A sufficient condition for counterexamples to the Nelson-Seiberg theorem

TL;DR

This paper establishes a specific sufficient condition involving R-charge assignments that identifies counterexamples to the Nelson-Seiberg theorem, which are models with supersymmetric vacua and spontaneous R-symmetry breaking.

Contribution

It provides a new, precise criterion for recognizing counterexample models to the Nelson-Seiberg theorem, including correct counting of field pairs with degenerated R-charges.

Findings

Counterexample models satisfy the new R-charge condition.

Correct counting method for degenerated R-charge field pairs.

Models exhibit supersymmetric vacua with spontaneous R-symmetry breaking.

Abstract

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