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The analytical $\mathcal{O}(a^4_s)$ perturbative QCD expression for theflavour non-singlet contribution to the Bjorken polarized sum rule in therather applicable at present gauge--dependent $\rm{miniMOM}$ scheme isobtained. 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 considerablysmaller than the gauge-independent $\rm{\overline{MS}}$ results. It is foundthat the fundamental property of the factorization of the QCD renormalizationgroup $\beta$-function in the generalized Crewther relation, which is valid inthe gauge-invariant $\rm{\overline{MS}}$ scheme up to$\mathcal{O}(a^4_s)$-level at least, is unexpectedly valid at the same level inthe $\rm{miniMOM}$-scheme for $\xi=0$, and for $\xi=-1$ and $\xi=-3$ in part.

journal of physics. conference series

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