Inverse seesaw and $(g-2)$ anomalies in $B-L$ extended two Higgs doublet model

We propose a gauged $U(1)_{B-L}$ extended two Higgs doublet model to explain both neutrino mass and lepton anomalous magnetic moments ($g-2$). Neutrino mass is generated via an inverse seesaw mechanism by introducing singlet fermions. Especially, we update the result of muon $g-2$ in light of the very recent report by E989 experiment at Fermilab, indicating $a_\mu^{\rm FNAL}=116592040(54)\times 10^{-11}$. Combining BNL result, we have the following deviation from the standard model prediction $\Delta a_\mu=(2.51\pm 5.9)\times10^{-10}$ at 4.2 $\sigma$. Thanks to an appropriate assignment for $U(1)_{B-L}\times Z_2$ symmetry and larger $(20\lesssim)\tan\beta$ that is favoured by type-X model, we realize natural hierarchies among neutral fermions. The lepton anomalous magnetic moments can be induced at the one loop level by introducing an iso-spin singlet singly-charged boson. This charged scalar plays a significant role in evading chiral suppression of these phenomenologies. We show sizable lepton ($g-2$) can be obtained after satisfying all the flavour constraints, such as $\mu\to e\gamma$ and flavour conserving leptonic $Z$ boson decays.