A Natural Solution to the Neutrino Mixing Problem

The combined requirements, of (i) a natural solution to the fermion mass hierarchy problem and (ii) an explanation of both the atmospheric and solar neutrino problems, lead to an essentially unique picture of neutrino masses and mixing angles. The electron and muon neutrinos are quasi-degenerate in mass with maximal mixing, giving $\nu_e - \nu_{\mu}$ vacuum oscillations. The overall neutrino mass scale is set by the atmospheric neutrino requirement $\Delta m^2 \sim 10^{-2}$ eV$^2$, implying a mass for $\nu_e$ and $\nu_{\mu}$ of order 1 eV in models with a natural mass hierarchy, whilst the tau neutrino is expected to be much lighter than this and only weakly mixed. We present an explicit example based on the anti-grand unification model of fermion masses.