Vacuum Neutrino Oscillations of Solar Neutrinos and Lepton Mass Matrices
  We consider the case that the solar neutrino deficit is due to the vacuum oscillation. The lepton mass matrices with nearly bi-maximal mixings are needed in order to explain both solar and atmospheric neutrino deficit. A texture with the symmetry of flavour democracy or $S_3$ has been investigated by taking account of the symmetry breaking terms of the charged lepton mass matrix. It is found that predicted mixings can be considerably changed from the neutrino mixings $\sin^2 2\th_\odot\simeq 1$ and   $\sin^2 2\th_{\rm atm}\simeq 8/9$ at the symmetric limit. The correlation between $|U_{e3}|$ and $|U_{e1}U_{e2}^*|$ is also presented. The test of the model is discussed by focusing on the three flavor analyses in the solar neutrinos, atmospheric neutrinos and long baseline experiments. 
Morimitsu Tanimoto