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

The vacuum oscillation (VO) solution to the solar anomaly requires an extre mely small neutrino mass splitting, Delta m(sol)(2) less than or similar to 10(-10) eV(2). We study under which circumstances this small splitting (wh atever its origin) is or is not spoiled by radiative corrections. The resul ts depend dramatically on the type of neutrino spectrum. If m(1)(2) similar to m(2)(2) greater than or similar to m(3)(2), radiative corrections alway s induce too large mass splittings. Moreover, if m(1) and m(2) have equal s igns, the solar mixing angle is driven by the renormalization group evoluti on to very small values, incompatible with the VO scenario (however, the re sults could be consistent with the small-angle MSW scenario). If m1 and m2 have opposite signs, the results are analogous, except for some small (thou gh interesting) windows in which the VO solution may be natural with modera te fine-tuning. Finally, for a hierarchical spectrum of neutrinos, m(1)(2) << m(2)(2) << m(3)(2), radiative corrections are not dangerous, and therefo re this scenario is the only plausible one for the VO solution.