Inverted Hierarchy Models of Neutrino Masses
  We study models of neutrino masses which naturally give rise to an inverted mass hierarchy and bi-maximal mixing. The models are based on the see-saw mechanism with three right-handed neutrinos, which generates a single mass term of the form $\nu_e(\nu_{\mu}+\nu_{\tau})$ corresponding to two degenerate neutrinos $\nu_e$ and $\nu_{\mu}+\nu_{\tau}$, and one massless neutrino $\nu_{\mu}-\nu_{\tau}$. Atmospheric neutrino oscillations are accounted for if the degenerate mass term is about $5\times 10^{-2}$ eV. Solar neutrino oscillations of the Large Mixing Angle MSW type arise when small perturbations are included leading to a mass splitting between the degenerate pair of about $(1.7-2.0)\times 10^{-4}$ eV for the successful cases. We study the conditions that such models must satisfy in the framework of a U(1) family symmetry broken by vector singlets, and catalogue the simplest examples. We then perform a renormalisation group analysis of the neutrino masses and mixing angles, assuming the supersymmetric standard model, and find modest radiative corrections of a few per cent, showing that the model is stable. At low energies we find $\sin^22\theta_{23}\approx 0.93-0.96$ and $\sin^22\theta_{12}\approx 0.9-1.0$. 
S. F. King and N. Nimai Singh