Nonanomalous Discrete R-Symmetry and Light Gravitino

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 ${10}^{-3}eV\text{lsim}{m}_{3\mathrm{/}2}\text{lsim}1TeV$ 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 $m_{3\mathrm{/}2}\sim {10}^{-3}eV$. If extra fields ${\bf 5}\oplus{\bf 5^}$or${\bf 10}\oplus{\bf 10^}$ are introduced, many solutions predicting $m_{3\mathrm{/}2}\text{gsim}{10}^{-3}eV$ 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.