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thesis/chapters/backmatter.tex
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\begin{colophon}
This thesis was made in \LaTeXe{} using the ``hepthesis'' class~\cite{hepthesis}.
\end{colophon}
%\begin{colophon}
%This thesis was made in \LaTeXe{} using the ``hepthesis'' class~\cite{hepthesis}.
%\end{colophon}
%% You're recommended to use the eprint-aware biblio styles which
%% can be obtained from e.g. www.arxiv.org. The file mythesis.bib
@@ -10,8 +10,8 @@
%% I prefer to put these tables here rather than making the
%% front matter seemingly interminable. No-one cares, anyway!
\listoffigures
\listoftables
%\listoffigures
%\listoftables
%% If you have time and interest to generate a (decent) index,
%% then you've clearly spent more time on the write-up than the

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thesis/chapters/chap1.tex
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\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)

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\chapter*{Introduction}
\thispagestyle{empty}
\addcontentsline{toc}{chapter}{Introduction}
\label{cha:introduction}
%% Restart the numbering to make sure that this is definitely page #1!
\pagenumbering{arabic}
%\begin{itemize}
%\item CLFV in 3 lines
%\item COMET in 3 lines
%\item structure of the thesis:
%\begin{itemize}
%\item physics motivation of CLFV, COMET (chap 1)
%\item overview of COMET, Phase-I, requirements for detectors (chap 2)
%\item details of the proton measurements:
%\begin{itemize}
%\item physics (chap 3)
%\item method (chap 4)
%\item experimental set up, calibration (chap 4? or 5 )
%\item data analysis (chap 5)
%\item results, impact (chap 6)
%\end{itemize}
%\end{itemize}
%\end{itemize}
%\begin{comment}
%The Standard Model (SM) is the most successful theory of particle physics
%as it could account for almost all experimental data from high energy
%experiments. The discovery of a Higgs-like boson at the LHC in 2012 is another
%triumph of the theory. However, it is known that the SM has its limitations,
%one example is there is no explanation for the existence of lepton
%flavours and flavour conservation.
%theory. For example, it does not explain the origin of mass, the nature of dark
%matter, or neutrino oscillations.
%The lepton flavour conservation in the SM is assured by assuming neutrinos are
%massless. But, extensive experiments with atmospheric, solar, accelerator,
%reactor neutrinos have shown that neutrinos have non-zero masses, and they do
%mix between flavours~\cite{BeringerArguin.etal.2012}. In other words, lepton
%flavour violation (LFV) does occur in neutrino oscillations.
%While lepton flavour is totally violated in the neutrino sector, no charged
%lepton flavour violation (CLFV) has ever been observed. Therefore, any
%experimental evidence of lepton flavour violation with charged lepton would be
%a breakthrough that leads to new physics beyond the SM.
%\end{comment}
%The Standard Model (SM) is the most successful theory of particle physics
%as it could account for almost all experimental data from high energy
%experiments. However, it is also known that the SM has its
%TODO: wording /duplicaitons
The COMET experiment~\cite{COMET.2007}, proposed at the Japan Proton
Accelerator Research Complex (J-PARC), is a next-generation-experiment that
searches for evidence of charged lepton flavour violation (CLFV) with muons.
The branching ratio of CLFV in the Standard Model, even with massive neutrinos,
is prohibitively small, at the order of $10^{-54}$. Therefore, any experimental
observation of CLFV would be a clear signal of new physics beyond the SM.
The COMET (\textbf{CO}herent \textbf{M}uon to \textbf{E}lectron
\textbf{T}ransition) Collaboration aims to probe the conversion of a muon to
an electron in a nucleus field at a sensitivity of $6\times10^{-17}$, pushing
for a four orders of magnitude improvement from the current limit set by the
SINDRUM-II~\cite{Bertl.etal.2006}. A staging approach is adopted at the COMET
to achieve an intermediate physics result, as well as to gain operational
experience. The first stage, COMET Phase I, is scheduled to start in 2016 with
the goal sensitivity of $3\times 10^{-15}$ after a three-month-running period.
A cylindrical drift chamber being developed by the Osaka University group
will be the main tracking detector in the COMET Phase I. It is anticipated that
the chamber will be heavily occupied by protons emitted after nuclear muon
capture in the stopping target, and thus an absorber will be installed to
reduce the proton hit rate to a tolerable level. A study of proton emission
following nuclear muon capture for optimisation of the proton absorber is
presented in this thesis.
The thesis is structured as follows:
firstly,
the physics motivation of the COMET experiment, with muon's normal decays and
CLFV decays, is described in Chapter~\ref{cha:clfv}.
Chapter~\ref{cha:comet_overview} gives an overview of the
COMET experiment: beam lines, detectors and their requirements, and expected
sensitivities. Details of the study on proton emission are described in
Chapters~\ref{cha:alcap_phys},~\ref{cha:the_alcap_run_2013},~\ref{cha:data_analysis}:
physics, method, experimental set up, data analysis. The results and impacts of
the study on COMET Phase-I design is discussed in
Chapter~\ref{cha:discussions}.
% chapter introduction (end)

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thesis/chapters/chap2.tex
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\chapter{The COMET Phase--I}
\label{cha:the_comet_phase_i}
\section{Overview of COMET and Phase--I}
\label{sec:overview_of_comet_and_phase_i}
\section{Experimental setup}
\label{sec:experimental_setup}
\section{CDC}
\label{sec:cdc}
% section cdc (end)
% section experimental_setup (end)
% section overview_of_comet_and_phase_i (end)
% chapter the_comet_phase_i (end)

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\chapter{Lepton flavour and $\mu-e$ conversion}
\thispagestyle{empty}
\label{cha:clfv}
\section{Lepton flavour}
\label{sec:lepton_flavour}
According to the SM, all matter is built from a small set of fundamental
spin one-half particles, called fermions: six quarks and six leptons.
The six leptons form three generations (or flavours), namely:
\begin{equation*}
\binom{\nu_e}{e^-}, \quad \binom{\nu_\mu}{\mu^-} \quad \textrm{ and } \quad
\binom{\nu_\tau}{\tau^-}
\end{equation*}
Each lepton is assigned a lepton flavour quantum number, $L_e$, $L_\mu$,
$L_\tau$, equals to $+1$ for each lepton and $-1$ for each antilepton of the
appropriate generation. The lepton flavour number is conserved in the SM, for
example in the decay of a positive pion:
\begin{align*}
&\pi^+ \rightarrow \mu^+ + \nu_\mu \\
L_\mu \quad &0\quad \textrm{ }-1 \quad +1
\end{align*}
or, the interaction of an electron-type antineutrino with a proton (inverse
beta decay):
\begin{align*}
&\quad \overline{\nu}_e + p \rightarrow e^+ + n \\
L_e \quad &-1 \quad \textrm{ }0 \quad -1 \textrm{ } \quad 0
\end{align*}
The decay of a muon to an electron and a photon, where lepton flavour numbers
are violated by one unit or more, is forbidden:
%(the limit
%on this branching ratio is \meglimit~at 90\% confidence level
%(C.L.)~\cite{Adam.etal.2013}).
\begin{equation}
\begin{aligned}
&\quad \mu^+ \rightarrow e^+ + \gamma\\
L_\mu \quad &-1 \qquad 0 \qquad 0\\
L_e \quad &\quad 0 \quad -1 \qquad 0
\end{aligned}
\label{eq:mueg}
\end{equation}
%One more decay?
%\hl{TODO: Why massless neutrinos help lepton flavour conservation??}
%\hl{TODO: copied from KunoOkada}
%In the minimal version of the SM, where only one Higgs doublet is included and
%massless neutrinos are assumed, lepton flavor conservation is an automatic
%consequence of gauge invariance and the renormalizability of the SM
%Lagrangian. It is the basis of a natural explanation for the smallness of
%lepton flavor violation (LFV) in charged lepton processes.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Muon and its decays in the Standard Model}
\label{sec:muon_decay_in_the_standard_model}
\subsection{Basic properties of the muon}
\label{sub:basic_properties_of_the_muon}
The muon is a charged lepton, its static properties have been measured with
great precisions and are summarised in the ``Review of Particle Physics'' of
the Particle Data Group (PDG)~\cite{BeringerArguin.etal.2012}. Some of the
basic properties are quoted as follows:
\begin{enumerate}
\item The muon mass is given by the muon to electron mass ratio,
\begin{align}
\frac{m_\mu}{m_e} &= 206.768 2843 \pm 0.000 0052\\
m_\mu &= 105.6583715 \pm 0.0000035 \textrm{ MeV/}c^2
\end{align}
\item The spin of the muon is determined to
be $\frac{1}{2}$ as the measurements of the muon's gyromagnetic give
$g_\mu = 2$ within an overall accuracy better than 1 ppm. It is common to
quoted the result of $g_\mu$ as muon magnetic moment anomaly:
\begin{equation}
\frac{g-2}{2} = (11659209 \pm 6)\times 10^{-10}
\end{equation}
\item The charge of the muon is known to be equal to that of the
electron within about 3 ppb,
\begin{equation}
\frac{q_{\mu^+}}{q_{e^-}} + 1 = (1.2 \pm 2.1)\times 10^{-9}
\end{equation}
\item Electric dipole moment:
\begin{equation}
d = \frac{1}{2}(d_{\mu^-} - d_{\mu^+})
= (-0.1 \pm 0.9) \times 10^{-19} \textrm{ }e\cdot\si{\centi\meter}
\end{equation}
\item The muon is not stable, average lifetime of the free muon is:
\begin{equation}
\tau_{\mu} = 2.1969811 \pm 0.0000022 \textrm{ }\si{\micro\second}
\end{equation}
\end{enumerate}
% subsection basic_properties_of_the_muon (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Decays of the muon}
\label{sub:decays_of_the_muon}
Because of charge and lepton flavour conservations, the simplest possible decay
of muons is:
\begin{equation}
\mu^- \rightarrow e^- \nu_\mu \overline{\nu}_e
\label{eq:micheldecay}
\end{equation}
Muons can also decay in the radiative mode:
\begin{equation}
\mu^- \rightarrow e^- \nu_\mu \overline{\nu}_e \gamma
\label{eq:mue2nugamma}
\end{equation}
or with an associated $e^+ e^-$ pair:
\begin{equation}
\mu \rightarrow e^- \nu_\mu \overline{\nu}_e e^+ e^-
\label{eq:mu3e2nu}
\end{equation}
The dominant process, \micheldecay is commonly called Michel decay. It can be
described by the V-A interaction which is a special case of a local,
derivative-free, lepton-number-conserving four-fermion interaction.
%using $V-A$
%inteaction, a special case of four-fermion interaction, by Louis
%Michel~\cite{Michel.1950}.
The model contains independent real parameters that can be determined from
measurements of muon life time, muon decay and inverse muon
decay. Experimental results from extensive measurements of Michel parameters
are consistent with the predictions of the V-A
theory~\cite{Michel.1950,FetscherGerber.etal.1986,BeringerArguin.etal.2012}.
The radiative decay~\eqref{eq:mue2nugamma} is treated as an internal
bremsstrahlung process~\cite{EcksteinPratt.1959}.
%It occurs at the rate of about 1\% of all muon decays.
Since it is not possible to clearly separated this mode
from Michel decay in the soft-photon limit, the radiative mode is regarded as
a subset of the Michel decay. An additional parameter is included to describe
the electron and photon spectra in this decay channel. Like the case of
Michel decay, experiments results on the branching ratio and the parameter are
in agreement with the SM's predictions~\cite{BeringerArguin.etal.2012}.
There is a small probability (order of $10^{-4}$~\cite{EcksteinPratt.1959})
that the photon in \muenng would internally convert to an
$e^+e^-$ pair, resulting in the decay mode \muennee.
%\hl{TODO: more?}
The branching ratios for decay modes of muons, compiled by the PDG, are
listed in Table~\ref{tab:SM_muon_decays}.
\begin{table}[htb!]
\begin{center}
\begin{tabular}{l l l}
\toprule
Decay mode & Branching ratio & Remarks\\
\midrule
\micheldecay & $\simeq 1$ & commonly called Michel decay\\
\muenng & $0.014 \pm 0.004$ &
subset of Michel decay, $E_\gamma > 10 \textrm{ MeV}$ \\
\muennee & $(3.4 \pm 0.2 \pm 0.3)\times 10^{-5}$ &
transverse momentum cut $p_T>17 \textrm{ MeV/c}$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Decay modes and branching ratios of muon listed by
PDG~\cite{BeringerArguin.etal.2012}}
\label{tab:SM_muon_decays}
\end{table}
%\hl{TODO: Michel spectrum}
% subsection decays_of_the_muon (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section muon_decay_in_the_standard_model (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Lepton flavour violated decays of muons}
\label{sec:lepton_flavour_violation}
%Historically, the ideas of lepton flavours and lepton flavour conservation
%emerged from null-result experiments, such as a series of searches for \mueg in
%1950s and 1960s
%The fact that there is no convincing fundamental symmetry that leads to the
%conservation, and
%The fact that no underlying symmetry leads to this
%conservation has been found, and mixing between generations does happen in the
%quark sector make experimental searches for lepton flavour violation (LFV)
%interesting.
%The decay \mueg and \mueee were of great interest in the 1950s and 1960s when
%it is believed that the muon is an excited state of the electron.
The existence of the muon has always been a puzzle. At first, people thought
that it would be an excited state of the electron. Therefore, the searches for
\mueg was performed by Hincks and Pontercorvo~\cite{HincksPontecorvo.1948}; and
Sard and Althaus~\cite{SardAlthaus.1948}. Those searches failed to find the
photon of about 50 MeV that would have accompanied the decay electron in case
the two-body decay \mueg had occurred. From the modern point of view, those
experiments were the first searches for charged lepton flavour violation (LFV).
Since then, successive searches for LFV with the muon have been carried out. All
the results were negative and the limits of the LFV branching ratios had been
more and more stringent. Those null-result experiments suggested the lepton
flavours - muon flavour $L_\mu$ and electron flavour $L_e$. The notion of lepton
flavour was experimentally verified in the Nobel Prize-winning experiment of
Danby et al. at Brookhaven National Laboratory
(BNL)~\cite{DanbyGaillard.etal.1962}. Then the concepts of generations of
particles was developed~\cite{MakiNakagawa.etal.1962}, and integrated into the
SM, in which the lepton flavour conservation is guaranteed by and exact
symmetry, owing to massless neutrinos.
Following the above LFV searches with muons, searches with various particles,
such as kaons, taus, and others have been done. The upper limit have been
improved at a rate of two orders of magnitude per decade. %TODO(Fig).
While all of those searches yielded negative results, LFV with neutrinos is
confirmed with observations of neutrino oscillations; i.e. neutrino
of one type changes to another type when it travels in space-time. The
phenomenon means that there exists a mismatch between the flavour and
mass eigenstates of neutrinos; and neutrinos are massive. Therefore, the SM
must be modified to accommodate the massive neutrinos.
With the massive neutrinos charged lepton flavour violation (CLFV) must occur
through oscillations in loops. But, CLFV processes are highly suppressed in the
SM.
For example, Marciano and Mori ~\cite{MarcianoMori.etal.2008} calculated the
branching ratio of the process \mueg to be \brmeg$<10^{-54}$. Other
CLFV processes with muons are also suppressed to similar practically
unmeasurable levels.%\hl{TODO: Feynman diagram}
Therefore, any experimental
observation of CLFV would be an unambiguous signal of the physics beyond the
SM. Many models for physics beyond the SM, including supersymmetric (SUSY)
models, extra dimensional models, little Higgs models, predict
significantly larger CLFV
~\cite{MarcianoMori.etal.2008, MiharaMiller.etal.2013, BernsteinCooper.2013}.
%\hl{TODO: DNA of CLFV charts}
%A comprehensive list of predictions from various models, compiled by
%Altmannshofer and colleagues ~\cite{AltmannshoferBuras.etal.2010a} is
%reproduced in Table~\ref{tab:clfv_dna}.
%\begin{table}[htb!]
%\begin{center}
%\begin{tabular}{l l l}
%\toprule
%Decay mode & Branching ratio & Remarks\\
%\midrule
%\micheldecay & $\simeq 1$ & commonly called Michel decay\\
%\muenng & $0.014 \pm 0.004$ &
%subset of Michel decay, $E_\gamma > 10 \textrm{ MeV}$ \\
%\muennee & $(3.4 \pm 0.2 \pm 0.3)\times 10^{-5}$ &
%transverse momentum cut $p_T>17 \textrm{ MeV/c}$\\
%\bottomrule
%\end{tabular}
%\end{center}
%\caption{CLFV rates from various models~\cite{AltmannshoferBuras.etal.2010a}}
%\label{tab:clfv_dna}
%\end{table}
%It can be seen from the table that there are two CLFV processes with muons are
%predicted to occur at large rates by all new physics models, namely \mueg and
%It is calculated that there are two CLFV processes that would
%occur at large rates by many new physics models,
Among the CLFV processes, the \mueg and
the \muec are expected to have large effect by many models. The current
experimental limits on these two decay modes are set by MEG
experiment~\cite{Adam.etal.2013} and SINDRUM-II
experiment~\cite{Bertl.etal.2006}:
\begin{equation}
\mathcal{B}(\mu^+ \rightarrow e^+ \gamma) < 5.7 \times 10^{-13}
\end{equation}
, and:
\begin{equation}
\mathcal{B} (\mu^- + Au \rightarrow e^- +Au) < 7\times 10^{-13}
\end{equation}
%\hl{TODO: mueg and muec relations, Lagrangian \ldots}
%The observation of one CLFV process may indicate the mass scale of the physics
%beyond the SM, but it would not be enough to distinguish between different
%models correspond to that physics.
% section lepton_flavour_violation (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Phenomenology of \mueconv}
\label{sec:phenomenoly_of_muec}
The conversion of a captured muon into an electron in the field of a nucleus
has been one of the most powerful probe to search for CLFV. This section
highlights phenomenology of the \muec.
\subsection{What is \mueconv}
\label{sub:what_is_muec}
When a muon is stopped in a material, it is quickly captured by atoms
into a high orbital momentum state, forming a muonic atom, then
it rapidly cascades to the lowest state 1S. There, it undergoes either:
\begin{itemize}
\item normal Michel decay: \micheldecay; or
\item weak capture by the nucleus: $\mu^- p \rightarrow \nu_\mu n$
\end{itemize}
In the context of physics beyond the SM, the exotic process of \mueconv where
a muon decays to an electron without neutrinos is also
expected, but it has never been observed.
\begin{equation}
\mu^{-} + N(A,Z) \rightarrow e^{-} + N(A,Z)
\end{equation}
The emitted electron in this decay
mode , the \mueconv electron, is mono-energetic at an energy far above the
endpoint
of the Michel spectrum (52.8 MeV):
\begin{equation}
E_{\mu e} = m_\mu - E_b - \frac{E^2_\mu}{2m_N}
\end{equation}
where $m_\mu$ is the muon mas; $E_b \simeq Z^2\alpha^2 m_\mu/2$ is the binding
energy of the muonic atom; and the last term is the nuclear recoil energy
neglecting high order terms. For Al ($Z = 13$), the target of choice in the new
\mueconv experiments, the outgoing electron has energy of $E_{\mu e} \simeq
104.96$ MeV.
% subsection what_is_muec (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Measurement of \mueconv}
\label{sub:measurement_of_mueconv}
The quantity measured in searches for \mueconv is the ratio between the rate of
\mueconv, and the rate of all muons captured:
\begin{equation}
R_{\mu e} =
\frac{\Gamma(\mu^-N \rightarrow e^-N)}{\Gamma(\textrm{capture})}
\label{eq:muerate_def}
\end{equation}
The normalisation to captures has advantages when one does calculation since
many details of the nuclear wavefunction cancel out in the ratio.
%Detailed
%calculations have been performed by Kitano et al.~\cite{KitanoKoike.etal.2002a,
%KitanoKoike.etal.2007}, and Cirigliano et al.~\cite{Cirig}
The muon capture rate can be measured by observing the characteristic X-rays
emitted when the muon stops, and cascades to the 1S orbit. Since the stopped
muon either decays or be captured, the stopping rate is:
\begin{equation}
\Gamma_{\textrm{stop}} = \Gamma_{\textrm{decay}} + \Gamma_{\textrm{capture}}
\end{equation}
The mean lifetime $\tau = 1/\Gamma$, then:
\begin{equation}
\frac{1}{\tau_{\textrm{stop}}} = \frac{1}{\tau_{\textrm{decay}}} +
\frac{1}{\tau_{\textrm{capture}}}
\end{equation}
The mean lifetimes of free muons and muons in a material are well-known,
therefore the number of captures can be inferred from the number of stops. For
aluminium, $\frac{\Gamma_{\textrm{capture}}}{\Gamma_{\textrm{stop}}} = 0.609$
and the mean lifetime of stopped muons is 864
ns~\cite{SuzukiMeasday.etal.1987}.
The core advantages of the \mueconv searches compares to other CLFV searches
(\mueg or \mueee) are:
\begin{itemize}
\item the emitted electron is the only product, so the measurement is simple,
no coincidence is required; and
\item the electron is mono-energetic, its energy is far above
the endpoint of the Michel spectrum (52.8 MeV) where the background is very
clean. Essentially, the only intrinsic physics background comes from decay
of the muon orbiting the nucleus.
\end{itemize}
% subsection measurement_of_mueconv (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\hl{TODO}
%\subsection{Signal and backgrounds of \mueconv experiments}
%\label{sub:signal_and_backgrounds_of_mueconv_experiments}
% subsection signal_and_backgrounds_of_mueconv_experiments (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section phenomenoly_of_muec (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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thesis/chapters/chap3.tex
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@@ -1,25 +0,0 @@
\chapter{The Alcap experiment}
\label{cha:the_alcap_experiment}
\section{Nuclear physics of muon capture}
\label{sec:physics_of_muon_capture}
% section physics_of_muon_capture (end)
\section{Experimental status}
\label{sec:experimental_status}
% section experimental_status (end)
\section{Experimental setup}
\label{sec:experimental_setup}
% section experimental_setup (end)
% chapter the_alcap_experiment (end)

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\chapter{The COMET experiment}
\label{cha:comet_overview}
\thispagestyle{empty}
This chapter describes the new experimental search for \mueconv, namely COMET -
(\textbf{CO}herent \textbf{M}uon to \textbf{E}lectron \textbf{T}ransition). The
experiment will be carried out at the Japan Proton Accelerator Research Complex
(J-PARC), aims at a sensitivity of \sn{6}{-17} i.e. 10,000 times better than the
current best limit.
%At the Japan Proton Accelerator Research Complex (J-PARC), an experiment to
%search for \muec~conversion, which is called
%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.
%\begin{itemize}
%\item present status of mueconv experiments
%\begin{itemize}
%\item SINDRUM-II description, results, short comings
%\item new ideas: MECO, Mu2e, COMET
%\end{itemize}
%\item Concepts of COMET
%\begin{itemize}
%\item highly intense muon beam
%\item pulsed proton beam
%\item curved solenoids
%\end{itemize}
%\item COMET's beam lines and detectors
%\begin{itemize}
%\item proton beam: energy, time structure, planned operations
%\item pion production: yields, target, capture solenoids
%\item muon transportation: requirements, field
%\item stopping target: material, geometry, field, energy loss
%\item electron transportation:
%\item detectors: electron tracker and calorimeter
%\item DAQ
%\end{itemize}
%\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental status of \mueconv searches}
\label{sec:experimental_status_of_mueconv_searches}
\subsection{Experimental history}
\label{sub:experimental_history}
The searches for \mueconv has been ongoing for more than 50 years, started in
1952 with cosmic rays~\cite{LagarriguePeyrou.1952} and then moved to
accelerators. The list in the Table~\ref{tab:mueconv_history} is reproduced
from a recent review of Bernstein and Cooper~\cite{BernsteinCooper.2013}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l l l c}
\toprule
\textbf{Year} & \textbf{Limit} (90\% C.L.) & \textbf{Material}
& \textbf{Reference}\\
\midrule
1952 & \sn{1.0}{-1} & Sn, Sb & \cite{LagarriguePeyrou.1952} \\
1955 & \sn{5.0}{-4} & Cu & \cite{SteinbergerWolfe.1955} \\
1961 & \sn{4.0}{-6} & Cu & \cite{SardCrowe.etal.1961}\\
1961 & \sn{5.9}{-6} & Cu & \cite{ConversiLella.etal.1961}\\
1962 & \sn{2.2}{-7} & Cu & \cite{ConfortoConversi.etal.1962}\\
1964 & \sn{2.2}{-7} & Cu & \cite{ConversiLella.etal.1961}\\
1972 & \sn{2.6}{-8} & Cu & \cite{ConversiLella.etal.1961}\\
1977 & \sn{4.0}{-10} & S & \cite{ConversiLella.etal.1961}\\
1982 & \sn{7.0}{-11} & S & \cite{ConversiLella.etal.1961}\\
1988 & \sn{4.6}{-12} & Ti & \cite{ConversiLella.etal.1961}\\
1993 & \sn{4.3}{-12} & Ti & \cite{ConversiLella.etal.1961}\\
1995 & \sn{6.5}{-13} & Ti & \cite{ConversiLella.etal.1961}\\
1996 & \sn{4.6}{-11} & Pb & \cite{ConversiLella.etal.1961}\\
2006 & \sn{7.0}{-13} & Au & \cite{ConversiLella.etal.1961}\\
\bottomrule
\end{tabular}
\end{center}
\caption{History of \mueconv experiments, reproduced
from~\cite{BernsteinCooper.2013}}
\label{tab:mueconv_history}
\end{table}
The most recent experiments were the SINDRUM and SINDRUM-II at the Paul
Scherrer Institute (PSI), Switzerland. The SINDRUM-II measured the branching
ratio of \mueconv on a series of heavy targets: Ti, Pb and Au. The proton beam
at PSI is a continuous wave beam, with a time structure of 0.3 ns bursts every
19.75 \si{\nano\second}. An 8-\si{\milli\meter}-thick CH$_2$ degrader was used to reduce
the radiative pion capture and other prompt backgrounds. Cosmic backgrounds are
rejected using a combination of
passive shielding, veto counters and reconstruction cuts. The momenta of muons
were 52 \si{\mega\electronvolt\per\cc} and 53 \si{\mega\electronvolt\per\cc}, and the
momentum spread was 2\%.
\begin{figure}[htbp] \centering
\includegraphics[width=0.85\textwidth]{figs/sindrumII_setup}
\caption{SINDRUM-II set up}
\label{fig:sindrumII_setup}
\end{figure}
Electrons emitted from the target were tracked in a 0.33 T solenoid field.
Detector system consisted of a superconducting solenoid, two plastic
scintillation hodoscopes, a plexiglass Cerenkov hodoscope, and two drift
chambers. In the latest measurement, the SINDRUM-II collaboration have not
found any conversion electron from captured muons in a gold target, hence set
the upper limit for
the branching ratio of \mueconv in gold with 90 \% C.L. at \sn{7.0}{-13}.
The reconstructed momenta of electrons around the signal region from SINDRUM-II
is shown in the Figure~\ref{fig:sindrumII_result}. It can be seen that the muon
decay in orbit background falls steeply near the endpoint as expected, but, the
prompt background induced by pions still remains even after the cut in timing
and track angle. This indicates the problem of pion contamination is very
important in probing lower sensitivity.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.55\textwidth]{figs/sindrumII_Au_result}
\caption{SINDRUM-II result}
\label{fig:sindrumII_result}
\end{figure}
% subsection experimental_history (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{New generation of \mueconv~experiments}
\label{sub:new_generation_of_mueconv_experiments}
A new generation of \mueconv experiments have been proposed with scenarios to
overcome pion induced background in the SINDRUM-II. Lobashev and collaborators
first suggested the basic idea for new \mueconv at the Moscow Muon Factory;
this idea was used to develop the MECO experiment at Brookhaven National
Laboratory. The MECO experiment was cancelled due to budget constraints. The two
modern experiments, COMET at J-PARC and Mu2e at Fermilab use the initial idea
with more upgrades and modifications.
The basic ideas of the modern experiments are:
\begin{enumerate}
\item Highly intense muon source: the total number of muons needed is of the
order of $10^{18}$ in order to achieve a sensitivity of $10^{-16}$. This
can be done by producing more pions using a high power proton beam, and
having a high efficiency pion collection system;
\item Pulsed proton beam with an appropriate timing: the proton pulse should
be short compares to the lifetime of muons in the stopping target material,
and the period between pulses should be long enough for prompt backgrounds
from pion to decay before beginning the measurement. It is also crucial
that there is no proton leaks into the measuring interval;
\item Curved solenoids for charge and momentum selection: at first, the curved
solenoids remove the line of sight backgrounds. A charged particle travels
through a curved solenoidal field will have the centre of the helical
motion drifted up or down depends on the sign of the charge, and the
magnitude of the drift is proportional to its momentum. By using this
effect and placing suitable collimators, charge and momentum selection can
be made.
\end{enumerate}
% subsection new_generation_of_mueconv_experiments (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section experimental_status_of_mueconv_searches (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Concepts of the COMET experiment}
\label{sec:concepts_of_the_comet_experiment}
This section elaborates the design choices of the COMET to realise the basic
ideas mentioned above. Figures and numbers, other than noted, are taken from
the COMET's documentations:
\begin{itemize}
%TODO citations
\item Conceptual design report for the COMET experiment~\cite{COMET.2009}
\item Proposal Phase-I 2012
\item TDR 2014
\end{itemize}
\subsection{Proton beam}
\label{sub:proton_beam}
A high power pulsed proton beam is of utmost importance to achieve the desired
sensitivity of the COMET experiment. A slow-extracted proton beam from
the J-PARC main ring (MR), which is designed to deliver \sn{3.6}{15} protons per
cycle at a frequency of 0.45 Hz, will be used for the COMET experiment. The
proton beam power of the current design is 8 GeV$\times$7 $\mu$A, or
\sn{4.4}{13} protons/s. The beam energy 8 \si{\giga\electronvolt} helps to minimise
the production of antiprotons.
The proton pulse width is chosen to be 100 ns, and the pulse period to be
$1 \sim 2 \textrm{ }\mu\textrm{s}$. This time structure is sufficient for the
search for \mueconv in an aluminium target where the lifetime of muons is 864
ns. A plan of accelerator operation to realise the scheme is shown in
the Figure~\ref{fig:comet_mr_4filled}, where 4 out of 9 MR buckets are filled.
As mentioned, it is very important that there is no stray proton arrives in the
measuring period between two proton bunches. An extinction factor is defined as
the ratio between number of protons in between two pulses and the number of
protons in the main pulse. In order to achieve the goal sensitivity of the
COMET, an extinction factor of \sn{}{-9} is required.
Requirements for the proton beam are summarised in the
Table~\ref{tab:comet_proton_beam}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{figs/comet_mr_4filled}
\caption{The COMET proton bunch structure in the RCS (rapid cycle
synchrotron) and MR where 4 buckets
are filled producing 100 \si{\nano\second} bunches separated by
1.2~\si{\micro\second}.}
\label{fig:comet_mr_4filled}
\end{figure}
\begin{table}[htb]
\begin{center}
\begin{tabular}{l l}
\toprule
Beam power & 56 \si{\kilo\watt}\\
Energy & 8 \si{\giga\electronvolt}\\
Average current & 7 \si{\micro\ampere}\\
Beam emittance & 10 $\pi\cdot$ mm$\cdot$ mrad\\
Protons per bunch & $<10^{11}$\\
Extinction & \sn{}{-9}\\
Bunch separation & $1 \sim 2$ \si{\micro\second}\\
Bunch length & 100 \si{\nano\second}\\
\bottomrule
\end{tabular}
\end{center}
\caption{Pulsed proton beam for the COMET experiment}
\label{tab:comet_proton_beam}
\end{table}
% subsection proton_beam (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pion production and capture solenoid}
\label{sub:pion_production_can_capture_solenoid}
Muons for the COMET experiment are produced by colliding the proton beam with
a pion production target, made of either platinum, gold or tungsten, collecting
pions and then letting them decay. To collect as many pions (and cloud muons)
as possible, the pions are captured
using a high solenoidal magnetic field with a large solid angle. Since muons
will be stopped in a conversion target, low energy muons, and thus low energy
pions, are preferred. It is known from other measurements that backward
scattered pions (with respect to proton beam direction) of high energy are
suppressed, and the yield of low energy pions in the backward direction is not
too low compares to that of the forward direction (see
Figure~\ref{fig:pion_yield}). For these reasons, the COMET
decided to collect backward pions.
\begin{figure}[htb]
\centering
\includegraphics[width=0.95\textwidth]{figs/pion_yield}
\caption{Comparison between backward and forward pions production in a gold
target.}
\label{fig:pion_yield}
\end{figure}
The pion capture system is composed of several superconducting solenoids:
capture solenoids and matching solenoids. The magnetic field distribution along
the beam axis of the COMET is shown in the Figure~\ref{fig:comet_Bfield}. The
peak field of 5 T is created by the capture solenoid, and the matching
solenoids provide a smooth transition from that peak field to the 3 T field in
the pions/muons transportation region. The superconducting solenoids are
cooled by liquid helium, and a radiation shield composed of copper and tungsten
will be installed inside the cryostat to reduce radiation heat load.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/comet_Bfield}
\caption{Magnetic field distribution along the COMET beam line.}
\label{fig:comet_Bfield}
\end{figure}
% subsection pion_production_can_capture_solenoid (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pions and muons transportation solenoids}
\label{sub:pion_and_muon_transportation}
Muons and pions are transported to the muon stopping target through a muon
beam line, which includes several curved and straight superconducting solenoid
magnets. A schematic layout of the muon beam line, include the capture and
detector sections, is shown in Figure~\ref{fig:comet_beamline_layout}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.95\textwidth]{figs/comet_beamline_layout}
\caption{Schematic layout of the COMET beam line.}
\label{fig:comet_beamline_layout}
\end{figure}
The requirements for the muon transportation beam line are:
\begin{itemize}
\item being long enough for pions to decay, for instance, the survival rate
of pions will be about \sn{2}{-3} after 20 m;
\item being able to select low momentum negative muons with momentum of around
40 MeV/$c$, and eliminate high momentum muons ($> 75\textrm{ MeV/}c$),
since they can decay in flight and produce spurious signals of $\sim$ 105
MeV electrons.
\end{itemize}
The selection of charge and momentum is done by the curved solenoids. It is
know that, in a curved solenoidal field, the centre of the helical trajectory
of a charged particle drifts perpendicularly to the curved plane. The magnitude
of the drift is given by:
\begin{align}
D &= \frac{1}{qB} \frac{s}{R} \frac{p_L^2 + \frac{1}{2}p_T^2}{p_L}\\
&= \frac{1}{qB} \frac{s}{R} \frac{p}{2}
\left( \textrm{cos}\theta + \frac{1}{\textrm{cos}\theta} \right)\\
&= \frac{1}{qB} \theta_{bend} \frac{p}{2}
\left( \textrm{cos}\theta + \frac{1}{\textrm{cos}\theta} \right)
\end{align}
where $q$ is the electric charge of the particle; $B$ is the magnetic field at
the axis; $s$ and $R$ are the path length and the radius of the curvature; $p$,
$p_T$ and $p_L$ are total momentum, transversal momentum and longitudinal
momentum of the particles, respectively; $\theta = \textrm{atan}(p_T/p_L)$ is
the pitch angle of the helical trajectory; and $\theta_{bend} = s/R$ is called
the bending angle.
It is clear that $D$ is proportional to $\theta_{bend}$, to total momentum $p$.
Charged particles with opposite signs move in opposite directions. Therefore it
is possible to select muons around 40 MeV/$c$ by using suitable collimator
after the curved solenoid.
In order to keep the centre of the helical trajectories of the muons with
a reference momentum $p_0$ in the vertical plane, a compensating dipole field
parallel to the drift direction is needed. In the COMET, the dipole fields are
produced by additional coils winded around the solenoid coils. The magnitude of
the compensating field is:
\begin{equation}
B_{\textrm{comp}} = \frac{1}{qR} \frac{p_0}{2}
\left( \textrm{cos}\theta_0 + \frac{1}{\textrm{cos}\theta_0} \right)
\end{equation}
where the trajectories of charged particles with momentum $p_0$ and pitch angle
$\theta_0$ are corrected to be on-axis. An average dipole field of 0.03 T is
needed to select 40 MeV/$c$ muons as required by the COMET design.
% subsection pion_and_muon_transportation (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Muon stopping target}
\label{sub:muon_stopping_target}
Muon stopping target is place at 180\si{\degree}~bending after the pion production
target (Figure~\ref{fig:comet_beamline_layout}) in its own solenoid. The target
is designed to maximise the muon stopping efficiency and minimise the energy
loss of signal electrons.
%\hl{TODO: Target choice: separation, product, lifetime, energy loss\ldots}
It is calculated that the branching ratio of \mueconv increases with atomic
number $Z$, and plateaus above $Z \simeq 30$, then decreases as $Z>60$. The
lifetime of muons inside a material decreases quickly as $Z$ increases.
Tracking wise, lower $Z$ material provides better reconstructed momentum
resolution. Therefore, light material is preferable as muon stopping target.
The first choice for the muon stopping target material in the COMET is
aluminium. A titanium target is also considered. Configuration of the target is
shown in the Table~\ref{tab:comet_al_target}. Monte Carlo studies with this
design showed that net stopping efficiency is 0.29, and average energy loss
of signal electrons is about 400 \si{\kilo\electronvolt}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l l}
\toprule
\textbf{Item} & \textbf{Specification}\\
\midrule
Material & Aluminium\\
Shape & Flat disks\\
Disk radius & 100 \si{\milli\meter}\\
Disk thickness & 200 \si{\micro\meter}\\
Number of disks & 17\\
Disk spacing & 50 \si{\milli\meter}\\
\bottomrule
\end{tabular}
\end{center}
\caption{Configuration of the muon stopping target.}
\label{tab:comet_al_target}
\end{table}
A graded magnetic field (reduces from 3 T to 1 T) is produced at the
location of the stopping target (see Figure~\ref{fig:comet_target_Bfield}) to
maximise the acceptance for \mueconv signals, since electrons emitted in the
backward
direction would be reflected due to magnetic mirroring. The graded field also
helps optimising the transmission efficiency to the subsequent electron
transport section.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/comet_target_Bfield}
\caption{The graded magnetic field near the stopping target region.}
\label{fig:comet_target_Bfield}
\end{figure}
% subsection muon_stopping_target (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Electron transportation beam line}
\label{sub:electron_transportation_beam_line}
The 180\si{\degree}~bending electron transport solenoids help remove line-of-sight
between the target and the detector system. It works similarly to the muon
transportation section, but is tuned differently to accept electrons of about
105~\si{\mega\electronvolt\per\cc}. A compensation field of 0.17 T along the
vertical direction will be applied. Electrons with momentum less than 80
\si{\mega\electronvolt\per\cc} are blocked at the exit of this section by
a collimator to reduce DIO electrons rate. The net acceptance of signals of
\mueconv is about 0.32, and the detector hit rate will be in the order of
1~\si{\kilo\hertz}~for \sn{}{11} stopped muons\si{\per\second}.
% subsection electron_transportation_beam_line (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Electron detectors}
\label{sub:electron_detectors}
The \mueconv signal electrons is measured by an electron detector system, which
consists of straw-tube trackers and an electromagnetic calorimeter - shown in
Figure~\ref{fig:comet_detector_system}. The
requirements for the detector system is to distinguish electrons from other
particles, and measure their momenta, energy and timings. The whole detector
system is in a uniform solenoidal magnetic field under vacuum. Passive and
active shielding against cosmic rays is considered.
The tracking detector has to provide a momentum resolution less than
350~\si{\kilo\electronvolt\per\cc} in order to achieve a sensitivity of
\sn{3}{-17}. There are five stations of straw-tube gas chambers, each provides
two
dimensional information. Each straw tube is 5~\si{\milli\meter} in diameter and has
a 25~\si{\micro\meter}-thick wall. According to a GEANT4 Monte Carlo simulation,
a position resolution of 250~\si{\micro\meter} can be obtained, which is enough for
350~\si{\kilo\electronvolt\per\cc} momentum resolution. The DIO background of 0.15
events is estimated.
The electromagnetic calorimeter serves three purposes: a) to measure electrons
energy with high energy resolution; b) to provide timing information and
trigger timing for the detector system; and c) to provide additional data on
hit positions. Two candidate crystals, GSO and LYSO, are under consideration.
\begin{figure}[htb]
\centering
\includegraphics[width=0.75\textwidth]{figs/comet_detector_system}
\caption{Layout of the electron detectors.}
\label{fig:comet_detector_system}
\end{figure}
The requirements for \mueconv signals are:
\begin{itemize}
\item from the 350~\si{\kilo\electronvolt\per\cc}~momentum resolution, the signal
region is determined to be 103.5~\si{\mega\electronvolt\per\cc}~to
105.2~\si{\mega\electronvolt\per\cc};
\item transversal momentum of signal electrons is required to be greater than
52~\si{\mega\electronvolt\per\cc} to remove backgrounds from beam electrons and
muons decay in flight;
\item timing wise, conversion electrons should arrive in the time window of
detection which is about 700~\si{\nano\second}~after each proton pulses
(Figure~\ref{fig:comet_meas_timing}). The acceptance in this detection
window is about 0.39 for aluminium.
\end{itemize}
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\textwidth]{figs/comet_meas_timing}
\caption{Timing window of detection.}
\label{fig:comet_meas_timing}
\end{figure}
% subsection electron_detectors (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Signal sensitivity and background estimation}
\label{sub:signal_sensitivity_and_background_estimation}
The single event sensitivity (SES) of the \mueconv search is defined as:
\begin{equation}
\mathcal{B}(\mu^-Al\rightarrow e^- Al) =
\frac{1}{N^{\textrm{stop}}_{\mu}\cdot f_{\textrm{cap}} \cdot A_e}
\label{eq:mue_sensitivity}
\end{equation}
where $N^{\textrm{stop}}_{\mu}$ is the number of muons stopping in the muon
target; $f_{\textrm{cap}}$ is the fraction of captured muons; and $A_e$ is the
detector acceptance. The total number of stopped muons is projected as
$N^{\textrm{stop}}_{\mu} = 2\times 10^{18}$ for a \sn{2}{7}\si{\second}~run time;
$f_{\textrm{cap}} = 0.61$ for aluminium; and the total acceptance for the COMET
detector system is $A_e =0.031$. Using these
numbers, the SES of the COMET is calculated to be
\sn{2.6}{-17}. The 90\% CL upper limit is given by $2.3\times\mathcal{B}$:
\begin{equation}
\mathcal{B}(\mu^-Al\rightarrow e^- Al) < 6 \times 10^{-17} \quad
\textrm{(90\% C.L.)}
\end{equation}
Potential backgrounds for the COMET are:
\begin{enumerate}
\item Intrinsic physics backgrounds: originates from muons stopped in the
stopping target, including muon decays in orbit, radiative muon capture and
particles such as protons and neutrons emitted after muon capture;
\item Beam related backgrounds: caused by particles (electrons, pions, muons
and antiprotons) in the beam. They are either prompt or late-arriving.
A beam pulsing with high proton extinction factor is required to reject
this type of backgrounds;
\item Accidental background from cosmic rays
\end{enumerate}
The expected background rates for the COMET at an SES of
\sn{3}{-17} is summarised in Table~\ref{tab:comet_background_estimation}.
\begin{table}[htb]
\begin{center}
%\begin{tabular}{l l}
\begin{tabular}{l r@{.}l}
\toprule
\textbf{Background} & \multicolumn{2}{l}{\textbf{Events}}\\
\midrule
%\end{tabular}{l l}
%\begin{tabular}{l r@{.}l}
Radiative pion capture & 0&05\\
Beam electrons & $<$0&1\\
Muon decay in flight & $<$0&0002\\
Pion decay in flight & $<$0&0001\\
Neutron induced & 0&024\\
Delayed pion radiative capture & 0&002\\
Antiproton induced & 0&007\\
Muon decay in orbit & 0&15\\
Radiative muon capture & $<$0&001\\
Muon capture with neutron emission & $<$0&001\\
Muon capture with proton emission & $<$0&001\\
Cosmic ray muons & 0&002\\
Electron cosmic ray muons & 0&002\\
\midrule
\textbf{Total} &0&34\\
\bottomrule
\end{tabular}
\end{center}
\caption{Backgrounds of the COMET experiment.}
\label{tab:comet_background_estimation}
\end{table}
% subsection signal_sensitivity_and_background_estimation (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section concepts_of_the_comet_experiment (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The COMET Phase-I}
\label{sec:the_comet_phase_i}
The techniques of beam pulsing and curved solenoids that the COMET will utilise
are believed to greatly reduce potential backgrounds, by several orders of
magnitude, for the \mueconv search. That also means that backgrounds are being
extrapolated over four orders of magnitude from existing data. In order to
obtain data-driven estimates of backgrounds, and inform the detailed design for
the ultimate COMET experiment, and initial phase is desirable. Also, the 5-year
mid-term plan from 2013 of J-PARC includes the construction of the COMET beam
line. For these reasons, the COMET collaboration considers a staged approach
with the first stage, so called COMET Phase-I, with a shorter muon
transportation solenoid, up to the first 90\si{\degree}.
%\begin{wrapfigure}{r}{0.5\textwidth}
%\centering
%\includegraphics[width=0.49\textwidth]{figs/comet_phase1_layout}
%\caption{Lay out of the COMET Phase-I, the target and detector solenoid are
%placed after the first 90\degree~bend.}
%\label{fig:comet_phase1_layout}
%\end{wrapfigure}
\begin{SCfigure}
\centering
\caption{Lay out of the COMET Phase-I, the target and detector solenoid are
placed after the first 90\si{\degree}~bend.}
\includegraphics[width=0.4\textwidth]{figs/comet_phase1_layout}
\label{fig:comet_phase1_layout}
\end{SCfigure}
The COMET Phase-I has two major goals:
\begin{enumerate}
\item Direct measurements of the proton extinction factor, and other potential
backgrounds for the full COMET experiment. These include backgrounds due to
beam particles such as pions, neutrons, antiprotons, photons and electrons;
and physics background from muon DIO. Straw tube trackers and crystal
calorimeter with the same technology in the full COMET will be used, thus
these detectors can be regarded as the final prototype.
\item Search for \mueconv with an intermediate sensitivity of \sn{3.1}{-15},
a two orders of magnitude improvement from the SINDRUM-II limit. To realise
this goal, two options for detectors are being considered, either a reused
of the detectors for background measurements, or a dedicated detector.
The latter will be described in detail later.
\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proton beam for the COMET Phase-I}
\label{sub:proton_beam_for_the_comet_phase_i}
Proton beam for the Phase-I differs only in beam power compares to that of the
full COMET. It is estimated that a beam power of
3.2~\si{\kilo\watt}~$=$~8~\si{\giga\electronvolt}~$\times$~0.4~\si{\micro\ampere}~(or
\sn{2.5}{12} protons\si{\per\second}) will be enough for beam properties
study and achieving the physics goal of this stage.
Starting from a lower intensity is also suitable for performing accelerator
studies that are needed to realise 8~\si{\giga\electronvolt} beam extraction from
the J-PARC main ring.
% subsection proton_beam_for_the_comet_phase_i (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Pion production and transportation solenoids}
\label{sub:pion_production_and_transportation_solenoids}
Since the beam power will be lower, it is proposed to use a graphite target in
the Phase-I. This will minimise the activation of the target station and heat
shield which will be easier for necessary upgrading for Phase-II operation.
A target length of 600~\si{\milli\meter}~(1.5 radiation length) and target radius of
20~\si{\milli\meter}~are chosen. The target is located at the centre of the pion
capture solenoid where the peak magnetic field of 5 T is achieved.
A correction dipole filed of 0.05 T is also applied to improve the pion yield.
The pion/muon beam line for COMET Phase-I consists of the pion capture solenoid
section (CS), muon transport solenoid section (TS) up to the first
90\si{\degree}~bending, and a set of matching solenoids (see
Figure~\ref{fig:comet_phase1_magnets}). At the end of the muon beam line, the
detectors and the detector solenoid (DS) are installed. To reduce beam
backgrounds, a beam collimator is placed upstream of the detector solenoid.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/comet_phase1_magnets}
\caption{A schematic view of the superconducting solenoid magnet system for
the COMET Phase-I. Prefix CS is for capture solenoids, MS is for matching
solenoids, and TS is for transport solenoids. BS and DS are beam collimation
system and detector solenoid, respectively.}
\label{fig:comet_phase1_magnets}
\end{figure}
% subsection pion_production_and_transportation_solenoids (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Detectors for \mueconv search in the Phase-I}
\label{sub:detectors_for_mueconv_search_in_the_phase_i}
As mentioned, two types of detectors are considered for physics measurements in
the Phase-I. The dedicated detector system consists of a cylindrical drift
chamber (CDC), a trigger hodoscope, a proton absorber and a detector solenoid
(Figure~\ref{fig:comet_phase1_cydet}).
The whole system is referred as cylindrical detector system (CyDet) in the
COMET's documentation. The CyDet has advantages that low momentum particles for
the stopping target will not reach the detector, thus the hit rates are kept
manageable even at high beam currents. Furthermore, the majority of beam
particles, except those scattering at large angles, will not directly hit the
CyDet.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/comet_phase1_cydet}
\caption{Schematic layout of the CyDet.}
\label{fig:comet_phase1_cydet}
\end{figure}
The CDC is the main tracking detector that provides information for
reconstruction of charged particle tracks and measuring their momenta. The key
parameters for the CDC are listed in the
Table~\ref{tab:comet_phase1_cdc_params}.
Trigger hodoscopes are placed at both upstream and downstream ends of the CDC.
An absorber is placed concentrically with respect to the CDC axis to
reduce potential high rates caused by protons emitted after nuclear muon
capture in the stopping target.
The CDC covers the region
from \SIrange{500}{831}{\milli\meter}~in the radial direction. The length
of the CDC is 1500~\si{\milli\meter}. The inner wall is made of
a 100~\si{\micro\meter}-thick aluminised Mylar. The end-plates will be conical
in shape and about 10~\si{\milli\meter}-thick to support the feedthroughs. The outer
wall is
made of 5~\si{\milli\meter}~carbon fibre reinforced plastic (CFRP).
The CDC is arranged in 20 concentric sense layers with alternating positive and
negative stereo angles. The sense wires are made of gold-plated tungsten,
30~\si{\micro\meter} in diameter, tensioned to 50~\si{\gram}. The field wires
are uncoated aluminium wires with a diameter of 80~\si{\micro\meter}, at the same
tension of \SI{50}{\gram}. A high voltage of $1700\sim1900$~\si{\volt} will be
applied to the sense wires with the field wires at ground potential, giving an
avalanche gain of
approximately \sn{4}{4}. A gas mixture of helium:isobutane(90:10) is preferred
since the CDC momentum resolution is dominated by multiple scattering. With
these configurations, an intrinsic momentum resolution of
197~\si{\kilo\electronvolt\per\cc} is achievable according to our tracking study.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l l l}
\toprule
\textbf{Inner wall} & Length & 1500 \si{\milli\meter}\\
& Radius & 500 \si{\milli\meter}\\
\midrule
\textbf{Outer wall} & Length & 1740.9 \si{\milli\meter}\\
& Radius & 831 \si{\milli\meter}\\
\midrule
\textbf{Sense wire} & Number of layers & 20\\
& Material & Gold-plated tungsten\\
& Diameter & 30 \si{\micro\meter}\\
& Number of wires & 4986\\
& Tension & 50 \si{\gram}\\
%& Radius of the innermost wire at the EP & 530 mm\\
%& Radius of the outermost wire at the EP & 802 mm\\
\midrule
\textbf{Field wire} & Material & Aluminium\\
& Diameter & 80 \si{\micro\meter}\\
& Number of wires & 14562\\
& Tension & 50 \si{\gram}\\
\midrule
\textbf{Gas} & & Helium:Isobutane (90:10)\\
\bottomrule
\end{tabular}
\end{center}
\caption{Main parameters of the CDC for the COMET Phase-I.}
\label{tab:comet_phase1_cdc_params}
\end{table}
The maximum usable muon beam intensity will be limited by the detector hit
occupancy. Charge particles with transversal momentum greater than 70
\si{\mega\electronvolt\per\cc} are expected to reach the CDC. Those particles are:
protons emitted from nuclear muon capture, and electrons from muon decay in
orbit. It is calculated that the hit rate due to proton emission dominates,
where the highest rate is 11~\si{\kilo\hertz\per}cell compares to
5~\si{\kilo\hertz\per}
cell contributing from DIO electrons. Another potential issue caused by protons
is the ageing effect on the CDC as they leave about a 100 times larger
energy deposit than the minimum ionisation particles.
For those reasons, we plan to install an absorber to reduce the rate of protons
reaching the CDC. However, there is no experimental data available for the rate
of protons emitted after muon capture in aluminium. In the design of the COMET
Phase-I, we use a conservative estimation of the rate of protons from energy
spectrum of charged particles emitted from muon capture in
$^{28}$Si~\cite{SobottkaWills.1968}. The baseline design for the proton
absorber is 1.0~\si{\milli\meter}-thick CFRP, which contributes
195~\si{\kilo\electronvolt\per\cc} to the momentum resolution of reconstructed
track.
In order to obtain a better understanding of the protons emission, and then
further optimisation of the CDC, a dedicated experiment to measure proton
emission rate and energy spectrum is being carried out at PSI. This experiment
is described in detail in next chapters.
% subsection detectors_for_mueconv_search_in_the_phase_i (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sensitivity of the \mueconv search in the Phase-I}
\label{sub:sensitivity_of_the_mueconv_search_in_the_phase_i}
The SES for the Phase-I is given by
the Equation~\ref{eq:mue_sensitivity}. Using $N_{\mu} = 1.3\times 10^{16}$,
$f_{\textrm{cap}} = 0.61$, and $A_e = 0.043$ from MC study for the Phase-I, the
SES becomes:
\begin{equation}
\mathcal{B}(\mu^-Al\rightarrow e^- Al) = 3.1\times 10^{-15}
\end{equation}
% subsection sensitivity_of_the_mueconv_search_in_the_phase_i (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Time line of the COMET Phase-I and Phase-II}
\label{sub:time_line_of_the_phase_i}
We are now in the construction stage of the COMET Phase-I, which is planned to
be finished by the end of 2016. We will carry out engineering run in 2016,
and subsequently, physics run in 2017. A beam time of 90 days is expected to
achieve the goal sensitivity of the Phase-I. An anticipated schedule for the
COMET, both Phase-I and Phase-II, is shown in Figure~\ref{fig:sched}.
\begin{figure}[tbh]
\centering
\includegraphics[width=0.8\textwidth]{figs/sched}
\caption{The anticipated schedule of the COMET experiment.}
\label{fig:sched}
\end{figure}
% subsection time_line_of_the_phase_i (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section the_comet_phase_i (end)

25
thesis/chapters/chap4.tex
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@@ -1,25 +0,0 @@
\chapter{MC study}
\label{cha:mc_study}
\section{Geometry}
\label{sec:geometry}
% section geometry (end)
\section{Response matrix?}
\label{sec:response_matrix_}
% section response_matrix_ (end)
\section{Proton spectrum? Cut?}
\label{sec:proton_spectrum_cut_}
% section proton_spectrum_cut_ (end)
% chapter mc_study (end)

901
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@@ -0,0 +1,901 @@
\chapter
[Proton emission following nuclear muon capture - The AlCap experiment]
{Proton emission following \\nuclear muon capture \\and the AlCap experiment}
\label{cha:alcap_phys}
\thispagestyle{empty}
As mentioned earlier, the emission rate of protons
following nuclear muon capture on aluminium is of interest to the COMET Phase-I
since protons can cause a very high hit rate on the proposed cylindrical drift
chamber. Another \mueconv experiment, namely Mu2e at Fermilab, which aims at
a similar goal sensitivity as that of the COMET, also shares the same interest
on proton emission. Therefore, a joint COMET-Mu2e project was formed to carry
out the measurement of proton, and other charged particles, emission. The
experiment, so-called AlCap, has been proposed and approved to be carried out
at PSI in 2013~\cite{AlCap.2013}. In addition to proton, the AlCap
experiment will also measure:
\begin{itemize}
\item neutrons, because they can cause backgrounds on other detectors and
damage the front-end electronics; and
\item photons, since they provide ways to normalise number of stopped muons
in the stopping target.
\end{itemize}
The emission of particles following muon capture in nuclei
%Historically, the emission of protons, as well as other particles, has
has been studied thoroughly for several nuclei in the context of ``intermediate
energy nuclear physics'' where it is postulated that the weak interaction is
well understood and muons are used as an additional probe to investigate the
nuclear structure~\cite{Singer.1974, Measday.2001}.
Unfortunately, the proton emission rate for aluminium in the energy range of
interest is not available. This chapter reviews the current knowledge on
emission of particles with emphasis on proton.
%theoretically and experimentally, hence serves as the motivation for the AlCap
%experiment.
\begin{comment}
\begin{itemize}
%\item Motivation: why looked for protons in COMET, what is the status in
%theory and experiment
%\begin{itemize}
%\item COMET Phase-I need
%\item lack of experimental data
%\item addition to protons: neutrons and photons
%\end{itemize}
\item Atomic capture of muon
\begin{itemize}
\item formation of the muonic atom
\end{itemize}
\item Nuclear muon capture
\begin{itemize}
\item physics: capture on proton
\item energy
\item de-excitation modes: mostly neutrons, other may occur
\end{itemize}
\item Charged particles/protons
\begin{itemize}
\item general
\item alpha, protons
\item
\end{itemize}
\item Plan and goals of the AlCap experiment
\end{itemize}
\end{comment}
\section{Atomic capture of the negative muon}
\label{sec:atomic_capture_of_the_negative_muon}
Theoretically, the capturing process can be described in the following
stages~\cite{FermiTeller.1947, WuWilets.1969}:
\begin{enumerate}
\item High to low (a few \si{\kilo\electronvolt}) energy: the muon velocity are
greater than the velocity of the valence electrons of the atom. Slowing
down process is similar to that of fast heavy charged particles. It takes
about \sn{}{-9} to \sn{}{-10} \si{\second}~to slow down from a relativistic
\sn{}{8}~\si{\electronvolt}~energy to 2000~\si{\electronvolt}~in condensed matter,
and about 1000 times as long in air.
\item Low energy to rest: in this phase, the muon velocity is less than that
of the valence electrons, the muon is considered to be moving inside
a degenerate electron gas. The muon rapidly comes to a stop either in
condensed matters ($\sim$\sn{}{-13}~\si{\second}) or in gases ($\sim$\sn{}{-9}
\si{\second}).
\item Atomic capture: the muon has no kinetic energy, it is captured by the
host atom into one of high orbital states, forming a muonic atom. The
distribution of initial states is not well known. The details depend on
whether the material is a solid or gas, insulator or material
\item Electromagnetic cascade: since all muonic states are unoccupied, the
muon cascades down to states of low energy. The transition is accompanied
by the emission of Auger electrons or characteristic X-rays, or excitation
of the nucleus. The time taken for the muon to enter the lowest possible
state, 1S, from the instant of its atomic capture is
$\sim$\sn{}{-14}\si{\second}.
\item Muon disappearance: after reaching the 1S state, the muons either
decays with a half-life of \sn{2.2}{-6}~\si{\second}~or gets captured by the
nucleus. In hydrogen, the capture to decay probability ratio is about
\sn{4}{-4}. Around $Z=11$, the capture probability is roughly equal to the
decay probability. In heavy nuclei ($Z\sim50$), the ratio of capture to
decay probabilities is about 25.
The K-shell muon will be $m_\mu/m_e \simeq 207$ times nearer the nucleus
than a K-shell electron. The close proximity of the K-shell muon in the
Coulomb field of a nuclear, together with its weak interaction with the
nucleus, allows the muon to spend a significant fraction of time (\sn{}{-7}
-- \sn{}{-6} \si{\second}) within the nucleus, serving as an ideal probe for the
distribution of nuclear charge and nuclear moments.
\end{enumerate}
% section atomic_capture_of_the_negative_muon (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nuclear capture of the negative muon}
\label{sec:nuclear_muon_capture}
The nuclear capture process is written as:
\begin{equation}
\mu^- + A(N, Z) \rightarrow A(N, Z-1) + \nu_\mu
\label{eq:mucap_general}
\end{equation}
The resulting nucleus can be either in its ground state or in an excited state.
The reaction is manifestation of the elementary ordinary muon capture on the
proton:
\begin{equation}
\mu^- + p \rightarrow n + \nu_\mu
\label{eq:mucap_proton}
\end{equation}
If the resulting nucleus at is in an excited state, it could cascade to lower
states by emitting light particles and leaving a residual heavy nucleus. The
light particles are mostly neutrons and (or) photons. Neutrons can also be
directly knocked out of the nucleus via the reaction~\eqref{eq:mucap_proton}.
Charged particles are emitted with probabilities of a few percent, and are
mainly protons, deuterons and alphas have been observed in still smaller
probabilities. Because of the central interest on proton emission, it is covered
in a separated section.
\subsection{Muon capture on the proton}
\label{sub:muon_capture_on_proton}
%It is theoretically
%very important in understanding the structure of the Lagrangian for the
%strangeness-preserving semileptonic weak interaction. But it is also the
%hardest one experimentally. The first reason is the rate is small ($\sim$460
%\reciprocal\second) compares to the decay rate
%($\sim$\sn{455}{3}~\reciprocal\second)~\cite{Measday.2001}. Secondly, the
%$\mu p$ atom is quite active, so it is likely to form muonic molecules like
%$p\mu p$, $p\mu d$ and $p\mu t$, which complicate the study of weak
%interaction.
The underlying interaction in proton capture in Equation~\eqref{eq:mucap_proton}
at nucleon level and quark level
are depicted in the Figure~\ref{fig:feyn_protoncap}. The flow of time is from
the left to the right hand side, as an incoming muon and an up quark
exchange a virtual $W$ boson to produce a muon neutrino and a down quark, hence
a proton transforms to a neutron.
\begin{figure}[htb]
\centering
\includegraphics[width=0.4\textwidth]{figs/mucap_proton}
\hspace{10mm}
\includegraphics[width=0.4\textwidth]{figs/mucap_quark}
\caption{A tree-level Feynman diagram of muon capture on the proton, at the
nucleon-level (left), and at the quark-level (right).}
\label{fig:feyn_protoncap}
\end{figure}
The four-momentum transfer in the interaction is fixed at
$q^2 = (q_n - q_p)^2 = -0.88m_\mu^2 \ll m_W^2$. The smallness of the momentum
transfer in comparison to the $W$ boson's mass makes it possible to treat the
interaction as a four-fermion interaction with Lorentz-invariant transition
amplitude:
\begin{equation}
\mathcal{M} = \frac{G_F V_{ud}}{\sqrt{2}}J^\alpha j_\alpha
\label{eq:4fermion_trans_amp}
\end{equation}
where $J$ is the nucleon current $p\rightarrow n$, and $j$ is the lepton
current $\mu \rightarrow \nu_\mu$, $G_F$ is the Fermi coupling constant, and
$V_{ud}$ is the matrix element of the Cabibbo-Kobayashi-Maskawa
(CKM) matrix. The lepton current is expressed as a purely $V-A$ coupling of
lepton states:
\begin{equation}
j_\alpha = i\bar{\psi}_\nu \gamma_\alpha (1 - \gamma_5) \psi_\mu
\label{eq:weakcurrent_lepton}
\end{equation}
The weak current of individual quarks is similar to that of leptons with the
only modification is an appropriate element of the CKM matrix ($V_{ud}$, which
is factored out in Eq.~\eqref{eq:4fermion_trans_amp}):
\begin{equation}
J^\alpha = i\bar{\psi}_d (1 - \gamma_5) \psi_u
\label{eq:weakcurrent_ud}
\end{equation}
If the nucleon were point-like, the nucleon current would have the same form as
in Eq.~\eqref{eq:weakcurrent_ud} with suitable wavefunctions of the proton and
neutron. But that is not the case, in order to account for the complication of
the nucleon, the current must be modified by six real form factors
$g_i(q^2), i = V, M, S, A, T, P$:
\begin{align}
J_\alpha &= i\bar{\psi}_n(V^\alpha - A^\alpha)\psi_p,\\
V^\alpha &= g_V (q^2) \gamma^\alpha + i \frac{g_M(q^2)}{2m_N}
\sigma^{\alpha\beta} q_\beta + g_S(q^2)q^\alpha,\\
A^\alpha &= g_A(q^2)\gamma^\alpha \gamma_5 + ig_T(q^2)
\sigma^{\alpha\beta} q_\beta\gamma_5 + \frac{g_P(q^2)}{m_\mu}\gamma_5
q^\alpha,
\end{align}
where the $V^\alpha$ and $A^\alpha$ are the vector and axial currents, $m_\mu$
and $m_N$ are the muon and nucleon mass, respectively. The scaling by the muon
and nucleon mass is by convention in Mukhopadhyay's
review~\cite{Mukhopadhyay.1977}.
Among the six form factors, the so-called second class currents, $g_T$ and
$g_S$, vanish under the symmetry of G-parity, which is the product of charge
conjugation and isospin rotation. Experimental limits for non-zero $g_T$ and
$g_S$ are not very tight, but are negligible with respect to other
uncertainties in muon capture~\cite{Measday.2001}.
The vector form factor $g_V$, and the weak-magnetic form factor $g_M$ are
equivalent to the electromagnetic form factors of the nucleon according the
conserved vector current (CVC) hypothesis. The values of these couplings are
determined from elastic electron-nucleon scattering experiments, then
extrapolated to the momentum transfer $q^2$.
Using $\mu - e$ universality, the axial form factor $g_A$ in this case is
related to that of electron as: $(g_A/g_V)^\mu = (g_A/g_V)^e$ at zero momentum
transfer. This equality has been checked using results from muon decay and beta
decay experiments. The $q^2$-dependence of $g_A$ is deducted from neutrino
scattering experiments.
The pseudoscalar form factor $g_P$ is determined by measuring the capture rate
of the process in Eq.~\eqref{eq:mucap_proton}. However, because of the smallness
capture rate in comparison to muon decay rate, and other complications due to
muonic molecules $p\mu p$, $d\mu p$ and $t\mu p$, $g_P$ is the least
well-defined form factor. Only recently, it is measured with a reasonable
precision~\cite{AndreevBanks.etal.2013a}.
The values of the six form factors at $q^2 = -0.88m^2_\mu$ are listed in
Table~\ref{tab:formfactors}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l l l}
\toprule
\textbf{Form factor} & \textbf{Value at $-0.88m^2_\mu$}\\
\midrule
$g_S$ & $0$\\
$g_T$ & $0$\\
$g_V$ & $0.976 \pm 0.001$\\
$g_M$ & $3.583 \pm 0.003$\\
$g_A$ & $1.247 \pm 0.004$\\
$g_P$ & $8.06 \pm 0.55$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Values of the weak form factors of the nucleon at $q^2
= -0.88m^2_\mu$}
\label{tab:formfactors}
\end{table}
%\hl{Radiative capture}
% subsection muon_capture_on_proton (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Total capture rate}
\label{sub:total_capture_rate}
The captured muon at the 1S state has only two choices, either to decay or to
be captured on the nucleus. Thus, the total capture rate for negative muon,
$\Lambda_t$ is given by:
\begin{equation}
\Lambda_t = \Lambda_c + Q \Lambda_d
\label{eq:mu_total_capture_rate}
\end{equation}
where $\Lambda_c$ and $\Lambda_d$ are partial capture rate and decay rate,
respectively, and $Q$ is the Huff factor, which is corrects for the fact that
muon decay rate in a bound state is reduced because of the binding energy
reduces the available energy.
%The total capture rates for several selected
%elements are compiled by Measday~\cite{Measday.2001},
%and reproduced in
%Table~\ref{tab:total_capture_rate}.
%\begin{table}[htb]
%\begin{center}
%\begin{tabular}{l l r@{.}l r@{.}l@{$\pm$}l l}
%\toprule
%\textbf{$Z$ ($Z_{\textrm{eff}}$)} &
%\textbf{Element} &
%\multicolumn{2}{l}{\textbf{Mean lifetime}} &
%\multicolumn{3}{l}{\textbf{Capture rate}} &
%\textbf{Huff factor}\\
%& &
%\multicolumn{2}{c}{\textbf{(\nano\second)}} &
%\multicolumn{3}{l}{\textbf{$\times 10^3$ (\reciprocal\second)}} &\\
%\midrule
%1 (1.00) & $^1$H & 2194&90 $\pm$0.07 & 0&450 &0.020 & 1.00\\
%& $^2$H & 2194&53 $\pm$0.11 & 0&470 &0.029 & \\
%2 (1.98) & $^3$He & 2186&70 $\pm$0.10 & 2&15 &0.020 & 1.00\\
%& $^4$He & 2195&31 $\pm$0.05 & 0&470&0.029 & \\
%\bottomrule
%\end{tabular}
%\end{center}
%\caption{Total capture rate of the muon in nuclei for several selected
%elements, compiled by Measday~\cite{Measday.2001}}
%\label{tab:total_capture_rate}
%\end{table}
Theoretically, it is assumed that the muon capture rate on a proton of the
nucleus depends only on the overlap of the muon with the nucleus. For light
nuclei where the point nucleus concept is applicable, there are $Z$ protons and
the radius of the muon orbital decreases as $Z^{-1}$, the probability of
finding the muon at the radius increases as $Z^3$, therefore the capture rate
increases as $Z^4$. Because the muon radius soon becomes comparable to that of
the nucleus, corrections are needed, so $Z_{\textrm{eff}}$ is used instead of
$Z$.
The effect of the nucleus for higher $Z$ is more profound, there is no
theoretical model that provides a satisfied explanation for all experimental
data. One simple formula from Primakoff gives a reasonable,
and of course not perfect, description of the existing data~\cite{Measday.2001}:
\begin{equation}
\Lambda_c(A,Z) = Z^4_{\textrm{eff}} X_1 \left[1
- X_2\left(\frac{A-Z}{2A}\right)\right]
\label{eq:primakoff_capture_rate}
\end{equation}
where $X_1 =$ \SI{170}{\second^{-1}}~is the muon capture rate for hydrogen, but
reduced because a smaller phase-space in the nuclear muon capture compares to
that of a nucleon; and $X_2 = 3.125$ takes into account the fact that it is
harder for protons to transforms into neutrons due to the Pauli exclusion
principle in heavy nuclei where there are more neutrons than protons.
% subsection total_capture_rate (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Neutron emission}
\label{sub:neutron_emission}
The average number of neutrons emitted per muon capture generally increases
with $Z$, but there are large deviations from the trend due to particular
nuclear structure effects. The trend is shown in Table~\ref{tab:avg_neutron}
and can be expressed by a simple empirical function
$n_{avg} = (0.3 \pm 0.02)A^{1/3}$~\cite{Singer.1974}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{c c}
\toprule
\textbf{Elements} & \textbf{Average number of }\\
& \textbf{neutrons per capture}\\
\midrule
Al & 1.262 $\pm$ 0.059\\
Si & 0.864 $\pm$ 0.072\\
Ca & 0.746 $\pm$ 0.032\\
Fe & 1.125 $\pm$ 0.041\\
Ag & 1.615 $\pm$ 0.060\\
I & 1.436 $\pm$ 0.056\\
Au & 1.662 $\pm$ 0.044\\
Pb & 1.709 $\pm$ 0.066\\
\bottomrule
\end{tabular}
\end{center}
\caption{Average number of neutrons emitted per muon capture compiled by
Measday~\cite{Measday.2001}}
\label{tab:avg_neutron}
\end{table}
The neutron emission can be explained by several mechanisms:
\begin{enumerate}
\item Direct emission follows reaction~\eqref{eq:mucap_proton}: these neutrons
have fairly high energy, from a few \si{\mega\electronvolt}~to as high as 40--50
\si{\mega\electronvolt}.
\item Indirect emission through an intermediate compound nucleus: the energy
transferred to the neutron in the process~\eqref{eq:mucap_proton} is 5.2
\si{\mega\electronvolt} if the initial proton is at rest, in nuclear
environment, protons have a finite momentum distribution, therefore the
mean excitation energy of the daughter nucleus is around 15 to 20
\si{\mega\electronvolt}~\cite{Mukhopadhyay.1977}. This is above the nucleon
emission threshold in all complex nuclei, thus the daughter nucleus can
de-excite by emitting one or more neutrons. In some actinide nuclei, that
excitation energy might trigger fission reactions. The energy of indirect
neutrons are mainly in the lower range $E_n \le 10$ \si{\mega\electronvolt}
with characteristically exponential shape of evaporation process. On top of
that are prominent lines might appear where giant resonances occur.
\end{enumerate}
Experimental measurement of neutron energy spectrum is technically hard, and it
is difficult to interpret the results. Due to these difficulties, only a few
energy spectrum measurements were made, none of them covers the full energy
range and mostly at high energy region~\cite{Measday.2001}.
% subsection neutron_emission_after_muon_capture (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[Proton emission]
{Proton emission}
\label{sec:proton_emission}
\subsection{Experimental status}
\label{sub:experimental_status}
The measurement of charged particle emission is quite difficult and
some early measurements with nuclear emulsion are still the best available
data. There are two reasons for that:
\begin{enumerate}
\item The emission rate is small: the de-excitation of the nucleus through
charged particle is possible, but occurs at very low rate compares to
neutron emission. The rate is about 15\% for light nuclei and
reduces to a few percent for medium and heavy nuclei.
\item The charged particles are short ranged: the emitted protons,
deuterons and alphas are typically low energy (2--20~\mega\electronvolt).
But a relatively thick target is normally needed in order to achieve
a reasonable muon stopping rate and charged particle statistics. Therefore,
emulsion technique is particularly powerful.
\end{enumerate}
The first study was done by Morigana and Fry~\cite{MorinagaFry.1953} where
24,000 muon tracks were stopped in their nuclear emulsion which contains silver,
bromine, and other light elements, mainly nitrogen, carbon, hydrogen and
oxygen. The authors identified a capture on a light element as it would leave
a recoil
track of the nucleus. They found that for silver bromide AgBr, $(2.2 \pm
0.2)\%$ of the captures produced protons and $(0.5 \pm 0.1)\%$ produced alphas.
For light elements, the emission rate for proton and alpha are respectively
$(9.5 \pm 1.1)\%$ and $(3.4 \pm 0.7)\%$. Subsequently, Kotelchuk and
Tyler~\cite{KotelchuckTyler.1968} had a result which was about 3 times more
statistics and in fair agreement with Morigana and Fry
(Figure~\ref{fig:kotelchuk_proton_spectrum})
\begin{figure}[htb]
\centering
\includegraphics[width=0.65\textwidth]{figs/kotelchuk_proton_spectrum}
\caption{Early proton spectrum after muon capture in silver bromide AgBr
recorded using nuclear emulsion. Image is taken from
Ref.~\cite{KotelchuckTyler.1968}}
\label{fig:kotelchuk_proton_spectrum}
\end{figure}
Protons with higher energy are technically easier to measure, but because of
the much lower rate, they can only be studied at meson facilities. Krane and
colleagues~\cite{KraneSharma.etal.1979} measured proton emission from
aluminium, copper and lead in the energy range above 40 \mega\electronvolt~and
found a consistent exponential shape in all targets. The integrated yields
above 40 \mega\electronvolt~are in the \sn{}{-4}--\sn{}{-3} range (see
Table~\ref{tab:krane_proton_rate}), a minor contribution to total proton
emission rate.
\begin{table}[htb]
\begin{center}
\begin{tabular}{c c r@{$\pm$}l@{$\times$}r}
\toprule
\textbf{Target} & \textbf{Exponential constant}&
\multicolumn{3}{c}{\textbf{Integrated yield}}\\
& \textbf{$E_0$ (MeV)}
& \multicolumn{3}{c}{\textbf{$E_p\ge 40$ MeV}}\\
\midrule
Al & $7.5 \pm 0.4$ & (1.38&0.09)&\sn{}{-3}\\
Cu & $8.3 \pm 0.5$ & (1.96&0.12)&\sn{}{-3}\\
Pb & $9.9 \pm 1.1$ & (0.171&0.028)&\sn{}{-3}\\
\bottomrule
\end{tabular}
\end{center}
\caption{Proton integrated yields and exponential constants measured by Krane
et al.~\cite{KraneSharma.etal.1979}. The yields are assumed to be
proportional to exp($-E/E_0$).}
\label{tab:krane_proton_rate}
\end{table}
Their result on aluminium, the only experimental data existing for this target,
is shown in Figure~\ref{fig:krane_proton_spec} in comparison with spectra from
neighbouring elements, namely silicon measured by Budyashov et
al.~\cite{BudyashovZinov.etal.1971} and magnesium measured Balandin et
al.~\cite{BalandinGrebenyuk.etal.1978}. The authors noted aluminium data and
silicon data are in reasonable agreement both in the yield and the energy
dependence, while magnesium data shows significant drop in intensity. They then
suggested the possibility of an interesting nuclear structure dependency that
might be at work in this mass range.
\begin{figure}[htb]
\centering
\includegraphics[width=0.65\textwidth]{figs/krane_proton_spec}
\caption{Yield of charged particles following muon capture in aluminium
target (closed circle) in the energy range above 40 MeV and an exponential
fit. The open squares are silicon data from Budyashov et
al.~\cite{BudyashovZinov.etal.1971}, the open triangles are magnesium data
from Balandin et al.~\cite{BalandinGrebenyuk.etal.1978}.}
\label{fig:krane_proton_spec}
\end{figure}
The aforementioned difficulties in charged particle measurements could be
solved using an active target, just like nuclear emulsion. Sobottka and
Wills~\cite{SobottkaWills.1968} took this approach when using a Si(Li) detector
to stop muons. They obtained a spectrum of charged particles up to 26
\mega\electronvolt~in Figure~\ref{fig:sobottka_spec}. The peak below 1.4
\mega\electronvolt~is due to the recoiling $^{27}$Al. The higher energy events
including protons, deuterons and alphas constitute $(15\pm 2)\%$ of capture
events, which is consistent with a rate of $(12.9\pm1.4)\%$ from gelatine
observed by Morigana and Fry. This part has an exponential
decay shape with a decay constant of 4.6 \mega\electronvolt. Measday
noted~\cite{Measday.2001} the fractions of events in
the 26--32 \mega\electronvolt~range being 0.3\%, and above 32
\mega\electronvolt~range being 0.15\%. This figure is in agreement with the
integrated yield above 40 \mega\electronvolt~from Krane et al.
In principle, the active target technique could be applied to other material
such as germanium, sodium iodine, caesium iodine, and other scintillation
materials. The weak point of this method is that there is no particle
identification like in nuclear emulsion, the best one can achieve after all
corrections is a sum of all charged particles. It should be noted here
deuterons can contribute significantly, Budyashov et
al.~\cite{BudyashovZinov.etal.1971} found deuteron components to be
$(34\pm2)\%$ of the charged particle yield above 18 \mega\electronvolt~in
silicon, and $(17\pm4)\%$ in copper.
\begin{figure}[htb]
\centering
\includegraphics[width=0.75\textwidth]{figs/sobottka_spec}
\caption{Charged particle spectrum from muon capture in a silicon detector,
image taken from Sobottka and Wills~\cite{SobottkaWills.1968}.}
\label{fig:sobottka_spec}
\end{figure}
Another technique had been used to study proton emission is the activation
method where the residual nucleus is identified by its radioactivity. This
method can provide the rate of charged particles emission by adding up the
figures from all channels such as $(\mu^-,\nu p)$, $(\mu^-,\nu p(xn))$,
$(\mu^-, \nu \alpha)$, $(\mu^-, \nu \alpha(xn))$. The number of elements that
can be studied using this method is limited by several requirements: (a)
mono-isotopic element is preferable; (b) the radioactive daughter should emit
gamma-rays with a reasonable half-life; (c) the $(\mu^-,\nu xn)$ reactions
should lead to either stable daughters, or daughters with very short
half-lives. The last condition is important in ensuring the dominating neutron
emission processes do not interfere with counting of the much less frequent
proton emission reactions.
Vil'gel'mova et al.~\cite{VilgelmovaEvseev.etal.1971} found the single proton
(unaccompanied by any neutron)
emission rates in the $^{28}\textrm{Si}(\mu^-,\nu p)^{27}\textrm{Mg}$ and
$^{39}\textrm{K}(\mu^-,\nu p)^{38}\textrm{Cl}$ reactions are $(5.3 \pm 1.0)$\%
and $(3.2 \pm 0.6)$\%, respectively.
Singer~\cite{Singer.1974} compared the figure for silicon and the result from
active target measurement and found that the reaction
$^{28}\textrm{Si}(\mu^-,\nu pn)^{26}\textrm{Mg}$ could occur at a similar rate
to that of the $^{28}\textrm{Si}(\mu^-,\nu p)^{27}\textrm{Mg}$. That also
indicates that the deuterons and alphas might constitute a fair amount in the
spectrum in Figure~\ref{fig:sobottka_spec}.
Wyttenbach et al.~\cite{WyttenbachBaertschi.etal.1978} studied $(\mu^-,\nu p)$,
$(\mu^-,\nu pn)$, $(\mu^-,\nu p2n)$, $(\mu^-,\nu p3n)$ and $(\mu^-,\nu\alpha)$
in a wide range of 18 elements from sodium to bismuth.Their results plotted
against the Coulomb barrier for the outgoing protons are given in
Figure~\ref{fig:wyttenbach_rate_1p}, ~\ref{fig:wyttenbach_rate_23p}. The
classical Coulomb barrier $V$ they used are given by:
\begin{equation}
V = \frac{zZe^2}{r_0A^{\frac{1}{3}} + \rho},
\label{eqn:classical_coulomb_barrier}
\end{equation}
where $z$ and $Z$ are the charges of the outgoing particle and of the residual
nucleus, values $r_0 = 1.35 \textrm{ fm}$, and $\rho = 0 \textrm{ fm}$ for
protons were taken.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/wyttenbach_rate_1p}
\caption{Activation results from Wyttenbach et
al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p)$ and
$(\mu^-,\nu pn)$ reactions.}
\label{fig:wyttenbach_rate_1p}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/wyttenbach_rate_23p}
\caption{Activation results from Wyttenbach et
al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p2n)$ and
$(\mu^-,\nu p3n)$ reactions.}
\label{fig:wyttenbach_rate_23p}
\end{figure}
Wyttenbach et al.\ saw that the cross section of each reaction decreases
exponentially with increasing Coulomb barrier. The decay constant for all
$(\mu^-,\nu pxn)$ is about 1.5 per \mega\electronvolt~of Coulomb barrier. They
also commented a ratio for different de-excitation channels:
\begin{equation}
(\mu^-,\nu p):(\mu^-,\nu pn):(\mu^-,\nu p2n):(\mu^-,\nu p3n) = 1:6:4:4,
\label{eqn:wyttenbach_ratio}
\end{equation}
The authors compared their results with many preceded works and rejected
the results from Vil'gel'mova et al.~\cite{VilgelmovaEvseev.etal.1971} as being
too high, but Measday~\cite{Measday.2001} noted it it is not
necessarily true since there has been suggestion from other experiments that
$(\mu^-, \nu p)$ reactions might become more important for light nuclei.
Measday also commented that the ratio~\eqref{eqn:wyttenbach_ratio} holds over
a broad range of mass, but below $A=40$ the $(\mu^-,\nu p)$ reaction can vary
significantly from nucleus to nucleus.
% subsection experimental_status (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theoretical models}
\label{sub:theoretical_models}
The first attempt to explain the result of Morigana and Fry was done by
Ishii~\cite{Ishii.1959}. He assumed a two-step scenario: firstly a compound
nucleus is formed, and then it releases energy by statistical emission of
various particles. Three models for momentum distribution of protons in the
nucleus were used: (I) the Chew-Goldberger distribution
$\rho(p) \sim A/(B^2 + p^2)^2$; (II) Fermi gas at zero temperature; and (III)
Fermi gas at a finite temperature ($kT = 9$ \mega\electronvolt).
A very good agreement with the experimental result for the alpha emission was
obtained with distribution (III), both in the absolute percentage and the energy
distribution (curve (III) in the left hand side of
Figure~\ref{fig:ishii_cal_result}). However, the calculated emission of protons
at the same temperature falls short by about 10
times compares to the data. The author also found that the distribution
(I) is unlikely to be suitable for proton emission, and using that distribution
for alpha emission resulted in a rate 15 times larger than observed.
\begin{figure}[htb]
\centering
\includegraphics[width=.49\textwidth]{figs/ishii_cal_alpha}
%\hspace{10mm}
\includegraphics[width=.49\textwidth]{figs/ishii_cal_proton}
\caption{Alpha spectrum (left) and proton spectrum (right) from Ishii's
calculation~\cite{Ishii.1959} in comparison with experimental data from
Morigana and Fry. Image is taken from Ishii's paper.}
\label{fig:ishii_cal_result}
\end{figure}
Singer~\cite{Singer.1974} noted that by assuming a reduced effective mass for
the nucleon, the average excitation energy will increase, but the proton
emission rate does not significantly improve and still could not explain the
large discrepancy. He concluded that the evaporation mechanism can account
for only a small fraction of emitted protons. Moreover, the high energy protons
of 25--50 \mega\electronvolt~cannot be explained by the evaporation mechanism.
He and Lifshitz~\cite{LifshitzSinger.1978, LifshitzSinger.1980} proposed two
major corrections to Ishii's model:
\begin{enumerate}
\item A new description of the nucleon momentum in the nucleus with more high
momentum components. This helps explaining the high momentum part of the
proton spectrum.
\item Pre-equilibrium emission of proton is included: both pre-equilibrium
and statistical emission were taken into account. The equilibrium state is
achieved through a series of intermediate states, and at each state there
is possibility for particles to escape from the nucleus.
\end{enumerate}
With these improvements, the calculated proton spectrum agreed reasonably with
data from Morigana and Fry in the energy range $E_p \le 30$ \mega\electronvolt.
Lifshitz and Singer noted the pre-equilibrium emission is more important for
heavy nuclei. Its contribution in light nuclei is about a few percent,
increasing to several tens of percent for $100<A<180$, then completely
dominating in very heavy nuclei. This trend is also seen in other nuclear
reactions at similar excitation energies. The pre-equilibrium emission also
dominates the higher-energy part, although it falls short at energies higher
than 30 \mega\electronvolt. The comparison between the calculated proton
spectrum and experimental data is shown in
Fig.~\ref{fig:lifshitzsinger_cal_proton}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/lifshitzsinger_cal_proton}
\caption{Proton energy spectrum from muon capture in AgBr, the data in
histogram is from Morigana and Fry, calculation by Lifshitz and
Singer~\cite{LifshitzSinger.1978} showed contributions from the
pre-equilibrium emission and the equilibrium emission.}
\label{fig:lifshitzsinger_cal_proton}
\end{figure}
The authors found their corrections accounts well for the observed data in
a wide range of elements $23 \le A \le 209$. They calculated both the single
proton emission rate $(\mu^-, \nu p)$ and the inclusive emission rate:
\begin{align*}
\sum(\mu^-, \nu p) = &(\mu^-, \nu p) + (\mu^-, \nu pn) + (\mu^-, \nu p2n)\\
&+ \ldots + (\mu^-, \nu d) + (\mu^-, \nu dn)) + \ldots
\end{align*}
The deuteron emission channels are included to comparisons with activation
data where there is no distinguish between $(\mu^-, \nu pn)$ and $(\mu^-,d)$,
\ldots Their calculated emission rates together with available experimental
data is reproduced in Table~\ref{tab:lifshitzsinger_cal_proton_rate}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{c c c c c}
\toprule
Target nucleus & Calculation & Experiment & Estimate & Comments \\
%\textbf{Col1}\\
\midrule
$^{27}_{13}$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) & \\
%item1\\
\bottomrule
\end{tabular}
\end{center}
\caption{Calculated of the single proton emission rate and the inclusive
proton emission rate. The experimental data are mostly from Wyttenbach et
al.\cite{WyttenbachBaertschi.etal.1978}}
\label{tab:lifshitzsinger_cal_proton_rate}
\end{table}
A generally good agreement between calculation and experiment can be seen from
Table~\ref{tab:lifshitzsinger_cal_proton_rate}. The rate of $(\mu^-,\nu p)$
reactions for $^{28}\textrm{Al}$ and $^{39}\textrm{K}$ are found to be indeed
higher than average, though not as high as Vil'gel'mora et
al.~\cite{VilgelmovaEvseev.etal.1971} observed.
For protons with higher energies in the range of
40--90 \mega\electronvolt~observed in the emulsion data as well as in later
experiments~\cite{BudyashovZinov.etal.1971,BalandinGrebenyuk.etal.1978,
KraneSharma.etal.1979}, Lifshitz and Singer~\cite{LifshitzSinger.1988}
suggested another contribution from capturing on correlated two-nucleon
cluster, an idea that had been proposed earlier by Singer~\cite{Singer.1961}.
In this calculation, the authors considered the captures on cluster in which
two nucleons interact with each other via meson exchange current. There is
experimental evidence that the nuclear surface is reach in nucleon clusters,
and it had been shown that the meson exchange current increases the total
capture rate in deuterons by 6\%. The result of this model was a mix, it
accounted well for Si, Mg and Pb data, but predicted rates about 4 times
smaller in cases of Al and Cu, and about 10 times higher in case of AgBr
(Table~\ref{tab:lifshitzsinger_cal_proton_rate_1988}).
\begin{table}[htb]
\begin{center}
\begin{tabular}{l l c}
\toprule
\textbf{Nucleus} & \textbf{Exp.$\times 10^3$} & \textbf{MEC cal.$\times
10^3$}\\
\midrule
Al & $1.38 \pm 0.09$ & 0.3\\
Si & $0.87 \pm 0.14$ & 0.5\\
Mg & $0.17 \pm 0.05$ & 0.2\\
Cu & $1.96 \pm 0.12$ & 0.5\\
AgBr & $(4.7 \pm 1.1)\times 10^{-2}$ & 0.4\\
Pb & $0.17 \pm 0.03$ & 0.3\\
\bottomrule
\end{tabular}
\end{center}
\caption{Probability of proton emission with $E_p \ge 40$
\mega\electronvolt~as calculated by Lifshitz and
Singer~\cite{LifshitzSinger.1988} in comparison with available data.}
\label{tab:lifshitzsinger_cal_proton_rate_1988}
\end{table}
% subsection theoretical_models (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Summary on proton emission from aluminium}
\label{sub:summary_on_proton_emission_from_aluminium}
There is no direct measurement of proton emission following
muon capture in the relevant energy for the COMET Phase-I of 2.5--10
\mega\electronvolt:
\begin{enumerate}
\item Spectrum wise, only one energy spectrum (Figure~\ref{fig:krane_proton_spec})
for energies above 40 \mega\electronvolt~is available from Krane et
al.~\cite{KraneSharma.etal.1979},
where an exponential decay shape with a decay constant of
$7.5 \pm 0.4$~\mega\electronvolt. At low energy range, the best one can get is
the charged particle spectrum, which includes protons, deuterons and alphas,
from the neighbouring element silicon (Figure~\ref{fig:sobottka_spec}).
This charged particle spectrum peaks around 2.5 \mega\electronvolt~and
reduces exponentially with a decay constant of 4.6 \mega\electronvolt.
\item The activation data from Wyttenbach et
al.~\cite{WyttenbachBaertschi.etal.1978} only gives rate of
$^{27}\textrm{Al}(\mu^-,\nu pn)^{25}\textrm{Na}$ reaction, and set a lower
limit for proton emission rate at $(2.8 \pm 0.4)\%$ per muon capture. If
the ratio~\eqref{eqn:wyttenbach_ratio} holds true for aluminium, then the
inclusive proton rate would be $7\%$, higher than the calculated rate of
$4\%$ by Lifshitz and Singer~\cite{LifshitzSinger.1980}.
Both activation technique and inclusive rate calculation do not distinguish
between different channels that give the same final state, such as between
$^{27}\textrm{Al}(\mu^-,\nu pn)^{25}\textrm{Na}$ and
$^{27}\textrm{Al}(\mu^-,\nu d)^{25}\textrm{Na}$ reactions.
\end{enumerate}
In short, the knowledge on proton emission from aluminium at low energy is
limited. The rate estimation does not separate protons from deuterons,
and experimentally, there is a lower limit of $(2.8\pm0.4)\%$ per muon capture.
A spectrum shape at this energy range is not available.
% subsection summary_on_proton_emission_from_aluminium (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section proton_emission (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The AlCap experiment}
\label{sec:the_alcap_experiment}
\subsection{Motivation of the AlCap experiment}
\label{sub:motivation_of_the_alcap_experiment}
As mentioned, protons from muon capture on aluminium might cause a very high
rate in the COMET Phase-I CDC. The detector is designed to accept particles
with momenta in the range of 75--120 \mega\electronvolt\per\cc.
Figure~\ref{fig:proton_impact_CDC} shows that protons with kinetic energies of
2.5--8 \mega\electronvolt~will hit the CDC. Such events are troublesome due to
their large energy deposition. Deuterons and alphas at that momentum range is
not of concern because they have lower kinetic energy and higher stopping
power, thus are harder to escape the muon stopping target.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/proton_impact_CDC}
\caption{Momentum-kinetic energy relation of protons, deuterons and alphas
below 10\mega\electronvolt. Shaded area is the acceptance of the COMET
Phase-I's CDC. Protons with energies in the range of 2.5--8
\mega\electronvolt~are in the acceptance of the CDC. Deuterons and alphas at
low energies should be stopped inside the muon stopping target.}
\label{fig:proton_impact_CDC}
\end{figure}
The COMET plans to introduce a thin, low-$Z$ proton absorber in between the
target and the CDC to produce proton hit rate. The absorber will be effective
in removing low energy protons. The high energy protons that are moderated by
the absorber will fall into the acceptance range of the CDC, but because of the
exponential decay shape of the proton spectrum, the hit rate caused by these
protons should be affordable.
The proton absorber solves the problem of hit rate, but it degrades the
reconstructed momentum resolution. Therefore its thickness and geometry should
be carefully designed. The limited information available makes it difficult to
arrive at a conclusive detector design. The proton emission rate could be 4\%
as calculated by Lifshitz and Singer~\cite{LifshitzSinger.1980}; or 7\% as
estimated from the $(\mu^-,\nu pn)$ activation data and the ratio
\eqref{eqn:wyttenbach_ratio}~\cite{WyttenbachBaertschi.etal.1978}; or as high
as 15-20\% from silicon and neon.
For the moment, design decisions in the COMET Phase-I are made based on
conservative assumptions: emission rate of 15\% and an exponential decay shape
are adopted follow the silicon data from Sobottka and Will
~\cite{SobottkaWills.1968}. The spectrum shape is fitted with an empirical
function given by:
\begin{equation}
p(T) = A\left(1-\frac{T_{th}}{T}\right)^\alpha e^{-(T/T_0)},
\label{eqn:EH_pdf}
\end{equation}
where $T$ is the kinetic energy of the proton, and the fitted parameters are
$A=0.105\textrm{ MeV}^{-1}$, $T_{th} = 1.4\textrm{ MeV}$, $\alpha = 1.328$ and
$T_0 = 3.1\textrm{ MeV}$. The baseline
design of the absorber is 1.0 \milli\meter~thick
carbon-fibre-reinforced-polymer (CFRP) which contributes
195~\kilo\electronvolt\per\cc~to the momentum resolution. The absorber also
down shifts the conversion peak by 0.7 \mega\electronvolt. This is an issue as
it pushes the signal closer to the DIO background region. For those reasons,
a measurement of the rate and spectrum of proton emission after muon capture is
required in order to optimise the CDC design.
% subsection motivation_of_the_alcap_experiment (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Experimental method for proton measurement}
\label{sub:experimental_method}
We planned to use a low energy, narrow momentum spread available at PSI to
fight the aforementioned difficulties in measuring protons. The beam momentum
is tunable from 28 to 45~\mega\electronvolt\ so that targets at different
thickness from 25 to 100 \micro\meter\ can be studied. The $\pi$E1 beam line
could provide about \sn{}{3} muons\per\second\ at 1\% momentum spread, and
\sn{}{4} muons\per\second\ at 3\% momentum spread. With this tunable beam, the
stopping distribution of the muons is well-defined.
The principle of the particle identification used in the AlCap experiment is
that for each species, the function describes the relationship between energy
loss per unit length (dE/dx) and the particle energy E is uniquely defined.
With a simple system of two detectors, dE/dx can be obtained by
measuring energy deposit $\Delta$E in one detector of known thickness
$\Delta$x, and E is the sum of energy deposit in both detector if the particle
is fully stopped.
In the AlCap, we realise the idea with a pair of silicon detectors: one thin
detector of 65~\micron\ serves as the $\Delta$E counter, and one thick detector
of 1500~\micron\ that can fully stop protons up to about 12~MeV. Since the
$\Delta \textrm{d}=65$~\micron\ is known, the function relates dE/dx to
E reduces to a function between $\Delta$E and E. Figure~\ref{fig:pid_sim} shows
that the function of protons can be clearly distinguished from other charged
particles in the energy range of interest.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.75\textwidth]{figs/pid_sim}
\caption{Simulation study of PID using a pair of silicon detectors}
\label{fig:pid_sim}
\end{figure}
The AlCap uses two pairs of detector with large area, placed symmetrically with
respect to the target provide a mean to check for muon stopping distribution.
The absolute number of stopped muons are inferred
from the number of muonic X-rays recorded by a germanium detector. For
aluminium, the $(2p-1s)$ line is at 346 \kilo\electronvolt. The acceptances of
detectors will be assessed by detailed Monte Carlo study using Geant4.
% subsection experimental_method (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Goals and plan of the experiment}
\label{sub:goals_of_the_experiment}
Our experimental program is organised in three distinct work packages (WP),
directed by different team leaders, given in parentheses.
\begin{itemize}
\item[WP1:] (Kammel (Seattle), Kuno(Osaka)) \textbf{Charged
Particle Emission after Muon Capture.}\\ Protons emitted after nuclear muon
capture in the stopping target dominate the single-hit rates in the tracking
chambers for both the Mu2e and COMET Phase-I experiments. We plan to measure
both the total rate and the energy spectrum to a precision of 5\% down to
proton energies of 2.5 MeV.
\item[WP2:] (Lynn(PNNL), Miller(BU))
\textbf{Gamma and X-ray Emission after Muon Capture.}\\ A Ge detector will
be used to measure X-rays from the muonic atomic cascade, in order to provide
the muon-capture normalization for WP1, and is essential for very thin
stopping targets. It is also the primary method proposed for calibrating the
number of muon stops in the Mu2e and COMET experiments. Two additional
calibration techniques will also be explored; (1) detection of delayed gamma
rays from nuclei activated during nuclear muon capture, and (2) measurement
of the rate of photons produced in radiative muon decay. The first of these
would use a Ge detector and the second a NaI detector. The NaI
calorimeter will measure the rate of high energy photons from radiative muon
capture (RMC), electrons from muon decays in orbit (DIO), and photons from
radiative muon decay (RMD), as potential background sources for the
conversion measurement. As these rates are expected to be extremely low near
the conversion electron energy, only data at energies well below 100 MeV will
be obtained.
\item[WP3:] (Hungerford(UH), Winter(ANL)) \textbf{Neutron
Emission after Muon Capture.}\\ Neutron rates and spectra after capture in
Al and Ti are not well known. In particular, the low energy region below 10
MeV is important for determining backgrounds in the Mu2e/COMET detectors and
veto counters as well as evaluating the radiation damage to electronic
components. Carefully calibrated liquid scintillation detectors, employing
neutron-gamma discrimination and spectrum unfolding techniques, will measure
these spectra. The measurement will attempt to obtain spectra as low or lower
than 1 MeV up to 10 MeV. \\
\end{itemize}
WP1 is the most developed
project in this program. Most of the associated apparatus has been built and
optimized. We are ready to start this experiment in 2013, while preparing and
completing test measurements and simulations to undertake WP2 and WP3.
The measurement of proton has been carried out in November and December 2013,
the details are described in following chapters.
% subsection goals_of_the_experiment (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section the_alcap_experiment (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section nuclear_muon_capture (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% chapter alcap_phys (end)

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\chapter{Data analysis}
\label{cha:data_analysis}
\section{Analysis strategy}
\section{Actual work ...}
% chapter data_analysis (end)

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\chapter{The AlCap Run 2013}
\label{cha:the_alcap_run_2013}
\thispagestyle{empty}
The first run of the AlCap experiment was performed at the $\pi$E1 beam line
area, PSI (Figure~\ref{fig:psi_exp_hall_all}) from November 26 to December 23,
2013. The goal of the run was to measure protons rate and spectrum following
muon capture on aluminium.
\begin{figure}[p]
\centering
\includegraphics[height=0.85\textheight]{figs/psi_exp_hall_all}
\caption{Layout of the PSI experimental hall, $\pi$E1 experimental area is
marked with the red circle. \\Image taken from
\url{http://www.psi.ch/num/FacilitiesEN/HallenplanPSI.png}}
\label{fig:psi_exp_hall_all}
\end{figure}
\section{Experimental set up}
\label{sec:experimental_set_up}
The low energy muons from the $\pi$E1 beam line were stopped in thin aluminium
and silicon targets, and charged particles emitted were measured by two pairs
of silicon detectors inside of a vacuum vessel
(Figure~\ref{fig:alcap_setup_detailed}). A stopped muon event is defined by
a group of upstream detectors and a muon veto plastic scintillator.
The number of stopped muons is monitored by a germanium detector placed outside
of the vacuum chamber. In addition, several plastic scintillators were used to
provide veto signals for the silicon and germanium detectors. Two liquid
scintillators for neutron measurements were also tested in this run.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.65\textwidth]{figs/alcap_setup_detailed}
\caption{AlCap detectors: two silicon packages inside the vacuum vessel,
muon beam detectors including plastic scintillators and a wire chamber,
germanium detector and veto plastic scintillators.}
\label{fig:alcap_setup_detailed}
\end{figure}
\subsection{Muon beam and vacuum chamber}
Muons in the $\pi$E1 beam line are decay products of pions created
as a 590~\mega\electronvolt\ proton beam hit a thick carbon target
(E-target in Figure~\ref{fig:psi_exp_hall_all}). The beam line was designed to
deliver muons with momenta ranging from 10 to 500~\mega\electronvolt\per\cc\
and
momentum spread from 0.26 to 8.0\%. These parameters can be selected by
changing various magnets and slits shown in
Figure~\ref{fig:psi_piE1_elements}~\cite{Foroughli.1997}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.7\textwidth]{figs/psi_piE1_elements}
\caption{The $\pi$E1 beam line}
\label{fig:psi_piE1_elements}
\end{figure}
One of the main requirements of the AlCap experiment was a low energy muon beam
with narrow momentum bite in order to achieve a high fraction of stopping muons
in the very thin targets. In this Run 2013, muons from 28 to
45~\mega\electronvolt\per\cc\ and momentum spread of 1\% and 3\%were used.
For part of the experiment the target was replaced with one of the silicon
detector packages allowed an accurate momentum and range calibration
%(via range-energy relations)
of the beam at the target. Figure~\ref{fig:Rates} shows the measured muon rates
as a function of momentum for two different momentum bites.
Figure~\ref{fig:Beam} shows an example of the resulting energy spectra.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\textwidth]{figs/Rates.png}
\caption{Measured muon rate (kHz) at low momenta. Momentum bite of 3 and 1 \%
FWHM, respectively.}
\label{fig:Rates}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.9\textwidth]{figs/beam.pdf}
\caption{Energy deposition at 36.4 MeV/c incident muon beam in an
1500-\micron-active
target. The peak at low energy is due to beam electrons, the
peaks at higher energies are due to muons. Momentum bite of 1 and 3\% FWHM
on left and right hand side, respectively.} \label{fig:Beam}
\end{figure}
The targets and charged particle detectors are installed inside the vacuum
chamber as shown in Figure~\ref{fig:alcap_setup_detailed}. The muon beam enters
from the right of the image and hits the target, which is placed at the
centre of the vacuum chamber and orientated at 45 degrees to the beam axis.
The side walls and bottom flange of the vessel provide several
vacuum-feedthroughs for the high voltage and signal cables for the silicon and
scintillator detectors inside the chamber.
In addition, the chamber is equipped with several lead collimators
%so that muons that are not captured in the target would quickly decay.
to quickly capture muons that do not stop in the actual target.
%\begin{figure}[htbp]
%\centering
%\includegraphics[width=0.55\textwidth]{figs/SetupOverview.jpg}
%\caption{Vacuum chamber in beam line}
%\label{fig:SetupOverall}
%\end{figure}
%It is known fact that there is a risk of sparkling between the electrodes of
%a silicon detector in the low vacuum region of $10^{-3}$ mbar.
%An interlock mechanism was installed to prevent the bias of the
%silicon detectors from being applied before the safe vacuum level.
For a safe operation of the silicon detector, a vacuum of $<10^{-4}$\,mbar was
necessary. With the help of the vacuum group of PSI, we could consistently
reach $10^{-4}$\,mbar within 45 minutes after closure of the chamber's top
flange.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Silicon detectors}
The main detectors for proton measurement in the Run 2013 were four large area
silicon detectors. The silicon detectors were grouped into two detector
packages located symmetrically at 90 degrees of the nominal muon beam path, SiL
and SiR in Figure~\ref{fig:alcap_setup_detailed}. Each arm consists of: one
$\Delta$E counter, a 65-\micro\meter-thick silicon detector, divided into
4 quadrants; one E counter made from 1500-\micron-thick silicon; and one
plastic scintillator to identify electrons or high energy protons that pass
through the silicon. The area of each of these silicon detectors and the
scintillators is $50\times50 \textrm{mm}^2$.
The detectors were named according to their positions relative to the muon
view: the SiL package contains the thin
detector SiL1 and thick detector SiL2; the SiR package has SiR1 and SiR2
accordingly. Each quadrant of the thin detectors were also numbered from 1 to
4, i.e. SiL1-1, SiL1-2, SiL1-3, SiL1-4, SiR1-1, SiR1-2, SiR1-3,
SiR1-4.
Bias for the four silicon detectors was supplied by an ORTEC 710 NIM module,
which has a vacuum interlock input to prevent biasing before the safe vacuum
level has been reached. Typical voltage to fully depleted the detectors were
-300~\volt\ and -10~\volt\ for the thick and thin silicon detectors
respectively. The leakage currents at the operating voltages are less than
1.5~\micro\ampere\ for the thick detectors, and about 0.05~\micro\ampere\
for the thin ones (see Figure~\ref{fig:si_leakage}).
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/si_leakage}
\caption{Leakage currents of the silicon detectors under bias.}
\label{fig:si_leakage}
\end{figure}
The fact that a detector were fully depleted was checked by putting
a calibration source $^{241}\textrm{Am}$ at its ohmic side, and observing the
output
pulse height on an oscilloscope. One would expect that the maximum pulse height
increases as the bias is raised until the voltage of fully depleted. The effect
can also be seen on the pulse height spectrum as in
Figure~\ref{fig:sir2_bias_alpha}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.75\textwidth]{figs/sir2_bias_alpha}
\caption{$^{241}\textrm{Am}$ spectra in cases of fully depleted (top), and
partly depleted (bottom).}
\label{fig:sir2_bias_alpha}
\end{figure}
%It is known that the noise level of a silicon detector increases linearly with
%its capacity. So both noise and pick-up suppression had been carefully
%optimised in the real PSI accelerator environment, particularly for the thin
%silicon detectors which have a large capacity of 1~\nano\farad~in each
%quadrant.
%After improving the feed-through flanges during the set-up phase of the
%experiment with isolated ground connections, good electronic resolution of
%55--76~\kilo\electronvolt\ FWHM was achieved in the thin silicon detectors.
%So achieving good energy resolution was particularly challenging
%for the thin silicon detector, as each quadrant had a large capacity of
%1~\nano\farad. Both
%noise and pick-up suppression had been carefully optimized in the real PSI
%accelerator environment.
%Optimization of the fast timing signals proved another challenge.
%The energy calibration for the silicon detectors were done
%by several means:
%\begin{enumerate}
%\item An $^{241}\textrm{Am}$ alpha source: the main alpha
%particles have energies of 5.484~\mega\electronvolt\ (85.2\%) and
%5.442~\mega\electronvolt\ (12.5\%). The source emits 79.5
%$\alpha\per\second$ in 2$\pi$~\steradian.
%\item Test pulse with a fixed amplitude: the preamplifiers used for the
%silicon detectors are come with the manufacturer's specification on the
%response, namely a 66 \milli\volt\ fed into the test input will produce an
%output equivalent to that of a 1 \mega\electronvolt\ energy deposition.
%\item Minimum ionisation particles
%(MIPs): electrons in the beam are MIPs with a nominal deposit energy of
%388~\electronvolt\per\micro\meter\ Si. This is only applicable for thick
%silicon detectors because the energy deposit is large enough and the energy
%resolution is good enough. During the run, this peak was observed to make
%sure the stability of the electronics.
%\item Muons with different momenta: the thick silicon detectors were placed
%at the target position during beam tuning period, allowed an accurate
%momentum and range calibration. This also only works with thick silicon
%detectors.
%\end{enumerate}
% subsection silicon_detectors (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Upstream counters}
\label{sub:upstream_counters}
The upstream detector consists of three counters: a 500~$\mu$m thick
scintillator muon trigger counter ($\mu$SC); a muon anti-coincidence counter
($\mu$SCA) surrounding the trigger counter with a hole
of 35 \milli\meter\ in diameter to define the beam radius; and a multi-wire
proportional chamber ($\mu$PC) that uses 24 X wires and 24 Y wires at
2~\milli\meter~intervals.
The upstream detectors provide signal of an incoming muon as coincident hits on
the muon trigger and the wire chamber in anti-coincident with the muon
anti-coincidence counter.
This set of detectors along with their read-out system
belong to the MuSun experiment, which operated at the same beam line just
before our run. Thanks to the MuSun group, the detectors were well-tuned and
ready to be used in our run without any modification.
% subsection upstream_counters (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Germanium detector}
%\begin{figure}[htbp]
%\centering
%\includegraphics[width=0.9\textwidth]{figs/neutron.png}
%\caption{Setup of two
%liquid scintillators outside the vacuum envelope for neutron detection.}
%\label{fig:neutron}
%\end{figure}
We used a germanium detector to normalise the number of stopped muons by
measuring characteristics muon X-rays from the target material. The primary
X-rays of interest are the 346.828~keV line for aluminium targets, and the
400.177 line for silicon targets. The energies and intensities of the X-rays
listed in Table~\ref{tab:xray_ref} follow measurement results from
Measday and colleagues~\cite{MeasdayStocki.etal.2007}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{c l l l l }
\toprule
\textbf{Elements} & \textbf{Transition}
& \textbf{Energy} & \textbf{Intensity}\\
\midrule
$^{27}\textrm{Al}$ & $2p-1s$ & $346.828 \pm 0.002$ & $79.8\pm 0.8$\\
& $3p-1s$ & $412.87 \pm 0.05$ & $7.62\pm 0.15$\\
\midrule
$^{28}\textrm{Si}$ & $2p-1s$ & $400.177 \pm 0.005$ & $80.3\pm 0.8$\\
& $3p-1s$ & $476.80 \pm 0.05$ & $7.40 \pm 0.20$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Reference values of major muonic X-rays from aluminium and silicon.}
\label{tab:xray_ref}
\end{table}
The germanium detector is
a GMX20P4-70-RB-B-PL, n-type, coaxial high purity germanium detector produced
by ORTEC. The detector was optimised for low energy gamma and X-rays
measurement with an ultra-thin entrance window of 0.5-mm-thick beryllium and
a 0.3-\micron-thick ion implanted contact (Figure~\ref{fig:ge_det_dimensions}).
This detector is equipped with a transistor reset preamplifier which,
according to the producer, enables it to work in an ultra-high rate environment
up to $10^6$ counts\per\second~ at 1~\mega\electronvolt.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\textwidth]{figs/ge_det_dimensions}
\caption{Dimensions of the germanium detector}
\label{fig:ge_det_dimensions}
\end{figure}
The detector was installed outside of the vacuum chamber at 32 cm from the
target, seeing the target through a 10-mm-thick aluminium window, behind
a plastic scintillator counter used to veto electrons. Liquid nitrogen
necessary for the operation of the detector had to be refilled every 8 hours.
A timer was set up in the data acquisition system to remind this.
\subsection{Plastic and liquid scintillators}
\label{sub:plastic_scintillators}
Apart from the scintillators in the upstream group, there were four other
plastic scintillators used as veto counters for:
\begin{itemize}
\item punch-through-the-target muons, ScVe
\item electrons and other high energy charged particles for germanium
detector (ScGe) and silicon detectors (ScL and ScR)
\end{itemize}
The ScL, ScR and ScVe were installed inside the vacuum vessel and were
optically connected to external PMTs by light-guides at the bottom flange.
We also set up two liquid scintillation counters for neutron measurements in
preparation for the next beam time where the neutron measurements will be
carried out.
% subsection plastic_scintillators (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Front-end electronics and data acquisition system}
The front-end electronics of the AlCap experiment was simple since we employed
a trigger-less read out system with waveform digitisers and flash ADCs
(FADCs). As shown in Figure~\ref{fig:alcapdaq_scheme}, all plastic
scintillators signals were amplified by PMTs, then fed into the digitisers. The
signals from silicon and germanium detectors were preamplified, and
subsequently shaped by spectroscopy amplifiers and timing filter amplifiers
(TFAs) to provide energy and timing information.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.99\textwidth]{figs/alcapdaq_scheme}
\caption{Schematic diagram of the electronics and DAQ used in the Run 2013}
\label{fig:alcapdaq_scheme}
\end{figure}
The germanium detector has its own transistor reset preamplifier
installed very close to the germanium crystal. Two ORTEC Model 142
preamplifiers were used for the thick silicon detectors. The timing outputs of
the preamplifiers were fed into three ORTEC Model 579 TFAs.
We used an ORTEC Model 673 to shape the germanium signal with 6~\micro\second
shaping time.
A more modern-style electronics was used for thin silicon detectors where the
preamplifier, shaping and timing amplifiers were implemented on one compact
package, namely a Mesytec MSI-8 box. This box has 8 channels, each channel
consists of one preamplifier board and one shaper-and-timing filter board which
can be fine-tuned independently. The shaping time was set to 1~\micro\second\
for all channels.
The detector system produced signals that differs significantly in time scale,
ranging from very fast (about 40~\nano\second\ from scintillators) to very slow
(several \micro\second\ from shaping outputs of semiconductor detectors). This
lead to the use of several sampling frequencies from 17~\mega\hertz\ to
250~\mega\hertz, and three types of digitisers were employed:
\begin{itemize}
\item custom-built 12-bit 170-MHz FADCs which was designed for the
MuCap experiment. Each FADC board has dimensions the same as those of
a single-width 6U VME module, but is hosted in a custom built crate due to
its different power supply mechanical structure. The FADC communicates with
a host computer through a 100-Mb/s Ethernet interface using a simple
Ethernet-level protocol. The protocol only allows detecting
incomplete data transfers but no retransmitting is possible due to the
limited size of the module's output buffer. The FADCs accept clock signal
at the frequency of 50~\mega\hertz\ then multiply that internally up to
170~\mega\hertz. Each channel on one board can run at different sampling
frequency not dependent on other channels. The FADC has 8 single-ended
LEMO inputs with 1~\volt pp dynamic range.
\item a 14-bit 100-MS/s CAEN VME FADC waveform digitiser model V1724. The
module houses 8 channels with 2.25~Vpp dynamic range on single-ended MCX
coaxial inputs. The digitiser features an optical link for transmission of
data to its host computer. All of 8 channels run at the same sampling
frequency and have one common trigger.
\item a 12-bit 250-MS/s CAEN desktop waveform digitizer model DT5720. This
digitiser is similar to the V1724, except for its form factor and maximum
sampling frequency. Although there is an optical link available, the module
is connected to its host computer through a USB 2.0 interface where data
transfer rate of 30 MB/s was determined to be good enough in our run
(actual data rate from this digitiser was typically about 5 MB/s during the
run). Communication with both CAEN digitisers was based on CAEN's
proprietary binary drivers and libraries.
\end{itemize}
All digitisers were driven by external clocks which were derived from the same
500-\mega\hertz\ master clock, a high precision RF signal generator Model SG382
of Stanford Research System.
The silicon detectors were read out by FADC boards feature network-based data
readout interface. To maximize the data throughput, each of the four FADC
boards was read out through separate network adapter.
The CAEN digitisers were used to read out
the germanium detector (timing and energy, slow signals) or scintillator
detectors (fast signals). For redundancy, all beam monitors ($\mu$SC, $\mu$SCA
and $\mu$PC) were also read out by a CAEN time-to-digital converter (TDC)
model V767 which was kindly provided by the MuSun experiment.
The Data Acquisition System (DAQ) of the AlCap experiment, so-called AlCapDAQ,
provided the readout of front-end electronics, event assembling, data logging,
hardware monitoring and control, and the run database of the experiment
(Figure~\ref{fig:alcapdaq_pcs}). It was based on MIDAS framework~\footnote{
MIDAS is a general purpose DAQ software system developed at PSI and TRIUMF:\\
\url{http://midas.triumf.ca}} and consisted of two circuits, {\em i})
a detector circuit for synchronous data readout from the front-end electronics
instrumenting detectors, and {\em ii}) a slow control circuit for asynchronous
periodic hardware monitoring (vacuum, liquid nitrogen
filling). The detector circuit consisted of three computers, two front-end
computers and one computer serving both as a front-end and as a back-end
processor. The slow circuit consisted of one computer. All computers were
running Linux operating system and connected into a private subnetwork.
%\hl{TODO: storage and shift monitor}
\begin{figure}[htb]
\centering
\includegraphics[width=0.95\textwidth]{figs/alcapdaq_pcs}
\caption{AlCapDAQ in the Run 2013. The {\ttfamily fe6} front-end is
a VME single board computer belongs to the MuSun group, reads out the
upstream detectors.}
\label{fig:alcapdaq_pcs}
\end{figure}
The data were collected as dead-time-free time segments of 110~ms, called
``block'', followed by about 10-ms-long time intervals used to complete data
readout and synchronize the DAQ. Such data collection approach was chosen to
maximize the data readout efficiency. During each 110-ms-long period, signals
from each detector were digitized independently by threshold crossing. The data
segment of each detector data were first written into on-board memories of
front-end electronics and either read out in a loop (CAEN TDCs and CAEN
digitizers) or streamed (FADCs) into the computer memories. The thresholds were
adjusted as low as possible and individually for each detector. The time
correlation between detectors would be established in the analysis stage.
At the beginning of each block, the time counter in each digitiser is reset to
ensure time alignment across all modules. The period of 110~ms was chosen to be:
{\em i} long enough compares to the time scale of several \micro\second\ of the
physics of interest, {\em ii} short enough so that there is no timer rollover
on any digitiser (a FADC runs at its maximum speed of 170~\mega\hertz\ could
handle up to about 1.5 \second\ with its 28-bit time counter).
To ease the task of handling data, the data collecting period was divided into
short runs, each run stopped when the logger had recorded 2 GB of data.
The data size effectively made each run last for about 5 minutes. The DAQ
automatically starts a new run with the same parameters after about 6 seconds.
The short period of each run also allows the detection, and helps to reduce the
influence of effects such as electronics drifting, temperature fluctuation.
\section{Data sets and statistics}
\label{sec:data_sets}
The main goal of this Run 2013 was to measure the rates and energy spectra of
protons following muon capture on aluminium. Also for normalisation and cross
checking against the existing experimental data, two types of measurements with
different targets were carried out for silicon targets:
\begin{itemize}
\item[(a)] an active, thick target similar to the set up
used by Sobottka and Wills~\cite{SobottkaWills.1968}. This provides
a cross-check against the existing experimental data. The silicon detector
package at the right hand side was moved to the target position with the
thick detector facing the muon beam in this set up.
\item[(b)] a passive, thin target and heavy charged particles were observed
by the two silicon packages. The measurement serves multiple purposes:
confirmation that the particle identification by dE/dx actually works,
separation of components of heavy charged particles emitted from the
silicon target.
\end{itemize}
As the emitted protons deposit a significant amount of energy in the target
material, thin targets and thus excellent momentum resolution of the low energy
muon beam are critical. Aluminium targets of 50-\micro\meter\ and
100~\micron\ thick were used. Although a beam with low momentum spread of
1\% is preferable, it was used for only a small portion of the run due to the
low beam rate (see Figure~\ref{fig:Rates}). The beam momentum for each target
was chosen to maximise the number of stopped muons. The collected data sets are
shown in Table~\ref{tb:stat}.
\begin{table}[htb!]
\begin{center}
\vspace{0.15cm}
\begin{tabular}{l c c c}
\toprule
\textbf{Target} &\textbf{Momentum} & \textbf{Run time} & \textbf{Number}\\
\textbf{and thickness}&\textbf{scaling factor} & \textbf{(h)} &\textbf{of muons}\\
\midrule
Si 1500 \micro\meter& 1.32& 3.07& $2.78\times 10^7$\\
& 1.30& 12.04& $2.89 \times 10^8$\\
& 1.10& 9.36& $1.37 \times 10^8$ \\
\midrule
Si 62 \micro\meter & 1.06& 10.29& $1.72 \times 10^8$\\
\midrule
Al 100 \micro\meter& 1.09& 14.37&$2.94 \times 10^8$\\
& 1.07& 2.56& $4.99 \times 10^7$\\
\midrule
Al 50 \micro\meter m & 1.07& 51.94& $8.81 \times 10^8$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Run statistics. Momentum scaling
normalized to 28 MeV/c.}
\label{tb:stat}
\end{table}
% section data_sets (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Analysis framework}
\subsection{Concept}
\label{sub:concept}
Since the AlCapDAQ is a trigger-less system, it stored all waveforms of the
hits occured in 100-ms-long blocks without considering their physics
significance The analysis code therefore must be able to extract parameters of
the waveforms, then organises the pulses into physics events correlated to
stopped muons (Figure~\ref{fig:muon_event}). In addition, the analyser is
intended to be usable as a real-time component of a MIDAS DAQ, where simple
analysis could be done online for monitoring and diagnostic during the run.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\textwidth]{figs/muon_event.pdf}
\caption{Concept of the AlCap analysis code: pulses from individual detector
in blocks of time are analysed, then sorted centred around stopped muons.}
\label{fig:muon_event}
\end{figure}
The analysis framework of the AlCap consists of two separate programs.
A MIDAS-based analyser framework, \alcapana{}, processes the raw data and
passes its ROOT data output to a second
stage, \rootana{}, where most of the physics analysis is performed.
Both programs were designed to be modularised, which allowed us to develop
lightweight analysis modules that were used online to generate plots quickly,
while more sophisticated modules can be applied in offline analysis.
The DAQ system generated MIDAS files which stores the data as a stream of MIDAS
``banks''. In the AlCapDAQ, each bank corresponds to a single channel on
a digitizer and was named according to a predefined convention. The map between
detector channels and MIDAS bank names was stored in the MIDAS online database
(ODB), along with other settings such as sampling frequencies, timing offsets,
thresholds and calibration coefficients of each channel.
%These can then be
%accessed by both \alcapana{} and \rootana{} for either online or offline
%analysis.
The first step
of the analysis framework is to convert the raw MIDAS data into waveforms,
series of digitised samples continuous in time corresponding to pulses from the
detector. The waveform is called \tpulseisland{}s, which contain the bank name,
the ADC values of the samples and the time stamp of the first sample. This
conversion is performed in \alcapana{} and the resulting objects are stored in
a ROOT output file as a {\ttfamily TTree}.
The next step of the analysis is to obtain summary parameters of the pulses
from the digitized samples. The parameters of primary interest are the
amplitude and time of the peak and the integral of the pulse. This extraction
of parameters is done by a \rootana{} module, and the objects produced by this
stage are called \tanalysedpulse{}s. Currently, we have a usable and simple
algorithm that takes the pulse parameters from the peak of the waveform. In
parallel, a pulse finding and template fitting code is being developed because
it would provide more accurate pulse information. The first iteration of this
code has been completed and is being tested.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/analysis_scheme}
\caption{Concept of the analysis framework in \rootana{}}
\label{fig:rootana_scheme}
\end{figure}
After obtaining pulse parameters for individual channel, the pairing up of
fast and slow pulses from the same physical detector needs to be done. This
entails looping through all fast and slow pulses from each detector,
checking for correlated pulses in time and amplitude, creating
{\ttfamily TDetectorPulse}s. The {\ttfamily TDetectorPulse}s allow better
understanding of the hits on the detector by combining timing information from
the fast channel and amplitude information from the slow channel. It also helps
reduce the impact of pile-up on the amplitude measurement, where the
improved time resolution of the fast channels can be used to separate the
overlapping amplitudes in the slow channels. The pulse pairing are applicable to
the silicon and germanium channels only. The scintillator channels provide only
fast timing signals which can be used as {\ttfamily TDetectorPulse}s directly.
The detector pulses are subsequently used to identify particles that hit the
detectors. These particle hits are still stored in the time-ordered tree
corresponds to the 110 ms block length from the AlCapDAQ. By iterating through
the tree to find stopped muons and taking any hits within a certain window
around this muon from every detector, a stopped-muon-centred tree shown in
Figure~\ref{fig:muon_event} can be produced. This will make it much easier to
look for coincidences and apply cuts, thereby bringing the end
goal of particle numbers and energy distributions.
% subsection concept (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Online analyser}
\label{sub:online_analyser}
The online analyser was developed and proved to be very useful during the run.
A few basic modules were used to produce plots for diagnostic purposes
including: persistency view of waveforms, pulse height
spectra, timing correlations with respect to the upstream counters. The
modules and their purposes are listed in Table~\ref{tab:online_modules}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l p{6cm}}
\toprule
\textbf{Module name} & \textbf{Functions}\\
\midrule
common/MUnCompressRawData & decompress raw MIDAS data\\
\midrule
FADC/MOctalFADCProcessRaw & \multirow{3}{6cm}{convert raw data to
{\ttfamily TPulseIsland}s}\\ v1724/MV1724ProcessRaw& \\
dt5720/MDT5720ProcessRaw&\\
\midrule
muSC\_muPC/MCaenCompProcessRaw& \multirow{4}{6cm}{decompress data from
{\ttfamily fe6}, make coincidence in upstream counters} \\
muSC\_muPC/MMuPC1AnalysisC&\\
muSC\_muPC/MMuPC1AnalysisMQL&\\
muSC\_muPC/MMuSCAnalysisMQL&\\
\midrule
diagnostics/MCommonOnlineDisplayPlots& produce plots of interest\\
\midrule
FADC/MOctalFADCBufferOverflow& \multirow{2}{6cm}{diagnostics for FADCs}\\
FADC/MOctalFADCPacketLoss&\\
\midrule
common/MExpectedIslands&\multirow{4}{6cm}{diagnostics in general}\\
common/MMuSCTimeDifferences&\\
common/MNumberIslands&\\
common/MPulseLengths&\\
\midrule
common/MTreeOutput& save {\ttfamily TPulseIsland}s tree\\
\bottomrule
\end{tabular}
\end{center}
\caption{Online analysis modules in the Run 2013.}
\label{tab:online_modules}
\end{table}
The \alcapana{} served the plots on port 9090 of the {\ttfamily abner}
via the ROOT socket protocol. We then used a ROOT-based program called
{\ttfamily online-display} to display the plots on the shift terminal
({\ttfamily alcap}). The {\ttfamily online-display} simply executed ROOT macros
which retrieved plots from the ROOT server, sorted then drew them in
groups such as upstream counters, silicon arms. It could also periodically
update the plots to reflect real-time status of the detector system.
%Screen
%shots of the {\ttfamily online-display} with several plots are shown in
%Figure~\ref{fig:online_display}.
%\hl{Screen shots}
\subsection{Offline analyser}
\label{sub:offline_analyser}
Some offline analysis modules has been developed during the beam time and could
provide quick feedback in confirming and guiding the decisions at the time. For
example, the X-ray spectrum analysis was done to confirm that we could observe
the muon capture process (Figure~\ref{fig:muX}), and to help in choosing optimal
momenta which maximised the number of stopped muons.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.7\textwidth]{figs/muX.png}
\caption{Germanium
detector spectra in the range of 300 - 450 keV with different setups: no
target, 62-\micron-thick silicon target, and 100-\micron-thick aluminium
target. The ($2p-1s$) lines from
aluminium (346.828 keV) and silicon (400.177 keV) are clearly visible,
the double peaks at 431 and 438 keV are from the lead shield, the peak at
351~keV is a background gamma ray from $^{211}$Bi.}
\label{fig:muX}
\end{figure}
Although the offline analyser is still not fully developed yet, several modules
are ready. They are described in detailed in the next chapter.
% subsection offline_analyser (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section analysis_strategy (end)
% chapter the_alcap_run_2013 (end)

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\chapter{Results and Discussions}
\label{cha:results_and_discussions}
% chapter results_and_discussions (end)

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\chapter{Discussions}
\label{cha:discussions}
\section{Thick aluminium target measurement}
\label{sub:active_target_measurement}
With a thick and active silicon target, I have tried to reproduce an existing
result from Sobottka and Wills~\cite{SobottkaWills.1968}. This is important in
giving confidence in our experimental method. The idea is the same as that of
the old measurement, where muons were stopped inside a bulk active target and
the capture products were measured. Due to the limitation of the
currently available analysis tool, a direct comparison with the result of
Sobottka and Wills is not practical at the moment.
But a partial comparison is available for a part of the spectrum from 8 to
10~MeV, where my result of $(1.22 \pm 0.19) \times 10^{-2} $ is consistent with
the derived value $(1.28\pm0.19)\times10^{-2}$ from the paper of Sobottka and
Wills. The agreement was partly because of large error bars in both results.
In my part, the largest error came from the uncertainty on choosing the
integration window. This can be solved with a more sophisticated pulse
finding/calculating algorithm so that the contribution of muons in the energy
spectrum can be eliminated by imposing a cut in pulse timing. The
under-testing pulse template fitting module could do this job soon.
The range of 8--10~MeV was chosen to be large enough so that the uncertainty of
integration window would not to be too great; and at the same time be small
enough so the protons (and other heavier charged particles) would not escape
the active target. This range is also more convenient for calculating the
partial rate from the old paper of Sobottka and Wills.
% section protons_following_muon_capture_on_silicon (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Thin silicon target measurement}
\label{sub:thin_and_passive_target_measurement}
The charged particles in the low energy region of 2.5--8~MeV were measured by
dE/dx method. The particle identification was good in lower energy part, but
losing its resolution power as energy increases. The current set up could do
the PID up to about 8~MeV for protons. This energy range is exactly the
relevant range to the COMET experiment (Figure~\ref{fig:proton_impact_CDC}).
In that useful energy range, the analysis showed a good separation of protons
from other heavy charged particles. The contribution of protons in the total
charged particles is 87\%. This is the high limit only since the heavier
particles at this energy range are most likely to stopped in the thin
detectors. More statistic would be needed to estimate the contributions from
other particles.
The effective emission rate of protons per muon capture in this measurement is
4.20\%, with a large uncertainty contribution comes from limitation of the
timing determination. The spectral integral in the region 2.5--8~MeV on
Figure~\ref{fig:sobottka_spec} is about 70\% of the spectrum from 1.4 to
26~\MeV, and corresponds to an emission rate of about 10\% per muon capture.
The two figures are not in disagreement.
In order to have a better comparison, a correction or unfolding for energy
loss and more MC simulation study are needed. I am on progress of these study.
% subsection thin_and_passive_target_measurement (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Aluminium target measurement}
\label{sec:aluminium_target_measurement}
The proton emission rate was derived as 2.37\%, but the problem on the SiL1-1
channel was not solved yet. One possible cause is the muons captured on other
lighter material inside the chamber. More investigation will be made on this
matter.
The rate of 2.37\% on aluminium appears to be smaller on that of silicon but
the two results are both effective rates, modified by energy loss inside the
target. Unfolding and MC study for the correction are ongoing.
% section aluminium_target_measurement (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% chapter discussions (end)

88
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%% Title
\titlepage[of Graduate School of Science]%
{A dissertation submitted to the Osaka University\\
for the degree of Doctor of Philosophy}
%\titlepage[\vspace{5mm}Department of Physics,\\
%Graduate School of Science]%
%{A dissertation submitted to the Osaka University\\
%for the degree of Doctor of Philosophy}
\titlepage[]{\vspace{5mm}Department of Physics, Graduate School of Science\\
Osaka University}
%% Abstract
\begin{abstract}%[\smaller \thetitle\\ \vspace*{1cm} \smaller {\theauthor}]
%\thispagestyle{empty}
\LHCb is a \bphysics detector experiment which will take data at
the \unit{14}{\TeV} \LHC accelerator at \CERN from 2007 onward\dots
\begin{abstract}
%[\smaller \thetitle\\ \vspace*{1cm} \smaller {\theauthor}]
\thispagestyle{empty}
COMET [1] is an experiment that aims to search for a charged lepton flavor
violation (CLFV) process, the muon-to-electron conversion in the presence of
a nucleus,
\muec. The process is forbidden in the Standard Model (SM), however is
predicted to occur in various extensions of SM, such as . Current experimental
upper limit of the branching ratio is $BR(\mu^{-} + Au \rightarrow e^{-} + Au)
< 7 \times 10^{-13}$, set by the SINDRUM II experiment [2].
Using the J-PARC proton beam and the pion capture by
a solenoidal field, COMET will have a single event sensitivity 10,000 times
better than the current limit. The COMET collaboration has taken a phased
approach in which the first phase, COMET Phase-I [3], starts in 2013 and
initial data taking in around 2017.
In order to optimize detector design for the Phase-I, backgrounds from nuclear
muon capture are crucial. We have proposed a dedicated experiment , namely
AlCap, at PSI, Switzerland to study the backgrounds, including: heavy charged
particles, neutrons and photons. The measurements of heavy charged particles
have been carried out in the 2013 run and the preliminary analysis results are
presented in this thesis.
\end{abstract}
%% Declaration
\begin{declaration}
This dissertation is the result of my own work, except where explicit
reference is made to the work of others, and has not been submitted
for another qualification to this or any other university. This
dissertation does not exceed the word limit for the respective Degree
Committee.
\vspace*{1cm}
\begin{flushright}
Andy Buckley
\end{flushright}
\thispagestyle{empty}
This dissertation is the result of my own work, except where explicit
reference is made to the work of others, and has not been submitted
for another qualification to this or any other university.
\vspace*{1cm}
\begin{flushright}
Nam Hoai Tran
\end{flushright}
\end{declaration}
%% Acknowledgements
\begin{acknowledgements}
Of the many people who deserve thanks, some are particularly prominent,
such as my supervisor\dots
\end{acknowledgements}
% Acknowledgements
%\begin{acknowledgements}
%\thispagestyle{empty}
%Of the many people who deserve thanks, some are particularly prominent,
%such as my supervisor Professor Yoshitaka Kuno.
%\end{acknowledgements}
%% Preface
\begin{preface}
This thesis describes my research on various aspects of the \LHCb
particle physics program, centred around the \LHCb detector and \LHC
accelerator at \CERN in Geneva.
\noindent
For this example, I'll just mention \ChapterRef{chap:SomeStuff}
and \ChapterRef{chap:MoreStuff}.
\end{preface}
%\begin{preface}
%\thispagestyle{empty}
%The thesis is about measurements of products of nuclear muon capture on an
%aluminum target, which is important for optimization of a tracking detector
%of a search for muon to electron conversion, the E21 experiment - so called
%COMET - at Japan Proton Accelerator Complex (J-PARC).
%\end{preface}
%% ToC
\tableofcontents
%% Strictly optional!
\frontquote{%
Writing in English is the most ingenious torture\\
ever devised for sins committed in previous lives.}%
{James Joyce}
%\frontquote{%
%Writing in English is the most ingenious torture\\
%ever devised for sins committed in previous lives.}%
%{James Joyce}