fermion representations. We discuss the properties of the $\{ \beta \}$-expansion for $D_A$ and $S^{Bjp}$ at higher orders which follow from the Crewther [1] and the Broadhurst-Kataev [2] relation.