Implications of the Super-K atmospheric data for the mixing angles $θ_{13}$ and $θ_{23}$

A three-neutrino analysis of oscillation data is performed using the recent, more finely binned Super-K oscillation data, together with the CHOOZ, K2K, and MINOS data. The solar parameters, $Δ_{21}$ and $θ_{12}$, are fixed from a recent analysis, and $Δ_{32}$, $θ_{13}$, and $θ_{23}$ are varied. We utilize the full three-neutrino oscillation probability and an exact treatment of the Earth's MSW effect with a castle-wall density. By including terms linear in $θ_{13}$ and $\varepsilon := θ_{23} - π/4$, we find asymmetric errors for these parameters $θ_{13}=-0.07^{+0.18}_{-0.11}$ and $\varepsilon=0.03^{+0.09}_{-0.15}$. For $θ_{13}$, we see that the lower bound is primarily set by the CHOOZ experiment while the upper bound is determined by the low energy $e$-like events in the Super-K atmospheric data. We find that the parameters $θ_{13}$ and $\varepsilon$ are correlated -- the preferred negative value of $θ_{13}$ permits the preferred value of $θ_{23}$ to be in the second octant, and the true value of $θ_{13}$ affects the allowed region for $θ_{23}$.

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