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

The type II seesaw is a popular mechanism for generating Majorana neutrino masses at the right scale, with well-known implications for Lepton Flavor Violation. In a non-abelian extension of the Standard Model featuring a Z' field, the type-II seesaw mechanism for Dirac neutrino masses naturally arises by imposing lepton number conservation and adding a soft-term breaking a $Z_2$ symmetry of the model. As expected, there are interesting outcomes for $\mu \rightarrow e \gamma$, but we noticed that the number of relativistic neutrinos, $N_{\text{eff}}$, has a greater impact driven by processes involving the $Z'$ boson and right-handed neutrinos. The presence of a gauge symmetry behind the physical processes, allows us to explore an interesting interplay between the type II seesaw, lepton flavor violation and $N_{\text{eff}}$. We find that using the indicated value by Planck $N_{\text{eff}}=2.99^{0.34}_{0.33}$ we impose constraints on the masses of the doubly charged scalar and Z' fields which are complementarity to direct searches at colliders.