Long Lived Superheavy Dark Matter with Discrete Gauge Symmetries
  The recently observed ultra-high energy (UHE) cosmic rays beyond the Greisen-Zatsepin-Kuzmin bound can be explained by the decays of some superheavy $X$ particles forming a part of dark matter in our universe. We consider various discrete gauge symmetries ${\bf Z}_N$ to ensure the required long lifetime ($\tau_X \simeq 10^{10}-10^{22} years$) of the $X$ particle to explain the UHE cosmic rays in the minimal supersymmetric standard model (MSSM) with massive Majorana neutrinos. We show that there is no anomaly-free discrete gauge symmetry to make the lifetime of the $X$ particle sufficiently long in the MSSM with the $X$ particle. We find, however, possible solutions to this problem especially by enlarging the particle contents in the MSSM. We show a number of solutions introducing an extra pair of singlets $Y$ and $\bar{Y}$ which have fractional ${\bf Z}_N$ (N=2,3) charges. The present experimental constraints on the $X$ particle are briefly discussed. 
K. Hamaguchi, Yasunori Nomura, and T. Yanagida