Where do IceCube neutrinos come from? Hints from the diffuse gamma-ray flux

Despite the spectacular discovery of an astrophysical neutrino flux by IceCube in 2013, its origin remains a mystery. Whatever its sources, we expect the neutrino flux to be accompanied by a comparable gamma-ray flux. These photons should be degraded in energy by electromagnetic cascades and contribute to the diffuse GeV-TeV flux precisely measured by the Fermi-LAT. Population studies have also permitted to identify the main classes of contributors to this flux, which at the same time have not been associated with major neutrino sources in cross-correlation studies. These considerations allow one to set constraints on the origin and spectrum of the IceCube flux, in particular its low-energy part. We find that, even accounting for known systematic errors, the Fermi-LAT data exclude to at least 95% C.L. any extragalactic transparent source class, irrespective of its redshift evolution, if the neutrino spectrum extends to the TeV scale or below. If the neutrino spectrum has an abrupt cutoff at similar to 10 TeV, barely compatible with current observations, the tension can be reduced, but this way out requires a significant modification to the current understanding of the origin of the diffuse extragalactic gamma-ray flux at GeV energies. In contrast, these considerations do not apply if a sizable fraction of IceCube data originates within the Galactic halo (a scenario however typically in tension with other constraints) or from a yet unidentified class of "opaque" extragalactic emitters, which do not let the high-energy gamma rays get out.