  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.
