Multiloop calculations and exact results in $N=1$ supersymmetric theories

The exact Novikov–Shifman–Vainshtein–Zakharov (NSVZ) solution for the beta-function of $N=1$ supersymmetric Yang–Mills theory relates the beta-function to the anomalous dimension of matter superfields. But explicit calculations performed to four-loop order in dimensional reduction reveal that this relation is not satisfied due to scheme dependence of both beta-function and anomalous dimension. In a three-loop calculation we show that, using regularization by higher covariant derivatives and subtracting only powers of divergent logarithms, it is possible to obtain an identity relating the beta-function to anomalous dimensions of all of the superfields, from which the NSVZ beta-function follows directly due to a non-renormalization theorem. We also apply the same technique to the calculation of the Adler $D$-function in $N=1$ supersymmetric quantum chromodynamics to three-loop order. There we demonstrate that higher covariant derivatives and minimal subtraction of logarithms lead to a relation between the $D$-function and the anomalous dimension of quark superfields similar to the NSVZ beta-function.