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: $\left(\Delta m^2{)}_{\mathrm{sol}}\right(\Delta m^2{)}_{\mathrm{atm}}/m_{\nu }^4({sin}^{2}2\theta {)}_{\mathrm{atm}}{=2I}^2$, where $m_{\nu }$ is the assumed common approximate mass of each neutrino (which may be suitable for hot dark matter) and $I=\left(3/16{\pi }^2\right)\left(G_F/\sqrt{2}\right)m_{\tau }^2\ln \left(m_{\xi }^2/m_W^2\right)$ comes from the radiative splitting of the degeneracy due to the charged leptons.

• Received 18 February 1999

DOI:https://doi.org/10.1103/PhysRevLett.83.2514