deletions | additions       diff --git a/kinematics.tex b/kinematics.tex  index 1a264a1..47506b8 100644  --- a/kinematics.tex  +++ b/kinematics.tex 
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\subsection{Kinematics of an Off-Axis Neutrino Beam}  The relevant decay for the NuMI beam is  \begin{equation}  \pi^{+} \rightarrow \mu^{+} \nu_{\mu}  \end{equation}  Which must conserved energy and momentum according to the 4 vector equation:  \begin{equation} \label{eq:energy-momentum}  \mathbf{\pi^{+}} = \mathbf{\mu^{+}} + \mathbf{\nu_{\mu}}  \end{equation}  Where $\mathbf{\pi^{+}}$, $\mathbf{\mu^{+}}$, and $\mathbf{\nu_{\mu}}$ are the energy-momentum 4-vectors.  Rearranging equation \ref{eq:energy-momentum} as  \begin{equation}  \mathbf{\mu^{+}} = \mathbf{\pi^{+}} - \mathbf{\nu_{\mu}}  \end{equation}  And squaring both sides, we get  \begin{equation} \label{eq:energy-momentum-expanded}  \mathbf{\mu^{+}}^2 = \mathbf{\pi^{+}}^2 - 2 \mathbf{\pi^{+}} \cdot \mathbf{\nu_{\mu}} - \mathbf{\nu_{\mu}}^2  \end{equation}  It is important to note that since the magnitude of a 4-vector is a Lorenz invariant, for any energy-momentum 4 vector, $\mathbf{p}$, where 
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\implies &\mathbf{\nu_{\mu}}^2 \approx 0  \end{split}  \end{equation}  The relevant decay for the NuMI beam is  \begin{equation}  \pi^{+} \rightarrow \mu^{+} \nu_{\mu}  \end{equation}  Which must conserved energy and momentum according to the 4 vector equation:  \begin{equation} \label{eq:energy-momentum}  \mathbf{\pi^{+}} = \mathbf{\mu^{+}} + \mathbf{\nu_{\mu}}  \end{equation}  Where $\mathbf{\pi^{+}}$, $\mathbf{\mu^{+}}$, and $\mathbf{\nu_{\mu}}$ are the energy-momentum 4-vectors.  Rearranging equation \ref{eq:energy-momentum} as  \begin{equation}  \mathbf{\mu^{+}} = \mathbf{\pi^{+}} - \mathbf{\nu_{\mu}}  \end{equation}  And squaring both sides, we get  \begin{equation} \label{eq:energy-momentum-expanded}  \mathbf{\mu^{+}}^2 = \mathbf{\pi^{+}}^2 - 2 \mathbf{\pi^{+}} \cdot \mathbf{\nu_{\mu}} - \mathbf{\nu_{\mu}}^2  \end{equation}