A counterexample to the Nelson-Seiberg theorem
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Springer
Received: August 18, 2020 Accepted: September 10, 2020 Published: October 12, 2020
Zheng Sun, Zipeng Tan and Lu Yang College of Physics, Sichuan University, 29 Wangjiang Road, Chengdu 610064, P.R. China
E-mail: sun [email protected], [email protected], [email protected] 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. Keywords: Global Symmetries, Supersymmetry Breaking, Supersymmetric Effective Theories ArXiv ePrint: 1904.09589
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)072
JHEP10(2020)072
A counterexample to the Nelson-Seiberg theorem
Contents 1 Introduction
1
2 The counterexample
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3 Discussions
4
Introduction
In 4-dimentinal N = 1 supersymmetry (SUSY) theories [1, 2], the relation between SUSY breaking and R-symmetries in Wess-Zumino models are described by the Nelson-Seiberg theorem and its extensions [3–6]. The original Nelson-Seiberg theorem [3] states that for SUSY breaking at a stable vacuum in a generic model, a necessary condition is to have an R-symmetric superpotentical, and a sufficient condition is to have the R-symmtery spontaneously broken at the vacuum. The statement can be extended to metastable SUSY breaking models with approximate R-symmetries [7–9]. A revised theorem [5] claims that with a polynomial superpotential, the neccessary and sufficient condition for SUSY breaking is to have an R-symmetric superpotentical and more R-charge 2 fields than R-charge 0 fields. The revised theorem can be generalized to include models with non-polynomial superpotentials [6], where the condition for SUSY breaking is the same as before for superpotentials smooth at the origin of the field space, and a singularity at the origin implies both R-symmetry breaking and supersymmetry breaking. Half of the revised theorem has even stronger claims [4]: if we have an R-symmetric superpotentical and less or equal R-charge 2 fields than R-charge 0 fields, the SUSY vacua claimed by the revised theorem have an additional property that the superpotential equals zero at the vacuum, and this part of claim is also true for non-Z2 discrete R-symmetries or non-Abelian discrete R-symmetries [10, 11]. The Nelson-Seiberg theorem and its extensions provides various tools for new physics model building. In SUSY phenomenology beyond the Standard Model, the SUSY breaking effect is medi
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