A neutrino mass matrix with seesaw mechanism and two-loop mass splitting

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