Nonrenormalization theorems in softly broken supersymmetric QED and the softβfunctions

The renormalization of softly broken supersymmetric QED (SQED) is related to supersymmetric QED by using the construction with a local gauge supercoupling and by taking into account softly broken anomalous axial $U\left(1\right)\mathrm{}$ symmetry. From this extended model one obtains the nonrenormalization theorems of scalar QED and the counterterms of the soft breaking parameters as functions of the supersymmetric counterterms. Because of the Adler-Bardeen anomaly of the axial vector current an invariant regularization scheme does not exist, and therefore the $\beta$ functions of soft breaking parameters are derived from an algebraic construction of the Callan-Symanzik equation and of the renormalization-group equation. We obtain the soft $\beta$ functions in terms of the gauge $\beta$ function and of the anomalous dimension of the supersymmetric matter mass. In particular, we find that the X term of the scalar mass $\beta$ function as well as the gauge $\beta$ function in $l>~2$ are due to the Adler-Bardeen anomaly of the axial symmetry.