Bilarge neutrino mixing andμ−τpermutation symmetry for two-loop radiative mechanism

The presence of approximate electron number conservation and $\mu -\tau$ permutation symmetry of $S_2$ is shown to naturally provide bilarge neutrino mixing. First, the bimaximal neutrino mixing together with $U_{e3\mathrm{}}=0\mathrm{}$ is guaranteed to appear owing to $S_2,$ and then, the bilarge neutrino mixing together with $|U_{e3\mathrm{}}|\ll 1$ arises as a result of a tiny violation of $S_2.$ The observed mass hierarchy of $\Delta m_{\odot }^2$ $\ll$ $\Delta m_{\mathrm{atm}}^2$ is subject to another tiny violation of the electron number conservation. This scenario is realized in a specific model based on ${\mathrm{SU}\left(3\right)\mathrm{}}_L\times {U\left(1\right)\mathrm{}}_N$ with two-loop radiative mechanism for neutrino masses. The radiative effects from heavy leptons contained in lepton triplets generate the bimaximal structure, and those from charged leptons, which break $S_2,$ generate the bilarge structure together with $|U_{e3\mathrm{}}|\ll 1.$ To suppress dangerous flavor-changing neutral current interactions due to Higgs exchanges especially for quarks, this $S_2$ symmetry is extended to a discrete $Z_8$ symmetry, which also ensures the absence of a one-loop radiative mechanism.