UntanglingCPviolation and the mass hierarchy in long baseline experiments

In the overlap region, for the normal and inverted hierarchies, of the neutrino-antineutrino bi-probability space for ${\nu }_{\mu }\to {\nu }_e$ appearance, we derive a simple identity between the solutions in the (${sin}^{2}2{\theta }_{13}$, $\sin \delta$) plane for the different hierarchies. The parameter ${sin}^{2}2{\theta }_{13}$ sets the scale of the ${\nu }_{\mu }\to {\nu }_e$ appearance probabilities at the atmospheric $\delta m_{\mathrm{a}\mathrm{t}\mathrm{m}}^2\approx 2.4\times {10}^{-3}{\mathrm{e}\mathrm{V}}^2$ whereas $\sin \delta$ controls the amount of $CP$ violation in the lepton sector. The identity between the solutions is that the difference in the values of $\sin \delta$ for the two hierarchies equals twice the value of $\sqrt{{sin}^{2}2{\theta }_{13}}$ divided by the critical value of $\sqrt{{sin}^{2}2{\theta }_{13}}$. We apply this identity to the two proposed long baseline experiments, T2K and NOvA, 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. This identity demonstrates the true complementarity of T2K and NOvA.