\section{Fast Rotation signal}

The details regarding how to produce the fast rotation signal can be found in~\cite{cornell} Sec. 3.

\subsection{Positron counts histogram}

The input to the fast rotation analysis is the same as the input to the anomalous spin precession frequency analysis: a histogram of the positron 
counts versus the time in the fill. This histogram was produced for each of the 24 calorimeters, and for each of the 8 bunches in the accelerator cycle.
The nominal value of the positron energy threshold is 1,500 MeV. 
The choice of the energy threshold will be a source of systematic uncertainty (see Sec.~\ref{sec:syst_eth}). 

\subsection{Combination of the 24 calorimeters}

The positron counts histograms of the 24 calorimeters are merged together for the nominal analysis (the 8 accelerator bunches are merged together for each calorimeter)
The histograms from calorimeters \#2 to \#24 are added to the histogram of calorimeter \#1 (taken arbitrarily as the reference)
time shifting them by $(\#-1) \times T_c/24$ where \# is the calorimeter number and $T_c$ the nominal cyclotron period 
of 149.14 ns corresponding to the so-called ``magic momentum''. The value of the cyclotron period is updated to the measured
value after completing the first round of the analysis and the analysis is performed again. The small variation of the cyclotron
frequency results in a small variation of the time shift constant (well below the ns level) and therefore yields a negligible change
in the fast rotation results.
Figure~\ref{fig:allcalo_intensity} shows the positron counts histogram for all the 24 calorimeters combined.
The time interval of the histogram is 1 ns.
The analysis is also performed per calorimeter and per bunch as presented in~\ref{sec:per_calobunch}.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{figure/intensity_allcalo_4us_5us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/intensity_allcalo_4us_14us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/intensity_allcalo_4us_104us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/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{Wiggle fit}

It is necessary to fit the positron counts histogram in order to factor out at the very least the muon life-time (exponential decay).
Section~\ref{sec:syst_fit} will show that the results change little fitting for more than the muon life-time.
The default fit is the 9-parameter fit that includes the muon life-time, anomalous spin precession and CBO modulation:

\begin{equation}
        N(t) = N_{0} \cdot e^{-t/\tau_{\mu}}[1+A \cdot cos(\wa 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 positron at $t=0$, $\tau_{\mu}$ is the muon boosted life-time of about 64 $\mu s$, $A$ called the asymmetry 
is the amplitude of the anomalous spin precession modulation, \wa\ the anomalous spin precession frequency (or spin tune), $\phi$ 
the phase of the modulation, $\tau_{cbo}$ the CBO life-time, $A_{cbo}$ the amplitude of the CBO modulation, $\omega_{cbo}$ the frequency
of the CBO modulation, and $\phi_{cbo}$ the phase of the modulation.

Figure~\ref{fig:allcalo_fit} shows the fit of the positron counts histogram starting at 30 $\mu s$ with respect to the beam injection.
The histogram was re-binned to a time interval of 149 ns to average out in good approximation the fast rotation feature.
Appendix~\ref{app:fitres} shows the fit residuals for different time ranges.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/fit_allcalo_4us_80us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/fit_allcalo_4us_130us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/fit_allcalo_4us_230us}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/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}

\subsection{Fast rotation signal}

The fast rotation signal is obtained by dividing out the fit function, given proper normalization to account for the 1 ns versus 149 ns time intervals, 
from the original positron counts histogram.
Figure~\ref{fig:allcalo_frs} shows the fast rotation signal for different time ranges.
The appendices~\ref{app:frbunches} and~\ref{app:frcalos} show the various fast rotation histograms for each calorimeter and each bunch.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/frs_4_5}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/frs_4_14}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/frs_4_54}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/frs_4_104}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/frs_4_204}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/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}
