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

Abstract

Abstract By an explicit calculation we demonstrate that the triple gauge-ghost vertices in a general renormalizable $${{\mathcal {N}}}=1$$ 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 $$\bar{c}^+ V c$$ c ¯ + V c -vertex proportional to $$(C_2)^2$$ ( C 2 ) 2 and use the finiteness of the two-loop contribution proportional to $$C_2 T(R)$$ C 2 T ( R ) which has been checked earlier. The theory under consideration is regularized by higher covariant derivatives and quantized in a manifestly $${{\mathcal {N}}}=1$$ N = 1 supersymmetric way with the help of $${{\mathcal {N}}}=1$$ 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 $$\xi $$ ξ -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 $$\beta $$ β -function.