Discovering Ultra–High‐Energy Neutrinos through Horizontal and Upward τ Air Showers: Evidence in Terrestrial Gamma Flashes?

We show that the expected $ \nu_{\tau} $ signals, by their secondary tau tracks, in Km^3 detectors at highest cosmic ray energy window $ 1.7\cdot 10^{21} eV \gt E_{\tau} \gt 1.6 x 10^{17} eV$, must overcome the corresponding $ \nu_{\mu} $ (or muonic) ones. Indeed, the Lorentz-boosted tau range length grows (linearly) above muon range, for $ E_{\tau} \RAISE 1.6 x 10^8 GeV$ and reaches its maxima extension, $ R_{\tau_{\max}} \simeq 191 km$, at energy $E_{\tau} \simeq 3.8 x 10^9 GeV$. At this peak the tau range is nearly 20 times the corresponding muon range (at the same energy) implying a similar ratio in $ \nu_{\tau} $ over $ \nu_{\mu} $ detectability. This dominance, however may lead (at present most abundant $ \nu_{\tau} $ model fluxes) to just a rare spectacular event a year (if flavor mixing occurs). Lower energetic $ \tau $ and $ \nu_{\tau} $ signals $ (\bar{\nu}_e e\to \bar{\nu}_{\tau} \tau, \nu_{\tau} N\to ...) $ at energy range ($ 10^5 \div 10^7 GeV$) may be more easily observed in km^3 detectors at a rate of a few $ (\bar{\nu}_e e\to \bar{\nu}_{\tau} \tau) $ to tens event $ (\nu_{\tau} N\to \tau + $ anything) a year.