Exact relations between running of $\alpha_s$ and $\alpha$ in ${\mathcal N}=1$ SQCD + SQED

In ${\mathcal N}=1$ supersymmetric QCD and QED theory we derive the (all-order) exact equations relating the renormalization group behaviour of the strong and electromagnetic couplings and prove that they are valid in the combined higher covariant derivative regularized and minimal subtraction of logarithms renormalized prescription. In particular, the $\beta$-function of ${\mathcal N}=1$ supersymmetric QCD can be expressed in terms of the Adler $D$-function. If all flavors have the same absolute value of the electromagnetic charges, it is also possible to write a simple relation between the $\beta$-functions for the strong and electromagnetic coupling constants. In this particular case there is a special renormalization group invariant relation.