Propagation of multi-TeV muons

We discuss the propagation of muons of energy above a TeV through rock and stress the importance of correctly accounting for the fluctuations of the energy loss in radiative processes. Accounting for these fluctuations affects the major types of underground muon fluxes in the opposite way from a naive treatment that neglects the muon straggling. The rates of downward atmospheric muons are increased, while the flux of upward neutrino-induced muons is decreased. The paper analyzes the causes of these effects and gives helpful parametrizations for the muon ranges applicable to the two types of muon rates. We also extend our calculations to muon energies of ${10}^6$ TeV and discuss the uncertainties in the muon energy loss at extremely high energy. An appendix gives a short review of the analytic techniques used to solve the problem of straggling and presents a toy model that displays the role of fluctuations in muon propagation. In another appendix we introduce an interesting technique for the generation of approximate energy, angular, and lateral distributions from the muon survival probability.