\section{\label{sec:exp_method} Experimental Method}

\subsection{\label{sec:muon_campus} Polarized Muon Production}
\textcolor{orange}{Diktys}
\medskip

The $\pi^+$ decay to $\mu^+ + \bar\nu_\mu$. The pion has zero spin and the neutrino is always right handed. In the pion rest frame the $\mu^+$ is also right handed. To conserved angular momentum. Boost to lab frame. The muons with momentum in the same direction as the pi in the lab frame will be all right handed. Muons with momentum antiparallel to the pi will end up left handed. And lower momentum. By selecting the highest momentum muons we get a polarized beam, order \SI{95}{\percent}.

\subsection{\label{sec:spin} Spin Precession}
\textcolor{orange}{Jason}
\medskip

\textcolor{red}{\textit{Elaborate on the spin precession.}}
\medskip

Muons decay to $\mu^+\rightarrow e^++\bar\nu_\mu +\nu_e$. The muon spin is correlated with the momentum of the electron in the lab frame. In the muon rest frame the spin is in the z direction. The total final state momentum is the sum of neutrinos and positron. The max momentum of the final state positron is when neutrinos have the same momentum (same direction) and the positron has equal but opposite. Since the spin of the neutrinos is equal but opposite, the positron carries off the angular momentum of the muon. The $e^+$, an anti-lepton is most likely to be right handed in a weak decay, and therefore with momentum in the same direction as that of the muon spin. The positron with minimum momentum in the muon rest frame will be when neutrinos are in opposite directions, with total spin 1 and the positron has near zero momentum and the spin opposite that of the muon. It is as before most likely to be right handed, and therefore to have momentum opposite to the muon spin. The result is that in the lab frame, positron momenta is correlated with muon spin. The measurement is by counting the time dependence of the number of high momentum decay positrons. The experiment is equipped with calorimeters that measure the number vs time of high momentum positrons. The muon has equal but opposite momentum to the neutrinos and necessarily the muon spin. The detector acceptance depends on the transverse position of the decay muon. Therefore acceptance varies with the betatron and dispersive variation of the distribution in phase space of the muon beam.

Momentum of the positron is correlated with the muon spin. Measurement of decay positron momentum as function of time is precession frequency. Focusing is the dipole field and electrostatic quads. So as not to introduce a field variation. But the E-field couples to the magnetic moment anyway. If the dimensionless g-factor of the muon were identically 2, then in any magnetic field the spin will stay synched with the trajectory independent of its momentum. In an electric field, if $g=2$, then at $\beta\ll 1$ there is no effect on the polarization and polarization is fixed independent of trajectory. At $\gamma \gg 1$ the magnetic field in the rest frame of the muon is $B\rightarrow cE$ and the spin is again pinned to the trajectory. Since $g>2$ the spin rotates more rapidly than the trajectory. There is a momentum, the so-called magic momentum, where the spin follows the trajectory. And for the muon that is at $\gamma=29.3$. The spin is frozen to the direction in an electric field. By operating at the magic momentum, the E-field the precession is decoupled from the E-field.

The expression for the anomalous precession frequency, in the limit of vanishing longitudinal magnetic field, is
\begin{align} % {} before the - sign turns it into a binary operator instead of a unary operator (spacing issues)
	\vec{\omega}_{a} &=-\frac{q}{m}\left[a_{\mu}\vec{B}-a_{\mu}\left(\frac{\gamma}{\gamma+1}\right)\left(\vec{\beta}\cdot\vec{B}\right)\right.\nonumber\\
	                             &\qquad\left.{}-\left(a_{\mu}-\frac{1}{\gamma^{2}-1}\right)\frac{\vec{\beta}\times\vec{E}}{c}\right].
	\label{eq:au} % get an error if last line has \nonumber. % manually put in parentheses for inline eq. references.
\end{align}
For muons at magic momentum ($\pmagic=\left.m\middle/\sqrt{a_{\mu}}\right.$) circulating on the design orbit,
\begin{equation}
	\vec{\omega}_{a}=-\frac{q}{m}a_{\mu}\vec{B}.\nonumber
%	\label{eq:au_simple}
\end{equation}

\subsection{\label{sec:bfield} Precision Magnetic Field}
\textcolor{orange}{Jason}
\medskip

\textcolor{red}{\textit{Elaborate on the high precision magnetic field, see Table~\ref{tab:ringparams} and Fig.~\ref{fig:bfield}.}}

\begin{table}[]
\caption{\label{tab:ringparams} \textcolor{orange}{Jason} Muon \gmtwo~storage ring magnet parameters.}
\begin{ruledtabular}
\begin{tabular}{lr}
\textnormal{Parameter}                                    & \textnormal{Value}\\
\colrule
\textnormal{Nominal Magnetic Field}               & \textnormal{\SI{1.4513}{\tesla}}\\
\textnormal{Nominal Current}                           & \textnormal{\SI{5200}{\ampere}}\\
\textnormal{Equilibrium Orbit (Magic) Radius} & \textnormal{\SI{7112}{\milli\meter}}\\
\textnormal{Muon Storage Region Radius}      & \textnormal{\SI{45}{\milli\meter}}\\
\textnormal{Magnet Gap Length}                     & \textnormal{\SI{180}{\milli\meter}}\\
\textnormal{Nominal Stored Energy}                & \textnormal{\SI{6}{\mega\joule}}\\
\end{tabular}
\end{ruledtabular}
\end{table}

\begin{figure}[]
	\centering
	\begin{subfigure}{\columnwidth}
		\begin{tikzpicture}
			\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig01a.pdf}};
			\draw (0, 0) node[rotate=45,opacity=0.4]{{\fontsize{45}{54}\selectfont\textcolor{red}{\textbf{Preliminary}}}}; % rule of thumb: \fontsize{size}{1.2*size}
		\end{tikzpicture}
		%\includegraphics[width=0.95\columnwidth]{fig01a.pdf}
		\caption{Dipole magnetic field.}\label{fig:dipole_bfield}
	\end{subfigure}
	%add desired spacing between images, e. g. ~, \quad, \qquad, \hfill etc (or a blank line to force the subfigure onto a new line).
	\begin{subfigure}{\columnwidth}
		\begin{tikzpicture}
			\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig01b.pdf}};
			\draw (0, 0) node[rotate=45,opacity=0.4]{{\fontsize{45}{54}\selectfont\textcolor{red}{\textbf{Preliminary}}}}; % rule of thumb: \fontsize{size}{1.2*size}
		\end{tikzpicture}
		%\includegraphics[width=0.95\columnwidth]{fig01b.pdf}
		\caption{Radial magnetic field.}\label{fig:radial_bfield}
	\end{subfigure}
	\caption{\textcolor{orange}{Mike} Preliminary storage ring magnetic field data. \textcolor{red}{\textit{Maybe we should show preliminary field data, as it is used to calculate closed orbit distortions.}}}\label{fig:bfield}
\end{figure}
