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

The problem of scheme and gauge dependence of the factorization property of the renormalization group $\beta$-function in the $SU\left(N_c\right)$ QCD generalized Crewther relation (GCR), which connects the flavor non-singlet contributions to the Adler and Bjorken polarized sum rule functions, is investigated at the $O\left(a_s^4\right)$ level of perturbation theory. It is known that in the gauge-invariant renormalization $\overline{MS}$-scheme this property holds in the QCD GCR at least at this order. To study whether this factorization property is true in all gauge-invariant schemes, we consider the $MS$-like schemes in QCD and the QED-limit of the GCR in the $\overline{MS}$-scheme and in two other gauge-independent subtraction schemes, namely in the momentum $MOM$ and the on-shell $OS$ schemes. In these schemes we confirm the existence of the $\beta$-function factorization in the QCD and QED variants of the GCR. The problem of the possible $\beta$-factorization in the gauge-dependent renormalization schemes in QCD is studied. To investigate this problem we consider the gauge non-invariant $mMOM$ and $MOMgggg$-schemes. We demonstrate that in the $mMOM$ scheme at the $O\left(a_s^3\right)$ level the $\beta$-factorization is valid for three values of the gauge parameter $\xi$ only, namely for $\xi =-3,-1$ and $\xi =0$. In the $O\left(a_s^4\right)$ order of PT it remains valid only for case of the Landau gauge $\xi =0$. The consideration of these two gauge-dependent schemes for the QCD GCR allows us to conclude that the factorization of RG $\beta$-function will always be implemented in any $MOM$-like renormalization schemes with linear covariant gauge at $\xi =0$ and $\xi =-3$ at the $O\left(a_s^3\right)$ approximation. It is demonstrated that if factorization property for the $MS$-like schemes is true in all orders of PT, as theoretically indicated in the several works on the subject, then the factorization will also occur in the arbitrary $MOM$-like scheme in the Landau gauge in all orders of perturbation theory as well.