Vacuum solutions of neutrino anomalies through a softly broken U(1) symmetry

We discuss an extended \(SU(2)\times U(1)\) 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 the highly suppressed radiative mass scale \(\Delta_S\sim 10^{-10}\,{\rm eV}^2\) 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. The possible embedding of the model into an SU(5) grand unified theory is discussed.