Search for U(1)$_{L_\mu-L_\tau}$ charged Dark Matter with neutrino telescope

We study a simple Dirac fermion dark matter model in U(1)$_{L_\mu-L_\tau}$ theory. The new light gauge boson $X$ plays important roles in both dark matter physics and the explanation for the muon $g-2$ anomaly. The observed dark matter relic density is realized by a large U(1)$_{L_\mu-L_\tau}$ charge without introducing a resonance effect of the $X$ boson. As a by-product of the model, characteristic neutrino signatures from sub-GeV dark matter $\psi$ are predicted depending on the mass spectrum. We formulate the analysis of $\psi\bar\psi\to \nu\bar\nu$, and of $\psi\bar\psi\to XX$ followed by $X\to\nu\bar\nu$, in a model independent way. The energy spectrum of neutrinos in the former process is monochromatic while in the latter process is bowl-shape. We also evaluate sensitivity at Super-Kamiokande and future Hyper-Kamiokande detectors. The analysis is finally applied to the U(1)$_{L_\mu-L_\tau}$ dark matter model.