# A counterexample to the Nelson-Seiberg theorem

## TL;DR

This paper provides a specific counterexample to the Nelson-Seiberg theorem, showing that certain R-charge assignments can lead to supersymmetric vacua that contradict the theorem's predictions, highlighting potential limitations in field counting methods.

## Contribution

The paper introduces a novel counterexample with a unique R-charge configuration that challenges the applicability of the Nelson-Seiberg theorem and its extensions.

## Key findings

- Counterexample with 4 chiral fields and R-charge 2, no R-charge 0
- Supersymmetric vacuum with one complex degeneracy and zero superpotential
- Discrepancy with the original Nelson-Seiberg theorem due to non-generic R-charge assignment

## Abstract

We present a counterexample to the Nelson-Seiberg theorem and its extensions. The model has 4 chiral fields, including one R-charge 2 field and no R-charge 0 filed. Giving generic values of coefficients in the renormalizable superpotential, there is a supersymmetric vacuum with one complex dimensional degeneracy. The superpotential equals zero and the R-symmetry is broken everywhere on the degenerated vacuum. The existence of such a vacuum disagrees with both the original Nelson-Seiberg theorem and its extensions, and can be viewed as the consequence of a non-generic R-charge assignment. Such counterexamples may introduce error to the field counting method for surveying the string landscape, and are worth further investigations.