  The latest results of anomalous muon magnetic moment at Fermilab show a
discrepancy of 4.2 $\sigma$ between the Standard Model (SM) prediction and
experimental value. In this work, we revisit $U(1)_{L_{\mu}-L_{\tau}}$ symmetry
with in the paradigm of scotogenic model which explains muon ($g-2$) and
neutrino mass generation, simultaneously. The mass of new gauge boson
$M_{Z_{\mu\tau}}$ generated after the spontaneous symmetry breaking of
$U(1)_{L_{\mu}-L_{\tau}}$ is constrained, solely, in light of the current
neutrino oscillation data to explain muon ($g-2$). In particular, we have
obtained two regions I and II, around 150 MeV and 500 MeV, respectively, in
$M_{Z_{\mu\tau}}-g_{\mu\tau}$ plane which explain the neutrino phenomenology.
Region I is found to be consistent with muon neutrino trident (MNT) bound
($g_{\mu\tau}$ $\leq$ $10^{-3}$) to explain muon ($g-2$), however, region II
violates it for mass range $M_{Z_{\mu\tau}}>300$ MeV. We, then, extend the
minimal gauged scotogenic model by a vector like lepton (VLL) triplet $\psi_T$.
The mixing of $\psi_T$ with inert scalar doublet $\eta$ leads to chirally
enhanced positive contribution to muon anomalous magnetic moment independent of
$Z_{\mu\tau}$ mass. Furthermore, we have, also, investigated the implication of
the model for $0\nu\beta\beta$ decay and $CP$ violation. The non-observation of
$0\nu\beta\beta$ decay down to the sensitivity of 0.01 eV shall refute the
model. The model, in general,is found to be consistent with both $CP$
conserving and $CP$ violating solutions.

High Energy Physics - Phenomenology