Long lived superheavy dark matter with discrete gauge symmetries

The recently observed ultrahigh 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 ${\mathbf{Z}}_N$ to ensure the required long lifetime $({\tau }_X\simeq {10}^{10}–{10}^{22}$ yr) 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 $\overline{Y}$ which have fractional ${\mathbf{Z}}_N\mathrm{}(N=2,3\mathrm{})\mathrm{}$ charges. The present experimental constraints on the X particle are briefly discussed.