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

TL;DR

This paper derives exact all-order relations between the running of the strong and electromagnetic couplings in ${ m f N}=1$ supersymmetric QCD-like theories, revealing deep connections between their renormalization group behaviors.

Contribution

It provides the first derivation of exact relations linking the beta functions of strong and electromagnetic couplings in ${ m f N}=1$ SQCD+SQED, valid in a specific renormalization scheme.

Findings

Beta function of ${ m f N}=1$ SQCD expressed via Adler D-function.

Existence of a simple relation between the beta functions when charges are equal.

Identification of a special RG-invariant relation in this case.

Abstract

In ${\cal N}=1$ supersymmetric QCD-like theories 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 HD+MSL renormalization prescription. In particular, the $\beta$-function of ${\cal N}=1$ SQCD can be expressed in terms of the Adler $D$-function. If all favors 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.