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Can the muon anomalous magnetic moment and the $B$ anomalies be simultaneously explained in a minimally extended $Z^{\prime}$ model?

We analyze the implications of an extended 2HDM theory in quark masses and mixings, electroweak precision tests, charged lepton flavor violating (CLFV) decays, neutrino trident production, B meson, $g-2$ muon and W anomalies. In the extended 2HDM theory considered in this work, the SM gauge symmetry is enlarged by the local $U(1)^{\prime}$ symmetry, whereas its scalar and fermion sectors are augmented by the inclusion of a gauge singlet scalar and vectorlike fermions, respectively. Within the framework of this model, we determine the allowed ranges of the $Z^{\prime}$ gauge boson mass consistent with the constraints arising from neutrino trident production, $R_{K^{*}}$ anomaly, $B_{s}$ meson oscillation. We conclude the $R_{K^{*}}$ and muon $g-2$ anomalies cannot be simultaneously explained by the same new physics under all the constraints. Besides that, we found that the new physics sources, the $Z^{\prime}$ and non SM scalars cannot be connected via the scalar potential. Furthermore, we have found that the model under consideration can successfully comply with the constraints arising from oblique $T,S,U$ parameters as well as with the $W$ mass and $g-2$ muon anomalies. In particular we found that the muon $g-2$ anomaly, whose dominant contribution arises from the one loop level scalar exchange, can be successfully accommodated within the $2σ$ experimentally allowed range.