A sufficient condition for counterexamples to the Nelson-Seiberg theorem

A bstract Several counterexample models to the Nelson-Seiberg theorem have been discovered in previous literature, with generic superpotentials respecting R-symmetry and non-generic R-charge assignments for chiral fields. This work present a sufficient condition such models: number of 2 fields, which is greater than 0 must be less or equal fields plus independent field pairs opposite R-charges satisfying some extra requirements. We give correct count when there are multiple degenerated R-charges. These supersymmetric vacua spontaneous breaking, thus counterexamples both its extensions.