Cascaded Gamma-ray Emission Associated with the KM3NeT Ultra-High-Energy Event KM3-230213A

A neutrino-like event with an energy of $\sim 220 \,{\rm PeV}$ was recently detected by the KM3NeT/ARCA telescope. If this neutrino comes from an astrophysical source, or from the interaction of an ultra-high-energy cosmic ray in the intergalactic medium, the ultra-high-energy gamma rays that are co-produced with the neutrinos will scatter with the extragalactic background light, producing an electromagnetic cascade and resulting in emission at GeV-to-TeV energies. In this paper, we compute the gamma-ray flux from this neutrino source considering various source distances and strengths of the intergalactic magnetic field (IGMF). We find that the associated gamma-ray emission could be observed by existing imaging air cherenkov telescopes and air shower gamma-ray observatories, unless the strength of the IGMF is $B\gtrsim 3\times 10^{-13}$ G, or the ultra-high-energy gamma-rays are attenuated inside of the source itself. In the latter case, this source is expected to be radio-loud.