NSVZ relation and the dimensional reduction in ${\cal N}=1$ SQED

TL;DR

This paper investigates the NSVZ relation in ${\ m N}=1$ SQED using dimensional reduction, showing it does not hold for bare couplings but can be restored with boundary conditions at three loops.

Contribution

It extends the understanding of the NSVZ relation to dimensional reduction regularization and proposes a scheme to restore it at three-loop order.

Findings

NSVZ relation does not hold for bare couplings with dimensional reduction.

Boundary conditions can restore the NSVZ scheme at three loops.

Factorization into double total derivatives is crucial in higher derivative regularization.

Abstract

It is known that factorization of the $\beta$-function loop integrals into integrals of double total derivatives is an important ingredient needed for deriving the NSVZ relation by direct perturbative calculations in ${\cal N}=1$ SQED regularized by the higher derivatives. It allows to relate the $\beta$-function and the anomalous dimension of the matter superfields defined in terms of the bare coupling constant. In this work we find the analog of this result in the case of using dimensional reduction regularization in the lowest orders. However, we demonstrate that in this case the NSVZ relation is not satisfied for the RG functions defined in terms of the bare coupling constant. Nevertheless, it is possible to impose boundary conditions to the renormalization constants determining the NSVZ scheme in the three-loop order for the RG functions defined in terms of the renormalized coupling constant.