  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.
