  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.
