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

The analytical $\mathcal{O}(a_s^4)$ perturbative QCD expression for the flavour non-singlet contribution to the Bjorken polarized sum rule in the rather applicable at present gauge–dependent $\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{M}\mathrm{O}\mathrm{M}$ 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 $\overline{\mathrm{M}\mathrm{S}}$ 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 $\overline{\mathrm{M}\mathrm{S}}$ scheme up to $\mathcal{O}(a_s^4)$-level at least, is unexpectedly valid at the same level in the $\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{M}\mathrm{O}\mathrm{M}$-scheme for $\xi =0$, and for $\xi =-1$ and $\xi =-3$ in part.