The diffuse gamma-ray flux associated with sub-PeV/PeV neutrinos from starburst galaxies

One attractive scenario for the excess of sub-PeV/PeV neutrinos recently reported by IceCube is that they are produced by cosmic rays in starburst galaxies colliding with the dense interstellar medium. These proton-proton ($pp$) collisions also produce high-energy gamma-rays, which finally contribute to the diffuse high-energy gamma-ray background. We calculate the diffuse gamma-ray flux with a semi-analytic approach and consider that the very high energy gamma-rays will be absorbed in the galaxies and converted into electron-position pairs, which then lose almost all their energy through synchrotron radiation in the strong magnetic fields in the starburst region. Since the synchrotron emission goes into energies below GeV, this synchrotron loss reduces the diffuse high-energy gamma-ray flux by a factor of about two, thus leaving more room for other sources to contribute to the gamma-ray background. For a $E_\nu^{-2}$ neutrino spectrum, we find that the diffuse gamma-ray flux contributes about 20% of the observed diffuse gamma-ray background in the 100 GeV range. However, for a steeper neutrino spectrum, this synchrotron loss effect is less important, since the energy fraction in absorbed gamma-rays becomes lower.