Supersymmetric gauged U(1)$_{L_{\mu}-L_{\tau}}$ model for neutrinos and the muon $(g-2)$ anomaly

The gauged $U(1)_{L_{\mu}-L_{\tau}}$ model can provide for additional contributions to the muon anomalous magnetic moment by means of a loop involving the $Z^{\prime}$ gauge boson. However, the parameter space of such models is severely constrained if one combines the latest muon $(g-2)$ data with various neutrino experiments, such as neutrino trident production, $\nu -e$ and $\nu -q$ elastic scattering, etc. In a supersymmetric $U(1)_{L_{\mu}-L_{\tau}}$ model, a larger region of parameter space opens up, thus enabling one to explore otherwise forbidden regions of parameter space in nonsupersymmetric models involving the new gauge coupling ($g_X$) and the mass of the $Z^\prime$ gauge boson ($M_{Z^{\prime}}$) . We show that the minimal model with the minimal supersymmetric Standard Model (MSSM) field content is strongly disfavored from $Z$-boson decay and neutrino data. We also show that the nonminimal model with two extra singlet superfields can lead to correct neutrino masses and mixing involving both tree-level and one-loop contributions. We find that, in this model, both muon $(g-2)$ and neutrino data may be simultaneously explained in a parameter region consistent with experimental observations. In addition, we observe that the muon $(g-2)$ anomaly can be accommodated even with higher values of electroweak sparticle masses compared to the MSSM. Charged lepton-flavor-violating processes (like $\mu\rightarrow e\gamma$, $\tau\rightarrow \mu\gamma$, etc.) may have potentially large branching ratios in this scenario. Depending on the magnitude of the supersymmetry contribution to these processes, they may constrain hitherto unconstrained regions of the $M_{Z^{\prime}}-g_X$ parameter space. However, we find that these branching fractions never exceed their upper bounds in a region where both muon $(g-2)$ and neutrino oscillation data can be simultaneously accommodated.