Mass Splitting of Three Seesaw Neutrinos

If a family symmetry exists so that all three neutrinos have equal Majorana masses (via the seesaw mechanism), then the breaking of this symmetry from charged-lepton masses (via two-loop double $W$ exchange) implies $\Delta m$ less than about $10^{-9}m_0$. With the common mass $m_0 \sim$ eV for hot dark matter, $\Delta m^2 \sim 10^{-10}$ eV$^2$ is natural for vacuum solar neutrino oscillations.