The structure of quantum corrections and exact results in supersymmetric theories from the higher covariant derivative regularization

We review some recent results of the studies of quantum corrections in supersymmetric theories derived using Slavnov's higher covariant derivative regularization. In particular, we demonstrate that the $\beta$-function of $\mathcal{N}=1$ supersymmetric theories is related to the anomalous dimensions of matter superfields by the NSVZ relation if the theory is regularized by higher covariant derivatives and the renormalization group functions are defined in terms of the bare couplings, because the corresponding loop corrections are given by integrals of double total derivatives in the momentum space. For the standard renormalization-group functions, we show that an all-loop NSVZ renormalization scheme is given by the HD$\,+\,$MSL renormalization prescription when the higher covariant derivative regularization is supplemented by minimal subtractions of logarithms. Applications of these results to the precise calculations in various supersymmetric theories are briefly described.