Exploring ultra-high energy neutrino experiments through the lens of the transport equation

We develop a first-principles formalism, based on the transport equation in the line-of-sight approximation, to link the expected number of muons at neutrino telescopes to the flux of neutrinos at the Earth's surface. We compute the distribution of muons inside Earth, arising from the up-scattering of neutrinos close to the detector, as well as from the decay of taus produced farther away. This framework allows one to account for systematic uncertainties, as well as to clarify the assumptions behind definitions commonly used in the literature, such as the effective area. We apply this formalism to analyze the high-energy muon event recorded by KM3NeT, with a reconstructed energy of $ 120^{+110}_{-60} \, \mathrm{PeV}$ and an elevation angle of $\left(0.54\pm 2.4\right)^\circ$, in comparison with the non-observation of similar events by IceCube. We find a $3.1\,\sigma$ tension between the two experiments, assuming a diffuse neutrino source with a power-law energy dependence. Combining both datasets leads to a preference for a very low number of expected events at KM3NeT, in stark contrast to the observed data. The tension increases both in the case of a diffuse source peaking at the KM3NeT energy and of a steady point source, whereas a transient source may reduce the tension down to $1.6\,\sigma$. The formalism allows one to treat potential beyond-the-Standard-Model sources of muons, and we speculate on this possibility to explain the tension.