Neutrino masses and mixings from an SMG times U(1)² model

A natural solution to the fermion mass hierarchy problem suggests the existence of a partially conserved chiral symmetry. We show that this can lead to a reasonably natural solution to the solar and atmospheric neutrino problems without fine-tuning or the addition of new low energy fermions. The atmospheric neutrino atmospheric neutrino anomaly is given by large mixing between $\nu_{\mu}$ and $\nu_{\tau}$, with $\Delta m^2_{atm} \sim 10^{-3} \eV^2$, and the solar neutrino deficit is due to nearly maximal electron neutrino vacuum oscillations. We present an explicit model for the neutrino masses which is an anomaly free Abelian extension of the standard model that also yields a realistic charged fermion spectrum.