Constraining dark matter from strong phase transitions in a \( \textrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} \) model: implications for neutrino masses and muon g − 2

In this paper, we study a non-minimal gauged $\mathrm{U}{\left(1\right)}_{L_{\mu }-L_{\tau }}$ \( \textrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} \) model, where we add two complex singlet scalars, three right-handed Majorana neutrinos (RHN), and a vector-like dark fermion to the Standard Model (SM), all non-trivially charged under the extra gauge symmetry. The model offers an easy resolution to the muon (g – 2) anomaly, which fixes the scale of spontaneous symmetry breaking. Furthermore, the two-zero minor structure in the RHN mass matrix provides successful predictions for neutrino oscillation parameters, including the Dirac phase. The extended scalar sector can easily induce first-order phase transitions. We identify all possible phase transition patterns in the three-dimensional field space. We quantify the associated gravitational waves from the sound wave source and demonstrate that the signatures can be observed in future space-based experiments. We find that strong first-order phase transitions require large values of scalar quartic couplings which constrain the scalar dark matter (DM) relic density to a maximum of 10⁻² and 10⁻⁵ when we consider the DM direct detection bound. Nonetheless, the model successfully explains the DM relic density via contribution from the vector-like dark fermion. We show the allowed range of the model parameters that can address all the beyond SM issues targeted in this study.