Simple Connection Between Atmospheric and Solar Neutrino Vacuum Oscillations

Extending the minimal standard model of particle interactions (without right-handed singlet neutrinos) to include a heavy scalar triplet $\xi$ to obtain nonzero Majorana neutrino masses, I derive the following simple realistic connection between atmospheric and solar neutrino vacuum oscillations: $(\Delta m^2)_{sol} (\Delta m^2)_{atm} / m_\nu^4 (\sin^2 2 \theta)_{atm} = 2 I^2$, where $m_\nu$ is the assumed common approximate mass of each neutrino (which may be suitable for hot dark matter) and $I = G_F m_\tau^2 \ln (m_\xi^2/m_W^2) / 4 \pi^2 \sqrt 2$ comes from the radiative splitting of the degeneracy due to the charged leptons.