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We investigate the NSVZ relations for ${\cal N}=1$ supersymmetric gaugetheories with multiple gauge couplings. As examples, we consider MSSM and theflipped $SU(5)$ model, for which they easily reproduce the results for thetwo-loop $\beta$-functions. For ${\cal N}=1$ SQCD interacting with the Abeliangauge superfield we demonstrate that the NSVZ-like equation for the Adler$D$-function follows from the NSVZ relations. Also we derive all-loop equationsdescribing how the NSVZ equations for theories with multiple gauge couplingschange under finite renormalizations. They allow describing a continuous set ofNSVZ schemes in which the exact NSVZ $\beta$-functions are valid for all gaugecoupling constants. Very likely, this class includes the HD+MSL scheme, whichis obtained if a theory is regularized by Higher covariant Derivatives anddivergences are removed by Minimal Subtractions of Logarithms. That is why wealso discuss how one can construct the higher derivative regularization fortheories with multiple gauge couplings. Presumably, this regularization allowsto derive the NSVZ equations for such theories in all loops. In this paper wemake the first step of this derivation, namely, the NSVZ equations for theorieswith multiple gauge couplings are rewritten in a new form which relates the$\beta$-functions to the anomalous dimensions of the quantum gauge superfields,of the Faddeev--Popov ghosts, and of the matter superfields. The equivalence ofthis new form to the original NSVZ relations follows from the extension of thenon-renormalization theorem for the triple gauge-ghost vertices, which is alsoderived in this paper.

The NSVZ relations for ${\cal N}=1$ supersymmetric theories with multiple gauge couplings