Detailed analysis of the decay spectrum of a super-heavy X particle

Decays of superheavy X particles with mass M_X ~ 10^12 - 10^16 GeV have been proposed as origin of the observed ultra high energy cosmic rays (UHECR). We describe in detail the physics involved in the different steps of the decay of such a particle. In particular, we give for the first time the complete set of splitting functions needed to model a parton shower in the minimal supersymmetric extension of the Standard Model (MSSM). We present our results in the form of fragmentation functions of any (s)particle of the MSSM into any final stable particle (proton, photon, electron, three types of neutrino, lightest superparticle LSP) at a virtuality Q = M_X, over a scaled energy range x = 2E/M_X in [10^{-13}, 1]. Extending the coverage to such small fractional energies is necessary since the energy region around 10^18 eV and below could be of considerable interest in testing this kind of model for generating UHECR. We explicitly demonstrate that our treatment conserves energy, and discuss the dependence of the final result on SUSY parameters. We also show that our results are essentially independent of the necessary extrapolation of the input fragmentation functions, which are known only for x >= 0.1, towards small x. Finally, we added a new treatment of the color coherence effects at very small x, using the analytic ``MLLA'' solution. Our computer code will soon be made available.