Radiative Splitting of Three Degenerate Neutrinos

We propose a radiative origin of the two mass splittings of three degenerate Majorana neutrinos. It can be achieved by extending the standard model to have the usual effective dimension 5 operators generating SO(3)-invariant tree-level masses, and a charged scalar singlet coupling with the leptons preserving the U(1) subgroup of SO(3). The mass splittings for the atmospheric and solar neutrino oscillations then arise from one-loop corrections due to the charged scalar singlet coupling and the usual tau Yukawa coupling, respectively.