Cosmic-Ray Constraints on the Flux of Ultrahigh-energy Neutrino Event KM3-230213A

The detection of a ≃220 PeV muon neutrino event by the KM3NeT telescope offers an unprecedented opportunity to probe the Universe at extreme energies. A photopion interaction origin of the neutrino requires a parent cosmic-ray energy of ≳4 EeV per nucleon. We analyze the origin of this event under three scenarios, i.e., a transient point source, diffuse astrophysical emission, and a line-of-sight interaction of an ultrahigh-energy cosmic ray (UHECR; E ≳ 0.1 EeV). Our analysis includes the flux from both a KM3NeT-only fit and a joint fit, incorporating data from KM3NeT, IceCube, and the Pierre Auger Observatory. If the neutrino event originates from transients, it requires a new population of transients that is energetic, γ-ray-dark, and more abundant than the known ones. In the framework of diffuse astrophysical emission, we compare the required local UHECR energy injection rate at ≳4 EeV with the rate derived from the flux measurements by Auger, across various models of source redshift evolution. This disfavors the KM3NeT-only fit, considering the source evolution up to high values of redshift, while the joint fit remains viable for sources contributing up to a maximum redshift z max ≳ 1 for the limiting case of photopion interaction efficiency, f_pγ = 0.1. For a cosmogenic origin from point sources, the luminosity obtained at redshifts z ≲ 1 from the joint fit is compatible with the Eddington luminosity of ∼10⁹ M_⊙ black holes in active galactic nuclei, assuming a proton composition and optimistic values of extragalactic magnetic field strength.