\chapter{Results and discussions}
\label{cha:results_and_discussions}
\section{Verification of the experimental method}
\label{sec:verification_of_the_experimental_method}
\subsection{Number of stopped muons calculation}
\label{sub:number_of_stopped_muons_normalisation}
The number of stopped muons calculated from the muonic X-ray spectrum is shown
to be consistent with that calculated from the active target spectrum. This
proves the validity of normalisation using muon X-ray measurement.
\subsection{Particle identification and unfolding}
\label{sub:particle_identification_and_unfolding}
The particle identification using specific energy loss using cut on
likelihood probability is shown in
\cref{sub:event_selection_for_the_passive_targets}. Since the distribution of
$\Delta E$ at a given $E$ is not Gaussian, the fraction of protons that do not
make the cut is 0.5\%, much larger than the threshold at \num{1E-4}. However,
that missing fraction is small compared to the statistical uncertainty of the
measurement (2.3\%) so the threshold is sufficient. 

The observed spectra on the two silicon arms reflect the muon stopping
distribution discussed in \cref{sub:momentum_scan_for_the_100_} where more
muons stopped at the downstream side of the target. The proton yields
calculated from two arms are consistent with each other, and show that the muon
stopping distribution used to generate the response matrices is reasonable.

\section{Emission rate of protons and the COMET Phase I's CDC}
\label{sec:emission_rate_of_protons_and_the_comet_phase_i_s_cdc}
The proton emission rate from the 100-\si{\um} aluminium target is 
$(3.5 \pm 0.2)$\%. This rate is significantly larger than the calculation rate
of 0.97\% by Lifshitz and Singer~\cite{LifshitzSinger.1978, LifshitzSinger.1980}.
The $(\mu^-,\nu p):(\mu^-,\nu pn)$ ratio is then roughly 1:1, not 1:6 as in
\eqref{eqn:wyttenbach_ratio}.
The rate smaller that the proton emission rate from silicon of
5.3\%~\cite{Measday.2001} which is expected since an odd-odd nucleus as
$^{28}$Al is less stable than an even-odd one.

For the COMET Phase I experiment, the emission rate of 3.5\% is about 5 times
smaller than the figure using to design the CDC. The measured spectrum shape
peaks around \SI{4}{\MeV} rather than \SI{2.5}{\MeV} in the silicon
spectrum(\cref{fig:sobottka_spec}). Therefore the proton hit rate on the CDC
should be smaller than the current estimation.

The CDC proton hit rate is calculated by a toy MC study. The protons with the
energy spectrum as the parameterisation in \cref{sub:proton_emission_rate} are
generated inside the COMET's muon stopping targets which are 17
200-\si{\um}-thick aluminium discs. A proton absorber made of CFRP is placed
\SI{5}{\cm} far from the inner wall of the CDC.
A muon stopping rate of \SI{1.3E9}{\Hz} is assumed as in the COMET Phase I's
TDR. The number of proton emitted is then $\num{1.3E9} \times 0.609 \times
0.035 = \SI{2.8E7}{\Hz}$. The hit rates on a single cell in the inner most
layer due to these protons with 
different absorber thickness are shown in \cref{tab:proton_cdc_hitrate}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{l r}
      \toprule
      \textbf{Absorber thickness} & \textbf{Hit rate}\\
      \midrule 
      \SI{1}{\mm} & \SI{2}{\Hz}\\
      \SI{0.5}{\mm} & \SI{126}{\Hz}\\
      \SI{0}{\mm} & \SI{1436}{\Hz}\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{CDC proton hit rates}
  \label{tab:proton_cdc_hitrate}
\end{table}

The proton hit rate even without the absorber is only \SI{1.4}{\kHz}, much
smaller than the current estimation of \SI{11}{\kHz} (using 1-mm-thick
absorber). Therefore a proton absorber is not needed for the COMET Phase I's
CDC.