Four Light Neutrinos in Singular Seesaw Mechanism with Abelian Flavor Symmetry

The four light neutrino scenario, which explains the atmosphere, solar and LSND neutrino experiments, is studied in the framework of the seesaw mechanism. By taking both the Dirac and Majorana mass matrix of neutrinos to be singular, the four neutrino mass spectrum consisting of two almost degenerate pairs separated by a mass gap $\sim 1$ eV is naturally generated. Moreover the right-handed neutrino Majorana mass can be at $\sim 10^{14}$ GeV scale unlike in the usual singular seesaw mechanism. Abelian flavor symmetry is used to produce the required neutrino mass pattern. A specific example of the flavor charge assignment is provided to show that maximal mixings between the $\nu_\mu-\nu_\tau$ and $\nu_e-\nu_s$ are respectively attributed to the atmosphere and solar neutrino anomalies while small mixing between two pairs to the LSND results. The implication in the other fermion masses is also discussed.