The higher derivative regularization and quantum corrections in N=2N=2 supersymmetric theories

We construct a new version of the higher covariant derivative regularization for a general N=2N=2 supersymmetric gauge theory formulated in terms of N=1N=1 superfields. This regularization preserves both supersymmetries of the classical action, namely, the invariance under the manifest N=1N=1 supersymmetry and under the second hidden on-shell supersymmetry. The regularizing N=2N=2 supersymmetric higher derivative term is found in the explicit form in terms of N=1N=1 superfields. Thus, N=2N=2 supersymmetry is broken only by the gauge fixing procedure. Then we analyze the exact NSVZ β-function and prove that in the considered model its higher loop structure is determined by the anomalous dimension of the chiral superfield Φ   in the adjoint representation which is the N=2N=2 superpartner of the gauge superfield V  . Using the background field method we find that this anomalous dimension is related with the anomalous dimension of the hypermultiplet and vanishes if the effective action is invariant under N=2N=2 background supersymmetry. As a consequence, in this case the higher loop contributions to β-function also vanish. The one-loop renormalization structure in the considered regularization is also studied by the explicit calculations of the one-loop renormalization constants.