Scheme independent consequence of the NSVZ relation for N=1 SQED with N_f flavors

Abstract

The exact NSVZ $\beta$-function is obtained for ${\cal N}=1$ SQED with $N_f$ flavors in all orders of the perturbation theory, if the renormalization group functions are defined in terms of the bare coupling constant and the theory is regularized by higher derivatives. However, if the renormalization group functions are defined in terms of the renormalized coupling constant, the NSVZ relation between the $\beta$-function and the anomalous dimension of the matter superfields is valid only in a certain (NSVZ) scheme. We prove that for ${\cal N}=1$ SQED with $N_f$ flavors the NSVZ relation is valid for the terms proportional to $(N_f)^1$ in an arbitrary subtraction scheme, while the terms proportional to $(N_f)^k$ with $k\ge 2$ are scheme dependent. These results are verified by an explicit calculation of a three-loop $\beta$-function and a two-loop anomalous dimension made with the higher derivative regularization in the NSVZ and MOM subtraction schemes. In this approximation it is verified that in the MOM subtraction scheme the renormalization group functions obtained with the higher derivative regularization and with the dimensional reduction coincide.