Understanding CP phase-dependent measurements at neutrino superbeams in terms of bi-rate graphs

We discuss the impact of the true value of the CP phase on the mass hierarchy, CP violation, and CP precision measurements at neutrino superbeams and related experiments. We we use a complete statistical experiment simulation including spectral information, systematics, correlations, and degeneracies to produce the results. However, since it is very complicated to understand the results in terms of a complete experiment simulation, we show the corresponding bi-rate graphs as useful tools to investigate the CP phase-dependencies qualitatively. Unlike bi-probability graphs, which are based upon oscillation probabilities, bi-rate graphs use the total event rates of two measurements simultaneously as a function of the CP phase. Since they allow error bars for direct quantitative estimates, they can be used for a direct comparison with a complete statistical experiment simulation. We find that one can describe the CP phase dependencies of the mentioned measurements at neutrino superbeam setups, as well as one can understand the role of the $\mathrm{sgn} (\Delta m_{31}^2)$-degeneracy. As one of the most interesting results, we discuss the dependence of the CP precision measurement as a function of the CP phase itself, which leads to ``CP patterns''. It turns out that this dependence is rather strong, which means that one has to be careful when one is comparing the CP precisions of different experiments.