Earth-Skimming Tau Neutrino Channel

• Earth-skimming tau neutrino channel is a detection technique that exploits tau leptons produced by neutrino interactions in the Earth to generate upward extensive air showers.
• Detailed models and Monte Carlo simulations quantify charged-current interactions, tau energy loss, and emergence probability to predict event rates and sensitivity.
• Detector arrays positioned near mountain edges optimize target mass, enabling effective identification of PeV–EeV neutrino fluxes and advancing ultra-high-energy astrophysics.

The Earth-skimming tau neutrino channel refers to the detection method where ultra-high-energy tau neutrinos ($\nu_\tau$) interact within the Earth at shallow angles, producing tau leptons ($\tau$) that escape the surface and decay in the atmosphere, yielding extensive air showers (EASs) observable by ground-based, airborne, or space-based detectors. This technique leverages the large target mass provided by the Earth and the spatially distinctive upward-going EAS signature, providing sensitivity to astrophysical and cosmogenic neutrino fluxes across the PeV–EeV range.

1. Physical Principles and Interaction Sequence

In the Earth-skimming scenario, a cosmic $\nu_\tau$ of energy $E_\nu$ arrives at the Earth at a zenith angle $\theta > 90^\circ$ (i.e., just below the horizon), entering either rock or water and traversing a chord of length $L(\theta)$. Along this chord, the $\nu_\tau$ may undergo a charged-current (CC) interaction at a slant depth $X_\text{int}$, producing a $\tau$ lepton with initial energy $E_\tau^\text{i} \approx (1-y) E_\nu$, where $y$ is the mean inelasticity (typically $\sim0.2-0.3$ at EeV energies). The $\tau$ propagates through the medium, experiencing continuous energy losses, and may emerge at or near the Earth's surface at a small emergence angle $\Delta\theta \approx \theta-90^\circ$. Provided the $\tau$ survives to exit, it decays in the atmosphere at altitude $h_\text{dec}$, initiating an upward-going EAS. The Cherenkov (and, depending on the energy and geometry, fluorescence or radio) emission from these showers forms the observable channel for experiments sited on mountain slopes, at high elevation, or at altitude (Gora et al., 2017, Gora et al., 2015, Hou, 2014).

2. Core Formalism: Cross Sections, Propagation, and Emergence Probability

The detection chain is quantitatively described by a set of coupled processes and key equations:

• Charged-current interaction probability:

$$P_\text{CC}(E_\nu,\theta) = 1 - \exp\left[-N_A \, \sigma_\text{CC}(E_\nu)\, X(\theta) \right]$$

where $N_A$ is Avogadro's number, $\sigma_\text{CC}(E_\nu)$ the νN CC cross section, and $X(\theta) = \int_0^{L(\theta)} \rho(\ell) d\ell$ is the column depth traversed (Gora et al., 2017, Gora et al., 2015).

• Tau energy loss (continuous-slowing-down):

$\frac{dE_\tau}{dX} = -[\alpha + \beta(E) E_\tau]$

where $\alpha$, $\beta$ depend on the medium; the radiative (photonuclear, pair production, bremsstrahlung) $\beta$ dominates above PeV energies. The value of $\beta$ is model-dependent, e.g., $\beta_0=1.2 \times 10^{-6}$ cm$^2$ g$^{-1}$, $\beta_1=0.16 \times 10^{-6}$ cm$^2$ g$^{-1}$ for rock (Boussaha et al., 2021, Alvarez-Muñiz et al., 2017), with significant uncertainties arising at the highest energies (Alvarez-Muñiz et al., 2017, Reno et al., 2019).

• Tau survival probability through matter:

$$P_\text{surv}(E_\tau, X) = \exp\left[ - \int_0^X \frac{dX'}{\lambda_\text{decay}(E_\tau(X'))} \right ] \approx \exp \left [ -\frac{m_\tau}{c \tau_\tau \rho} \int_0^X \frac{dX'}{E_\tau(X')} \right ]$$

with $\lambda_\text{decay}(E) = c\tau_\tau (E/m_\tau)$ the decay length, and $\rho$ the density (Gora et al., 2017).

• Emergence (exit) probability:

$$P_\text{em}(E_\nu, \theta) = \int_{E_\text{th}}^{E_\nu} dE_\tau \; P_\text{CC}(E_\nu,\theta)\; P_\text{surv}(E_\tau, X_\text{int}\to 0)$$

where $E_\text{th}$ is the minimum $\tau$ energy required for detection (Gora et al., 2017, Gora et al., 2015, Boussaha et al., 2021).

This formalism is embedded in full transport equations for $\nu_\tau\leftrightarrow\tau$ fluxes, including regeneration by $\tau$ decay and neutral-current (NC) scattering, as detailed in (Alvarez-Muñiz et al., 2017, Reno et al., 2019).

3. Monte Carlo Realizations and Detector Simulation

Realistic predictions of event rates and acceptance rely on comprehensive Monte Carlo (MC) chains that implement:

• Neutrino propagation: MC tools (e.g., extended ANIS, NuTauSim) propagate $\nu_\tau$ through stratified Earth models, including surface layers (water, ice, rock), simulating CC/NC interactions and subsequent $\tau$ transport and regeneration (Gora et al., 2017, Alvarez-Muñiz et al., 2017, Boussaha et al., 2021, Reno et al., 2019).
• $\tau$-induced air shower development: EASs from $\tau$ decay are simulated with codes such as CORSIKA (with CURVED EARTH and TAUOLA for $\tau$ decays) or CONEX, incorporating electromagnetic and hadronic models (Gora et al., 2015, Boussaha et al., 2021).
• Cherenkov, fluorescence, or radio signal modeling: Detector-specific frameworks (e.g., MARS for MAGIC, proprietary codes for Trinity/NTA/Ashra) simulate Cherenkov photon propagation, mirror/camera response, trigger logic, and background discrimination (Gora et al., 2017, Brown et al., 2021, Asaoka et al., 2012).
• Reconstruction and analysis: Event selection leverages Hillas moment-based parameters, geometrical constraints (e.g., FOV, impact distance), timing/topology, and likelihood or machine-learning classifiers to separate $\tau$-induced signatures from hadronic or atmospheric backgrounds (Gora et al., 2017, Yilmaz et al., 2021, Asaoka et al., 2012).

4. Detector Implementations and Performance Metrics

A range of experimental platforms exploit the Earth-skimming method, each with distinctive geometry, observational window, and performance:

Experiment	Energy Range (PeV–EeV)	Technique	Key Features
MAGIC	2–1000	Imaging Cherenkov	3.5° FOV, stereo imaging, $\sim$few–100 km$^2$
NTA	10–1000	Cherenkov+Fluorescence	Multi-site stereoscopy, $\sim$100 km$^3$ vol.
Ashra-1	1–1000	Cherenkov Imaging	42° FOV, arcmin res., background-free
Trinity	1–10,000	Imaging Cherenkov	Horizon-pointing, $\sim$1–10 km$^2$ aperture
TAUWER	10–1000	Scintillator array	Mountain chain geometry, $<$1° res.
HAWC	1–100	WCD (Water Cherenkov)	Volcano-shielded, track-based, $\sim$km$^2$

For MAGIC, point-source aperture rises from $\sim10^3$ m$^2$ at 100 TeV to $\sim10^5$ m$^2$ at 1 EeV, with sensitivity $$E_\nu^2\,\Phi^\text{PS}(E_\nu) < 2.0 \times 10^{-4}$$ GeV cm$^{-2}$ s$^{-1}$ in 30 h of sea-on observation, comparable to Auger's down-going channel scaled to the same exposure (Gora et al., 2017).

Event-rates are model-dependent, but for a three-year, 10 km$^2$ array (TAUWER), 13 events are expected in the 10–1000 PeV range, assuming present upper limits on the $\nu_\tau$ flux (Yilmaz et al., 2021).

NTA simulations reach differential point-source sensitivities of $E^2\Phi \sim 10^{-9}$ GeV cm$^{-2}$ s$^{-1}$ at $10^{18}$ eV, with effective water-equivalent mass $>100$ km$^3$ (Hou, 2014).

5. Air Shower Observables and Background Discrimination

The $\tau$-induced upward EAS develops in the atmosphere with a geometry and composition distinct from downward or nearly-horizontal cosmic ray showers:

• Shower profiles: For PeV–EeV $\tau$-decay showers, the longitudinal development is parameterized by Gaisser–Hillas functions, with $X_\text{max}$ scaling roughly $\sim\!\ln E_\tau$ (Boussaha et al., 2021).
• Cherenkov emission: The upward-going EAS's Cherenkov cone ($\alpha_\text{Cher}\sim1.35^\circ$) defines the geometrical trigger window. The photon yield at $d\sim20$ km is $\sim2\times10^5$ photons/PeV, attenuated by atmospheric conditions (Gora et al., 2015, Brown et al., 2021).
• Reconstruction: Discrimination against hadronic backgrounds exploits image morphology (e.g., Hillas parameter cuts), FOV geometry, and directionality. Cuts on combinations like $$Y' = \log_{10}(\text{Size})\cos\alpha - \log_{10}(\text{Length})\sin\alpha$$ (with optimized $\alpha$) efficiently remove backgrounds while retaining $30$–$50\%$ of $\tau$-signal showers in MAGIC (Gora et al., 2017).
• Angular/energy resolution: Under optimal conditions, Cherenkov arrays (Ashra-1, NTA) reach sub-degree or arcminute pointing, while scintillator or water arrays achieve $<1^\circ$ using timestamp and footprint information (Hou, 2014, Yilmaz et al., 2021).

Cosmic ray albedo and anthropogenic light are generally negated by the upward event geometry, pattern analysis, and imaging track criteria (Asaoka et al., 2012, Brown et al., 2021).

6. Physics Reach, Sensitivity, and Theoretical Uncertainties

Earth-skimming $\nu_\tau$ detection enables stringent limits and potential discovery reach for astrophysical and cosmogenic fluxes in the hitherto weakly probed PeV–EeV regime:

• Sensitivity scaling: Sensitivity improves with exposure (e.g., $K_{90\%} = 2.44 / N_\text{ev}(\Phi_0)$ for 90% C.L. upper limits) and is set by the effective acceptance, which rises steeply with energy due to the increasing interaction cross-section and longer $\tau$ decay lengths (Gora et al., 2017, Gora et al., 2015, Hou, 2014).
• Comparisons: NTA and Ashra-1 aim to fill the “gap” between IceCube (E ≲ 10 PeV) and radio observatories such as Auger or GRAND (E ≳ 100 PeV–1 EeV), with effective water-equivalent masses and apertures exceeding $100$ km$^3$ and $10^3$–$10^4$ km$^2$ sr, respectively (Hou, 2014, Asaoka et al., 2012).
• Uncertainties:
  • Dominant systematic uncertainties include those associated with the high-energy extrapolation of $\nu$N cross sections ($\pm10$–$30\%$), the $\tau$ photonuclear energy loss models ($\lesssim30$–$50\%$ at EeV), precise density profiles (ice/water/rock layering), and atmospheric propagation (Alvarez-Muñiz et al., 2017, Hou, 2014, Reno et al., 2019).
  • Model dependencies (e.g., ALLM vs. ASW energy loss), regeneration effects, and the stochastic nature of $\tau$ propagation can result in order-unity variations in predicted event rates for a given flux.

7. Prospects, Optimization, and Experimental Landscapes

Design choices and site selection critically impact channel performance:

• Terrain optimization: Placing detectors near mountain edges or in valleys with thick rock/mountain screens maximizes effective target mass and escape probability, with optimal mountain thicknesses $\sim5$–$10$ km and valley widths $\sim9$ km for $10^{17}$–$10^{19}$ eV (Boussaha et al., 2021).
• Array configuration: High-density arrays (e.g., 640 stations over $0.6$ km$^2$ in TAUWER) exploit spatially-coincident hits, timestamping, and topological cuts for reconstructing EAS directionality and energy (Yilmaz et al., 2021).
• Duty cycle and exposure: Imaging Cherenkov telescopes have limited duty cycles ($\sim10$–$20\%$), but water Cherenkov and radio-based arrays can operate near-continuously. Duty cycle and observation geometry dictate the cumulative exposure and hence the achievable flux limits (Brown et al., 2021, Asaoka et al., 2012, Yilmaz et al., 2021).
• Future directions: Continued MC refinement, exploitation of hybrid Cherenkov/radio/fluorescence observations, and the deployment of networked wide-field arrays (NTA, Trinity, GRAND) promise order-of-magnitude gains in acceptance and the ability to probe both steady and transient (e.g., GRB) neutrino sources with precise angular and temporal correlation (Hou, 2014, Brown et al., 2021, Asaoka et al., 2012).

In summary, the Earth-skimming tau neutrino channel constitutes a robust and technically mature strategy for extending ultra-high-energy neutrino searches, with demonstrated effectiveness in leveraging large inactive mass for target conversion, discriminating signal from background by distinctive geometry and shower topology, and achieving sensitivities competitive with and complementary to established in-ice, in-water, and radio-detection techniques. The method underpins cutting-edge efforts in neutrino astronomy for mapping and identifying both steady and transient astrophysical sources in the extreme-energy regime (Gora et al., 2017, Boussaha et al., 2021, Hou, 2014, Brown et al., 2021, Gora et al., 2015, Yilmaz et al., 2021, Asaoka et al., 2012).