\section{Systematic uncertainties}
\label{sec:syst}

The philosophy of the systematic uncertainties estimation is to vary or turn on/off the various analysis parameters and take the value of the uncertainties as half the maximum
variation in the results in order to obtain a symmetric $\pm$ uncertainty.

\subsection{\tz\ systematic}

As shown in~\cite{tmcperf} Sec. 4.1, the sub-ns knowledge of the \tz\ parameter is essential in order to reach the dozens of ppb uncertainty on $C_E$.
Figure~\ref{fig:t0scan} shows how $x_e$, $\sigma$ and $C_E$ vary as a function of \tz.
The small change in $\sigma$ drives the limited change of 50 ppb in $C_E$. The large relative variation of $x_e$ does not significantly impact $C_E$
given that $x_e$ and $\sigma$ contribute with the same weight t0 the $C_E$ calculation and \mbox{$\sigma = 9.2 > x_e=-6.4$ mm}.
The source of uncertainty from \tz, as detailed in~\cite{tmcperf} Sec. 4.1, is estimated by performing its optimization, and then the remaining analysis, using 
three different fit functions for the background: cardinal sine, error function, triangle-based function. 
Table~\ref{tab:t0syst} shows the analysis results using the three functions. The systematic uncertainty is estimated as half the maximum difference in the 
results from the three background fit functions.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/xe_vs_t0}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/width_vs_t0}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/ce_vs_t0}}
\caption{Results of the fast rotation analysis as a function of \tz\ for a 1 ns range: (a) $x_e$, (b) $\sigma$, and (c) $C_E$.}
\label{fig:t0scan}
\end{figure}

\begin{table}
\begin{center}
\begin{tabular}{lcccc} \hline\hline
& $t_0$ [ns] & $x_{e}$ [mm] & $\sigma$ [mm] & $C_{E}$ [ppb]\\ \hline
cardinal sine           & 128.091 & 6.35 & 9.19 & -519 \\
error function          & 128.091 & 6.35 & 9.19 & -519 \\
triangle function       & 127.993 & 6.42 & 9.23 & -526 \\
\hline
systematic & 0.05 & 0.04 & 0.02 & 3.5 \\
\hline
\hline
\end{tabular}
\caption{Estimation of the \tz\ systematic uncertainty. The systematic uncertainty is estimated as half the maximum variation between the results from the three background fit functions.}
\label{tab:t0syst}
\end{center}
\end{table}

\clearpage

\subsection{\ts\ systematic}

In an ideal world, the fast rotation Fourier analysis would be performed from $t_s = t_0$ in order to avoid the complication due to missing early data (see Sec.~\ref{sec:tsopt}).
The ideal world would also allow to start the analysis at $t_s = 30~\mu s$, time at which the anomalous spin precession frequency analysis starts. 
The main reason for skipping the first 30 $\mu s$ is to leave time for the beam to stabilize after the scraping period ends at about 25 $\mu s$  
(the first-to-second step scraping transition happens at 7 $\mu s$). Scraping shifts the closed orbit of the beam both vertically and horizontally. Therefore
using data before 25 $\mu s$ might bias the reconstructed radial distribution that would end up not being fully representative of the distribution of the
stored beam after 25 $\mu s$. There are also other effects that are more important early time in the fill versus later in the fill: instantaneous pile-up rate, gain correction, muon loss.
It is thus essential to show that the fast rotation results change within an acceptable range between the earliest possible $t_s$ and $t_s = 30~\mu s$.

The \ts\ scan (see~\cite{tmcperf} Sec. 4.2) is performed using the triangle-based background fit function. The \tz\ value is fixed for all the \ts\ values to the one optimized
for the nominal analysis ($t_s = 4~\mu s$) given the earliest the start time the more the \tz\ optimization performs well. The definition of the background is fixed to the one defined 
by the nominal analysis. 
Figure~\ref{fig:tsscan} shows the results of the analysis as a function of \ts\ between 3 and 30 $\mu s$.
Figure~\ref{fig:tsscan2} shows the same results only up to 25 $\mu s$.
The statistical uncertainty on each point is not shown but is increasing
exponentially with time since the fast rotation statistics decreases exponentially (muon life-time).
Figure~\ref{fig:bkgfit_ts} shows the background fit using the triangle-based function for six different \ts\ values.
Figure~\ref{fig:tmc3_bkgfitinfo_ts} shows the information related to the background fit quality: $\chi^2$/d.o.f. and residuals.

The trend of the results cannot currently be explain with satisfaction. 
Tentative explanations could be made using arguments related to beam dynamics and scraping.
For instance, the first step scraping moves the radial closed orbit at early time and scrapes the radial tail of the beam, thus shifts the equilibrium radius and shrinks the
width of the beam. The second step then re-centers the beam and allows its width to grow and its equilibrium radius to move.
These kind of by-hand arguments need to be thoroughly investigated with full scale high statistics simulations using {\tt BMAD} and {\tt GM2RINGSIM}.
For now, and for the scope of estimating a systematic uncertainty, half the maximum variation in the results for the \ts\ scan from 3 to 25 $\mu s$ is used as the
systematic uncertainty and is under control for the \runo. The upper limit of 25 $\mu s$ is motivated by studies done with toy Monte Carlo simulations~\cite{tmcperf}
and might be sufficient since scraping is over by that time.
The last 5 $\mu s$, between 25 and 30 $\mu s$, to reach the analysis start time of the anomalous spin precession frequency remain for now un-accounted for.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/xe_vs_ts}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/width_vs_ts}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/ce_vs_ts}}
\caption{Results of the fast rotation analysis as a function of \ts\ for a 30 $\mu s$ range using the triangle-based background fit function: (a) $x_e$, (b) $\sigma$, and (c) $C_E$.} 
\label{fig:tsscan}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/xe_vs_ts2}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/width_vs_ts2}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/ce_vs_ts2}}
\caption{Results of the fast rotation analysis as a function of \ts\ for a 25 $\mu s$ range using the triangle-based background fit function: (a) $x_e$, (b) $\sigma$, and (c) $C_E$.}
\label{fig:tsscan2}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_ts5}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_ts10}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_ts15}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_ts20}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_ts25}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_ts30}}
\caption{Background fit of the cosine Fourier transform using the triangle-based function: (a) $t_s=5~\mu s$, (b) $t_s=10~\mu s$, (c) $t_s=15~\mu s$, (d) $t_s=20~\mu s$,
(e) $t_s=25~\mu s$, (f) $t_s=30~\mu s$.}
\label{fig:bkgfit_ts}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/chi2_vs_ts}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/residuals_vs_ts}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/chi2_vs_residuals}}
\caption{Background fit information for the triangle-based function: (a) $\chi^2$/d.o.f. as a function of \ts,
(b) fit residuals as a function of \ts, and (c) $\chi^2$/d.o.f. as a function of fit residuals.}
\label{fig:tmc3_bkgfitinfo_ts}
\end{figure}

\clearpage

\subsection{\tm\ systematic}
\label{sec:syst_tm}

As explained in~\cite{cornell} Sec. 7.3 and~\cite{tmcperf} Sec. 4.2, there is a trade-off increasing the length of the 
fast rotation signal between improving the resolution and adding increasing noise.
The interest of performing a \tm\ scan is to optimize this trade-off.
Figure~\ref{fig:tmscan} shows the fast rotation analysis results as a function of \tm\ for \ts\ fixed to $t_s=4~\mu s$. 
Figure~\ref{fig:tmscan2} shows the same information but for $t_m>150~\mu s$.
For each \tm\ value, the analysis is performed with its nominal configuration.
Figure~\ref{fig:tmscan_res} shows the background fit residuals and $\chi^2$ per degrees of freedom as a function of \tm.
Figure~\ref{fig:tmscan_res2} shows the same information but for $t_m>150~\mu s$.
Overall, the results appear the most stable for \tm\ values between 150-300 $\mu s$. Before 150 $\mu s$, the resolution
is too low and leads to a poor fidelity of the frequency distribution. After 300 $\mu s$, the increasing statistical
noise of the fast rotation signal (see Fig.~\ref{fig:allcalo_frs}(f)) distorts the cosine Fourier transform.
Figure~\ref{fig:bkgfit_tm} shows the background fit to the cosine Fourier transform for six values of \tm.
The systematic uncertainty is taken as half the maximum variation of the results between 150 and 300 $\mu s$.
The variation in the results might mainly be due to spectral leakage given the small size of the statistical uncertainty
and the very high statistical correlation between consecutive scan data points.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/xe_vs_tm}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/width_vs_tm}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/ce_vs_tm}}
\caption{Results of the fast rotation analysis for $t_s=4~\mu s$ as a function of \tm:
(a) $x_e$, (b) $\sigma$, and (c) $C_E$.}
\label{fig:tmscan}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/xe_vs_tm2}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/width_vs_tm2}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/ce_vs_tm2}}
\caption{Results of the fast rotation analysis for $t_s=4~\mu s$ as a function of \tm\ for $t_m>150$ $\mu s$:
(a) $x_e$, (b) $\sigma$, and (c) $C_E$.}
\label{fig:tmscan2}
\end{figure}


\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/chi2_vs_tm}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/residuals_vs_tm}}
\caption{Background fit quality as a function of \tm: (a) $\chi^2$/d.o.f., and (b) fit residuals.}
\label{fig:tmscan_res}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/chi2_vs_tm2}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/residuals_vs_tm2}}
\caption{Background fit quality as a function of \tm\ for $t_m>150$ $\mu s$: (a) $\chi^2$/d.o.f., and (b) fit residuals.}
\label{fig:tmscan_res2}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/cosine_tm50}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/cosine_tm100}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/cosine_tm150}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/cosine_tm200}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/cosine_tm400}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/cosine_tm500}}
\caption{Background fit to the cosine Fourier transform for different \tm\ values: (a) 50, \mbox{(b) 100}, (c) 150, (d) 200, (e) 400, and (f) 500 $\mu s$.} 
\label{fig:bkgfit_tm}
\end{figure}

\clearpage

\subsection{Frequency interval}

As explained in~\cite{cornell} Sec. 7.3 and~\cite{tmcperf} Sec. 4.4, the nominal frequency interval used to produce the cosine Fourier transform
is 2 kHz. This 2 kHz value is smaller than the intrinsic frequency resolution that is set by the number of bins and time interval of the fast
rotation signal. For $t_s=4~\mu s$, $t_m = 300 ~\mu s$, and a 1 ns time interval, the intrinsic frequency resolution is:

\begin{align*}
1/(\text{time~interval} \times \text{number~of~bins})= 1/(10^{-9} \times 296000)=3.34~\text{kHz}.
\end{align*}

Using a frequency interval of 2 kHz leads to over-sampling responsible for the modulation seen in Fig.~\ref{fig:bkgfit_tm}(a) for instance.
Over-sampling was shown in toy Monte Carlo studies (see \cite{tmcperf} Sec. 4.4) to be sound. Nonetheless, a frequency interval scan is
performed to ensure the same behavior in data. 
Figure~\ref{fig:freqscan} shows the fast rotation results as a function of frequency interval. 
The allowed values for the frequency interval are integer numbers of the frequency window used for the cosine Fourier transform.
The results appear stable for a frequency interval up to 2.5 kHz. This behavior is the same as the one observed in the toy Monte Carlo studies (see~\cite{tmcperf} Sec. 4.4).
The systematic uncertainty is taken as half the maximum variation in the results for the upper limit of the frequency interval of 2.5 kHz.
Above 2.5 kHz, the fidelity of the frequency/radial distribution cannot be restored well enough due to the small number of data points.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/xe_vs_df}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/width_vs_df}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/ce_vs_df}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/t0_vs_df}}
\caption{Results of the fast rotation analysis as a function of the frequency interval:
\mbox{(a) $x_e$,} (b) $\sigma$, and (c) $C_E$.}
\label{fig:freqscan}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_df025}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_df1}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_df15}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_df25}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_df30}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_df375}}
\caption{Background fit to the cosine Fourier transform for different frequency interval values: (a) 0.25, (b) 1.0, (c) 1.5, (d) 2.5, (e) 3.0, and (f) 3.75 kHz.}
\label{fig:bkgfit_freqscan}
\end{figure}

\clearpage

\subsection{Background}

The definition of the background is key for optimizing \tz\ and correcting the cosine Fourier transform.
The functional form of the background is already part of the \tz\ systematic uncertainty. The other sources of uncertainty from the background
have to do with how it is defined, and how its statistical fluctuation can affect the analysis results via a lever arm effect, the data point
being far from the mean of the distribution.

\subsubsection{Background definition}

The background of the cosine Fourier transform is defined as the data point that are within $\pm N \cdot \sigma_{bkg}$ of the fit with $\sigma_{bkg}$ being the statistical noise
of the background estimated from the fit residuals for the optimal \tz\ value. The parameter that can be varied is $N$, its nominal value being $N=2$. The larger $N$ the closer
to the mean of the distribution the background will get.
Figure~\ref{fig:bkgfit_n1n5} shows the background fit for two values of $N$: $N=1$ and $N=5$.
Figure~\ref{fig:bkgth_scan} shows the results of the background threshold scan.
Similarly to what is observed in toy Monte Carlo studies (see \cite{tmcperf} Sec. 4.5), the width of the radial distribution decreases with $N$ increasing. 
It is due to the fact that the background definition moves closer to the mean of the distribution for increasing $N$.
It results in the background fit to be slightly shifted upward to account for the extra background data points, and therefore the corrected frequency 
distribution is shifted downward slightly, reducing the width in consequence.
The systematic uncertainty is taken as half the maximum variation observed in the scan results. 

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_th1}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgfit_th5}}
\caption{Background fit to the cosine Fourier transform for two background definition thresholds: (a) $N=1$, and (b) $N=5$.}
\label{fig:bkgfit_n1n5}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/xe_vs_t0bkgth}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/width_vs_t0bkgth}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/ce_vs_t0bkgth}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/t0_vs_t0bkgth}}
\caption{Results of the fast rotation analysis as a function of the background threshold definition $N$:
(a) $x_e$, (b) $\sigma$, (c) $C_E$, and (d) \tz.}
\label{fig:bkgth_scan}
\end{figure}

\subsubsection{Background removal}

The statistical fluctuation in the tail (background) of the radial distribution can bias the extraction of the equilibrium radius and width given a potential large lever arm effect.
This effect is addressed by removing (zero-ing out) the background from the distribution.
The nominal fast rotation analysis is performed once with $N=2$. The value of $N$ is then varied and the data point within $\pm N \cdot \sigma_{bkg}$ of the nominal fit function 
(i.e. the data point tagged ``background'') are removed.
Figure~\ref{fig:bkgrmv} shows the radial distributions for $N=1,~5$ with background removal.
Figure~\ref{fig:bkgrmvth_scan} shows the results of the background removal scan.
The systematic uncertainty is taken as half the maximum variation observed in the scan results.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgrmvlth1}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/bkgrmvlth5}}
\caption{Radial distribution with background removal for: (a) $N=1$, and (b) $N=5$.}
\label{fig:bkgrmv}
\end{figure}

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/xe_vs_bkgrmvlth}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/width_vs_bkgrmvlth}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/ce_vs_bkgrmvlth}}
\caption{Results of the fast rotation analysis as a function of the background removal threshold $N$:
(a) $x_e$, (b) $\sigma$, and (c) $C_E$.}
\label{fig:bkgrmvth_scan}
\end{figure}

\clearpage

\subsection{Wiggle fit}
\label{sec:syst_fit}

The nominal fit to the positron counts histogram is the 9-parameter fit: muon life-time, anomalous spin precession and CBO modulation.
The anomalous spin precession frequency performs a fit to the positron counts histograms with many more parameters: pile-up, vertical waist, muon loss etc.
In order to estimate the importance of the accuracy of the fit to the data, the analysis is performed on fast rotation signals
produced using a different number of parameters in the wiggle fit: 2 (muon life-time only), 5 (muon life-time and anomalous spin precession) and 9
(muon life-time, anomalous spin precession and CBO modulation).
Figure~\ref{fig:syst_wiggle} shows the radial distribution for the three cases. The results appear to be consistent within the statistical uncertainty.
This is not surprising because both the anomalous spin precession and CBO modulation frequencies (and their beating/aliasing frequencies) are not overlapping
with the frequency range of the cyclotron revolution. It is important to note that fitting at least for the muon life-time is essential to avoid the artificial
broadening of the cyclotron frequency peak due to the exponential decay in the fast rotation spectrum\footnote{The fast rotation exponential feature will
enter the Fourier transform calculation and will yield a frequency broadening.}.
One question remains: what of the potential effects of pile-up, muon loss and gain correction not included in the fit? The \ts\ scan provides a good handle on it
because the size of the pile-up, muon loss and gain correction effects significantly decreases over the first 30 $\mu s$ of the fill and therefore are part of
the systematic uncertainty associated with the \ts\ scan.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/rad_wiggle2p}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/rad_wiggle5p}}
\subfigure[]{\includegraphics[width=0.55\textwidth]{fig/rad_wiggle9p}}
\caption{Radial distributions for fast rotation signals produced from a wiggle fit with: \mbox{(a) 2 parameters,} (b) 5 parameters, and (d) 9 parameters.}
\label{fig:syst_wiggle}
\end{figure}

\clearpage

\subsection{Positron energy threshold}
\label{sec:syst_eth}

The nominal positron energy threshold used to produce the positron counts histogram is \mbox{1,500 MeV}.
Given the anomalous spin precession frequency analysis is performed for different energy threshold values and with different methods (T-method, A-method, E-method, Q-method etc.),
it is important to perform the fast rotation analysis as a function of the energy threshold. 
The nominal fast rotation analysis is therefore performed\footnote{The only exception being the background definition being fixed to avoid the associated systematic effect.} 
for positron counts histograms produced with a wide range of positron energy threshold. Appendix~\ref{app:frs_eth} shows fast rotation histograms for different positron energy thresholds.
Figure~\ref{fig:eth_scan} shows the results of the fast rotation analysis as a function of the positron energy threshold.
The statistical uncertainty on each data point is not shown, but increases with increasing energy threshold.
There is a clear trend of $x_e$ and $\sigma$ increasing with the energy threshold. This effect is believed to be related
to a calorimeter acceptance effect, and not to a beam dynamics effect. It needs to be thoroughly studied with
full scale high statistics simulations with {\tt BMAD} and {\tt GM2RINGSIM}. The value of the systematic uncertainty
is taken as half the maximum variation and is under control for this \runo\ data set.

\begin{figure}[bt]
\centering
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/xe_vs_eth}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/width_vs_eth}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/ce_vs_eth}}
\subfigure[]{\includegraphics[width=0.49\textwidth]{fig/t0_vs_eth}}
\caption{Results of the fast rotation analysis as a function of the positron energy threshold:
(a) $x_e$, (b) $\sigma$, (c) $C_E$, and (d) \tz.}
\label{fig:eth_scan}
\end{figure}

\clearpage

\subsection{Summary}

The correlation between the various systematic uncertainties has not been estimated.
Table~\ref{tab:syst} summarizes the estimation of the various systematic uncertainties and the corresponding total assuming
0\% correlation (quadratic sum) and 100\% of correlation (linear sum) between all the systematic uncertainties.
The final systematic uncertainty numbers are taken as the average of the 0\% and 100\% correlation cases.

\begin{table}
\begin{center}
\begin{tabular}{lcccc} \hline\hline
& $t_0$ [ns] & $x_{e}$ [mm] & $\sigma$ [mm] & $C_{E}$ [ppb]\\ \hline
\tz\ & 0.05 & 0.04 & 0.02 & 4 \\ %done
\ts\ & -- & 0.06 & 0.04 & 6 \\ %done
\tm\ &  0.13 & 0.11 & 0.09 & 10 \\ %done
\tz$\pm T_c$ & -- & 0.02 & 0.07 & 5 \\ %done
frequency interval & 0.02 & 0.03 & 0.05 & 3 \\ %done
background definition & 0.08 & 0.06 & 0.11 & 8 \\ %done
background removal & -- & 0.03 & 0.06 & 5 \\ %done
wiggle fit & 0.01 & 0.01 & 0.01 & 1 \\ %done
energy threshold & 0.08 & 0.15 & 0.04 & 10 \\ %done
\hline
systematic (0\% correlation)    & 0.18 & 0.24 & 0.19 & 19 \\
systematic (100\% correlation)  & 0.37 & 0.51 & 0.49 & 52 \\
\hline
\hline
\end{tabular}
\caption{Summary of the estimation of the systematic uncertainties and the total uncertainty
assuming 0\% and 100\% correlation..}
\label{tab:syst}
\end{center}
\end{table}
