Structure of three-loop contributions to the beta-function of N=1 SQED with N_f flavors, regularized by the dimensional reduction

In the case of using the higher derivative regularization for $N=1$ SQED with $N_f$ flavors the loop integrals giving the $β$-function are integrals of double total derivatives in the momentum space. This feature allows to reduce one of the loop integrals to an integral of the $δ$-function and to derive the NSVZ relation for the renormalization group functions defined in terms of the bare coupling constant. In this paper we consider $N=1$ SQED with $N_f$ flavors regularized by the dimensional reduction in the $\overline{\mbox{DR}}$-scheme. Evaluating the scheme-dependent three-loop contribution to the $β$-function proportional to $(N_f)^2$ we find the structures analogous to integrals of the $δ$-singularities. After adding the scheme-independent terms proportional to $(N_f)^1$ we obtain the known result for the three-loop $β$-function.

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