Type-II Seesaw Mechanism for Dirac Neutrinos and its Implications on $N_{\text{eff}}$ and Lepton Flavor Violation in a 3-3-1 model

Abstract: In this study, we implement the type-II seesaw mechanism for Dirac neutrino masses within the framework of a 3-3-1 model. To this end, we introduce a scalar sextet and impose both lepton number conservation and invariance under a discrete $Z_2$ symmetry in the Lagrangian. This mechanism naturally generates small Dirac neutrino masses by allowing the soft breaking of the $Z_2$ symmetry through a unique term in the scalar potential, while preserving lepton number. Specifically, we explore the realization of this model at low-energy scales. Regarding flavor implications, we analyze its contributions to the rare decay processes $\mu \to e \gamma$ and $\mu \to \bar e ee$. In the cosmological context, we analyze the influence of right-handed neutrinos on the effective number of neutrino species, $N_\text{eff}$, through interactions mediated by the $Z^{\prime}$ boson. Our findings establish a lower bound of $m_{Z^{\prime}} > 4.4$ TeV, which slightly exceeds the current lower limit set by the Large Hadron Collider (LHC).