Finiteness of the triple gauge-ghost vertices in ${\cal N}=1$ supersymmetric gauge theories: the two-loop verification

TL;DR

This paper explicitly verifies at two loops that the triple gauge-ghost vertices in ${ m extbf{N}=1}$ supersymmetric gauge theories are UV finite, supporting the all-order nonrenormalization theorem and the derivation of the NSVZ beta-function.

Contribution

It provides a two-loop calculation confirming the finiteness of gauge-ghost vertices in ${ m extbf{N}=1}$ supersymmetric gauge theories, supporting the nonrenormalization theorem.

Findings

Two-loop gauge-ghost vertices are UV finite.

Finiteness holds for all gauge choices and external line configurations.

Supports the all-order derivation of the NSVZ beta-function.

Abstract

By an explicit calculation we demonstrate that the triple gauge-ghost vertices in a general renormalizable ${\cal N}=1$ supersymmetric gauge theory are UV finite in the two-loop approximation. For this purpose we calculate the two-loop divergent contribution to the $\bar c^+ V c$-vertex proportional to $(C_2)^2$ and use the finiteness of the two-loop contribution proportional to $C_2 T(R)$ which has been checked earlier. The theory under consideration is regularized by higher covariant derivatives and quantized in a manifestly ${\cal N}=1$ supersymmetric way with the help of ${\cal N}=1$ superspace. The two-loop finiteness of the vertices with one external line of the quantum gauge superfield and two external lines of the Faddeev--Popov ghosts has been verified for a general $\xi$-gauge. This result agrees with the nonrenormalization theorem proved earlier in all orders, which is an important step for the all-loop derivation of the exact NSVZ $\beta$-function.