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The scheme and gauge dependence of the factorization property of the RG$\beta$-function in the $SU(N_c)$ QCD generalized Crewther relation (GCR),which connects the non-singlet contributions to the Adler and Bjorken polarizedsum rule functions, is investigated at the $\mathcal{O}(a^4_s)$ level. In thegauge-invariant $\rm{\overline{MS}}$-scheme this property holds at least atthis order. To study whether this property is true in all gauge-invariantschemes, we consider the $\rm{MS}$-like schemes in QCD and the QED-limit of theGCR in the $\rm{\overline{MS}}$-scheme and in the $\rm{MOM}$ and the $\rm{OS}$schemes. In these schemes we confirm the existence of the $\beta$-functionfactorization in the QCD and QED variants of the GCR. The problem of thepossible $\beta$-factorization in the gauge-dependent renormalization schemesin QCD is studied. We consider the gauge non-invariant $\rm{mMOM}$ and$\rm{MOMgggg}$-schemes and demonstrate that in the $\rm{mMOM}$ scheme at the$\mathcal{O}(a^3_s)$ level the $\beta$-factorization is valid for three valuesof the gauge parameter $\xi$ only, namely for $\xi=-3, -1$ and $\xi=0$. In the$\mathcal{O}(a^4_s)$ order of PT it remains valid only for case of the Landaugauge $\xi=0$.  The consideration of these two schemes for the QCD GCR allows us to concludethat the factorization of RG $\beta$-function will always be implemented in any$\rm{MOM}$-like schemes with linear covariant gauge at $\xi=0$ and $\xi=-3$ atthe $\mathcal{O}(a^3_s)$ level. It is demonstrated that if factorizationproperty for the $\rm{MS}$-like schemes is true in all orders of PT, astheoretically indicated, then the factorization will also occur in thearbitrary $\rm{MOM}$-like scheme in the Landau gauge in all orders of PT aswell.

the journal of high energy physics/the journal of high energy physics

Renormalization scheme and gauge (in)dependence of the generalized  Crewther relation: what are the real grounds of the $\beta$-factorization  property?