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

Abstract We verify a recently derived equations relating the renormalization group running of two gauge couplings in $$\mathcal{N}=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> 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 $$\overline{\text{ DR }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mrow> <mml:mspace/> <mml:mtext>DR</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> 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.