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Alec Aivazis edited kinematics.tex
almost 10 years ago
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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
...
\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}