Detecting the $L_{\mu}-L_{\tau}$ gauge boson at Belle II

We discuss the feasibility of detecting the gauge boson of the $U(1)_{L_{\mu}-L_{\tau}}$ symmetry, which possesses a mass in the range between MeV and GeV, at the Belle-II experiment. The kinetic mixing between the new gauge boson $Z'$ and photon is forbidden at the tree level and is radiatively induced. The leptonic force mediated by such a light boson is motivated by the discrepancy in muon anomalous magnetic moment and also the gap in the energy spectrum of cosmic neutrino. Defining the process $e^{+} e^{-} \rightarrow \gamma Z' \rightarrow \gamma \nu \bar{\nu}~(missing~energy)$ to be the signal, we estimate the numbers of the signal and the background events and show the parameter region to which the Belle-II experiment will be sensitive. The signal process in the $L_{\mu}-L_{\tau}$ model is enhanced with a light $Z'$, which is a characteristic feature differing from the dark photon models with a constant kinetic mixing. We find that the Belle-II experiment with the design luminosity will be sensitive to the $Z'$ with the mass $M_{Z'} \lesssim 1 $ GeV and the new gauge coupling $g_{Z'} \gtrsim 8\cdot 10^{-4}$, which covers a half of the unconstrained parameter region that explains the discrepancy in muon anomalous magnetic moment. The possibilities to improve the significance of the detection are also discussed.