Softly broken supersymmetric Yang-Mills theories: Renormalization and nonrenormalization theorems

We present a minimal version for the renormalization of softly broken super Yang-Mills theories using the extended model with a local gauge coupling. It is shown that the nonrenormalization theorems of the case with unbroken supersymmetry are valid without modifications and that the renormalization of the soft-breaking parameters is completely governed by the renormalization of the supersymmetric parameters. The symmetry identities in the present context are peculiar, since the extended model contains two anomalies: the Adler-Bardeen anomaly of the axial vector current and an anomaly of supersymmetry in the presence of the local gauge coupling. From the anomalous symmetries we derive the exact all-order expressions for the $\beta$ functions of the gauge coupling and of the soft-breaking parameters. They generalize earlier results to arbitrary normalization conditions and imply the NSVZ expressions for a specific normalization condition on the coupling.