\section{\label{sec:system} Systematics}

\subsection{\label{sec:fit_model} Detector Acceptance \& Coherent Beam Motion}
\textcolor{orange}{Jason}
\medskip

The acceptance of the calorimeters for the decay positron depends on the radial position of the parent muon. The acceptance therefore varies as the centroid of the distribution oscillates and the width is modulated over the course of the fill. The coherent motion of the centroid, that results from errors in injection angle and kicker field, and the coherent modulation of the beam width, arising from the phase space mismatch at injection, are characterized by decoherence times. (The coherent motion and modulation early in the fill is evident for example in Figs.~\ref{fig:fibercentroid} and~\ref{fig:fiberwidth}, and the decoherence at later times in Fig.~\ref{fig:trackercentroid}). The beam decoheres because of the amplitude and momentum dependence of tunes associated with quadrupole nonlinearity (discussed above) and the chromaticity. But the leading source of decoherence is the large momentum spread and the strong dependence of path length on momentum. The change in circumference due to fractional momentum offset is
\begin{equation}
	\frac{\Delta C}{C}\sim\frac{\eta}{R}\frac{\Delta p}{p}\Rightarrow\frac{\Delta p}{p}\nonumber
%	\label{eq:rel_delta_c}
\end{equation}
in the weak focusing limit. The momentum acceptance of the ring is $\left.\Delta p\middle/p\right.\sim\SI{\pm0.125}{\percent}$. The distribution will lap itself in as few as four hundred turns or \SI{60}{\micro\second}, effectively mixing the distribution. Indeed a measure of the decoherence time is an indication of the width of the momentum distribution. 

\medskip
\textcolor{red}{\textit{Elaborate on the CBO...}}
\medskip

\textcolor{red}{\textit{Talk about beam frequencies getting into the wiggle plots, see Table~\ref{tab:beamfreq} and Fig.~\ref{fig:residfft}.}}

\begin{table*}[]
\caption{\label{tab:beamfreq} \textcolor{orange}{Jason} Beam frequencies that can affect the decay positron spectra via detector acceptance for $n=0.137$ \textcolor{red}{\textit{[update with $V_{\textnormal{quad}}=\SI{18.3}{\kilo\volt}$ values]}}.}
\begin{ruledtabular}
\begin{tabular}{lcccc}
\textnormal{Frequency}                               & \textnormal{Variable}   & \textnormal{Expression}                                                                               & \textnormal{Value~[\SI{}{\mega\hertz}]} & \textnormal{Corresponding Period~[\SI{}{\micro\second}]}\\
\colrule
\textnormal{Anomalous Precession}           & $f_{a}$                         & $\left.\left(ea_{\mu}\dipbfield\right)\middle/\left(2\pi m_{\mu}\right)\right.$   & 0.23                                                        & 4.37\\
\textnormal{Cyclotron}                                 & $f_{c}$                         & $\left.\pmagic\middle/\left(2\pi m_{\mu}\gammamagic\rmagic\right)\right.$ & 6.71                                                        & 0.149\\
\textnormal{Horizontal Betatron}                 & $f_{x}$                         & $\nu_{x}f_{c}$                                                                                               & 6.23                                                        & 0.160\\
\textnormal{Vertical Betatron}                      & $f_{y}$                         & $\nu_{y}f_{c}$                                                                                              & 2.48                                                        & 0.402\\
\textnormal{Coherent Betatron Oscillation} & $f_{\textnormal{cbo}}$ & $f_{c}-f_{x}$                                                                                                 & 0.48                                                        & 2.10\\
\textnormal{Vertical Waist}                           & $f_{\textnormal{vw}}$  & $f_{c}-2f_{y}$                                                                                               & 1.74                                                        & 0.57\\
\end{tabular}
\end{ruledtabular}
\end{table*}

\begin{figure}[]
	\centering
	\includegraphics[width=0.95\columnwidth]{fig22}
	\caption{\textcolor{orange}{Jason} FFT of 5 parameter fit residuals for a decay positron time spectrum.}\label{fig:residfft}
\end{figure}

\subsection{\label{sec:pitch}Pitch Correction}
\textcolor{orange}{David R.}
\medskip
\vskip 0.1in
\input{tex/PitchCorrection-dlr}

\subsection{\label{sec:efield} E-field Correction}
\textcolor{orange}{David R.}
\medskip
\input{tex/EfieldCorrection-dlr}

\subsection{\label{sec:early-to-late} Early-to-late Effects}
\textcolor{orange}{Bill}
\medskip

The clock starts at $t_{0}$. The muons enter the ring with some distribution of times with respect to $t_{0}$. We measure the polarization as a function of time. If we measure polarization at time $t$, polarization of muons that entered the ring at the head of the muon pulse (that enter the ring at times $t_{ini}<t_0$ will have precessed a bit more than those at the tail of the bunch. The difference in their precession will correspond to a phase shift. What is fitted is the oscillation of the distribution. If different parts of the distribution have different phase it will not effect the fitted frequency. But suppose the particles at the head and tail of the pulse have different energies. High momentum muons survive longer than low momentum muons. Now the average precession phase of the distribution will vary with time, introducing a systematic error in the measurement of the frequency. There are a couple of mechanisms for introducing an early to late momentum dependence. Energy acceptance depends on kicker field. And the kicker field is not uniform over the length of the bunch. That gives us a correlation with energy along the bunch. Also, there may be a momentum spin correlation introduced upstream as part of the pi-mu production mechanism \textcolor{red}{\textit{(see docdb 3693)}}. $\Delta t_{ini}$ 

\medskip
\textcolor{red}{Say something about differential decay???}

\subsection{\label{sec:lost_muons} Spin Resonances}
\textcolor{orange}{Volodya}
\medskip

\textcolor{red}{\textit{Someone should say something about the spin resonance lines, see Fig.~\ref{fig:tuneplanespin}.}}

\begin{figure}[]
	\centering
	\begin{tikzpicture}
		\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig27}};
		\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]{fig27}
	\caption{\textcolor{orange}{Jason} Betatron tune plane with spin resonance lines.}\label{fig:tuneplanespin}
\end{figure}

\subsection{\label{sec:lost_muons} Lost Muons}
\textcolor{orange}{Sudeshna}
\medskip

During the first couple of hundred turns after injection, muons outside the aperture defined by the collimators are scraped out, and ideally particles that survive scraping will be lost only to decay. There may however be some particles with long time scale unstable motion, perhaps associated with a resonance (Fig.~\ref{fig:lostmuonsscan}), that eventually exit. Particles may also be lost to scattering from residual gas. If there is any correlation of these lost muons with the parameters of the muon phase space, the distribution will change with time and confuse our corrections. If the lost muons populate a portion of the phase space that is correlated with momentum or position, the phase space of muons contributing to the measurement of polarization will be distorted. the difference frequency will change as the lost particles are lost.

\begin{figure}[]
	\centering
	\begin{tikzpicture}
		\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig28.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]{fig28.pdf}
	\caption{\textcolor{orange}{Sudeshna} Fraction of lost muons as a function of EQS storage set-point voltage for calorimeter 1. An increased lost muon rate is observed from the betatron resonances centered near 18.8 and \SI{21.2}{\kilo\volt}. These resonances are due to the $3\nu_{y}=1$ and $\nu_{x}+2\nu_{y}=2$ lines in the tune plane.}\label{fig:lostmuonsscan}
\end{figure}

The phase space volume of the injected muon distribution overwhelms the ring acceptance. As a result, the distribution that is stored fills the aperture. Muons at the edge of the momentum or transverse aperture may be lost over the course of a fill. If those losses correlate with any particular region of the muon phase space or polarization, there will be a systematic distortion on the measurement of momentum and/or phase space distribution and/or $\omega_a$ phase~\cite{lostmuons}. If for example, high momentum muons are lost early in the fill by hitting an aperture, there will be a systematic shift of the $\omega_a$ phase over the course of the fill as the lower momentum muons in the distribution decay away. Or if particles with large vertical oscillation are lost, the pitch correction can be skewed. 

In order to insure that all circulating muons are well within the aperture defined by the collimators, muons at the edge of the aperture are scraped at the start of the fill. The scraping is accomplished by adjusting the voltages on the quadrupole plates asymmetrically, thereby introducing an electrostatic dipole moment, that displaces the closed orbit by about \SI{2}{\milli\meter} both vertically and radially from the center of the aperture. The asymmetric voltages  are sustained for \SI{7}{\micro\second} (many betatron periods) and then the closed orbit is restored adiabatically. The vertical displacement of the distribution is measured by the traceback detectors (see Fig.~\ref{fig:injvert}) at a point in the ring \SI{180}{\degree} beyond the injection point and over the course of the scraping cycle. The remaining distribution on the restored closed orbit will nominally have at least \SI{1}{\milli\meter} clearance with respect to the collimators. Muons that are lost are identified as a double (or triple) coincidence in adjacent calorimeters so that the effectiveness of the scraping procedure can be confirmed.

\begin{figure}[]
	\centering
	\begin{tikzpicture}
		\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig29.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]{fig29.pdf}
	\caption{\textcolor{orange}{David R.} Vertical displacement of the centroid at the \SI{180}{\degree} tracker during the scraping cycle.}\label{fig:injvert}
\end{figure}
