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

Abstract

By an explicit calculation we demonstrate that the triple gauge-ghost vertices in a general renormalizable $\mathcal{N}=1$ supersymmetric gauge theory are UV finite in the two-loop approximation. For this purpose we calculate the two-loop divergent contribution to the ${\bar{c}}^{+}Vc$-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 $\mathcal{N}=1$ supersymmetric way with the help of $\mathcal{N}=1$ 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.

arXiv comment19 pages, 8 figures

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• A condition for the reduction of couplings in the $P=\frac{1}{3}Q$ supersymmetric theories