New Weinberg operator for neutrino mass and its seesaw origin

We consider a new Weinberg operator for neutrino mass of the form $H_u\tilde{H_d}L_iL_j$ involving two different Higgs doublets $H_u, H_d$ with opposite hypercharge, where $\tilde{H_d}$ is the charge conjugated doublet. It may arise from a model where the two Higgs doublets carry the same charge under a $U(1)'$ gauge group which forbids the usual Weinberg operator but allows the mixed one. The new Weinberg operator may be generated via two right-handed neutrinos oppositely charged under the $U(1)'$, which may be identified as components of a fourth vector-like family in a complete model. Such a version of the type I seesaw model, which we refer to as type Ib to distinguish it from the usual type Ia seesaw mechanism which yields the usual Weinberg operator, allows the possibility of having potentially large violations of unitarity of the leptonic mixing matrix whose bounds we explore. We also consider the relaxation of the unitarity bounds due to the further addition of a single right-handed neutrino, neutral under $U(1)'$, yielding a usual type Ia seesaw contribution.