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Interplay between Appearance and Disappearance Channels for Precision Measurements of $θ_{23}$ and $δ$

We discuss how the CP violating phase $δ$ and the mixing angle $θ_{23}$ can be measured precisely in an environment where there are strong correlations between them. This is achieved by paying special attention to the mutual roles and the interplay between the appearance and the disappearance channels in long-baseline neutrino oscillation experiments. We analyze and clarify the general structure of the $θ_{23} - θ_{13} - δ$ degeneracy for both the appearance and disappearance channels in a more complete fashion than what has previously been discussed in the literature. A full understanding of this degeneracy is of vital importance if $θ_{23}$ is close to maximal mixing. The relative importance between the appearance and disappearance channels depends upon the particular setup and how close to maximal mixing Nature has chosen the value for $θ_{23}$. For facilities that operate with a narrow band beam or a wide band beam centered on the first oscillation extremum, the contribution of the disappearance channel depends critically on the systematic uncertainties assumed for this channel. Whereas for facilities that operate at energies above the first oscillation extremum or at the second oscillation extremum the appearance channels dominate. On the other hand, for $δ$ we find that the disappearance channel usually improves the sensitivity, modestly for facilities around the first oscillation extremum and more significantly for facilities operating at an energy above the first oscillation extremum, especially near $δ\sim \pm π/2$.