The soft volume of ultra-high energy neutrinos experiments

We develop a semi-analytical framework to map ultra-high-energy neutrino fluxes onto event rates at neutrino telescopes. The formulation is based on the Boltzmann equation for the distribution of secondary muons produced by neutrino interactions in matter, and naturally accounts for the effective target volume relevant for through-going tracks. This "soft volume" is controlled by muon propagation and stochastic energy losses, and can be significantly larger than the instrumented detector volume. Exploiting the dominance of soft energy losses, we derive a controlled second-order expansion of the collision operator, reducing the transport problem to a drift-diffusion equation in energy space, with rare hard scatterings treated perturbatively. The resulting master formula provides a fast alternative to full Monte Carlo simulations while retaining a direct connection to the microscopic muon energy-loss processes. We apply the formalism to IceCube through-going muon data, marginalizing over theoretical uncertainties in the transport coefficients, and obtain a diffuse-flux fit compatible with the experimental result. We also revisit the interpretation of the ultra-high-energy track event reported by KM3NeT, assessing its consistency with IceCube non-observation in the same energy range. Our results provide a first-principles bridge between neutrino-flux models and track-event observables, with direct applications to precision neutrino astronomy and searches for physics beyond the Standard Model.