Survival probability of charged particles dissipating their energies by radiation and ionization

The survival probabilities of charged particle traversing through matters are evaluated by solving the diffusion equation, taking account of radiation and ionization losses. The solutions corresponding to high and low incident energies are proposed respectively, and inaccuracies of the saddle point method appearing in case of large probabilities are removed by deriving the results via the complementary probabilities. The enough-accurate results will be valuable to improve the resolution of radiographic analyses with atmospheric muons, and others.