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 (L) over bar =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 (L) over bar 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 (L) over bar breaking leads exclusively to a non-zero ee matrix element.