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 normalisation and detection cross-sections and efficiencies. A ${\chi }^2$ analysis using these quantities is performed yielding $3\sigma$ ranges which are approximately given by $(0.725 - 1.0, 4 \times 10^{-4} - 2 \times 10^{-2} eV^2)$and$(0.74 - 1.0, 1 \times 10^{-3} - 2 \times 10^{-2} eV^2)$for$(\sin^2 2\theta,\Delta m^2)$forthe$\nu_{\mu} \to \nu_{\tau}$and$\nu_{\mu} \to \nu_s$solutions,respectively\mathrm{.}Valuesof$\Delta m^2$smallerthanabout$2 \times 10^{-3}$eV$^2$aredisfavouredforthe$\nu_{\mu} \to \nu_s$ solution, suggesting that future long baseline experiments should see a positive signal if this scenario is the correct one.