Three flavor neutrino oscillation analysis of atmospheric neutrinos in Super-Kamiokande

We report on the results of a three-flavor oscillation analysis using Super-Kamiokande I atmospheric neutrino data, with the assumption of one mass scale dominance ($\Delta m_{12}^2=0$). No significant flux change due to matter effect, which occurs when neutrinos propagate inside the Earth for ${\theta }_{13}\ne 0$, has been seen either in a multi-GeV ${\nu }_e$-rich sample or in a ${\nu }_{\mu }$-rich sample. Both normal and inverted mass hierarchy hypotheses are tested and both are consistent with observation. Using Super-Kamiokande data only, 2-dimensional 90% confidence allowed regions are obtained: mixing angles are constrained to ${sin}^2{\theta }_{13}<0.14$ and $0.37<{sin}^2{\theta }_{23}<0.65$ for the normal mass hierarchy. Weaker constraints, ${sin}^2{\theta }_{13}<0.27$ and $0.37<{sin}^2{\theta }_{23}<0.69$, are obtained for the inverted mass hierarchy case.