Relation between two-point Green functions of ${\cal N}=1$ SQED with $N_f$ flavors, regularized by higher derivatives, in the three-loop approximation

TL;DR

This paper verifies a key identity relating two-point Green functions in ${ m extbf{N}=1}$ SQED with $N_f$ flavors, using three-loop calculations with higher derivative regularization, to derive the NSVZ $eta$-function exactly.

Contribution

It explicitly confirms the identity at three loops, elucidates why the gauge superfield Green function involves double total derivatives, and derives the NSVZ $eta$-function from bare coupling functions.

Findings

The identity holds at three loops, confirming theoretical predictions.

The gauge superfield Green function is expressed as integrals of double total derivatives.

The derived integrals match the sums of three-loop supergraphs.

Abstract

We verify the identity which relates the two-point Green functions of ${\cal N}=1$ SQED with $N_f$ flavors, regularized by higher derivatives, by explicit calculations in the three-loop approximation. This identity explains why in the limit of the vanishing external momentum the two-point Green function of the gauge superfield is given by integrals of double total derivatives in the momentum space. It also allows to derive the NSVZ $\beta$-function exactly in all loops if the renormalization group functions are defined in terms of the bare coupling constant. In order to verify the considered identity we use it for constructing integrals giving the three-loop $\beta$-function starting from the two-point Green functions of the matter superfields in the two-loop approximation. Then we demonstrate that the results for these integrals coincide with the sums of the corresponding three-loop supergraphs.