One-loop polarization operator of the quantum gauge superfield for ${\cal N}=1$ SYM regularized by higher derivatives

TL;DR

This paper calculates the one-loop polarization operator for the quantum gauge superfield in ${ m extbf{N}=1}$ supersymmetric gauge theories with higher derivative regularization, providing explicit expressions and finite constants in the massless limit.

Contribution

It derives the unrenormalized two-point Green function in ${ m extbf{N}=1}$ SYM with matter, using higher covariant derivatives, and analytically evaluates integrals in the Feynman gauge at zero momentum.

Findings

Explicit one-loop polarization operator expression obtained.

Finite constants identified alongside divergent terms.

Analytic results for specific gauge and regulator choices.

Abstract

We consider the general ${\cal N}=1$ supersymmetric gauge theory with matter, regularized by higher covariant derivatives without breaking the BRST invariance, in the massless limit. In the $\xi$-gauge we obtain the (unrenormalized) expression for the two-point Green function of the quantum gauge superfield in the one-loop approximation as a sum of integrals over the loop momentum. The result is presented as a sum of three parts: the first one corresponds to the pure supersymmetric Yang--Mills theory in the Feynman gauge, the second one contains all gauge dependent terms, and the third one is the contribution of diagrams with a matter loop. For the Feynman gauge and a special choice of the higher derivative regulator in the gauge fixing term we analytically calculate these integrals in the limit $k\to 0$. In particular, in addition to the leading logarithmically divergent terms, which are determined by integrals of double total derivatives, we also find the finite constants.