Exact $\beta$-function in Abelian and non-Abelian $\mathcal{N} = 1$ supersymmetric gauge models and its analogy with the QCD $\beta$-function in the $\mathrm{C}$-scheme

For $\mathcal{N} = 1$ supersymmetric Yang–Mills theory without matter it is demonstrated that there is a class of renormalization schemes, in which the exact Novikov, Shifman, Vainshtein, and Zakharov (NSVZ) formula for the renormalization group $\beta$-function, defined in terms of the renormalized coupling constant, is valid. These schemes are related with each other by finite renormalizations forming a one-parameter commutative subgroup of general renormalization group transformations. The analogy between the exact $\beta$-function in $\mathcal{N} = 1$ supersymmetric Yang–Mills theory without matter and the $\beta$-function of quantum chromodynamics in the $\mathrm{C}$-scheme is discussed.