Untangling CP Violation and the Mass Hierarchy in Long Baseline Experiments

Abstract

In the overlap region, for the normal and inverted hierarchies, of the neutrino-antineutrino bi-probability space for $ν_μ\to ν_e$ appearance, we derive a simple identity between the solutions in the ($\sin^2 2θ_{13}$, $\sin δ$) plane for the different hierarchies. The parameter $\sin^2 2θ_{13}$ sets the scale of the $ν_μ\to ν_e$ appearance probabilities at the atmospheric $δm^2_{atm} \approx 2.4 \times 10^{-3}$ eV$^2$ whereas $\sin δ$ controls the amount of CP violation in the lepton sector. The identity between the solutions is that the difference in the values of $\sin δ$ for the two hierarchies equals twice the value of $\sqrt{\sin^2 2θ_{13}}$ divided by the {\it critical} value of $\sqrt{\sin^2 2θ_{13}}$. We apply this identity to the two proposed long baseline experiments, T2K and NO$ν$A, and we show how it can be used to provide a simple understanding of when and why fake solutions are excluded when two or more experiments are combined. The identity demonstrates the true complimentarity of T2K and NO$ν$A.