  The energy spectrum of cosmic neutrinos, which was recently reported by the
IceCube collaboration, shows a gap between 400 TeV and 1 PeV. An unknown
neutrino interaction mediated by a field with a mass of the MeV scale is one of
the possible solutions to this gap. We examine if the leptonic gauge
interaction L_{\mu} - L_{\tau} can simultaneously explain the two phenomena in
the lepton sector: the gap in the cosmic neutrino spectrum and the unsettled
disagreement in muon anomalous magnetic moment. We illustrate that there
remains the regions in the model parameter space, which account for both the
problems. Our results also provide a hint for the distance to the source of the
high-energy cosmic neutrinos.
