Nonanomalous Discrete R-Symmetry and Light Gravitino

Abstract

We discuss nonanomalous R-symmetry in the supersymmetric grand unified theories. In particular, we explore anomaly-free solutions predicting the gravitino mass in the range of $10^{-3}eV\text{lsim}{m}_{3/2}\text{lsim}1TeV$ when the $\mu$-parameter is fixed to be $\mu \simeq 1TeV$. In the minimal SU(5) GUT, we have shown that $\mu \simeq 1TeV$ is obtained only if the gravitino is ultralight with mass $m_{3/2}\sim 10^{-3}eV$. If extra fields $5\oplus 5^{\ast }$ or $10\oplus 10^{\ast }$ are introduced, many solutions predicting $m_{3/2}\text{gsim}10^{-3}eV$ are found. The R-parity is violated due to the vacuum expectation value of the superpotential, but it is controlled by the discrete R-symmetry. We find that the R-parity violating couplings are naturally suppressed much below the experimental bounds for some charge assignments. These charge assignments predict light gravitino with masses of order $\mathcal{O}(10^{-3}eV)$--$\mathcal{O}(1MeV)$. These discrete R-symmetries can be considered as solutions to the $\mu$-problem in low energy supersymmetry breaking models such as the gauge mediation.

Comments: 20 pages, no figure. v2: minor corrections, references added, "Note Added" in Summary added