Remark on neutrino masses and oscillations in anSU(3)L×U(1)Nmodel with the radiative mechanism

We have discussed how neutrino masses and oscillations are radiatively generated in an ${\mathrm{SU}\left(3\right)\mathrm{}}_L\times {U\left(1\right)\mathrm{}}_N$ gauge model with a symmetry based on $L_e-L_{\mu }-L_{\tau }$ $\left(\equiv L^{\prime }\right).\mathrm{}$ The model is characterized by lepton triplets ${\psi }^i=\left({\nu }^i{,l}^{-i}{,E}^{-i}\right),$ where $E^{-i}$ are negatively charged heavy leptons, an ${\mathrm{SU}\left(3\right)\mathrm{}}_L$ triplet Higgs scalar $\xi ,$ and a singlet Higgs scalar $k^{++}.$ These Higgs scalars can be interpreted as a Zee and Zee-Babu scalar for radiative mechanisms. We have demonstrated that the mass hierarchy of $\Delta m_{\mathrm{atm}}^2\gg \Delta m_{\odot }^2$ arises as a consequence of the dynamical hierarchy between $L^{\prime }$-conserving one-loop effects and $L^{\prime }$-violating two-loop effects, where $L^{\prime }$ is broken softly in the Higgs potential, and that our model is relevant to yield a quasivacuum solution for the solar neutrino problem.