Seesaw Neutrino Mass and New U(1) Gauge Symmetry

The three electroweak doublet neutrinos $\nu_{e,\mu,\tau}$ of the Standard Model may acquire small seesaw masses, using either three Majorana fermion singlets $N$ or three Majorana fermion triplets $(\Sigma^+,\Sigma^0,\Sigma^-)$. It is well-known that the former accommodates the U(1) gauge symmetry $B-L$. It has also been shown some years ago that the latter supports a new $U(1)_X$ gauge symmetry. Here we study two variations of this $U(1)_X$, one for two $N$ and one $\Sigma$, the other for one $N$ and two $\Sigma$. Phenomenological consequences are discussed.