An analytical derivation of the survival probability of muon penetrating through matters

The survival probabilities of muon after penetrating through matters are evaluated analytically by solving the diffusion equation, taking account of not only bremsstrahlung but also positron-electron pair production and photonuclear interactions, together with ionization loss. Inaccuracy of probabilities derived by the ordinary saddle point method, reaching to e/root 2 pi similar or equal to 1.08, is corrected by applying the complementary-probability method. Accuracies of the results are discussed by comparing them with those derived by a Monte Carlo method.