Comparing and contrasting the ${\nu }_{\mu }\to {\nu }_{\tau }$ and ${\nu }_{\mu }\to {\nu }_s$ solutions to the atmospheric neutrino problem with SuperKamiokande data

The ${\nu }_{\mu }\to {\nu }_{\tau }$ and ${\nu }_{\mu }\to {\nu }_s$ solutions to the atmospheric neutrino problem are compared with SuperKamiokande data. The differences between these solutions due to matter effects in the Earth are calculated for the ratio of $\mu$-like to $e$-like events and for up-down flux asymmetries. These quantities are chosen because they are relatively insensitive to theoretical uncertainties in the overall neutrino flux normalization and detection cross sections and efficiencies. A ${\chi }^2$ analysis using these quantities is performed yielding $3\sigma$ ranges which are approximately given by (0.725–1.0, $4\times {10}^{-4}$–$2\times {10}^{-2}$ ${\mathrm{eV}}^2)$ and (0.74–1.0, $1\times {10}^{-3}$–$2\times {10}^{-2}$ ${\mathrm{eV}}^2)$ for $({\sin }^{2}2\theta ,\Delta m^2)$ for the ${\nu }_{\mu }\to {\nu }_{\tau }$ and ${\nu }_{\mu }\to {\nu }_s$ solutions, respectively. Values of $\Delta m^2$ smaller than about $2\times {10}^{-3}$ ${\mathrm{eV}}^2$ are disfavored for the ${\nu }_{\mu }\to {\nu }_s$ solution, suggesting that future long baseline experiments should see a positive signal if this scenario is the correct one.