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

\textcolor{red}{\textit{Bill: People want to know how this affects the physics. The quad non-linearities (mainly 20-pole) make the CBO de-cohere faster. Jason has shown that this gives a smaller CBO systematic error. I don't think the 20-pole affects the E-field correction too much. Those are the two physics issues I can think of for the 20-pole quad non-linearity. Also, there are 20-pole betatron resonances which give higher lost muon rates (see David Tarazona's docdbs), but we avoid them.}}

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

The muon beam exhibits coherent motion inside of the storage ring due to several effects. The non-ideal magnetic kicker pulses produce a horizontal (radial) modulation of the beam centroid with characteristic frequency $f_{x}$, as the centroid is not kicked onto the the central orbit (magic) radius. Injection angle errors produce a vertical modulation of the beam centroid with characteristic frequency $f_{y}$, but these errors and corresponding vertical modulation are very small. There is a mismatch between the $\SI{18}{\milli\meter}~\textnormal{(width)}\times\SI{56}{\milli\meter}~\textnormal{(height)}$ Inflector aperture~\cite{Yamamoto:2002bb,Froemming:IPAC2018-WEPAF014} and circular cross section of the storage region, which has a \SI{90}{\milli\meter} diameter. The Inflector aperture imposes a minimum beam spot (waist) at injection, and so there is a modulation of the beam width as the collection of particles rotates in phase space.

Figure~\ref{fig:radialdist} shows a decay muon radial distribution with a peak near \SI{20}{\milli\meter} and a large low-side tail that arises from the non-ideal magnetic kicker pulses. The muons rotate in $x$-$x'$-phase space about different $x_{e}$, e.g. $\delta=\SI{0}{\percent}$ (magic momentum) muons rotate around $x\approx\SI{0}{\milli\meter}$ and $\delta=\SI{0.125}{\percent}$ muons with a quadrupole high voltage at \SI{18.3}{\kilo\volt} will rotate around $x\approx\SI{9.96}{\milli\meter}$. \textcolor{red}{\textit{[Update with \runtwo~based value]}} Hence, the ``wide-asymmetric'' distribution completes one revolution in $x$-$x'$-phase space per \SI{360}{\degree} betatron phase advance, and so $f_{x}$ is the dominant frequency of the radial width modulation. Figure~\ref{fig:verticaldist} shows a Gaussian-like decay muon vertical distribution with a peak near \SI{0}{\milli\meter}. The muon distribution mean and peak are near the $y$-$y'$-phase space origin, and so the ``narrow-symmetric'' distribution completes one revolution in phase space per \SI{180}{\degree} betatron phase advance. Hence, $2f_{y}$ is the dominant frequency of the vertical width modulation.

\begin{figure}[]
	\centering
	\begin{subfigure}{\columnwidth}
		\begin{tikzpicture}
			\draw (0, 0) node[inner sep=0]{\includegraphics[width=0.95\columnwidth]{fig26a.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]{fig26a.pdf}
		\caption{Horizontal (radial) distribution.}\label{fig:radialdist}
	\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]{fig26b.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]{fig26b.pdf}
		\caption{Vertical distribution.}\label{fig:verticaldist}
	\end{subfigure}
	\caption{\textcolor{orange}{David R.} Decay muon distributions measured by the~\textcolor{red}{\textit{??}} straw tracker station for a subset of preliminary \runone~data. The data is not corrected for acceptance or resolution. \textcolor{red}{\textit{Should the y-axis label read $\left.dN\middle/dy\right.$ or counts; remove timestamp and directory information; check with tracker experts.}}}\label{fig:positiondist}
\end{figure}

Muon bunches are injected into the storage ring and the bunches have a cyclotron motion with characteristic frequency $f_{c}$, where the longitudinal bunch width increases over time (debunching) due to the muon momentum spread. The coherent motion of the beam centroid and width are characterized by decoherence times. The coherent motion and modulation at early times is evident in Figs.~\ref{fig:fibercentroid} and~\ref{fig:fiberwidth} and decoherence at later times in Fig.~\ref{fig:trackercentroid}. The beam decoheres in part due to the amplitude and momentum dependence of the tunes associated with the quadrupole nonlinearities, see Section~\ref{sec:optics}, and the chromaticity. But, the leading source of decoherence is the large momentum spread and the strong dependence of path length on momentum. The relative change in circumference is given by
\begin{equation}
	\frac{\Delta C}{\cmagic}\approx\frac{\eta_{x}}{\rmagic}\delta\approx\delta\nonumber
%	\label{eq:rel_delta_c}
\end{equation}
in the weak focusing limit, where $\Delta C$ is change in circumference, $\cmagic$ is the design orbit (magic) circumference, and the storage ring momentum acceptance is $\delta\sim\SI{\pm0.125}{\percent}$. The distribution will lap itself in as few as 400 turns (\SI{\sim60}{\micro\second}), effectively mixing the distribution. Indeed, a measure of the decoherence time is an indication of the momentum distribution width. 

The detectors are at fixed locations and provide finite coverage around the storage ring, and so the cyclotron motion of the muon bunches creates an effective sample rate. The Nyquist frequency for the detectors is given by $\left.f_{c}\middle/2\right.$. The $f_{x}$ and $2f_{y}$ frequencies are larger than the Nyquist frequency, and so the data contains the corresponding Coherent Betatron Oscillation (CBO) and Vertical Waist (VW) alias frequencies respectively. The CBO and VW frequencies are physical alias frequencies, as they arise from the cyclotron motion and detector acceptance effects (an example of the the stroboscopic effect), as opposed to alias frequencies that can arise from histogram binning and associated discrete Fourier transforms. Table~\ref{tab:beamfreq} gives important beam frequencies that affect the \runone~data that was taken at the \SI{18.3}{\kilo\volt} EQS storage set-point.

\begin{table*}[]
\caption{\label{tab:beamfreq} \textcolor{orange}{Jason} Important beam frequencies that affect the \runone~decay positron spectra for the \SI{18.3}{\kilo\volt} EQS storage set-point \textcolor{red}{\textit{[Update with \runtwo~based values]}}. The tune values used to calculate the frequencies are taken from Table~\ref{tab:n_nu_estim}.}
\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}           & $f_{a}$                         & $\left.\left(ea_{\mu}\dipbfield\right)\middle/\left(2\pi m_{\mu}\right)\right.$   & 0.2291                                                    & 4.365\\
\textnormal{Cyclotron}                                 & $f_{c}$                         & $\left.\pmagic\middle/\left(2\pi m_{\mu}\gammamagic\rmagic\right)\right.$ & 6.705                                                      & 0.1491\\
\textnormal{Horizontal Betatron}                 & $f_{x}$                         & $\nu_{x}f_{c}$                                                                                               & 6.334                                                      & 0.1579\\
\textnormal{Vertical Betatron}                      & $f_{y}$                         & $\nu_{y}f_{c}$                                                                                              & 2.201                                                      & 0.4543\\
\textnormal{Coherent Betatron Oscillation} & $f_{\textnormal{cbo}}$ & $f_{c}-f_{x}$                                                                                                 & 0.371                                                      & 2.70\\
\textnormal{Vertical Waist}                           & $f_{\textnormal{vw}}$  & $f_{c}-2f_{y}$                                                                                               & 2.303                                                      & 0.4342\\
\end{tabular}
\end{ruledtabular}
\end{table*}

Equation~(\ref{eq:wiggle_func}) is used to describe the decay positron time spectra after applying a suitable energy threshold cut to the positron events, where the \runone~data is expected to have a $N$ (normalization) dependance on the CBO and VW frequencies, $A$ (asymmetry) dependance on the CBO frequency, and $\phi$ (\gmtwo~phase) dependance on the CBO frequency. The normalization dependence on coherent beam motion is due to detector acceptance, as the likelihood of a positron striking a detector depends on the muon decay position. The asymmetry dependence on coherent beam motion is due to the modulation of the average detected positron energy, as there is a positron energy and muon decay position correlation for detected positrons. A detector does not measure the muon decay time, but the positron arrival time at the detector. The difference in times corresponds to a shift in the \gmtwo~phase. Hence, the \gmtwo~phase dependence on coherent beam motion is due to the detected positron time of flight, as there is a positron trajectory and muon decay position correlation for detected positrons.

Figure~\ref{fig:residfft} shows CBO and VW effects in the frequency spectrum for fit residual from a decay positron time spectrum. A so called ``5-parameter fit'' is used to generate the fit residuals for Fig.~\ref{fig:residfft}, where the 5-parameter fit sets each of the $N$, $A$, and $\phi$ terms to a single fit parameter (constant function). The subset of \runone~data used to produce Fig.~\ref{fig:residfft} is probably too small to show the vertical betatron and twice CBO frequency effects, as these effects are expected to be very small~\cite{Bennett:2006fi}.

\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} The Fast Fourier transform (FFT) frequency spectrum for 5-parameter fit residuals from a decay positron time spectrum. Specifically, the magnitude of the FFT amplitude is shown. This subset of \runone~data was taken at $V_{Q}=\SI{18.3}{\kilo\volt}$. \textcolor{red}{\textit{Final unless David S. wants to make an updated version.}}}\label{fig:residfft}
\end{figure}

A typical decay positron time spectrum fit function for the BNL level of statistics is given by Eq.~\ref{eq:wiggle_func} with
\begin{align}
	&N\left(t\right)                                =N_{0}N_{\textnormal{cbo}}\left(t\right)N_{\textnormal{vw}}\left(t\right)\Lambda\left(t\right),\nonumber\\
	&N_{\textnormal{cbo}}\left(t\right) =1+A_{\textnormal{cbo}}^{N}e^{-\left.t\middle/\tau_{\textnormal{cbo}}\right.}\cos\left(\omega_{\textnormal{cbo}}t+\phi_{\textnormal{cbo}}^{N}\right),\nonumber\\
	&N_{\textnormal{vw}}\left(t\right)   =1+A_{\textnormal{vw}}^{N}e^{-\left.t\middle/\tau_{\textnormal{vw}}\right.}\cos\left(\omega_{\textnormal{vw}}t+\phi_{\textnormal{vw}}^{N}\right),\nonumber\\
	&\Lambda\left(t\right)                     =1-\kappa_{\textnormal{loss}}\int_{0}^{t}dt'\,e^{\left.t'\middle/\tau\right.}L\left(t'\right),\nonumber\\
	&A\left(t\right)                                =A_{0}\left[1+A_{\textnormal{cbo}}^{A}e^{-\left.t\middle/\tau_{\textnormal{cbo}}\right.}\cos\left(\omega_{\textnormal{cbo}}t+\phi_{\textnormal{cbo}}^{A}\right)\right],\nonumber\\
	&\phi\left(t\right)                             =\phi_{0}+A_{\textnormal{cbo}}^{\phi}e^{-\left.t\middle/\tau_{\textnormal{cbo}}\right.}\cos\left(\omega_{\textnormal{cbo}}t+\phi_{\textnormal{cbo}}^{\phi}\right).
	\label{eq:bnlfitfunc} % get an error if last line has \nonumber. % manually put in parentheses for inline eq. references.
\end{align}
The $\Lambda$ term in Eq.~(\ref{eq:bnlfitfunc}) is used to model the muon exponential decay correction that arises from the so called ``lost  muons'', which are lost during the measurement period due to beam interactions with storage ring materials. The $\Lambda$ function is written in terms of the muon loss per lab frame lifetime function ($L$), which is determined by measuring the coincident minimum ionizing particles (MIPs) in the calorimeters that are produced by lost muons, see Section~\ref{sec:lost_muons}. The $\kappa_{\textnormal{loss}}$ fit parameter gives the absolute scale of the muon losses after applying calorimeter gain and positron event pileup corrections. The $N_{0}$, $A_{0}\sim\SI{0.37}{}$, and $\phi_{0}$ fit parameters give the underlying normalization, asymmetry, and \gmtwo~phase respectively. The CBO and VW decoherence are modeled by exponential terms with characteristic time constants $\tau_{\textnormal{cbo}}\SI{\sim180}{\micro\second}$ and $\tau_{\textnormal{vw}}\SI{\sim20}{\micro\second}$ respectively. The $A_{\textnormal{cbo}}^{N}\sim\SI{0.004}{}$, $A_{\textnormal{cbo}}^{A}\sim\SI{0.0007}{}$, and $A_{\textnormal{cbo}}^{\phi}\sim\SI{0.0004}{}$ fit parameters give the amplitudes and $\phi_{\textnormal{cbo}}^{N}$, $\phi_{\textnormal{cbo}}^{A}$, and $\phi_{\textnormal{cbo}}^{\phi}$ fit parameters give the corresponding phases of the CBO modulations, while the $A_{\textnormal{vw}}^{N}\sim\SI{0.003}{}$ fit parameter gives the amplitude and $\phi_{\textnormal{vw}}^{N}$ fit parameter gives the corresponding phase of the VW modulation. The $\omega_{\textnormal{vw}}\sim\SI[per-mode=symbol]{14.4}{\radian\per\micro\second}$ fit parameter gives the angular frequency of the VW modulation \textcolor{red}{\textit{[check values]}}. 

The $\omega_{\textnormal{cbo}}$ fit parameter gives the angular frequency of the CBO modulation. Several of the EQS potted resistors, which are used to set the $RC$ time constant for the transition from beam scraping to stable storage, were found to be damaged/defective after the completion of \runone. The damage caused the effective resistance to be larger than expected, where two of the resistors have an $RC$ time constant that is significantly larger than the nominal \SI{5}{\micro\second}. This causes the $\omega_{\textnormal{cbo}}$ to drift during the measurement period for the \runone~data. A phenomenological model based on straw tracker data has been developed to model the $\omega_{\textnormal{cbo}}$ drift for the \runone~decay positron fit functions, which is given by
\begin{equation}
	\omega_{\textnormal{cbo}}=\omega_{0}\left(1+\Delta\omega t+T_{1}e^{-\left.t\middle/\tau_{1}\right.}+T_{2}e^{-\left.t\middle/\tau_{2}\right.}\right),
	\label{eq:w_cbo_drift}
\end{equation}
where $\omega_{0}\sim\SI[per-mode=symbol]{2.31}{\radian\per\micro\second}$, $\Delta\omega\sim\SI[per-mode=symbol]{1.9e-5}{\per\micro\second}$, $T_{1}\sim\SI{-0.051}{}$, $\tau_{1}\sim\SI{73}{\micro\second}$, $T_{2}\sim\SI{-0.13}{}$, and $\tau_{2}\sim\SI{17}{\micro\second}$ \textcolor{red}{\textit{[check values; include 9-day and end-game models]}}. The numerical values given for the Eqs.~(\ref{eq:bnlfitfunc}) and~(\ref{eq:w_cbo_drift}) fit parameters are for a subset of \runone~data taken at the \SI{18.3}{\kilo\volt} EQS storage set-point, where the data from all of the calorimeters has been added together. The damaged potted resistors have been replaced with properly functioning resistors for \runtwo~data collection, and so $\omega_{\textnormal{cbo}}$ should be able to be treated as a single fit parameter (constant function) for \runtwo.  

The Fermilab experiment has the goal of collecting 20 times the total statistics collected by BNL ($8.6348\times10^{9}$ decay positrons and electrons~\cite{Bennett:2006fi}). The increased statics may lead to the detection of small, as of yet unseen, coherent beam motion effects. The Eq.~(\ref{eq:bnlfitfunc}) CBO models are based on the radial modulation of the beam centroid, but detector acceptance effects may lead to the $N$, $A$, and $\phi$ terms having small dependences on higher-order moments of the beam. The $N$, $A$, and $\phi$ can be parametrized in terms of the decay muon phase space ($x$, $x'$, $y$, $y'$, $s$, $\dot{s}$, and $t$, where $\dot{s}\equiv\left.ds\middle/dt\right.$) for each individual detector. The primary concern for the Fermilab experiment at full statistics is small effects from higher-order radial beam moments, and so the following discussion will only consider a dependance of $N$, $A$, and $\phi$ on $x$ and $x'$. Furthermore, the problem will be simplified by only considering an explicit dependence on $x$, as $x$ and $x'$ are coupled variables, i.e. muon radial motion is described at leading-order by simple harmonic motion in $x$-$x'$-phase space.

Assume that the instantaneous dependence of the $N$, $A$, and $\phi$ terms on $x$ for an individual detector may be expressed as Taylor series given by
\begin{align}
	&N_{I}\left(x\right)=\sum_{i=0}^{\infty}a_{i}\left(x-d\right)^{i},\nonumber\\
	&A_{I}\left(x\right)=\sum_{i=0}^{\infty}b_{i}\left(x-d\right)^{i},\nonumber\\
	&\phi_{I}\left(x\right)=\sum_{i=0}^{\infty}c_{i}\left(x-d\right)^{i}.
	\label{eq:inst_par_dep} % get an error if last line has \nonumber. % manually put in parentheses for inline eq. references.
\end{align}
The $a_{i}$, $b_{i}$, and $c_{i}$ series coefficients depend on the $d$ expansion point and $E_{\textnormal{th}}$. The Eq.~(\ref{eq:inst_par_dep}) coefficients are effective parameters that also depend on the integration of the decay muon distribution over the $x'$, $y$, $y'$, $s$, $\dot{s}$, and $t$ variables, and these coefficients correspond to detector acceptance effects. Non-zero higher order terms ($i\geq2$) can arise from the interaction of decay positrons with storage ring materials, e.g. kicker and quadrupole plates.

The time dependent decay muon moments with respect to $d$ can be written as
\begin{align}
	&\mu_{i}^{d}\left(t\right)=\int_{-\infty}^{\infty}dx\,\left(x-d\right)^{i}\frac{\mathcal{B}\left(x,t\right)}{\mathcal{B}\left(t\right)},\nonumber\\
	&\int_{0}^{\infty}dt\,\int_{-\infty}^{\infty}dx\,\mathcal{B}\left(x,t\right)=1,\nonumber\\
	&\mathcal{B}\left(t\right)=\int_{-\infty}^{\infty}dx\,\mathcal{B}\left(x,t\right),
	\label{eq:moments} % get an error if last line has \nonumber. % manually put in parentheses for inline eq. references.
\end{align}
where $\mathcal{B}\left(x,t\right)$ is a joint, $\mathcal{B}\left(t\right)$ is a marginal, and $\left.\mathcal{B}\left(x,t\right)\middle/\mathcal{B}\left(t\right)\right.$ is a conditional probability density function (PDF) for the decay muons ($\mathcal{B}$ dependence on the $x'$, $y$, $y'$, $s$, and $\dot{s}$ variables has been integrated and normalized away) \textcolor{red}{\textit{[check statistics lingo]}}. The simplest way to think about this problem is to use $d=0$ (raw moments), but the raw moments can be written in terms of the mean and central moments via Eq.~(\ref{eq:moments}) by letting $d$ be equal to the time dependent mean and expanding out the $\left(x-d\right)^{i}$ term into a finite Maclaurin series before carrying out the integration. The generic form of the final fit function models should not matter on whether raw or central moments are used, as both types of moments should lead to a collection of equilibrium terms and periodic terms that decay away.

Equations~(\ref{eq:moments}) and~(\ref{eq:inst_par_dep}) can then be used to write $N$, $A$, and $\phi$ as functions of $t$:
\begin{align}
	&N\left(t\right)=\int_{-\infty}^{\infty}dx\,N_{I}\left(x\right)\frac{\mathcal{B}\left(x,t\right)}{\mathcal{B}\left(t\right)}=a_{0}+\sum_{i=1}^{\infty}a_{i}\mu_{i}^{d}\left(t\right),\nonumber\\
	&A\left(t\right)=\int_{-\infty}^{\infty}dx\,A_{I}\left(x\right)\frac{\mathcal{B}\left(x,t\right)}{\mathcal{B}\left(t\right)}=b_{0}+\sum_{i=1}^{\infty}b_{i}\mu_{i}^{d}\left(t\right),\nonumber\\
	&\phi\left(t\right)=\int_{-\infty}^{\infty}dx\,\phi_{I}\left(x\right)\frac{\mathcal{B}\left(x,t\right)}{\mathcal{B}\left(t\right)}=c_{0}+\sum_{i=1}^{\infty}c_{i}\mu_{i}^{d}\left(t\right).
	\label{eq:avg_par_dep} % get an error if last line has \nonumber. % manually put in parentheses for inline eq. references.
\end{align}
The $\mu_{0}^{d}\left(t\right)=1$ by construction, but the $i\geq1$ decay muon moments are expected to have the form of an equilibrium term added to a periodic (modulation) term that decays away (decoheres), because the muons spread out and fill up the storage ring admittance as the beam motion decoheres over the measurement period due to nonlinear effects. The $i\geq1$ decay muon moments in Eqs.~(\ref{eq:moments}) and~(\ref{eq:avg_par_dep}) can be written as
\begin{align}
	&\mu_{i}^{d}\left(t\right)=E_{i}^{d}\left(t\right)+\nonumber\\
	&\qquad D_{i}^{d}\left(t\right)\left[A_{i,0}^{d}+\sum_{j=1}^{\infty}A_{i,j}^{d}\cos\left(j\omega_{\textnormal{cbo}}t+\phi_{i,j}^{d}\right)\right],
	\label{eq:form_of_moments} % get an error if last line has \nonumber. % manually put in parentheses for inline eq. references.
\end{align}
where $E_{i}^{d}\left(t\right)$ is the equilibrium term, $D_{i}^{d}\left(t\right)$ is the decay envelope, and the cosine Fourier series inside of the brackets is the periodic term. The BNL fit model given in Eq.~(\ref{eq:bnlfitfunc}) only considers the $i\leq1$ and $j\leq1$ case for Eqs.~(\ref{eq:avg_par_dep}) and~(\ref{eq:form_of_moments}), where $E_{1}^{d}\left(t\right)=\textnormal{constant value}$, $D_{1}^{d}\left(t\right)=\exp\left(-\left.t\middle/\tau_{\textnormal{cbo}}\right.\right)$, and $A_{1,0}^{d}=0$. Furthermore, the $\phi_{1,1}^{d}$ is treated as a separate fit parameter for each of the $N$, $A$, $\phi$ terms, because the corresponding cosine amplitude and phase fit parameters are correlated at some level and the different cosine amplitude fit parameters are neither equal to $A_{1,1}^{d}$ nor each other. The $D_{1}^{d}\left(t\right)=\exp\left(-\left.t\middle/\tau_{\textnormal{cbo}}\right.\right)$ model in particular and the fit model as a whole are not expected to be perfectly accurate, and letting the $\phi_{1,1}^{d}$ be a separate fit parameter for each of the $N$, $A$, $\phi$ terms allows the fit function to be more ``flexible with the data''.

The BNL model can be used to fit data from individual detectors or a collection of detectors, where the CBO effects are significantly reduced if the positron data from all of the calorimeters is added together. The reduction in CBO effects occurs because the CBO modulations are \SI{180}{\degree} out of phase for the detectors that are on opposite sides of the storage ring. Perfect cancellation does not occur because the detectors are not perfectly aligned and they have different decay positron acceptances. The BNL fit model uses the same decohrence model for the $N$, $A$, and $\phi$ terms, but this may not be adequate for the full Fermilab statistics. Furthermore, new decohrence models may need to be developed given the different individual detector acceptances and potential contributions from higher order beam moments.

\subsection{\label{sec:pitch}Pitch Correction}
\textcolor{orange}{David R.}
\medskip

Our measurement of the anomalous precession frequency of the muon depends on the correlation of muon velocity and muon polarization. The projection of polarization onto velocity is given by (Jackson 11.171)
\begin{equation}
\frac{d}{dt}({\bf\hat\beta\cdot s})=-\frac{e}{mc}{\bf s_\perp \cdot}\left[a_\mu({\bf\hat\beta\times B})+\left(a_\mu-\frac{1}{\gamma^2-1}\right)\beta{\bf E}\right]
\label{eq:betadots}.
\end{equation}  
The dependence of $\omega_a$ on the amplitude of vertical betatron oscillations appears in the term ${\bf\hat\beta\times B}$ in Eq.~(\ref{eq:betadots}). If ${\bf B} = B{\bf\hat y}$ and ${\bf\hat \beta} = \sin\psi_{yz} {\bf\hat y}+\cos\psi_{yz}{\bf\hat z}$ then
${\bf\hat\beta\times B} \sim \beta_z B = \beta(\cos\psi_{yz})B$, where $\psi_{yz}$ is the angle of the trajectory in the $y-z$ plane. The contribution to $\omega_a$ due to the pitch for a particular trajectory is $\frac{\Delta\omega_a}{\omega_a}=(1-\cos\psi_{yz})$ and for $\psi_{yz}\ll 1$, $$\frac{\Delta\omega_a}{\omega_a}\sim \frac{1}{2}\psi^2_{yz}.$$ The contribution from the pitch angle averaged over all $\phi$ for each amplitude $a$ and then over all possible amplitudes $a$, is $$\langle\langle\frac{\Delta\omega_a}{\omega_a}\rangle_{\phi}\rangle_a \sim \frac{1}{2}\langle\langle\psi^2_{yz}\rangle_\phi\rangle_a.$$ In the limit of strictly linear betatron motion, the displacement $y$ is related to amplitude $a$ according to $y=\sqrt{a\beta_y}\cos\phi_y$, where $\phi_y$ is the betatron phase advance. Then the angle $$\psi_{yz}= \psi_0\sin\phi=\sqrt{\frac{a}{\beta_y}}\sin\phi_y.$$ The average of $\psi_{yz}^2$ over all $\phi_y$ for amplitude $a$ is $$\langle\psi^2_{yz}(a)_\phi =\frac{1}{2}\psi_0^2(a)=\frac{\langle y^2(a)\rangle_\phi}{\beta^2}.$$ Next, average over all amplitudes $a$. We suppose that the distribution of amplitudes is given by some function $P(a)$ defined over the physical aperture. Then
\begin{eqnarray*}
\langle\langle y^2\rangle_\phi\rangle_a &=& \frac{1}{2}\beta\frac{\int_{aper}aP(a)da}{\int_{aper}P(a) da}\\
\langle\langle \psi^2\rangle_\phi\rangle_a &=& \frac{1}{2}\langle\psi_0^2\rangle_a =\frac{a}{\beta^2}\langle\langle y^2\rangle_\phi\rangle_a\\
C_p &\equiv& \frac{\Delta \omega_a}{\omega_a} = \frac{1}{4}\langle\psi_0^2\rangle_a = -\frac{n \langle y^2\rangle}{2R_0^2},
\end{eqnarray*}
where $\beta_y = \frac{\sqrt{n}}{R_0}$.

The vertical distribution of decay positrons is measured by the straw tracking chambers. Tracks of positrons are extrapolated back to the parent muon decay point to determine the vertical distribution of the muons as shown for a subset of the data in Fig.~\ref{fig:verticaldist}.

\subsection{\label{sec:efield} E-field Correction}
\textcolor{orange}{David R.}
\medskip

The E-field systematic is associated with the third term on the right hand size of Eq.~(\ref{eq:au}) and depends on momentum and radial electric field. The contribution from the electric field is minimized by operating near the magic momentum, namely where $a_\mu = \frac{1}{\gamma^2-1}.$ The field of the ring magnet is chosen so that the closed orbit of the magic momentum muon corresponds to the design trajectory. Then the contribution to $\omega_a$ is
\begin{equation}
C_e\sim-2\frac{\Delta p}{p_m}\frac{\vec\beta\times\vec E}{Bc},
\end{equation}
where $p_m$ is the magic momentum $\gamma$ and $\beta$ are evaluated at $p_m$.

The electric field depends on transverse displacement in the quadrupoles. Insofar as the field is with displacement, the average electric field along the trajectory is proportional to the average radial displacement $x_e$, namely the closed orbit. The  average radial electric field is given by $$\langle E_r\rangle = n\left(\frac{v_s B}{R_0}\right)x_e,$$ where $n$ is the focusing index. The average radial displacement is $$x_e = \eta\frac{\Delta p}{p}.$$ Then the contribution to $\omega_a$ due to electric field is $$C_e \sim -2\beta^2n(1-n)\frac{x_e^2}{R_0^2},$$ where $\eta =\sim\frac{1-n}{R_0}$. Determination of the E-field correction thus depends on measurement of the equilibrium radial distribution. (The magnitude of the E-field correction is reduced by a few percent due to the nonlinearity of the quadrupole field.)

The radial distribution is determined by a ``fast rotation analysis'', that exploits the connection between the revolution frequency and the radial displacement. A technique~\cite{Orlov:2002ag} based on Fourier transform yields a frequency spectrum that can be correlated with radius (circumference) and momentum. An alternative method extracts the radial distribution from the measured debunching of the muon beam. The fast rotation data is provided by the calorimeters which measure the time dependence of the intensity of the distribution. An example of radial (closed orbit) distribution extracted by both the Fourier method and the debunching analysis are shown in Fig.~\ref{fig:fourierdebunching}. The dependence of the E-field contribution to $\omega_a$, on the radial offset of the closed orbit, superimposed on the the measured distribution is shown in Fig.~\ref{fig:efield_cor}.

\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}{David R.} Fourier analysis of the fast rotation signal and parameterized fit give the muon equilibrium radial (closed orbit) distribution revolution. \textcolor{red}{\textit{Harmonize notation; do we want lines connecting the points.}}}\label{fig:fourierdebunching}
\end{figure}

\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.} The contribution to $\omega_a$ ($C_e$) due to the electric field is computed by spin tracking as a function of muon momentum and plotted in terms of the closed orbit. The measured radial distribution is superimposed. The average correction is the convolution of the two. \textcolor{red}{\textit{Harmonize notation; should right-side y-axis label read $\left.dN\middle/dx_{e}\right.$ or counts; should x-axis range go from \SI{\pm45}{\milli\meter}.}}}\label{fig:efield_cor}
\end{figure}

\subsection{\label{sec:early-to-late} Spin Correlations \& Early-to-late Effects}
\textcolor{orange}{Jason}
\medskip

The total spin phase (argument) of the cosine function in Eq.~(\ref{eq:wiggle_func}) can be written in the following general form:
\begin{equation}
	\Phi\left(t\right)=\left[\int_{0}^{t}dt'\,\omega_{a}\left(t'\right)\right]+\phi\left(t\right)
	\label{eq:cos_phase}.
\end{equation}
The difference between Eqs.~(\ref{eq:wiggle_func}) and~(\ref{eq:cos_phase}) is that $\omega_{a}$ is allowed to be a function of time. For example, $\omega_{a}$ will become time dependent if the magnetic or electric fields are changing with time during the measurement period. Equation~(\ref{eq:cos_phase}) illustrates the ambiguity between the $\omega_{a}$ and $\phi$ in a fit, where the $\omega_{a}$ and $\phi$ fit terms are treated as constant values under ideal conditions without experimental systematic effects. The $\Phi$ is the quantity being measured, while the $\omega_{a}$ and $\phi$ require accurate models to precisely extract the $\omega_{a}$ due to muon precession. There will be a shift in the constant $\omega_{a}$ fit value if the fit function does not accurately model the time dependence of $\phi$. This can be illustrated by expanding $\phi$ into a Maclaurin series,
\begin{equation}
	\phi\left(t\right)=\phi\left(0\right)+\frac{d\phi\left(0\right)}{dt}t+\frac{1}{2}\frac{d^{2}\phi\left(0\right)}{dt^{2}}t^{2}+\cdots\nonumber
	\label{eq:phase_series},
\end{equation}
and noting that the linear term has the same functional form as the $\omega_{a}t$ term in Eq.~(\ref{eq:wiggle_func}).

The combination of spin correlations and corresponding so called ``early-to-late effects'' form an important class of systematic effects that need to be considered. In this context, a decay muon property $F$ (e.g. $x$, $y$, $p$, etc.) is said to have correlations with spin if
\begin{equation}
	\frac{\partial\phi_{0}}{\partial\langle F\rangle}\neq0\nonumber
	\label{eq:spin_corr_def},
\end{equation} 
where $\langle F\rangle$ is the mean value of the decay muon property $F$ and $\phi_{0}$ is the ``constant'' component of the \gmtwo~phase. The $\phi_{0}$ is modeled as a constant fit parameter, which is added to the temporal modulation of $\phi$ due to the coherent beam motion, e.g. see Eq.~(\ref{eq:bnlfitfunc}). Furthermore, the decay muon property $F$ is said to have early-to-late effects if
\begin{equation}
	\frac{d\langle F\rangle}{dt}\neq0\nonumber
	\label{eq:early_to_late_def}.
\end{equation} 

If spin correlations and corresponding early-to-late effects combine together, then this implies that $\phi_{0}$ is really a function of time, since
\begin{align} % {} before the + sign turns it into a binary operator instead of a unary operator (spacing issues)
	&\frac{d}{dt}\phi_{0}\left(\langle F\rangle\left(t\right),\langle G\rangle\left(t\right),\langle H\rangle\left(t\right),\cdots\right)=\frac{\partial\phi_{0}}{\partial\langle F\rangle}\frac{d\langle F\rangle}{dt}\nonumber\\
	&\qquad{}+\frac{\partial\phi_{0}}{\partial\langle G\rangle}\frac{d\langle G\rangle}{dt}+\frac{\partial\phi_{0}}{\partial\langle H\rangle}\frac{d\langle H\rangle}{dt}+\cdots\neq0,
%	\label{eq:spin_corr_and_early_to_late_def} % get an error if last line has \nonumber. % manually put in parentheses for inline eq. references.
\end{align}
where $F,G,H,\cdots$ are decay muon properties and the equality is obtained from the chain rule for multivariable functions with one independent variable. \textcolor{red}{\textit{Double check chain rule!}} The dependence of $\phi_{0}$ on $\langle F\rangle,\langle G\rangle,\langle H\rangle,\cdots$ can be obtained from a decay positron fit over a very small time window centered on time $t$ for each corresponding set of $\langle F\rangle,\langle G\rangle,\langle H\rangle,\cdots$ values at time $t$. The decay positron fit function is assumed to accurately model the $\omega_{a}$ term, because the $\omega_{a}$ and $\phi$ terms are coupled in the fit as indicated by Eq.~(\ref{eq:cos_phase}). The $\langle F\rangle,\langle G\rangle,\langle H\rangle,\cdots$ values at time $t$ can be obtained by taking the mean of the $F,G,H,\cdots$ values in the very small time window centered on time $t$.

Nonuniform spin-momentum distributions are generated in the Muon Campus beamline and muon storage ring, which will then produce spin-momentum correlations,
\begin{equation}
	\frac{\partial\phi_{0}}{\partial\langle p\rangle}\neq0\nonumber
	\label{eq:spin_mom_corr_def},
\end{equation}
in the muon storage ring. The majority of the muons are produced from pion decays in the M2-line and M3-line, which are primarily straight sections that transport positive pions and collect decay muons. The M2-line and M3-line FODO cells (alternating focusing and defocusing magnets) and dipole bending magnets will generate some spin-momentum distribution for the muons produced in these sections of the Muon Campus. In particular, a dipole bending magnet will change the angle between the muon spin and momentum by
\begin{equation}
	\Delta\phi_{sp}\approx a_{\mu}\gamma\phi_{b}
	\label{eq:precess_bend_mag},
\end{equation}
where $\phi_{b}$ is the angle at which the muon bends through the dipole field. Equation~(\ref{eq:precess_bend_mag}) indicates that dipole bending magnets will lead to nonuniform spin-momentum distributions, as the Lorentz factor can be written as a function of momentum:
\begin{equation}
	\gamma=\sqrt{1+\left(\frac{p}{m_{\mu}c}\right)^{2}}\nonumber
	\label{eq:lorentz_factor}.
\end{equation}

The initial decay muon transverse component of spin, $x'$, and $y'$ all depend on the direction of the initial muon momentum when the pion decays. A FODO cell maximum $x$-$x'$ admittance and $y$-$y'$ admittance occur at the horizontal-defocusing-vertical-focusing and horizontal-focusing-vertical-defocusing magnets respectively. Thus, most of the muons with a large horizontal spin component come from pion decays near the horizontal-defocusing-vertical-focusing magnets, and most muons with a large vertical spin component come from pion decays near the horizontal-focusing-vertical-defocusing magnets. The dependence between spin and the transverse coordinates decreases as the distance between the magnets decreases, which creates smaller $\beta$ functions. The dependence of the muon storage ring $x_{e}$ on $p$ and the use of circular collimators in the muon storage ring leads to a dependance of $x$, $x'$, $y$, and $y'$ on $p$. Hence, the FODO cells will also lead to nonuniform spin-momentum distributions.

Some of the muons are also produced from kaon and pion decays that occur before the beam reaches the M2-line. A momentum spectrometer is used to inject the \SI[per-mode=symbol]{\sim3.1}{\giga\eV\per\clight} positive particles from the Inconel-600 target into the M2-line. The momentum spectrometer is composed of the pulsed dipole magnet (PMAG), water-cooled collimator that protects the PMAG, and beam dump for the off-momentum particles. Some of the muons that are produced before and inside of the spectrometer will have a momentum that is \SI[per-mode=symbol]{\sim3.1}{\giga\eV\per\clight}, and so they will be injected into the M2-line. These so called ``target muons'' will have a different spin-momentum distribution than the muons from the M2-line and M3-line, as the target muons are produced and collected under different conditions.

Muons are also produced from pion decays in the Delivery Ring (DR). The DR is a storage ring that is used to remove the proton contamination from the beam and to provide time for the remaining pions to decay into muons. The spin tune for a storage ring with a dipole magnetic field is given by
\begin{align}
	&\omega_{c}=\frac{1}{\gamma}\frac{qB}{m},\nonumber\\
	&\omega_{a}\approx a_{\mu}\frac{qB}{m},\nonumber\\
	&\nu_{a}=\frac{\omega_{a}}{\omega_{c}}\approx a_{\mu}\gamma,
	\label{eq:spin_tune} % get an error if last line has \nonumber. % manually put in parentheses for inline eq. references.
\end{align}
where $\omega_{c}$ is the cyclotron frequency, $\omega_{a}$ is the anomalous precession frequency, and $B$ is the magnitude of the dipole magnetic field. Equation~(\ref{eq:precess_bend_mag}) can be derived from Eq.~(\ref{eq:spin_tune}), because one can imagine a dipole bending magnet as being a component in some storage ring (charged particles travel in circular arcs). The DR is made out of FODO cells and bending dipole magnets, and so the muons on the design orbit should only experience the net bending field. Thus, the approximation given in Eq.~(\ref{eq:precess_bend_mag}) is valid for the DR, where the angle between the muon spin and momentum changes by
\begin{equation}
	\Delta\phi_{sp}=\nu_{a}\phi_{b}\approx a_{\mu}\gamma\phi_{b}=2\pi a_{\mu}\gamma M
	\label{eq:precess_ring},
\end{equation}
where $M$ is the number of turns the muon has gone around the storage ring, and the $\gamma$ dependence on $p$ will lead to nonuniform spin-momentum distributions. The DR is expected to affect the spin-momentum distribution somewhere around the \SI{}{\milli\radian} level, as Eq.~(\ref{eq:precess_ring}) leads to $\Delta\phi_{sp}\approx\SI{0.214669}{\radian}$ per 1 turn for $\delta=\SI{0}{\percent}$ and $\Delta\phi_{sp}\approx\SI{0.214937}{\radian}$ per 1 turn for $\delta=\SI{0.125}{\percent}$, where $a_{\mu}\approx\SI{0.00116592}{}$, $m\approx\SI[per-mode=symbol]{0.105658}{\giga\eV\per\clight\squared}$, and $\pmagic\approx\SI[per-mode=symbol]{3.09435}{\giga\eV\per\clight}$ \textcolor{red}{\textit{Check arithmetic}}. The muon campus beam undergoes 4 turns around the DR for physics quality data before it is extracted and sent to the muon storage ring. The so called ``DR muons'' will have a different spin-momentum distribution than the other muons, as the DR muons are produced and collected under different conditions than the other muons. Figure~\ref{fig:beamline_spin_distr} gives estimated spin $x$- and $y$-component distributions for muons born in the M2-line, M3-line, and DR.

\begin{figure*}[]
	\centering
	\begin{tikzpicture}
		\draw (0, 0) node[inner sep=0]{\includegraphics[width=1.95\columnwidth]{fig30.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=1.95\columnwidth]{fig30.pdf}
	\caption{\textcolor{orange}{Mike} Horizontal and vertical muon spin component distributions near the entrance to the Storage Ring for \SI{\sim e5}{} muons born in the Muon Campus M2-line, M3-line, and Delivery Ring (DR). These distributions have been generated using the G4beamline model of the Muon Campus lattice. \textcolor{red}{\textit{Harmonize notation; do the spin units make sense.}}}\label{fig:beamline_spin_distr}
\end{figure*}

There is also a nonuniform spin-momentum distribution that arises from the muon storage ring beam injection and kick, which is negligible for the BNL level of statistics, but should be studied for the full Fermilab experiment statistics. The muon bunches extend along the longitudinal direction (\SI{\sim150}{\nano\second} full bunch width), where the muons at the head of the bunch have spent more time precessing in the storage ring magnetic field than the muons at the tail of the bunch. Thus, the beam injection leads to a gradient in $\Delta_{sp}$ across the longitudinal bunch length. The storage ring Kicker System magnetic field is nonuniform over the longitudinal length of the bunch, and the muon momentum acceptance depends on the integrated Kicker field seen by a muon. Thus, the beam kick leads to a gradient in $p$ across the longitudinal bunch length. Hence, there is nonuniform spin-momentum distribution that depends on the longitudinal position across a bunch.

The average $\Delta\phi_{sp}$ per time interval $\Delta t$ in the storage ring depends on the muon trajectories, as a muon trajectory will determine what magnetic and electric fields are seen by the muon. Since $x_{e}$ depends on $p$, then a trajectory also depends on $p$. There are two equivalent toy models that can be used to determine the average $\Delta\phi_{sp}$ per $\Delta t$. One could assume that there is a time independent $\omega_a$ and time dependent initial phase $\phi_{0}$. One could instead assume that there is a time dependent $\omega_a$ and time independent $\phi_{0}$. These two models are both consistent with Eq.~(\ref{eq:cos_phase}). This leads to the question of how one should classify the different spin-momentum based effects. The convention that has arisen over time is that a nonuniform spin-momentum distribution is considered to affect $\phi_{0}$ when the distribution is created before the decay positron fit start time (\SI{\sim30}{\micro\second} after beam injection into the ring), and the distribution is consider to affect $\omega_{a}$ when it is created during the fit time window, e.g. the E-field and pitch corrections.

The lost muon spin systematic effect is significant for the \runone~dataset, where it arises from the spin-momentum correlation due to the nonuniform spin-momentum distributions discussed in this section and the lost muon early-to-late effect,
\begin{equation}
	\frac{d\langle p\rangle}{dt}\neq0\nonumber
	\label{eq:mom_early_to_late_def_a},
\end{equation} 
which is caused by the loss of muons during the measurement period, see Section~\ref{sec:lost_muons} for more details. There is also the differential decay early-to-late effect,
\begin{equation}
	\frac{d\langle p\rangle}{dt}\neq0\nonumber
	\label{eq:mom_early_to_late_def_a},
\end{equation} 
which is caused by the muon lab-frame lifetime dependence on $\gamma$, where $\gamma$ is in turn dependent on $p$. The nonuniform spin-momentum distributions discussed in this section also couple with the differential decay early-to-late effect, which is negligible for the BNL level of statistics, but should be studied for the full Fermilab Experiment statistics.

\subsection{\label{sec:spin_res} Spin Resonances}
\textcolor{orange}{Jason}
\medskip

Blah blah blah ....

\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]{fig31.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]{fig31.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. \textcolor{red}{\textit{Harmonize notation; update with \runtwo~calo data; do we need to compare to \runone~data; add COSY curve.}}}\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]{fig32.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]{fig32.pdf}
	\caption{\textcolor{orange}{David R.} Vertical displacement of the centroid at the \SI{180}{\degree} tracker during the scraping cycle. \textcolor{red}{\textit{Check with tracker experts.}}}\label{fig:injvert}
\end{figure}
