Comparing and contrasting the $ u_{mu} to u_{tau}$ and $ u_{mu} to u_s$ solutions to the atmospheric neutrino problem with SuperKamiokande data

The $ u_{mu} to u_{tau}$ and $ u_{mu} to u_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 $3sigma$ 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 $(sin^2 2theta,Delta m^2)$ for the $ u_{mu} to u_{tau}$ and $ u_{mu} to u_s$ solutions, respectively. Values of $Delta m^2$ smaller than about $2 times 10^{-3}$ eV$^2$ are disfavoured for the $ u_{mu} to u_s$ solution, suggesting that future long baseline experiments should see a positive signal if this scenario is the correct one.