By using $L_e-L_{\mu }-L_{\tau }$ symmetry, we construct an ${\mathrm{SU}\left(3\right)\mathrm{}}_L\times {U\left(1\right)\mathrm{}}_N$ gauge model that provides two-loop radiative neutrino masses as well as one-loop radiative neutrino masses. The generic smallness of the two-loop neutrino masses leading to $\Delta m_{\odot }^2$ compared with the one-loop neutrino masses leading to $\Delta m_{\mathrm{atm}}^2$ successfully explains $\Delta m_{\mathrm{atm}}^2\gg \Delta m_{\odot }^2$ by invoking $L_e-L_{\mu }-L_{\tau }$ breaking. The Higgs scalar ${(h}^{+})$ that initiates the radiative mechanisms is unified into a Higgs triplet together with the standard Higgs scalar $({\phi }^{+},{\phi }^0)$ to form $({\phi }^{+},{\phi }^0,h^{+}),$ which calls for three families of lepton triplets: $({\nu }_L^i,l_L^i,{\omega }_L^i)\mathrm{}(i=1,2,3\mathrm{}),$ where ${\omega }^i$ denote heavy neutral leptons. The two-loop radiative mechanism is found possible by introducing a singly charged scalar, which couples to $l_R^i{\omega }_R^j\mathrm{}(i,j=2,3\mathrm{}).$

• Received 3 December 2000

DOI:https://doi.org/10.1103/PhysRevD.63.095006