Adler Function, Bjorken Sum Rule, and the Crewther Relation to Order alpha_s^4 in a General Gauge Theory

TL;DR

The paper computes the first ${\cal O}(\alpha_s^4)$ contributions to the Bjorken sum rule and the non-singlet Adler function for a general gauge group. It verifies the generalized Crewther relation at this order, providing a nontrivial cross-check against the five-loop quenched QED beta function and confirming the appearance of the irrational constant $\zeta_3$ in that context. The authors present complete ${\cal O}(\alpha_s^4)$ results for the relevant coefficient functions $d_4$ and $b_4$ across all color structures, satisfying all six Crewther-consistency constraints. They also derive a commensurate scale relation between the effective couplings extracted from Bjorken sum rule and Adler function, enabling cross-validation of observables and informing higher-twist analyses.

Abstract

We compute, for the first time, the order alpha_s^4 contributions to the Bjorken sum rule for polarized electron-nucleon scattering and to the (non-singlet) Adler function for the case of a generic colour gauge group. We confirm at the same order a (generalized) Crewther relation which provides a strong test of the correctness of our previously obtained results: the QCD Adler function and the five-loop beta-function in quenched QED. In particular, the appearance of an irrational contribution proportional to zeta_3 in the latter quantity is confirmed. We obtain the commensurate scale equation relating the effective strong coupling constants as inferred from the Bjorken sum rule and from the Adler function at order alpha_s^4.