Neutrino oscillation in dark matter with $L_mu-L_tau$

In this paper, we study the phenomenology of a Dirac dark matter in the $L_mu-L_tau$ model and investigate the neutrino oscillation in the dark halo. Since dark matter couples to the muon neutrino and the tau neutrino with opposite sign couplings, it contributes effective potentials, $pm A_chi$, to the evolution equation of the neutrino flavor transition amplitude, which can be significant for high energy neutrino oscillations in a dense dark matter environment. We discuss neutrino masses, lepton mixing angles, Dirac CP phase, and neutrino oscillation probabilities in the dark halo using full numerical calculations. Results show that neutrinos can endure very different matter effects. When the potential $A_chi$ becomes ultra-large, three neutrino flavors decouple from each other.