Breaking eightfold degeneracies in neutrinoCPviolation, mixing, and mass hierarchy

We identify three independent twofold parameter degeneracies $(\delta ,{\theta }_{13}),$ $\mathrm{sgn}\left(\delta m_{31}^2\right)$ and $({\theta }_{23},\pi /2\mathrm{}-{\theta }_{23})$ inherent in the usual three-neutrino analysis of long-baseline neutrino experiments, which can lead to as much as an eightfold degeneracy in the determination of the oscillation parameters. We discuss the implications these degeneracies have for detecting $\mathrm{CP}$ violation and present criteria for breaking them. A superbeam facility with a baseline at least as long as the distance between Fermilab and Homestake (1290 km) and a narrow band beam with energy tuned so that the measurements are performed at the first oscillation peak can resolve all the ambiguities other than the $({\theta }_{23},\pi /2\mathrm{}-{\theta }_{23})$ ambiguity (which can be resolved at a neutrino factory) and a residual $(\delta ,\pi -\delta )$ ambiguity. However, whether or not $\mathrm{CP}$ violation occurs in the neutrino sector can be ascertained independently of the latter two ambiguities. The $(\delta ,\pi -\delta )$ ambiguity can be eliminated by performing a second measurement to which only the $\cos \delta$ terms contribute. The hierarchy of mass eigenstates can be determined at other oscillation peaks only in the most optimistic conditions, making it necessary to use the first oscillation maximum. We show that the degeneracies may severely compromise the ability of the proposed SuperJHF-HyperKamiokande experiment to establish $\mathrm{CP}$ violation. In our calculations we use approximate analytic expressions for oscillation probabilitites that agree with numerical solutions with a realistic Earth density profile.