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Discrete R symmetries for the MSSM and its singlet extensions

Hyun Min Lee, Stuart Raby, Michael Ratz, Graham G. Ross, Roland Schieren, Kai Schmidt-Hoberg, Patrick K. S. Vaudrevange

TL;DR

The paper classifies anomaly-free discrete $R$-symmetries that commute with grand-unified groups and address the MSSM’s $\mu$ problem and proton decay, showing the order $M$ must divide $24$ and identifying five viable symmetries, with $Z_4^R$ being the simplest that also commutes with $SO(10)$. It constructs a string-derived MSSM realizing $Z_4^R$ and analyzes non-perturbative effects that generate the $\mu$ term at the SUSY-breaking scale while preserving matter parity, thus yielding a realistic phenomenology. The work extends the framework to singlet extensions (notably the NMSSM), where $Z_4^R$ and $Z_8^R$ can solve the hierarchy problem and, in a $Z_{24}^R$-based model, simultaneously address the strong CP problem via an axion. It also discusses how discrete anomalies are canceled through the Green–Schwarz mechanism, and how string constructions realize the remnant $Z_M^R$ to produce the exact MSSM spectrum below the GUT scale. Overall, the study provides a concrete, anomaly-consistent route to embedding MSSM physics in string-inspired setups with controlled proton decay, a viable $\mu$ term, and potential axion solutions.

Abstract

We determine the anomaly free discrete R symmetries, consistent with the MSSM, that commute with SU(5) and suppress the $μ$ parameter and nucleon decay. We show that the order M of such $Z_M^R$ symmetries has to divide 24 and identify 5 viable symmetries. The simplest possibility is a $Z_4^R$ symmetry which commutes with SO(10). We present a string-derived model with this $Z_4^R$ symmetry and the exact MSSM spectrum below the GUT scale; in this model $Z_4^R$ originates from the Lorentz symmetry of compactified dimensions. We extend the discussion to include the singlet extensions of the MSSM and find $Z_4^R$ and $Z_8^R$ are the only possible symmetries capable of solving the $μ$ problem in the NMSSM. We also show that a singlet extension of the MSSM based on a $Z_{24}^R$ symmetry can provide a simultaneous solution to the $μ$ and strong CP problem with the axion coupling in the favoured window.

Discrete R symmetries for the MSSM and its singlet extensions

TL;DR

The paper classifies anomaly-free discrete -symmetries that commute with grand-unified groups and address the MSSM’s problem and proton decay, showing the order must divide and identifying five viable symmetries, with being the simplest that also commutes with . It constructs a string-derived MSSM realizing and analyzes non-perturbative effects that generate the term at the SUSY-breaking scale while preserving matter parity, thus yielding a realistic phenomenology. The work extends the framework to singlet extensions (notably the NMSSM), where and can solve the hierarchy problem and, in a -based model, simultaneously address the strong CP problem via an axion. It also discusses how discrete anomalies are canceled through the Green–Schwarz mechanism, and how string constructions realize the remnant to produce the exact MSSM spectrum below the GUT scale. Overall, the study provides a concrete, anomaly-consistent route to embedding MSSM physics in string-inspired setups with controlled proton decay, a viable term, and potential axion solutions.

Abstract

We determine the anomaly free discrete R symmetries, consistent with the MSSM, that commute with SU(5) and suppress the parameter and nucleon decay. We show that the order M of such symmetries has to divide 24 and identify 5 viable symmetries. The simplest possibility is a symmetry which commutes with SO(10). We present a string-derived model with this symmetry and the exact MSSM spectrum below the GUT scale; in this model originates from the Lorentz symmetry of compactified dimensions. We extend the discussion to include the singlet extensions of the MSSM and find and are the only possible symmetries capable of solving the problem in the NMSSM. We also show that a singlet extension of the MSSM based on a symmetry can provide a simultaneous solution to the and strong CP problem with the axion coupling in the favoured window.

Paper Structure

This paper contains 34 sections, 95 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Discrete symmetries of the MSSM
  3. Non--$\boldsymbol{R}$ discrete symmetries
  4. Discrete $\boldsymbol{R}$--symmetries
  5. A constraint on the order $\boldsymbol{M}$
  6. Classification
  7. Dimension five nucleon decay operators
  8. The gravitational anomaly constraint
  9. Compatibility with $\boldsymbol{\mathrm{SO}(10)}$
  10. A simple $\boldsymbol{\mathbbm{Z}_{4}^R}$ symmetry in the MSSM
  11. Non--perturbative terms
  12. Dilaton stabilization and supersymmetry breaking
  13. Phenomenology
  14. $\boldsymbol{\mathbbm{Z}_{4}^R}$ literature
  15. String theory realization
  16. ...and 19 more sections

Figures (2)

  • Figure 1: Absence of induced $QQQL$ operators. Either one vertex or the mass term is forbidden by $\mathbbm{Z}_{4}^R$.
  • Figure 2: Dilaton potential for $\{c_1,c_2,p_1,p_2,q\}=\{-28.8292,22.6129,2,3,4\}$.