A class of the NSVZ renormalization schemes for N=1{\cal N}=1 SQED

For the ${\cal N}=1$ supersymmetric electrodynamics we investigate renormalization schemes in which the NSVZ equation relating the $\beta$-function to the anomalous dimension of the matter superfields is valid in all loops. We demonstrate that there is an infinite set of such schemes. They are related by finite renormalizations which form a group and are parameterized by one finite function and one arbitrary constant. This implies that the NSVZ $\beta$-function remains unbroken if the finite renormalization of the coupling constant is related to the finite renormalization of the matter superfields by a special equation derived in this paper. The arbitrary constant corresponds to the arbitrariness of choosing the renormalization point. The results are illustrated by explicit calculations in the three-loop approximation.Comment: 12 page

The MSˉ{\rm{\bar{MS}}}-scheme αs5\alpha_s^5 QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold {β}\{\beta\}-expanded representation

We consider the two-fold expansion in powers of the conformal anomaly and of the strong coupling $\alpha_s$ for the non-singlet contributions to Adler $D$-function and Bjorken polarized sum rule calculated previously in the \MSbar-scheme at the four-loop level. This representation provides relations between definite terms of different loop orders appearing within the $\{\beta\}$-expansion of these quantities. Supposing the validity of this two-fold representation at the five-loop order and using these relations, we obtain some $\mathcal{O}(\alpha_s^5)$ corrections to the $D$-function, to the $R$-ratio of $e^+e^-$-annihilation into hadrons and to Bjorken polarized sum rule. These corrections are presented both analytically in the case of the generic simple gauge group and numerically for the $SU(3)$ color group. The arguments in the favor of validity of the two-fold representation are given at least at the four-loop level. Within the $\{\beta\}$-expansion procedure the analytical Riemann $\zeta_4$-contributions to the five-loop expressions for the Adler function and Bjorken polarized sum rule are also fixed for the case of the generic simple gauge group.Comment: 30 pages, 1 Figure, 6 Tables, text modified, the results unchanged, list of references modified, accepted for publication in JHE