Vacuum Solutions of Neutrino Anomalies Through a Softly Broken U(1)   symmetry
  We discuss an extended SU(2)XU(1) model which naturally leads to mass scales and mixing angles relevant for understanding both the solar and atmospheric neutrino anomalies in terms of the vacuum oscillations of the three known neutrinos. The model uses a softly broken L_e-L_{\mu}-L_{\tau} symmetry and contains a heavy scale M_H\sim 10^{15} GeV. The L_e-L_{\mu}-L_{\tau} symmetric neutrino masses solve the atmospheric neutrino anomaly while breaking of L_e-L_{\mu}-L_{\tau} generates highly suppressed radiative mass scale \Delta_S\sim 10^{-10} eV^2 needed for the vacuum solution of the solar neutrino problem. All the neutrino masses in the model are inversely related to M_H, thus providing seesaw-type of masses without invoking any heavy right-handed neutrinos. Possible embedding of the model into an SU(5) grand unified theory is discussed. 
Anjan S. Joshipura and Saurabh D. Rindani