Theoretical constraints on the vacuum oscillation solution to the solar neutrino problem

The vacuum oscillation (VO) solution to the solar anomaly requires an extremely small neutrino mass splitting, Δm_sol² ≲ 10^-10 eV². We study under which circumstances this small splitting (whatever its origin) is or is not spoiled by radiative corrections. The results depend dramatically on the type of neutrino spectrum. If m₁ ² ∼ m₂ ², ≳ m₃ ², radiative corrections always induce too large mass splittings. Moreover, if m₁ and m₂ have equal signs, the solar mixing angle is driven by the renormalization group evolution to very small values, incompatible with the VO scenario (however, the results could be consistent with the small-angle MSW scenario). If m₁ and m₂ have opposite signs, the results are analogous, except for some small (though interesting) windows in which the VO solution may be natural with moderate fine-tuning. Finally, for a hierarchical spectrum of neutrinos, m₁ ² ≪ m₂ ² ≪ m₃ ², radiative corrections are not dangerous, and therefore this scenario is the only plausible one for the VO solution.