The absence of QCD $\beta$-function factorization property of the generalized Crewther relation in the 't Hooft $\bar{MS}$-based scheme

Abstract

We apply the 't Hooft $\bar{MS}$-based scheme to study the scheme-dependence of the QCD generalization of Crewther relation for the product of the normalised non-singlet perturbative contributions to the $e^+e^-$-annihilation Adler function and to the Bjorken sum rule of the polarized lepton-nucleon deep-inelastic scattering process. We prove that after the transformations from the pure $\bar{MS}$-scheme to the 't Hooft scheme the characteristic $\bar{MS}$-scheme theoretical property of this relation, namely the factorization of the $\beta$-function in its conformal symmetry breaking part, disappears. Another "non-comfortable" theoretical consequence of the application of this prescription in $\mathcal{N}=1$ SUSY QED model is mentioned. It is shown, that within the 't Hooft scheme the expansions of Green functions in terms of the Lambert function is simplified in high orders of perturbation theory. This may be considered as the attractive feature of the 't Hooft scheme, which manifest itself in high-order perturbative phenomenological applications.