The Singular Seesaw Mechanism with Hierarchical Dirac Neutrino Mass

The singular seesaw mechanism can naturally explain the atmospheric neutrino deficit by the maximal oscillation between $\nu_{\mu_L}$ and $\nu_{\mu_R}$. This mechanism can also induce three different scales of neutrino mass squared differences, which can explain the neutrino deficits of three independent experiments (solar, atmospheric, and LSND) by neutrino oscillations. In this paper we show that the realistic mixing angles among neutrinos can be obtained by introducing the hierarchy in the Dirac neutrino mass. In the case where Majorana neutrino mass matrix has rank 2, the solar neutrino deficit is explained by the vacuum oscillation between $\nu_e$ and $\nu_\tau$. We also consider the case where Majorana neutrino mass matrix has rank 1. In this case, the mater enhanced Mikheyev-Smirnov-Wolfenstein solar neutrino solution is prefered as the solution of the solar neutrino deficit.