Finiteness of the two-loop matter contribution to the triple gauge-ghost vertices in ${\cal N}=1$ supersymmetric gauge theories regularized by higher derivatives

TL;DR

This paper proves that the two-loop matter contributions to certain gauge-ghost vertices in ${\ m N}=1$ supersymmetric gauge theories are UV finite, supporting the all-order finiteness theorem crucial for the NSVZ relation.

Contribution

It explicitly verifies the two-loop UV finiteness of matter contributions to gauge-ghost vertices using higher covariant derivative regularization, confirming a key theorem in supersymmetric gauge theories.

Findings

Two-loop matter contributions are UV finite in general gauges.

Explicit calculations support the all-order finiteness theorem.

Results reinforce the derivation of the NSVZ relation.

Abstract

For a general renormalizable ${\cal N}=1$ supersymmetric gauge theory with a simple gauge group we verify the ultraviolet (UV) finiteness of the two-loop matter contribution to the triple gauge-ghost vertices. These vertices have one leg of the quantum gauge superfield and two legs corresponding to the Faddeev--Popov ghost and antighost. By an explicit calculation made with the help of the higher covariant derivative regularization we demonstrate that the sum of the corresponding two-loop supergraphs containing a matter loop is not UV divergent in the case of using a general $\xi$-gauge. In the considered approximation this result confirms the recently proved theorem that the triple gauge-ghost vertices are UV finite in all orders, which is an important ingredient of the all-loop perturbative derivation of the Novikov-Shifman-Vainshtein-Zakharov relation.