  We study the possibilities on the search of the light and weakly interacting
gauge boson in the gauged $L_\mu - L_\tau$ model. Introducing the kinetic
mixing at the tree-level, the allowed parameter regions for the gauge coupling
and kinetic mixing parameter are presented. Then, we analyze one photon plus
missing event within the allowed region and show that search for the light
gauge boson will be possible at Belle-II experiment. We also analyze neutrino
trident production process in neutrino beam experiments.
