\clearpage
\section{Nominal analysis}
\label{sec:nominal}

This section will detail the nominal analysis of the \runo\ \sixty. Section~\ref{sec:stat} and~\ref{sec:syst} will present respectively
the statistical and systematic uncertainties estimation. The details of the analysis can be found in~\cite{cornell}.

\subsection{Choice of the \ts\ parameter}
\label{sec:tsopt}

The \ts\ parameter is the start time of the analysis. The ideal case would be $t_s=t_0$ where \tz\ corresponds to the time when the centroid of the longitudinal
profile of the beam is detected by calorimeter \#1 right after injection (first turn of the beam into the ring). This ideal scenario is unfortunately not possible
for two reasons. The first is the saturation of the calorimeter electronics during the first $\mu s$ of the fill due to the high intensity of the incoming beam.
The second is the contamination by beam-line positrons of the incoming muon beam. The positrons are lost due to synchrotron radiation after about 3-4 $\mu s$.
Figure~\ref{fig:frs_0_10} shows the fast rotation signal for the first 10 $\mu s$. The first $\mu s$ is not available due to the saturation, and the signal stabilizes
at 3-4 $\mu s$ after the positrons are lost.
The \ts\ value, because of the reasons explained above, is set to $t_s=4~\mu s$. This value is slightly optimized such as the optimized \ts\ value corresponds to
the fast rotation signal intensity of 1. This is done in order to minimize the effect from spectral leakage (see~\cite{cornell} Sec. 7.1 and 7.2).
The optimized value is $t_s =3.9765~\mu s$.

\begin{figure}[bt]
\centering
\includegraphics[width=0.6\textwidth]{fig/frs_0_10}
\caption{Fast rotation signal as a function of time as seen by all the calorimeters combined for the \runo\ \sixty\ for the time ranges 0-10 $\mu s$ with respect to the beam injection. 
The time interval is 1 ns.}
\label{fig:frs_0_10}
\end{figure}

\subsection{Choice of the \tm\ parameter}

The \tm\ parameter is the end time of the analysis. The nominal choice for the \sixty\ is $t_m=300~\mu s$. This value is optimized
performing a \tm\ scan (see Sec.~\ref{sec:syst_tm}). As explained in~\cite{cornell} Sec. 7.3 and in~\cite{tmcperf} Sec. 4.3, there is a trade off
increasing the length of the fast rotation signal between improving the frequency resolution and adding exponentially growing
statistical noise at late time. The \tm\ scan allows to optimized this trade off. The exact value of \tm\ is optimized in the
same fashion as \ts\ in order to match the fast rotation signal intensity with 1.

\subsection{Choice of the \tz\ parameter}

The \tz\ parameter corresponds to the time when the centroid of the longitudinal profile of the beam is detected by calorimeter \#1 
right after injection (first turn of the beam into the ring). Given the saturation and the beam-line positron contamination, the data
corresponding to the first turn is not recorded. The \tz\ value therefore needs to be optimized. The iterative optimization procedure is explained in~\cite{cornell} Sec. 6.
It relies on a $\chi^2$ minimization fitting for the background of the cosine Fourier transform of the fast rotation signal.
Figure~\ref{fig:t0opt} shows the results of the four iterations: the optimum background fit for each iteration and the $\chi^2$ distribution of the background fit as a function of \tz.
After the fourth iterations, the optimized \tz\ value is 128.091 ns. This value is consistent with the work presented in~\cite{aaron}.
Figure~\ref{fig:bkgfit} shows the optimum background fit after the \tz\ optimization procedure.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.37\textwidth]{fig/tz1_cosine}}
\subfigure[]{\includegraphics[width=0.37\textwidth]{fig/tz1_chi2}}
\subfigure[]{\includegraphics[width=0.37\textwidth]{fig/tz2_cosine}}
\subfigure[]{\includegraphics[width=0.37\textwidth]{fig/tz2_chi2}}
\subfigure[]{\includegraphics[width=0.37\textwidth]{fig/tz3_cosine}}
\subfigure[]{\includegraphics[width=0.37\textwidth]{fig/tz3_chi2}}
\subfigure[]{\includegraphics[width=0.37\textwidth]{fig/tz4_cosine}}
\subfigure[]{\includegraphics[width=0.37\textwidth]{fig/tz4_chi2}}
\caption{Results of the four iterations (from top to bottom) of the \tz\ optimization procedure. The figures on the left show the cosine Fourier transform
with its cardinal sine background fit for the optimum \tz\ value. The figures on the right show the $\chi^2$ distribution of the background fit as a function of \tz.
The optimized \tz\ value after four iterations is 128.091 ns.}
\label{fig:t0opt}
\end{figure}

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

\subsection{Frequency distribution}

Once the \tz\ optimization is performed, the optimum cosine Fourier transform is available. 
It can be corrected for its background using the background fit. 
The correction corresponds to subtracting the background fit to the cosine Fourier transform. 
Figure~\ref{fig:freq} shows the cosine Fourier transform and its corrected version limited to the collimator aperture.
Figure~\ref{fig:freq2} shows the zoomed-out version for the full frequency range of 150 kHz used for the analysis.
The measured average and width of the cyclotron frequency distribution are respectively \mbox{6,698.99 kHz} and \mbox{8.65 kHz}.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/CosineCollAperture}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/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]{fig/Cosine}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/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{Radial distribution}

The frequency distribution can be converted to the radial distribution as explained in~\cite{cornell} \mbox{Sec. 8}.
Figure~\ref{fig:rad_beam} shows the radial distribution in the beam coordinate (the so-called ``magic radius'' 
of 7112 mm corresponds to r=0, positive (negative) r values correspond to radially outward (inward) positions) limited to the collimator aperture ($\pm$45 mm).
Figure~\ref{fig:rad_ring} shows the same distribution in the ring coordinate limited and not limited to the collimator aperture.
The equilibrium radius (average of the radial distribution) is 6.35 mm and the width of the distribution is 9.19 mm.

\begin{figure}[bt]
\centering
\includegraphics[width=0.99\textwidth]{fig/RadialBeamCoordinate}
\caption{Radial distribution in the beam coordinate limited to the collimator aperture.}
\label{fig:rad_beam}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/RadialCollAperture}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/Radial}}
\caption{Radial distribution in the ring coordinate: (a) limited to the collimator aperture, and (b) not limited to the collimator aperture but to the full range of the analysis.}
\label{fig:rad_ring}
\end{figure}

\subsection{Electric field correction estimation}

The electric field correction is estimated from eq. (\ref{eq:efield}), given the radial distribution in Fig.~\ref{fig:rad_beam}, to be $C_E=-519$ ppb.
The field index in eq. (\ref{eq:efield}) is expressed (for the continuous quad approximation) as:

\begin{equation}
n = 1-\nu_x^2,
\end{equation}

where $\nu_x$ is the radial tune expressed as:

\begin{equation}
\nu_x = 1- f_{cbo} / f_c,
\end{equation}

where \mbox{$f_{cbo}=413.66$ kHz} is the CBO frequency extracted from the 9-parameter wiggle fit and $f_c=6,698.99$ kHz is the average cyclotron frequency.
The resulting field index used to estimate the electric field correction is $n=0.1197$. This value is in very good agreement with the field index measurement done by
the tracker and calorimeter teams.

\clearpage
