Feng, Fisher, Wilczek and Yu (Phys. Rev. Lett. 88, 161102 (2002)) proposed detecting Earth-skimming tau neutrinos through upward air showers from emerging tau decays, observed by ground-level fluorescence detectors, and estimated event rates with an analytic kernel built on explicit approximations: a tau energy-loss rate dominated by an uncertain photonuclear component; neglected tau-neutrino regeneration and multibang chains; neglected neutral-current energy degradation with the charged lepton assigned the full neutrino energy despite a mean inelasticity of about 0.2; cross-section details deferred to a companion paper that never appeared; and idealized detector apertures. Later literature treated these approximations extensively: dedicated tau-transport Monte Carlo, regeneration studies, topography-based aperture calculations, and the operational Auger Earth-skimming searches implemented most of the requested refinements, so the rate-prediction machinery now exists. A precise open core survives: (i) the photonuclear contribution to tau energy loss at tau energies around 10810^8-101210^{12} GeV is experimentally unconstrained and propagates as a 20-50% systematic in tau emergence probabilities and into every published Earth-skimming flux limit; and (ii) the proposed optical detection strategy has never been validated by a published fluorescence-based Earth-skimming search or a confirmed detection. The task is: a rate prediction for the Earth-skimming channel whose dominant photonuclear systematic is reduced or constrained to an uncertainty below 20% on the tau emergence probability; the absence of any fluorescence-based Earth-skimming search (item (ii)) is the experimental status motivating the prediction, not a deliverable of it.

Background

Ultrahigh-energy (UHE) neutrinos, with energies above roughly 101710^{17} eV, are expected from interactions of the highest-energy cosmic rays with the cosmic microwave background (the Greisen–Zatsepin–Kuzmin, or GZK, mechanism) and are predicted in many speculative source models. They are uniquely penetrating astronomical messengers, and uniquely hard to detect: at these energies the neutrino–nucleon interaction length in rock falls below about 2000 km water equivalent (km w.e., the traversed matter column expressed as an equivalent thickness of water), so the Earth is opaque to upward-going neutrinos, while the atmosphere, at most a few hundred meters water equivalent deep even for quasi-horizontal trajectories, is nearly transparent, so downward-going neutrinos rarely interact before reaching the ground.

The Earth-skimming tau channel circumvents both limits. A tau neutrino ντ\nu_\tau entering the Earth at a grazing, near-horizontal angle may undergo a charged-current (CC) interaction — one in which the neutrino converts into its charged partner lepton — in the crust; the produced τ\tau can emerge from the surface before decaying, because time dilation stretches the tau decay length from meters at PeV energies to tens of kilometers at EeV energies. An emerging tau decays in the lower atmosphere and initiates an upward-going extensive air shower (EAS), the cascade of secondary particles produced when a high-energy particle interacts in air. Feng, Fisher, Wilczek and Yu (FFWY) proposed in 2002 that such showers be observed with ground-level fluorescence detectors — telescopes imaging the ultraviolet light emitted by atmospheric nitrogen excited by the shower — using the Earth as a neutrino-to-tau converter and the atmosphere as the detector volume.

FFWY computed the emerging-lepton flux with an analytic kernel combining a neutrino survival probability, a CC conversion probability, a tau survival factor, and an energy-conservation delta function, assuming a constant crust density ρs=2.65 g/cm3\rho_s = 2.65\ \mathrm{g/cm^3} above a standard Earth density profile. Tau energy loss was treated in the continuous-loss approximation dEτ/dX=−(ατ+βτEτ)dE_\tau/dX = -(\alpha_\tau + \beta_\tau E_\tau), where XX is the traversed matter column and βτ≈0.8×10−6 cm2/g\beta_\tau \approx 0.8\times 10^{-6}\ \mathrm{cm^2/g} (Dutta et al.); at these energies βτ\beta_\tau is dominated by photonuclear interactions — tau scattering from nuclear matter — whose strength rests on extrapolating hadronic structure functions, the functions that parametrize lepton–nucleon scattering, into a kinematic region (very small Bjorken-xx, the fraction of the nucleon momentum carried by the struck quark or gluon, at high momentum transfer Q2Q^2) without accelerator data. With near-maximal νμ\nu_\mu–ντ\nu_\tau oscillations implying equal νμ\nu_\mu and ντ\nu_\tau fluxes at Earth, FFWY folded this kernel with effective apertures for the Fly's Eye, HiRes and the proposed Telescope Array fluorescence detectors, derived from the Fly's Eye signal-to-background analysis of Baltrusaitis et al., and a 10% duty cycle for clear, moonless nights. They found AGN and topological-defect neutrinos marginally detectable at HiRes (for example ∼0.66{\sim}0.66 AGN events in three years) and rates enhanced by more than two orders of magnitude at the proposed Telescope Array.

The paper was explicit that these estimates rested on approximations left for later work: (1) the photonuclear part of βτ\beta_\tau is uncertain, and a smaller true energy loss would significantly enhance the rates; (2) multibang events — chains of repeated ντ↔τ\nu_\tau\leftrightarrow\tau conversions (tau regeneration, in which the tau decays back to a tau neutrino that can reconvert), producing several separated shower vertices — were neglected; (3) neutral-current (NC) interactions, which degrade the neutrino energy while preserving its flavor, were neglected, and the charged lepton was assigned the full neutrino energy even though the mean CC inelasticity — the fraction of the neutrino energy transferred to the hadronic shower, ⟨1−Eℓ/Eν⟩\langle 1-E_\ell/E_\nu\rangle — is about 0.2 (Gandhi, Quigg, Reno and Sarcevic 1996); (4) the cross-section details were deferred to a companion paper 'in preparation' that never appeared; and (5) the apertures were idealized — overlapping fields of view, actual site topography (such as detector placement in valleys), horizon-optimized optics, and the relative sensitivity of the Auger fluorescence telescopes were all flagged as needing dedicated study. Later work implemented items (2)-(4) and the aperture machinery of (5): dedicated tau-neutrino regeneration and multibang studies (Blanch Bigas, Deligny, Payet and Van Elewyck 2008; Huang 2022), propagation chains that sample event-by-event charged- and neutral-current inelasticity in place of the full-energy-lepton delta function (the operational Pierre Auger analyses, on cross-section inputs from Gandhi, Quigg, Reno and Sarcevic 1998), and a topography-based aperture calculation from a digital elevation map of the Auger site (Miele, Pastor and Pisanti 2006); the surviving open core is the reduction of (1), while the fluorescence-based configuration itself remains experimentally unvalidated.

Problem Statement

Starting from the Feng–Fisher–Wilczek–Yu (2002) formulation, replace every approximation in the prediction of Earth-skimming tau-neutrino detection rates at ground-level optical detectors — the fluorescence telescopes of the original proposal and the optical Cherenkov designs that pursue the same Earth-skimming geometry — with an explicit, uncertainty-quantified treatment. Concretely, determine: (i) the tau energy-loss rate in crustal rock at tau energies of 10810^8-101210^{12} GeV — in particular its photonuclear component, which dominates the loss but rests on structure-function extrapolation into a kinematic region with no accelerator data — and propagate it into tau emergence probabilities and event rates; (ii) the effect of tau-neutrino regeneration and multibang (multiple-conversion) chains on the emerging tau flux across the full range of emergence angles; (iii) the effect of neutral-current energy degradation and of the true event-by-event charged-current inelasticity distribution, in place of assigning the charged lepton the full neutrino energy; (iv) the uncertainty of the ultrahigh-energy neutrino–nucleon cross section entering the conversion and survival probabilities; and (v) realistic detector apertures — actual site topography, horizon-optimized coverage, overlapping fields of view, and the ∼10%{\sim}10\% duty cycle of fluorescence observation. The completion boundary is a rate prediction for the Earth-skimming channel in which every one of these components is treated explicitly and in which the dominant residual systematic, the photonuclear tau energy loss, is reduced or constrained to an uncertainty below 20% on the tau emergence probability — the lower edge of the present 20-50% spread among structure-function extrapolations — with the resulting uncertainty propagated into the predicted rate, while the 40% systematic assigned to tau energy loss in the operational Auger Earth-skimming exposure is a separate, downstream quantity.

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