A Comprehensive Approach to Tau-Lepton Production by High-Energy Tau Neutrinos Propagating Through Earth

There has been a recent surge in interest in the detection of $\tau$ lepton-induced air showers from detectors at altitude. When a $\tau$ neutrino ($\nu_\tau$) enters the Earth it produces $\tau$ leptons as a result of nuclear charged current interactions. In some cases, this process results in a $\tau$ lepton exiting the surface of the Earth, which can subsequently decay in the atmosphere and produce an extensive air shower. These upward-going air showers can be detected via fluorescence, optical Cherenkov, or geomagnetic radio emission. Several experiments have been proposed to detect these signals. We present a comprehensive simulation of the production of $\tau$ leptons by $\nu_\tau$'s propagating through Earth to aid the design of future experiments. These simulations for $\nu_\tau$'s and leptons in the energy range from $10^{15}$ eV to $10^{21}$ eV treat the full range of incidence angles from Earth-skimming to diametrically-traversing. Propagation of $\nu_\tau$'s and leptons include the effects of rock and an ocean or ice layer of various thicknesses. The interaction models include $\nu_\tau$ regeneration and account for uncertainties in the Standard Model neutrino cross-section and in the photo-nuclear contribution to the $\tau$ energy loss rate.