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

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 $β$-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 $β$-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.