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 $10^{-3} eV \lsim m_{3/2} \lsim 1 TeV$ when the $\mu$-parameter is fixed to be $\mu \simeq 1 TeV$. In the minimal SU(5) GUT, we have shown that $\mu \simeq 1 TeV$ is obtained only if the gravitino is ultralight with mass $m_{3/2} \sim 10^{-3} eV$. If extra fields ${\bf 5}\oplus{\bf 5^*}$ or ${\bf 10}\oplus{\bf 10^*}$ are introduced, many solutions predicting $m_{3/2} \gsim 10^{-3} eV$ 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 ${\cal O}(10^{-3} eV)$--${\cal O}(1 MeV)$. These discrete R-symmetries can be considered as solutions to the $\mu$-problem in low energy supersymmetry breaking models such as the gauge mediation.