Propagation of Tau Neutrinos and Tau Leptons through the Earth and their Detection in Underwater/Ice Neutrino Telescopes

If muon neutrinos produced in cosmological sources oscillate, neutrino telescopes can have a chance to detect $\tau$-neutrinos. In contrast to $\nu_{mu}$'s the Earth is completely transparent for $\nu_{\tau}$'s thanks to the short life time of $\tau$-leptons produced in charged current interactions. $\tau$-lepton decays in flight producing another $\nu_{\tau}$ (regeneration chain). Thus, $\nu_{\tau}$'s cross the Earth without being absorbed, though loosing energy both in regeneration processes and in neutral current interactions. Neutrinos of all flavors can be detected in deep underwater/ice detectors by means of \u{C}erenkov light emitted by charged leptons produced in $\nu$ interactions. Muon and $\tau$-leptons have different energy loss features, which provide opportunities to identify $\tau$-events among the multitude of muons. Some signatures of $\tau$-leptons that can be firmly established and are background free have been proposed in literature, such as 'double bang' events. In this paper we present results of Monte Carlo simulations of $\tau$-neutrino propagation through the Earth accounting for neutrino interactions, $\tau$ energy losses and $\tau$ decays. Parameterizations for hard part and corrections to the soft part of the photonuclear cross-section (which contributes a major part to $\tau$ energy losses) are presented. Different methods of $\tau$-lepton identification in large underwater/ice neutrino telescopes are discussed. Finally, we present a calculation of $\nu_{\tau}$ double bang event rates in km$^3$ scale detectors.