Three-loop $\beta$-functions and NSVZ relations for the MSSM regularized by higher covariant derivatives

The three-loop $\beta$-functions are obtained for the Minimal Supersymmetric Standard Model (MSSM) regularized by higher covariant derivatives for an arbitrary supersymmetric renormalization prescription. The two-loop anomalous dimensions defined in terms of the bare couplings are found for all MSSM chiral matter superfields. Using the NSVZ relations, the three-loop $\beta$-functions also defined in terms of the bare couplings are constructed. This is possible because the NSVZ equations are satisfied in all orders for the renormalization group functions defined in terms of the bare couplings under the chosen regularization. Using this result, the same two-loop anomalous dimensions and $\beta$-functions standardly defined in terms of the renormalized couplings are obtained for an arbitrary supersymmetric renormalization prescription. It is also verified that for a certain renormalization prescription, the result coincides with the $\overline{DR}$ scheme one obtained earlier, and can therefore be considered its independent confirmation.