The Tau Neutrino Double-Bang Window: Decay Lengths and Expected Event Rates

A tau neutrino interacting in ice creates a tau lepton whose decay, displaced from the interaction point, produces a second cascade — the double-bang signature that identifies the tau flavour event by event. This paper computes the energy window in which that signature is resolvable in a cubic-kilometre array and the event rate inside it. The tau decay length grows as 49 metres per PeV of tau energy, reaching 294 metres at 6 PeV. Requiring a separation between 20 metres and 1 kilometre places the window between 0.408 and 20.4 PeV. Integrating the diffuse flux over that window with a schematic charged-current cross section and a cubic kilometre of ice gives 0.0912 double-bang events per year — of order one per decade. Tau identification in current data therefore rests mainly on subtler signatures, and the double-bang becomes a working channel only for larger future arrays. The study is computed from stated assumptions, not measured.