Accommodating solar and atmospheric neutrino deficits, hot dark matter, and a double β decay signal

Neutrino mass explanations of the solar and atmospheric neutrino deficits and a hot dark matter component require one of three patterns of those masses, as already pointed out by us. Recently there have been indications of a nonvanishing amplitude for neutrinoless double β decay. If this additional hint of neutrino mass is true, it would make even likely the one unfavored pattern (a sterile neutrino giving warm, rather than hot dark matter), which would alter another by making the ${\nu }_{\mathit{e}}$ a contributor to the hot dark matter, and would make the third (${\nu }_{\mathit{e}}$, ${\nu }_{\mathrm{\mu }}$, and ${\nu }_{\mathrm{\tau }}$ approximately degenerate) much more likely than previously. For this third case we construct a gauge model consistent with other weak interaction data. This model utilizes a more general version of the seesaw mechanism, which is very likely to be the source of neutrino mass, if this degenerate pattern is correct. A new supernova constraint is ulilized, and implications and tests of the different mass matrices are noted.