Neutrino and $Z$ phenomenology in an anomaly-free $mathbf{U}(1)$ extension: role of higher-dimensional operators

We consider an anomaly-free $mathrm{U}(1)$ extension of the Standard Model with three right-handed neutrinos (RHNs) and two complex scalars, wherein the charge assignments preclude all tree-level mass terms for the neutrinos. Considering this setup, in turn, to be only a low-energy effective theory, we introduce higher-dimensional terms {em a la} Froggatt-Nielsen to naturally generate tiny neutrino masses. One of the RHNs turns out to be very light, thereby constituting the main decay mode for the $Z$ and hence relaxing the LHC dilepton resonance search constraints. This very RHN has a lifetime comparable to or bigger than the age of the Universe, and, hence, could account for a non-negligible fraction of the dark matter.