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We study an extension of the minimal gauged $L_{\mu}-L_{\tau}$ model in orderto explain the anomalous magnetic moments of muon and electron simultaneously.Presence of an additional scalar doublet $\eta$ and an in-built $Z_2$ symmetryunder which the right handed singlet fermions and $\eta$ are odd, leads tolight neutrino mass in scotogenic fashion along with a stable dark mattercandidate. In spite of the possibility of having positive and negativecontributions to $(g-2)$ from vector boson and charged scalar loopsrespectively, the minimal scotogenic $L_{\mu}-L_{\tau}$ model can not explainmuon and electron $(g-2)$ simultaneously while being consistent with otherexperimental bounds. We then extend the model with a vector like lepton doubletwhich not only leads to a chirally enhanced negative contribution to electron$(g-2)$ but also leads to the popular singlet-doublet fermion dark matterscenario. With this extension, the model can explain both electron and muon$(g-2)$ while being consistent with neutrino mass, dark matter and other directsearch bounds. The model remains predictive at high energy experiments likecollider as well as low energy experiments looking for charged lepton flavourviolation, dark photon searches, in addition to future $(g-2)$ measurements.

Lepton Anomalous Magnetic Moment with Singlet-Doublet Fermion Dark Matter in Scotogenic $U(1)_{L_{\mu}-L_{\tau}}$ Model