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.
Highlights
The rest part of this paper is arranged as following
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
Summary
Taking the non-existence of a solution to (2.1) as the criteria for SUSY breaking, the Nelson-Seiberg theorem and its extensions [1,2,3] give the conditions for SUSY breaking, expressed in terms of R-symmetries under which W has R-charge 2 These theorems are as follows: Theorem 1 (The Nelson-Seiberg theorem) In a Wess-Zumino model with a generic superpotential, assuming the existence of a vacuum at the global minimum of the scalar potential, an R-symmetry is a necessary condition, and a spontaneously broken R-symmetry is a sufficient condition for SUSY breaking at the vacuum. At least one pair of oppositely R-charged fields satisfy the condition that they both appear only linearly in the superpotential, and are not involved in any quadratic term The product of such a field pair gets a non-zero VEV, and acts effectively like an R-charge 0 field to help solving the F-term equations.
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