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

The problem of scheme and gauge dependence of the factorization property of the renormalization group beta-function in the SU(N-c) 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(a(s)(4)) level of perturbation theory. It is known that in the gauge-invariant renormalization 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 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 (a(s)(3)) 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 (a(s)(4)) 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 (a(s)(3)) 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.