Renormalization of the Fayet-Iliopoulos term in softly broken supersymmetric gauge theories

It is shown that renormalization of the Fayet-Iliopoulos (FI) term in a softly broken supersymmetric gauge theory, in full analogy with all the other soft term renormalizations, is completely defined in a rigid or an unbroken theory. However, contrary to the other soft renormalizations, there is no simple differential operator that acts on the renormalization functions of a rigid theory and allows one to get the renormalization of the FI term. One needs an analysis of the superfield diagrams and some additional diagram calculations in the components. The method is illustrated by the four loop calculation of some part of the renormalization proportional to the soft scalar masses and the soft triple couplings.