
\section {Fast Rotation}
The contribution to the measured precession frequency of the electric field depends on the momentum distribution of the stored muon beam, or equivalently 
the equilibrium radial distibution. The muon revolution frequency is inversely proportional to its momentum. 
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 decay positron distribution. 
The positron counts from the 24 calorimeters are merged together with time offset of $T/24$/calorimeter where $T$ is the revolution period of the magic momentum muon.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/intensity_allcalo_4us_5us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/intensity_allcalo_4us_14us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/intensity_allcalo_4us_104us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/intensity_allcalo_4us_500us}}
\caption{Positron counts as a function of time as seen by all the calorimeters combined for the \runo\ \sixty\ for the time ranges: (a) 4-5, (b) 4-14, (d) 4-104 and
(e) \mbox{4-500 $\mu s$} with respect to the beam injection. The time interval is 1 ns.}
\label{fig:allcalo_intensity}
\end{figure}
\subsection{Fourier method}
The intensity profile of a distribution with zero emittance, zero momentum spread and zero bunch length, measured at a fixed point in the ring, is
\begin{equation}
I(t)=\sum_{n=0}^\infty \delta(t-t_0-nT) \label{eq:comb}
\end{equation}
where $n$ includes all nonnegative integers. (For the moment we ignore the finite lifetime of the muon.) If we extend Equation ~\ref{eq:comb} to include
all integers then $I(t)$ is symmetric about $t=t_0$ and the Fourier transform is simply a cosine transform.  The frequency spectrum is the cyclotron frequency 
$\omega_c = 2\pi/T$
and all of its harmonics.
\begin{equation}
F(\omega) = \sum_{n=-\infty}^\infty \cos n\omega T\label{eq:cosnwt}
\end{equation}
The cosine transform of the signal for $n\ge 0$, is equivalent to the Fourier transform of the signal that includes a symmetric extension to negative times. 
The revolution period is $T(1+\Delta)$ for particles with fractional momentum offset $\Delta$ and
$$I(t,\Delta) = \sum_n\delta(t-t_0-mT(1+\Delta)$$ and if $\rho(\Delta)$ is the distribution of momentum offsets,
$$S(t) = \sum_n\int(\rho(\delta)\delta(t-t_0-nT(1+\Delta) d\Delta$$
The fast rotation signal is linear in particle momentum, and the cosine transform linear in frequency so that
$$\hat S(\omega) = \int S(t)\cos\omega t =  \int \cos\omega t \sum_n\int(\rho(\delta)\delta(t-t_0-nT(1+\Delta) d\Delta dt$$
The distribution of muons injected into the ring extends over some finite time duration. Suppose that some fraction of the distribution
corresponds to $$S(t)_{early} = \sum\delta(t-t_0-\Delta t-nT).$$
$S(t)_{early}$ is not symmetric about $t=t_0$ and the fourier transform of the signal symmetrized about $t=t_0$ will include a sine as well as a cosine term.
However, if the injected distribution is symmetric about $t=t_0$ such that $S(t-t_0-\Delta t)_{late} = S(t-t_0+\Delta t)_{early}$, 
that is $$S(t) = \sum\delta(t-t_0-\Delta t-nT) + \sum\delta(t-nT) + \sum\delta(t-t_0+\Delta t-nT),$$
then the sine term vanishes once again. In practice we fit $t_0$ to be the time that best symmetrizes the frequency distribution.
\subsubsection{Start time}
The fast rotation signal is contaminated during the first few microseconds (a few tens of revolutions) of the fill by background from positrons, protons, and deuterons.
As a result $$S(t) = \sum_{n=n_{start}}\int \rho(\Delta)\delta(t-t_0-nT(1+\Delta))d\Delta$$ where $n_{start} \sim 25$, and unphysical frequencies
appear in the cosine transform. The particles were indeed circulating beginning at t0, but we werer unable to see them. We refer to the distortion of the spectrum that
arises from the gap in our measurement of the fast rotation signal as 'background'. 
We write $$\hat S(\omega) = \sqrt{\frac{2}{\pi}}\int_{t_s}^\infty S(t)\cos\omega t dt + \sqrt{\frac{2}{\pi}}\int_{t_0}^{t_s} S(t)\cos\omega t dt$$
where $S(t)$ is measured over the interval $t_s\rightarrow t_\infty$. Define
\begin{eqnarray*}
\hat S^\prime(\omega) &=& \sqrt{\frac{2}{\pi}}\int_{t_s}^\infty S(t)\cos\omega t dt\\
\Delta(\omega)&=&\sqrt{\frac{2}{\pi}}\int_{t_0}^{t_s} S(t)\cos\omega t dt
\end{eqnarray*}
To a first approximation 
$$S_0(t)=\int_{\omega_-}^{\omega_+} \hat S^\prime(\omega^\prime)\cos\omega^\prime t d\omega^\prime$$
where the limits of integration correspond to the range of momenta within the physical aperture of the ring.
Then 
\begin{eqnarray*}
\Delta(\omega)&=&\sqrt{\frac{2}{\pi}}\int_{t_0}^{t_s} S_0(t)\cos\omega t dt\\
&=&{\frac{2}{\pi}}\int_{t_0}^{t_s} \int_{\omega_-}^{\omega_+} \hat S^\prime(\omega^\prime)\cos\omega^\prime t\cos\omega t  d\omega^\prime dt\\
&=&{\frac{1}{\pi}} \int_{\omega_-}^{\omega_+} \hat S^\prime(\omega^\prime)\left(\frac{\sin(\omega^\prime+\omega)(t_s-t_0)}{\omega^\prime+\omega}
+\frac{\sin(\omega^\prime-\omega)(t_s-t_0)}{\omega^\prime-\omega}\right)  d\omega^\prime \\
&\sim&{\frac{1}{\pi}} \int_{\omega_-}^{\omega_+} \hat S^\prime(\omega^\prime)\frac{\sin(\omega^\prime-\omega)(t_s-t_0)}{\omega^\prime-\omega}  d\omega^\prime \\
\end{eqnarray*}
In practice we guess an appropriate functional form for $\hat S^\prime(\omega)$ and fit the parameters of the function to the background. 
If the spectrum is a Dirac comb at the revolution frequency then from Equation~\ref{eq:cosnwt} we see that a first approximation might be
$$\hat S^\prime(\omega)\sim \sum_{n=0}^{n_{start}}\cos n\omega T= \sum_{n=0}^{n_{start}}\cos n(\omega_c+\delta\omega)T= \sum_{n=0}^{n_{start}}\cos(n\delta\omega T) $$
As long as $n_{start}\delta\omega T\ll 1\rightarrow n_{start} $, then we can expand the cosine in an even power series, and in that limit we find that a polynomial gives a good fit to the background. 
The spread in frequencies is limited by the momentum acceptance of the storage ring to $\delta\omega/\omega_0 < \pm 0.6\%\rightarrow n_{start}\ll 25$ turns or $\sim 4\mu$s. 
More sophisticated functional forms have been demonstrated to give good background fits for start times up to $t_{start}\sim 25\mu$s in Monte Carlo simulation,
and wee find that 
the fitted spectrum
is very nearly independent of start time for $4\mu{\rm s}<t_{start}<25\mu$s.
\subsection{9 parameter fit}
In addition to revolution frequencies,  the intensity spectrum of the positrons will include the horizontal betatron oscillation frequency, the muon precession frequency, and the muon decay time.
In order to mitigate contamination of the fourier analysis of revolution frequencies, we divide the fast rotation data by a nine parameter fitted function
\begin{equation}
N(t) = N_0e^{-t/\tau_\mu}[1+A\cdot\cos(\omega_a t+\phi)]\cdot e^{-t/\tau_{cbo}}[1+A_{cbo}\cdot\cos(\omega_{cbo}t+\phi_{cbo})].
\end{equation}
where $N_0$ is the number of detected positrons at $t=0$, $\tau_\mu$ is the muon boosted life-time of about $64\mu$s, $A$ is the assymmetry of the spin precession modulation, $\omega_a$ the anomalous spin precession frequency and
$\phi$ the phase of the modulation,$A_{cbo}$, $\omega_{cbo}$ and $\tau_{cbo}$, and $\phi_{cbo}$ the amplitude, frequency, 'lifetime' and phase of the coherent betatron oscillation. 
The fitted function represents the low frequency part of the signal.   

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/fit_allcalo_4us_80us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/fit_allcalo_4us_130us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/fit_allcalo_4us_230us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/fit_allcalo_4us_500us}}
\caption{Fit of the positron counts histogram as a function of time as seen by all the calorimeters combined for the \runo\ \sixty\ for the time ranges: (a) 4-80, (b) 4-130, (d) 4-230 and
(e) 4-500 $\mu s$ with respect to the beam injection. The time interval is 149 ns.}
\label{fig:allcalo_fit}
\end{figure}

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{comment}
\begin{figure}[h]
	\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}[h]
	\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}
\end{comment}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/frs_4_5}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/frs_4_14}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/frs_4_54}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/frs_4_104}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/frs_4_204}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{figures/frs_4_500}}
\caption{Fast rotation signal as a function of time as seen by all the calorimeters combined for the \runo\ \sixty\ for the time ranges: (a) 4-5, (b) 4-14, (d) 4-54,
(e) 4-104, \mbox{(f) 4-204} and (g) 4-504 $\mu s$ with respect to the beam injection. The time interval is 1 ns. The modulation with a 35 $\mu s$ period corresponds to the
beam partially and slowly re-bunching due to its asymmetric momentum distribution. This could potentially also be a sign of time-momentum correlation in the incoming beam profile.}
\label{fig:allcalo_frs}
\end{figure}


\begin{figure}[bt]
\centering
\includegraphics[width=0.55\textwidth]{figures/bkgfit}
\caption{Optimum cardinal sine background fit to the cosine Frequency distribution.}
\label{fig:bkgfit}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{figures/CosineCollAperture}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{figures/CorrectedCosineCollAperture}}
\caption{Frequency distributions: (a) cosine Fourier transform, and (b) corrected cosine Fourier transform limited to the collimator aperture.}
\label{fig:freq}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{figures/Cosine}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{figures/CorrectedCosine}}
\caption{Frequency distributions: (a) cosine Fourier transform, and (b) corrected cosine Fourier transform for the full frequency range used in the analysis.}
\label{fig:freq2}
\end{figure}


\subsection{Momentum time correlation} 
In the above we assume that the momentum distribution in the captured muon pulse is uncorrelated with longitudinal position in the pulse.
If the momentum and time are uncorrelated, then the distribution is 'maximally' bunched when it enters the ring. The distribution debunches as the
low momentum muons at inner radii move ahead, and higher momentum muons at outer radii fall behind. The peak intensity of the fast rotation signal 
decreases with time. The behavior is symmetric with respect to time reversal. We imagine that the peak intensity likewise decreases moving
backwards in time. This symmetry, as noted above, is essential requirement of the fourier method. 

We find however, that momentum-time correlation is introduced by the injection kicker. The efficiency with which muons are captured in the ring
depends on momentum and the amplitude of the kicker. If the kicker field is low, acceptance is higher for high momentum muons. And if the kicker
field is high, lower momenta muons are favored. Since the magnetic field of the kicker varies over the time duration of the muon pulse,
so does the momentum of muons that are stored in the ring.

We depend on simulation to characterize the momentum-time correlation of the captured muons, and to estimate the systematic 
uncertainty in the measurement of the momentum distribution from the fast rotation analysis.  
