For the N = 1 supersymmetric electrodynamics we investigate renormalization schemes in which the NSVZ equation relating the β-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 β-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.

At the three-loop level we analyze, how the NSVZ relation appears for N = 1 SQED regularized by the dimensional reduction. This is done by the method analogous to the one which was earlier used for the theories regularized by higher derivatives. Within the dimensional technique, the loop integrals cannot be written as integrals of double total derivatives. However, similar structures can be written in the considered approximation and are taken as a starting point. Then we demonstrate that, unlike the higher derivative regularization, the NSVZ relation is not valid for the renormalization group functions defined in terms of the bare coupling constant. However, for the renormalization group functions defined in terms of the renormalized coupling constant, it is possible to impose boundary conditions to the renormalization constants giving the NSVZ scheme in the three-loop order. They are similar to the all-loop ones defining the NSVZ scheme obtained with the higher derivative regularization, but are more complicated. The NSVZ schemes constructed with the dimensional reduction and with the higher derivative regularization are related by a finite renormalization in the considered approximation.

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Two renormalization group invariant quantities in quantum chromodinamics (QCD), defined in Euclidean space, namely, Adler Dfunction of electron-positron annihilation to hadrons and Bjorken polarized deep-inelastic scattering sum rule, are considered. It is shown, that the 5-th order corrections to them in MS-like renormalization prescriptions, proportional to Riemann ζ-function ζ (4), can be restored by the transition to the C-scheme, with the β-function, analogous to Novikov, Shifman, Vainshtein and Zakharov exact β-function in N = 1 supersymmetric gauge theories. The general analytical expression for these corrections in SU (N c ) QCD is deduced and their scale invariance is shown. The β-expansion procedure for these contributions is performed and mutual cancellation of them in the 5-th order of the generalized Crewther identity are discussed.

We consider the two-fold expansion in powers of the conformal anomaly and of the strong coupling αs for the non-singlet contributions to Adler D-function and Bjorken polarized sum rule calculated previously in the $$ \overline{\mathrm{MS}} $$ MS ¯ -scheme at the four-loop level. This representation provides relations between definite terms of different loop orders appearing within the {β}-expansion of these quantities. Supposing the validity of this two-fold representation at the five-loop order and using these relations, we obtain some $$ \mathcal{O} $$ O ($$ {\alpha}_s^5 $$ α 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 {β}-expansion procedure the analytical Riemann ζ4-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.