  Very recently, the Fermi-Lab reported the new experimental combined results
on the magnetic momentum of muon with a 4.2$\sigma$ discrepancy compared with
the expectation of the Standard Model \cite{Fermi_Lab}. A new light gauge boson
$X$ in the $L_{\mu}-L_{\tau}$ model provides a good explanation for the $g-2$
anomaly. A Dirac fermion dark matter with a large $L_{\mu}-L_{\tau}$ charge can
explain both the $g-2$ anomaly and the dark matter relic density
\cite{Asai_2021}. In this work, we focus on the case that the mass of the dark
matter is larger than the mass of muon (i.e. $m_{\Psi} > m_{\mu}$) for which
the channel $\Psi \Psi \rightarrow \mu^- \mu^+$ opens. Although the cross
section $(\sigma v)_{\mu^{-}\mu^{+}}$ is smaller by a factor of $1/q_{\Psi}^2$
($q_{\Psi}$ represents the $L_{\mu}-L_{\tau}$ charge of the dark matter)
compared with the channel $\Psi\Psi \rightarrow XX \rightarrow
\nu\nu\bar{\nu}\bar{\nu}$, the resulting secondary electrons and positrons
could imprint on their spectra above GeV energies due to the reacceleration
effect of cosmic ray propagation. We use the AMS-02 measurements of electrons
and positrons to constrain the annihilation cross section of the channel
$\Psi\Psi \rightarrow \mu^{-}\mu^{+}$, which rules out part of the parameter
space of the large $L_{\mu}-L_{\tau}$ charged dark matter model to account for
the muon $g-2$ anomaly.
