Quantum Corrections and Exact Results in Supersymmetric Theories Regularized by Higher Covariant Derivatives

Some properties of quantum corrections in various supersymmetric theories revealed with the help of the higher covariant derivative regularization are briefly reviewed. In particular, we discuss how the exact NSVZ $$\beta$$ -function appears, and why it is valid in all orders of the perturbation theory in the HD+MSL renormalization prescription. It is demonstrated that this fact can be used for essential simplifying some multiloop calculations. In particular, we present the expression for the three-loop $$\beta$$ -function of an arbitrary $$\mathcal{N}=1$$ supersymmetric gauge theory with a simple gauge group. For $$\mathcal{N}=2$$ supersymmetric theories written in terms of $$\mathcal{N}=1$$ superfields we formulate the conditions imposed on a subtraction scheme which are needed for the finiteness beyond the one-loop approximation. In particular, the renormalization prescription should be $$\mathcal{N}=2$$ supersymmetric, so that the ratio of Yukawa and gauge couplings should not depend on the renormalization point $$\mu$$ . A possibility of achieving a similar condition for $$\mathcal{N}=1$$ supersymmetric theories satisfying the so-called $$P=\frac{1}{3}Q$$ condition and its consequences are discussed.