Radiatively induced neutrino masses and oscillations in anSU(3)L×U(1)Ngauge model

We have constructed an $SU(3{)}_L\times U(1{)}_N$ gauge model utilizing a $U(1{)}_{L^{\prime }}$ symmetry, where $L^{\prime }{=L}_e-L_{\mu }-L_{\tau },$ which accommodates tiny neutrino masses generated by $L^{\prime }$-conserving one-loop and $L^{\prime }$-breaking two-loop radiative mechanisms. The generic smallness of two-loop radiative effects compared with one-loop radiative effects describes the observed hierarchy of $\Delta m_{\mathrm{atm}}^2$ $\gg \Delta m_{\odot }^2.$ A key ingredient for radiative mechanisms is a charged scalar ${(h}^{+})$ that couples to charged lepton-neutrino pairs and $h^{+}$ together with the standard Higgs scalar $\left(\phi \right)$ can be unified into a Higgs triplet as $({\phi }^0,$ ${\phi }^{-},$ $h^{+}{)}^T.$ This assignment in turn requires lepton triplets $\left({\psi }_L^i\right)$ with heavy charged leptons $\left({\kappa }_L^{+i}\right)$ as the third member: ${\psi }_L^i=({\nu }_L^i{,l}_L^i,{\kappa }_L^{+i}{)}^T,$ where i $\mathrm{}(=1,2,3\mathrm{})\mathrm{}$ denotes three families. It is found that our model is relevant to yield quasivacuum oscillations for solar neutrinos.