Mass matrix for atmospheric, solar, and LSND neutrino oscillations

Abstract

We construct a mass matrix for the four neutrino flavors, three active and one sterile, needed to fit oscillations in all three neutrino experiments: atmospheric, solar, and LSND, simultaneously. It organizes the neutrinos into two doublets whose central values are about $1 \mathrm{eV}$ apart, and whose splittings are of the order of ${10}^{\ensuremath{-}3} \mathrm{eV}.$ Atmospheric neutrino oscillations are described as maximal mixing within the upper doublet, and solar as the same within the lower doublet. Then LSND is a weak transition from one doublet to the other. We comment on the Majorana versus Dirac nature of the active neutrinos and show that our mass matrix can be derived from an ${S}_{2}\ifmmode\times\else\texttimes\fi{}{S}_{2}$ permutation symmetry plus an equal splitting rule.