Neutrino mass matrix with the seesaw mechanism and two-loop mass splitting

We propose a model which uses the seesaw mechanism and the lepton number $\overline{L}{=L}_e-L_{\mu }-L_{\tau }$ to achieve the neutrino mass spectrum $m_1{=m}_2$ and $m_3=0,$ together with a lepton mixing matrix U with $U_{e3\mathrm{}}=0.\mathrm{}$ In this way, we accommodate atmospheric neutrino oscillations. A small mass splitting $m_1>m_2$ is generated by breaking $\overline{L}$ spontaneously and using Babu’s two-loop mechanism. This allows us to incorporate either the “just so” or the LOW solution of the solar-neutrino problem, with maximal mixing, into the model. The resulting mass matrix has three parameters only, since $\overline{L}$ breaking leads exclusively to a non-zero $\mathrm{ee}$ matrix element.