Sterile Neutrinos in E6 and a Natural Understanding of Vacuum Oscillation Solution to the Solar Neutrino Puzzle

If Nature has chosen the vacuum oscillation solution to the Solar neutrino puzzle, a key theoretical challenge is to understand the extreme smallness of the ∆m2νe−νX (∼ 10 −10 eV2) required for the purpose. We find that in a class of models such as [SU(3)]3 or its parent group E6, which contain one sterile neutrino, νis for each family, the ∆m 2 νi−νis is proportional to the cube of the lepton Yukawa coupling. Therefore fitting the atmospheric neutrino data then predicts the νe−νes mass difference square to be ∼ ( me mμ )3 ∆matmos, where the atmospheric neutrino data is assumed to be solved via the νμ−νμs oscillation. This provides a natural explanation of the vacuum oscillation solution to the solar neutrino problem.