The NSVZ scheme for ${\cal N}=1$ SQED with $N_f$ flavors, regularized by the dimensional reduction, in the three-loop approximation

TL;DR

This paper investigates the appearance of the NSVZ relation in ${ m extbf{N=1}}$ SQED with $N_f$ flavors at the three-loop level using dimensional reduction, revealing differences from higher derivative regularization and establishing a connection between schemes.

Contribution

It demonstrates how the NSVZ relation can be imposed in the three-loop approximation for dimensional reduction regularization, and compares it with higher derivative regularization schemes.

Findings

NSVZ relation is not valid for bare coupling functions in dimensional reduction.

NSVZ scheme can be imposed for renormalized coupling functions with boundary conditions.

Dimensional reduction and higher derivative regularization schemes are related by finite renormalization.

Abstract

At the three-loop level we analyze, how the NSVZ relation appears for ${\cal N}=1$ SQED regularized by the dimensional reduction. This is done by the method analogous to the one which was earlier used for the theories regularized by higher derivatives. Within the dimensional technique, the loop integrals cannot be written as integrals of double total derivatives. However, similar structures can be written in the considered approximation and are taken as a starting point. Then we demonstrate that, unlike the higher derivative regularization, the NSVZ relation is not valid for the renormalization group functions defined in terms of the bare coupling constant. However, for the renormalization group functions defined in terms of the renormalized coupling constant, it is possible to impose boundary conditions to the renormalization constants giving the NSVZ scheme in the three-loop order. They are similar to the all-loop ones defining the NSVZ scheme obtained with the higher derivative regularization, but are more complicated. The NSVZ schemes constructed with the dimensional reduction and with the higher derivative regularization are related by a finite renormalization in the considered approximation.