The analytical $\mathcal{O}(a^4_s)$ expression for the polarized Bjorken sum rule in the miniMOM scheme and the consequences for the generalized Crewther relation

TL;DR

This paper derives an analytical expression for the polarized Bjorken sum rule in the miniMOM scheme at four-loop order, revealing scheme and gauge dependence and exploring implications for the generalized Crewther relation.

Contribution

It provides the first four-loop analytical expression for the polarized Bjorken sum rule in the miniMOM scheme, including gauge-dependent coefficients and their comparison with the MS-bar scheme.

Findings

Scheme-dependent coefficients are smaller than in MS-bar scheme.

Factorization property of the Crewther relation holds in miniMOM scheme for certain gauges.

The gauge dependence affects the scheme coefficients but not the fundamental factorization in some gauges.

Abstract

The analytical $\mathcal{O}(a^4_s)$ perturbative QCD expression for the flavour non-singlet contribution to the Bjorken polarized sum rule in the rather applicable at present gauge--dependent $\rm{miniMOM}$ scheme is obtained. For the considered three values of the gauge parameter, namely $\xi=0$ (Landau gauge), $\xi=-1$ (anti--Feynman gauge) and $\xi=-3$ (Stefanis--Mikhailov gauge), the scheme-dependent coefficients are considerably smaller than the gauge-independent $\rm{\overline{MS}}$ results. It is found that the fundamental property of the factorization of the QCD renormalization group $\beta$-function in the generalized Crewther relation, which is valid in the gauge-invariant $\rm{\overline{MS}}$ scheme up to $\mathcal{O}(a^4_s)$-level at least, is unexpectedly valid at the same level in the $\rm{miniMOM}$-scheme for $\xi=0$, and for $\xi=-1$ and $\xi=-3$ in part.