  The renormalization of softly broken SQED is related to the one of
supersymmetric QED by using the construction with a local gauge supercoupling
and by taking into account softly broken anomalous axial U(1) symmetry. From
this extended model one obtains the non-renormalization theorems of SQED and
the counterterms of the soft breaking parameters as functions of the
supersymmetric counterterms. Due to the Adler-Bardeen anomaly of the axial
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 $l\geq 2$ are due to
the Adler-Bardeen anomaly of the axial symmetry.
