The exact NSVZ relation between a β-function of N = 1 SQED and an anomalous dimension of the matter superfields is studied within the Slavnov higher derivative regularization approach. It is shown that if the renormalization group functions are defined in terms of the bare coupling constant, this relation is always valid. In the renormalized theory the NSVZ relation is obtained in the momentum subtraction scheme supplemented by a special finite renormalization. Unlike the dimensional reduction, the higher derivative regularization allows to fix this finite renormalization. This is made by imposing the conditions Z 3 (α, µ = Λ) = 1 and Z(α, µ = Λ) = 1 on the renormalization constants of N = 1 SQED, where Λ is a parameter in the higher derivative term. The results are verified by the explicit three-loop calculation. In this approximation we relate the DR scheme and the NSVZ scheme defined within the higher derivative approach by the finite renormalization.

For N = 1 supersymmetric quantum electrodynamics, regularized by higher derivatives, a method for summation of all Feynman diagrams defining the βfunction is presented. Using this method we prove that the β-function is given by an integral of a total derivative, which can be easily calculated. It is shown that surviving terms give the exact NSVZ β-function. The results are compared with the explicit three-loop calculation.

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Three-loop quantum corrections to the effective action are calculated for N = 1 supersymmetric electrodynamics, regularized by higher derivatives. Using the obtained results we investigate the anomaly puzzle in the considered model.

We briefly review the calculations of quantum corrections related with the exact NSVZ β-function in N = 1 supersymmetric theories, paying especial attention to the scheme dependence of the results. It is explained, how the NSVZ relation is obtained for the renormalization group functions defined in terms of the bare coupling constant if a theory is regularized by higher derivatives. Also we describe, how to construct a special renormalization prescription which gives the NSVZ relation for the renormalization group functions defined in terms of the renormalized coupling constant exactly in all orders for Abelian supersymmetric theories, regularized by higher derivatives. The scheme dependence of the NSVZ β-function (for the renormalization group functions defined in terms of the renormalized coupling constant) is discussed in the non-Abelian case. It is shown that in this case the NSVZ β-function leads to a certain scheme-independent equality.Dedicated to the 75-th Birthday of Prof. A.A.Slavnov