Analytical Singlet αs4 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(3)(s) corrections to the Gross-Llewellyn Smith sum rule S-GLS of nu N deep inelastic scattering and the singlet alpha(3)(s) correction to the e(+)e(-)-annihilation Adler function in the product of the corresponding perturbative series. Taking into account the Baikov-Chetyrkin-Kuhn fourth order result for SGLS and the perturbative effects of the violation of the conformal symmetry in the generalized Crewther relation, we obtain the analytical contribution to the singlet correction to the D-A(V) function. Its a-posteriori comparison with the recent result of direct diagram-by-diagram evaluation of the singlet fourth order corrections to D-A(V) function demonstrates the coincidence of the predicted and obtained zeta(2)(3)-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 perturbative series in quantum field theory gauge models does not contradict to the property of the conformal symmetry and can be considered as regular feature. The Banks-Zaks motivated relation between our predicted and the obtained directly fourth 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.