Quantum corrections and exact relations for the renormalization group functions in $N=1$ supersymmetric theories regularized by higher covariant derivatives

The talk is devoted to research on testing exact relations between renormalization group functions in the $N=1$ supersymmetric theories, regularized by higher covariant derivatives, using explicit two- and three- loop calculations in which the scheme dependence is essential. In $N=1$ non- Abelian supersymmetric theories, such relations include the finiteness of the triple gauge-ghost vertices and a new form of the NSVZ equation relating the $\bata$-function to the anomalous dimensions of quantum superfields in the previous order of perturbation theory. Also we consider $N=1$ supersymmetric theories satisfying the so-called $P=1/3$ $Q$ condition and analyze a possibility of obtaining the renormalization group invariant ratio of the Yukawa couplings to the gauge coupling. It turned out that the renormalization group invariance of this ratio is equivalent to an exact relation between the anomalous dimensions of quantum superfields. This relation is verified in the one- and two-loop approximation, and its validity in higher orders of perturbation theory is also discussed.