= Perturbative renormalization of a Fayet–Iliopoulos term
{title2=$\delta\xi\propto\operatorname{Tr}q$}
In an exactly supersymmetric Abelian theory a one-loop auxiliary-field tadpole can generate a cutoff-dependent <Fayet–Iliopoulos term> proportional to the trace of the matter charges. Gauge invariance with arbitrary background <spurions> forbids a general coupling-dependent full-superspace coefficient multiplying the vector superfield. In the supersymmetry-preserving Wilsonian setting the additive perturbative correction is exhausted by the one-loop term; it vanishes for a traceless charge generator. Canonical vector normalization and soft breaking require separate treatment. For a simple non-Abelian group the term itself is forbidden.
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