Calculating the anomalous supersymmetry breaking in super Yang-Mills theories with local coupling

Supersymmetric Yang-Mills theories with local gauge coupling have a new type of anomalous breaking, which appears as a breaking of supersymmetry in the Wess-Zumino gauge. The anomalous breaking generates the two-loop order of the gauge $\beta$ function in terms of the one-loop $\beta$ function and the anomaly coefficient. We determine the anomaly coefficient in the Wess-Zumino gauge by solving the relevant supersymmetry identities. For this purpose we use a background gauge and show that the anomaly coefficient is uniquely determined by convergent one-loop integrals. When evaluating the one-loop diagrams in the background gauge, it is seen that the anomaly coefficient is determined by the Feynman-gauge value of the one-loop vertex function to $G^{\mu \nu }{G}_{\mu \nu }$ at vanishing momenta.