A gauged non-universal $U(1)_{X}$ model to study muon $g-2$ and $B$ meson anomalies

We study a non-universal $U(1)_{X}$ extension of the Standard Model with an extended scalar sector of two doublets and one singlet plus three additional exotic quarks and two exotic charged leptons on the fermionic sector. In order to obtain the observed fermion mass hierarchy, an additional $\mathbb{Z}_{2}$ discrete symmetry is imposed, where the heaviest fermions acquire their masses from three different scales determined by two Higgs doublets and one singlet, whereas the lightest fermions obtain their masses from effective operators up to dimension seven. From chiral anomalies cancellation, the model also includes heavy right-handed neutrinos which get massive via an inverse see-saw mechanism, reproducing the observed mass differences for the active neutrinos. We analyze phenomenological consequences of the model in view of the so-called flavour anomalies, namely the latest measurement made by Fermilab of the anomalous magnetic moment of the muon $g-2$, and additionally, the fit to semi-leptonic $B$ meson decays made by different flavour groups, dominated mainly by the LHCb 2020 data. We obtain that the model can explain the former at the $1\sigma$ level by means of contributions coming from charged $W^{+}$ bosons interacting with exotic Majorana neutrinos at one-loop level, with the $Z'$ boson contribution itself coming from the $U(1)_X$ symmetry being negligible. However, we find that the model is able to accommodate the $b\to s\ell^{+}\ell^{-}$ transitions associated to the $B$ meson anomalies only at the $2\sigma$ level via tree-level $Z'$ boson exchange, while simultaneously respecting various constraints from the recent $R_{K^{(*)}}$ measurements made by the LHCb, neutrino trident production and $B-\bar B$ oscillations.