\chapter{Introduction}
\label{chap:intro}

%% Restart the numbering to make sure that this is definitely page #1!
\pagenumbering{arabic}

\section{Muon to electron conversion}
\label{sec:_mu_e_conversion}
Charged lepton flavor violation (CLFV) belongs to the class of flavor-changing
neutral currents, which are suppressed at tree level in the Standard Model
(SM) where they are mediated by $\gamma$ and $Z^0$ bosons, but arise at loop
level via weak charged currents mediated by the $W^{\pm}$ boson. Because flavor
violation requires mixing between generations, CLFV exactly vanishes in the SM
with massless neutrinos. Even in the framework of the SM with massive neutrinos
and their mixing, branching ratio of CLFV is still very small - for example, in
case of \mueg~\cite{marciano}:
\begin{equation}
	\mathcal{B}(\mu^{+} \rightarrow e^{+}\gamma) \simeq 
	10^{-54} \left( \frac{sin^{2}2\theta_{13}}{0.15}\right)
\end{equation}
This is an unobservably tiny
branching ratio so that any experimental evidence of CLFV would be a clear
sign of new physics beyond the SM.

One of the most prominent CLFV processes is
a process of coherent muon-to-electron conversion ($\mu
- e$ conversion) in the field of a nucleus: \muecaz. When muons are stopped in
a target, they are quickly
captured by atoms ($~10^{-10}$ s) and cascade down to the 1S orbitals. There,
they can undergo:
(a) ordinary decay, (b) weak capture, $\mu^- p \rightarrow \nu_\mu n$, or (c)
$\mu - e$ conversion, \muec. The last of these reactions is a CLFV process
where lepton flavor numbers, $L_\mu$ and $L_e$, are violated by one unit.

The $\mu - e $ conversion is attractive both from theoretical and experimental
points of view. Many extensions of the SM predict that it would has sizeable
branching ratio~\cite{altman}. One possible supersymmetric contribution to the
$\mu - e$ conversion is shown in Fig.~\ref{fig:susy_contr}. Experimentally, the
simplicity and distinctive signal, a mono-energetic electron of energy $E_{e}$:
$	
	E_{e} = m_{\mu} - B_{\mu}(Z, A) - R(A) \simeq \textrm{105 MeV},
$
where $m_\mu$ is the muon mass, $B_\mu(Z, A)$ is the muonic atom binding
energy, and $R(A)$ is the nuclear recoil energy, allow experimental searches
without accidentals and thus in extremely high rates. As a result, one of the
best upper limits of CLFV searches comes from a search for $\mu - e$ conversion
in muonic gold done by the SINDRUM--II collaboration:
\sindrumlimit~\cite{sindrumii}.

\begin{figure}[tbh]
	\centering
	\includegraphics[width=\textwidth]{figs/susy_contr}
	\caption{Possible SUSY contributions to the CLFV processes \mueg
	(left) and \muec (right).}
	\label{fig:susy_contr}
\end{figure}

%\section{Motivation}
%\label{sec:motivation}

\subsection{COMET experiment}

At the Japan Proton Accelerator Research Complex (J-PARC), an experiment to
search for \muec~conversion, which is called COMET (COherent Muon to Electron
Transition), has been proposed~\cite{comet07}. The experiment received Stage--1
approval in
2009. Utilising a proton beam of 56 kW (8 GeV $\times$ 7 $\mu$A) from the
J-PARC main ring, the COMET aims for a single event sensitivity of $3 \times
10^{-17}$, which is 10000 times better than the current best
limit at SINDRUM--II. As of April 2013, the COMET collaboration has 117
members in 27 institutes from 12 countries.

The COMET experiment is designed to be carried out at the Hadron
Experimental Facility using a bunched proton beam that is
slowly-extracted from the J-PARC main ring. The experimental set-up consists of
a dedicated proton beam line, a muon beam transport section, and a detector
section. The muon beam section is composed of superconducting magnets: pion
capture solenoid and a pion/muon transport solenoid. The
detector section has a multi-layered muon stopping target, an electron
transport beam line for $\mu - e$ conversion signals,
followed by detector systems. 

The COMET collaboration has adopted a staging approach with two
phases~\cite{comet12}. COMET Phase--I  is scheduled to
have an engineering run in 2016, followed by a physics run in 2017. Phase--I
should achieve a sensitivity
of $3 \times 10^{-15}$, 100 times better than that of SINDRUM--II; while
Phase--II will reach a sensitivity of $2.6 \times 10^{-17}$, which is
competitive with the Mu2e project at Fermilab~\cite{mu2e08}.
A schematic layout of the COMET experiment with its two phases is
shown in Fig.~\ref{fig:comet_phase1}, and a schedule for two phases is shown in
Fig.~\ref{fig:sched}.
\begin{figure}[tbh]
	\centering
\includegraphics[width=\textwidth]{figs/comet_phase1}
\caption{Schematic layout of the COMET experiment with two phases: Phase--I
(left) and Phase--II (right).}
\label{fig:comet_phase1}
\end{figure}

\begin{figure}[tbh]
	\centering
\includegraphics[width=0.8\textwidth]{figs/sched}
\caption{The anticipated schedule of the COMET experiment.}
\label{fig:sched}
\end{figure}

COMET Phase--I has two major goals:
\begin{itemize}
	\item Background study for the COMET Phase--II by using the actual COMET beam
		line constructed at Phase--I, 
	\item Search for $\mu-e$ conversion with a single event sensitivity of $3
		\times 10^{-15}$.
\end{itemize}

In order to realize the goals, COMET Phase--I proposes to have two systems of
detector. A straw tube detector and an electromagnetic calorimeter will be used
for the background study. For the $\mu-e$ conversion search, a cylindrical
drift chamber (CDC) will be built.

\subsection{Proton emission issue}
We, as a jointed force between Mu2e and COMET, would like to measure rates and
energy spectrum of charged particle emission after nuclear muon capture on
aluminum. The rates and spectra of charged particle emission, in particular
protons, is very important to optimize the detector configuration both for the
Mu2e and COMET Phase-I experiments. 

\noindent The tracking chambers of COMET Phase-I and Mu2e are designed to be
measure charged particles of their momenta greater than 70 MeV/$c$ and 53
MeV/$c$ respectively. In that momentum ranges, it turns out that single hit
rates of the tracking chambers would be dominated by protons after nuclear muon
capture.
The second source of the hit rate will be electrons from muon decays in orbit
(DIO). In order to limit the single hit rate of the tracking chamber to an
acceptable level, both experiments are considering to place proton absorbers in
front of the tracking chambers to reduce proton hit rates. However, the proton
absorber would deteriorate the reconstructed momentum resolution of electrons
at birth. And similarly the rate of proton emission is important to determine
thickness of the muon stopping target made of aluminum. Therefore it is
important to know the rate so that the detector system can be optimized in
terms of both hit rate and momentum resolution.

\noindent Unfortunately the yield, energy spectrum and composition of the
charged particles emitted in muon capture on Al and Ti have not been measured
in the relevant energy range for COMET Phase-I and Mu2e.
Figure~\ref{fg:silicon-proton} shows the spectrum of charged particle emission
from muons being stopped and captured in a silicon detector \cite{sobo68}. The
peak below 1.4 MeV is from the recoiling heavy ions, mainly $^{27}$Al, when no
charged particles were emitted. Hungerford~\cite{hung34} fitted the silicon
spectrum in Fig.~\ref{fg:silicon-proton} with an empirical function given by
%
\begin{equation} p(T) = A(1-{T_{th} \over T})^{\alpha} e^{-(T/T_0)}
	\label{eq:protons} \end{equation}
%
where $T$ is the kinetic energy and the fitted parameters are $A=0.105$
MeV$^{-1}$, $T_{th}$ = 1.4 MeV, $\alpha$=1.328 and $T_0$ = 3.1 MeV. The
spectrum is normalized to 0.1 per muon capture. Some other results in the past
experiments are summarized in Table~\ref{tb:proton}. 

%\begin{figure}[htb] 
	%\centering
	%\includegraphics[width=0.7\textwidth]{figs/si-proton.pdf} 
	%\caption{Charged particle spectrum from muons stopping and being captured in
	%a silicon detector~\cite{sobo68}.} 
	%\label{fg:silicon-proton} 
%\end{figure}


\begin{table}[htb] 
	\centering \caption{Probabilities in unites of $10^{-3}$ per
	muon capture for inclusive proton emission calculated by Lifshitz and
	Singer~\cite{lifshitz80}.
	The numbers in crescent parenthesis are estimates for the total inclusive
	rate derived from the measured exclusive channels by the use of the
	approximate regularity, such as $(\mu, \nu p):(\mu, \nu p n):(\mu, \nu p
	2n):(\mu, \nu p 3n) = 1:6:4:4$.}
	\label{tb:proton} 
	\vskip 3mm
	\begin{tabularx}{\textwidth}{ccccX}
		\toprule
		Target nucleus & Calculation & Experiment & Estimate & Comments \\ 
		\midrule
		%$_{10}$Ne 			& 		& $200\pm 40$ & 			& \\
		$^{27}_{13}$Al 	& 40 	& $>28 \pm 4$ & (70) 	& 7.5 for $T>40$ MeV \\
		$^{28}_{14}$Si 	& 144 & $150\pm30$ 	& 			& 3.1 and 0.34 $d$ for $T>18$ MeV \\ 
		$^{31}_{15}$P 	& 35 	& $>61\pm6$ 	& (91) 	& \\ 
		$^{46}_{22}$Ti 	& 		& 						& 			& \\
		$^{51}_{23}$V 	& 25 	& $>20\pm1.8$ & (32) 	& \\ 
		\bottomrule
	\end{tabularx}
\end{table}

\noindent The limited information available at present makes it difficult to
draw quantitative conclusive detector design. From Table~\ref{tb:proton}, the
yield for Al can be taken from experiment to be $>$3\% for $T>40$ MeV, or from
theory to be 4\%, or estimated based on the ratio of exclusive channels from
other nuclei to be 7\%, or speculated to be as high as Si 
%or Ne
, namely 15-20\%. The
energy spectrum can only be inferred from the Si data or from
Ref.~\cite{bala67}. At this moment, for both COMET Phase-I and Mu2e, this
analytical spectrum has been used to estimate proton emission. And also the $p,
d, \alpha$ composition is not known. The Ti proton yield can only be estimated
from V to be around 3\%.

\noindent It might be worth to present how proton emission affects a single
rate of the tracking chambers.  As an example for COMET Phase-I, single rates
of the tracking chamber (cylindrical drift chamber) have been simulated based
on the spectrum given in Eq.(\ref{eq:protons}). To reduce protons entering the
tracking chamber, in addition to the inner wall of the drift chamber (of 400
$\mu$m) a cylindrical proton absorber of different thickness is located in
front of the tracking chamber. Monte Carlo simulations were done with three
different thickness of proton degrader, namely 0~mm, 5~mm, and 7.5~mm.
%Figure~\ref{fig:protongenerated} shows a proton momentum spectrum generated
(larger than 50 MeV/$c$) in the simulation study, and regions in red show
protons reaching the first layer.  The results are summarized in
Table~\ref{tb:protonhits}, where the proton emission rate of 0.15 per muon
capture is assumed. If we assume the number of muons stopped in the
muon-stopping target is $5.8 \times 10^{9}$/s, the number of muon capture on
aluminum is about $3.5 \times 10^{9}$/s since the fraction of muon capture in
aluminum is $f_{cap}=0.61$. Therefore the total number of hits in all the cells
in the first layer is estimated to be 530 kHz (1.3 MHz) for the case of a
proton degrader of 5 mm (0 mm) thickness. This example present the importance
to understand the proton emission, rate and spectrum, from nuclear muon capture
on aluminum for COMET Phase-I and Mu2e.
%
\begin{table}[htb] 
	\begin{center} 
		\caption{Total numbers of hits in the first
	layer by protons emitted from muon capture for different trigger counter
	thickness. 100 k proton events were generated for COMET Phase-I. 15 \%
	protons per muon capture is assumed.} 
	\label{tb:protonhits} 
	\vspace{5mm}
	\begin{tabular}{lccc} 
		\toprule
		Proton degrader thickness & 0 mm & 5 mm& 7.5 mm\\
		\midrule
%		number of 1 hit events & 2467 & 87 & 28 \cr\hline number of 2 hit events &
%		73 & 8 & 1 \cr\hline number of 3 hit events & 9 & 0 & 0 \cr\hline\hline
%		number of 4 hit events & 1 & 0 & 0 \cr\hline\hline
		Hits & 2644 & 103 & 30 \cr
		Hits per proton emission & 2.6 \% & 0.1 \% & 0.03 \% \cr
		Hits per muon capture & $3.9\times10^{-3}$  & $1.5\times10^{-4}$ & $4.5\times10^{-5}$ \cr
		\bottomrule
	\end{tabular}
	\end{center} 
\end{table} 
\subsection{Any physics implication??}

% section _mu_e_conversion (end)