The analytical singlet $\alpha_s^4$ QCD contributions into the $e^+e^-$-annihilation Adler function and the generalized Crewther relations

The generalized Crewther relations in the channels of the non-singlet and vector quark currents are considered. These relations follow from the double application of the operator product expansion approach to the same axial vector–vector–vector triangle amplitude in two regions, adjoining to the angle sides $(x,y)$ (or $p^2,q^2$). We assume that the generalized Crewther relations in these two kinematic regimes result in the existence of the same perturbation expression for two products of the coefficient functions of annihilation and deep-inelastic scattering processes in the non-singlet and vector channels. This feature explains the conformal symmetry motivated cancellations between the singlet $\alpha_s^3$ corrections to the Gross–Llewellyn Smith sum rule $S_{\rm GLS}$ of ${\nu N}$ deep inelastic scattering and the singlet $\alpha_s^3$ correction to the $e^+e^-$-annihilation Adler function $D_A^{V}$ in the product of the corresponding perturbative series. Taking into account the Baikov–Chetyrkin–Kuhn 4-th order result for $S_{\rm GLS}$ and the perturbative effects of the violation of the conformal symmetry in the generalized Crewther relation, we obtain the analytical contribution to the singlet $\alpha_s^4$ correction to the $D_A^{V}$-function. Its a-posteriori comparison with the recent result of direct diagram-by-diagram evaluation of the singlet 4-th order corrections to $D_A^{V}$- function demonstrates the coincidence of the predicted and obtained $\zeta_3^2$-contributions to the singlet term. They can be obtained in the conformal invariant limit from the original Crewther relation. Therefore, on the contrary to previous belief, the appearance of $\zeta_3$-terms in the perurbative series in quantum field theory gauge models does not contradict to the property of the the conformal symmetry and can be considered as regular feature. The Banks–Zaks motivated relation between our predicted and the obtained directly 4-th order corrections is mentioned. It confirms the expectation, previously made by Baikov–Chetykin–Kuhn, that at the 5-loop level the generalized Crewther relation in the channel of vector currents may receive additional singlet contribution, which in this order of perturbation theory is proportional to the first coefficient of the QCD $\beta$-function.