In this work, we have tried to explain a few unsolved beyond standard model problems namely electron and muon ($g-2$), neutrino mass and dark matter.
Moreover, we have also explained the implications of the involved
particles at the collider. For this purpose, in the present work, we have proposed a suitable
extension of the minimal $L_{\mu}-L_{\tau}$ model to address the aforementioned drawbacks of SM.
In our model,
a new Yukawa interaction involving electrons, a
singlet vector-like fermion ($\chi^{\pm}$) and a scalar (either a complex singlet
$\Phi^\prime_4$ or an SU(2)$_{L}$ doublet $\Phi^\prime_2$) provides the additional one loop contribution to electron (g-2).
On the other hand $(g-2)_{\mu}$ can be satisfied with the
$Z_{\mu\tau}$ gauge boson. The judicious choice of $L_{\mu}-L_{\tau}$
charges of the additional fields make the gauge model anomaly free.
The lightest component among the neutral parts of the
additional scalar fields is a DM candidate freezes out by the $3 \rightarrow 2$ processes. Our DM can also be detected at the direct detection experiments mediated by the SM Z boson and the process is suppressed by the
$Z-Z_{\mu\tau}$ mixing angle. We have also
satisfied the bound from DM relic density,
unitarity and self-interaction of DM which are applicable to the present DM
study. Finally, our proposed model can also be tested at the future
lepton collider through the process $e^{+} e^{-} \rightarrow \chi^{+} \chi^{-} \rightarrow e^{+} e^{-} \cancel{E}_T$.