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 $({\mathrm{\Sigma }}^{+},{\mathrm{\Sigma }}^0,{\mathrm{\Sigma }}^{-})$. 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 $\mathrm{\Sigma }$, the other for one $N$ and two $\mathrm{\Sigma }$. Phenomenological consequences are discussed.