Three-Loop Gauge Beta Functions in Supersymmetric Theories with Exponential Higher Covariant Derivative Regularization

We study the three-loop gauge $\beta$-functions in general $\mathcal{N}=1$ supersymmetric gauge theories regularized by higher covariant derivatives (HCD) supplemented with Pauli--Villars subtraction. The all-structure three-loop form of the $\beta$-functions in the HCD framework is known and involves regulator-dependent parameters. Here we evaluate these parameters explicitly for the exponential regulators $R(x)=e^{x^n}$ and $F(x)=e^{x^m}$. We obtain the constants $A(n)$ and $B(m)$ in closed form, together with their large-$n,m$ asymptotics, and substitute them into the general three-loop expressions. This yields fully explicit, regulator-parameterized $\beta$-functions and a systematic expansion in $1/n$ and $1/m$ that organizes finite, scheme-dependent terms. We then exhibit finite coupling redefinitions that map the renormalized $\overline{\mathrm{DR}}$ result to a scheme compatible with the Novikov--Shifman--Vainshtein--Zakharov relation. Our analysis clarifies how exponential higher-derivative regulators preserve this relation at the bare level and illustrates the regulator-driven structure of supersymmetric renormalization group flows.