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The confirmed probability of a muon neutrino to "oscillate" into an electron neutrino is given by a function of the form,  \begin{align} \begin{equation}  P(\nu_{\mu} \rightarrow \nu_{e}) &= =  f \left(\sin^2 \theta_{23} \sin^2 2\theta_{13},\\  &  \frac{\Delta_{31}}{\Delta_{31} \mp a L} \sin(\Delta_{31} \mp aL), \sin \delta_{CP}\right) \end{align}  where $i$ and $j$ carry values 1,2,3 and stand for electron, muon, and tau quantities respectively; $\theta_{ij}$ is the "mixing angle" between the various flavors; $\Delta_{ij} = \Delta m_{ij}^2 L / 4E $; L is the length over which the oscillations occur; $a = G_F N_e \sqrt{2} \simeq (4000 \rm{km})^{-1}$; the $\delta_{CP}$ is a measure of the amount by which charge and parity conversation are allowed to be violated; the $\mp$ correspond to neutrinos and antineutrinos respectively. It is important to note that the aL term comes about because of the initial momentum of the neutrinos in the beam and allows for close study of the mass hierarchy since a larger L gives a bigger probability for muon antineutrinos to oscillate \cite{Paley_2012}.