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$$=$0). No significant flux change due to matter effect, which occurs when neutrinos propagate inside the Earth for $\theta_{13}$$\neq$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.