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

Abstract

We verify the identity which relates the two-point Green’s functions of \(\mathcal{N} = 1\) 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’s function of the gauge superfield is given by integrals of double total derivatives in the momentum space. It also makes it possible to exactly derive the NSVZ β-function 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 to construct integrals giving the three-loop β-function starting from the two-point Green’s functions of 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.