\section{\label{sec:ring} Storage Ring}
\textcolor{orange}{Renee}
\medskip

Figure~\ref{fig:schematic} shows the layout of the Muon \gmtwo~storage ring. Muons are injected into the circular storage ring, which has a \SI{7.112}{\meter} central radius ($\rmagic$), \SI{44.69}{\meter} circumference ($\cmagic$), and \SI{0.149}{\micro\second} revolution period. A \SI{1.4513}{\tesla} vertical magnetic dipole field~\cite{Grange:2015fou,Danby:2001eh} provides horizontal (radial) focussing, while an electrostatic quadrupole system~\cite{Grange:2015fou,Semertzidis:2003zs,Crnkovic:IPAC2018-WEPAF015} provides vertical focusing. The Muon \gmtwo~storage ring is a weak focusing machine with a single adjustable parameter, namely the electrostatic quadrupole voltage ($V_{Q}$), which determines the horizontal tune ($\nu_{x}$), $\beta$ function ($\beta_{x}$), and dispersion function ($D_{x}$), as well as the vertical tune ($\nu_{y}$), $\beta$ function ($\beta_{y}$), and dispersion function ($D_{y}$).

\begin{figure}[]
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		%\includegraphics[width=0.95\columnwidth]{fig03a.jpeg}
		\caption{\textcolor{orange}{Jason} Picture of the storage ring. \textcolor{red}{\textit{Make a final decision on which picture to use.}}}\label{fig:picture}
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		\caption{\textcolor{orange}{David T.} Diagram of the storage ring. \textcolor{red}{\textit{Finalize diagram with David T. and Leah.}}}\label{fig:diagram}
	\end{subfigure}
	\caption{The muon storage ring used by the Muon \gmtwo~Experiment.}\label{fig:schematic}
\end{figure}

The injected beam enters the storage ring through a hole in the back-leg iron, and it emerges from the back-leg hole and enters into the superconducting inflector magnet~\cite{Grange:2015fou,Yamamoto:2002bb,Froemming:IPAC2018-WEPAF014}. During injection the muons are shielded from the dipole storage field by the inflector, where the double-cosine-theta design minimizes the flux leakage into the storage region. Although the inflector axis is tangent to the central orbit, the exit window is offset by \SI{77}{\milli\meter} from the center of the storage region, resulting in orbits that would circle directly back to the inflector after one revolution. Three pairs of kicker plates~\cite{Grange:2015fou,Schreckenberger:IPAC2018-THPML093}, positioned approximately \SI{90}{\degree} downstream of the inflector exit, create a sharp magnetic pulse (\SI{\sim0.2}{\micro\second}) that serves to kick the muon beam on the order of \SI{10}{\milli\radian}, centering it on the equilibrium orbit. The sub-optimal kick and mismatch between the inflector exit and storage region results in coherent beam motion, see Section~\ref{sec:fit_model}. 

Two techniques are used to quickly remove muons living on the edge of the stored beam phase space distribution. First, the four quadrupole plates are ramped asymmetrically to full voltage. This process, known as scraping, leads to a systematic increase in the average vertical and radial position of the beam. As the beam shifts the muons at the edges of the beam distribution will scatter from one of the five copper collimators distributed around the ring, lose energy and spiral inward and out of the storage region.  By the end of the scraping period the beam storage region is defined by the \SI{90}{\milli\meter} diameter of the collimator aperture.

The positrons from the muon decays spiral inward and deposit their energy in the 24 calorimeters that are evenly spaced (every \SI{15}{\degree}) around the inside of the storage ring, where each calorimeter is made out of a $6\times9$ segmented array of lead(II) fluoride (PbF$_{2}$) crystals~\cite{{Grange:2015fou,Fienberg:2014kka,Kaspar:2016ofv}}. Straw tube tracker modules~\cite{Grange:2015fou} installed in front of 2 calorimeter stations (located \SI{180}{\degree} and \SI{270}{\degree} from the injection point) detect the daughter electron trajectories, allowing for the reconstruction of the vertical and horizontal muon decay distributions. Muons injected on the central radius with precisely the magic momentum ($\pmagic=\SI[per-mode=symbol]{3.094}{\giga\eV\per\clight}$) and no transverse momentum will simply follow the circular design orbit. In reality, the beam has a momentum width of $\sim\delta{p}/{p}=\SI{0.1}{\percent}$ and therefore a distribution of muon orbit radii. The muons with paths deviating from the magic radius are perturbed by the electric field gradient, inducing radial and vertical betatron oscillations that depend on the electric potential of the quadrupole plates.  

Several detector packages were installed to measure beam injection parameters and oscillations in the storage ring. As the muons exit the M5-line final focus upstream of the ring, they pass through two scintillator paddles used to register the time distribution of the incoming beam pulses. The relative intensities of the eight beam pulses are directly related to the integral of the pulse from $T_{0}$ detector~\cite{T0}. The phase space of the injected beam is measured by the inflector beam monitoring system (IBMS)~\cite{IBMS}. The IBMS is composed of three sets of scintillating fiber planes. The two sets placed upstream of the inflector are composed of a horizontal and a vertically strung plane,  allowing for real time feedback while tuning the beam position and momentum. The third set has only a single vertically strung plane that is inserted downstream of the inflector and directly upstream of the first calorimeter after the inflector exit. The third IBMS detector is retracted during data-taking so that the wires and frame do not destructively interfere with the beam.

Two sets of retractable fiber harp beam monitors~\cite{Bennett:2006fi,Grange:2015fou} are located \SI{180}{\degree} and \SI{270}{\degree} downstream of the inflector exit. Each set has a horizontal and vertical plane constructed from seven parallel \SI{0.5}{\milli\meter} diameter scintillating fibers with \SI{13}{\milli\meter} spacing. The fiber harps directly measure the beam $x$-$y$ beam motion after injection and through the period of kicker pulsing and quadrupole scraping. They are also used to measure the CBO oscillations of the beam centroid position and width at early times in the muon storage fill, see Figs.~\ref{fig:fibercentroid} and~\ref{fig:fiberwidth}. Ultimately the fiber harps destructively interfere with the beam, and so they are retracted during production running. The beam characteristics are measured by the straw tracker detectors during production running, see Fig.~\ref{fig:trackercentroid}. 

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	\caption{\textcolor{orange}{Dat} Horizontal position of centroid measured with the \SI{180}{\degree} fiber harp (blue) and \SI{270}{\degree} harp (red) for the first \SI{30}{\micro\second} of the fill. A discrete Fourier transform yields the horizontal tune. \textcolor{red}{\textit{Check with fiber harp experts.}}}\label{fig:fibercentroid}
\end{figure}

\begin{figure}[]
	\centering
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	\caption{\textcolor{orange}{Dat} Horizontal width at \SI{180}{\degree} harp (blue) and \SI{270}{\degree} harp (red) over the first \SI{30}{\micro\second} of the fill. The deep modulation is a combination of $\beta$ and $D$ mismatch. \textcolor{red}{\textit{Check with fiber harp experts.}}}\label{fig:fiberwidth}
\end{figure}

\begin{figure}[]
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	\caption{\textcolor{orange}{David R.} Radial centroid as measured with the \SI{180}{\degree} tracker over the first \SI{220}{\micro\second} of the fill. Red curve is a fitted damped sinusoid. Measurement is increasingly noisy at longer times as statistics are limited. \textcolor{red}{\textit{Check with tracker experts.}}}\label{fig:trackercentroid}
\end{figure}
