Three-loop verification of the equations relating running of the gauge couplings in ${\cal N}=1$ SQCD+SQED

TL;DR

This paper confirms the validity of specific gauge coupling relations in ${ m N}=1$ SQCD+SQED at three-loop order within the HD+MSL scheme through explicit calculations, highlighting scheme dependence of these relations.

Contribution

It demonstrates that the gauge coupling relations hold in the HD+MSL scheme at three loops and shows how they can be derived from this scheme via finite renormalizations.

Findings

Relations are valid in HD+MSL scheme at three loops.

In $ar{ ext{DR}}$ scheme, relations do not hold at three loops.

Finite renormalizations connect different schemes.

Abstract

We verify a recently derived equations relating the renormalization group running of two gauge couplings in ${\cal N}=1$ SQCD+SQED by the explicit three-loop calculation. It is demonstrated that these equations are really valid in the HD+MSL scheme. In other words, if a theory is regularized by higher covariant derivatives and the renormalization is made by minimal subtractions of logarithms, the analogs of the strong and electromagnetic gauge couplings do not run independently. However, in the $\overline{\mbox{DR}}$ scheme the considered equations do not hold starting from the three-loop order, where the scheme dependence becomes essential. Therefore, they are valid only for a certain set of the renormalization prescriptions. We prove that all of them can be obtained from the HD+MSL scheme by finite renormalizations which satisfy a special constraint and illustrate how this works in the three-loop approximation.