$\mu \to e \gamma$ and $\mu \to 3e$ processes with polarized muons and supersymmetric grand unified theories

$\mu^{+} \to e^{+} \gamma$ and $\mu^{+} \to e^{+}e^{+}e^{-}$ processes are analyzed in detail with polarized muons in supersymmetric grand unified theories. We first present Dalitz plot distribution for $\mu^{+} \to e^{+}e^{+}e^{-}$ decay based on effective Lagrangian with general lepton-flavor-violating couplings and define various P- and T-odd asymmetries. We calculate branching ratios and asymmetries in supersymmetric SU(5) and SO(10) models taking into account complex soft supersymmetry breaking terms. Imposing constraints from experimental bounds on the electron, neutron and atomic electric dipole moments, we find that the T-odd asymmetry for $\mu^{+} \to e^{+}e^{+}e^{-}$ can be 15% in the SU(5) case. P-odd asymmetry with respect to muon polarization for $\mu^{+} \to e^{+} \gamma$ varies from -20% to -100% for the SO(10) model while it is $+100%$ in the SU(5) case. We also show that the P-odd asymmetries in $\mu^{+} \to e^{+}e^{+}e^{-}$ and the ratio of $\mu^{+} \to e^{+}e^{+}e^{-}$ and $\mu^{+} \to e^{+} \gamma$ branching fractions are useful to distinguish different models.