252 lines
12 KiB
TeX
252 lines
12 KiB
TeX
\chapter{Introduction}
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\label{chap:intro}
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%% Restart the numbering to make sure that this is definitely page #1!
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\pagenumbering{arabic}
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\section{Muon to electron conversion}
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\label{sec:_mu_e_conversion}
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Charged lepton flavor violation (CLFV) belongs to the class of flavor-changing
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neutral currents, which are suppressed at tree level in the Standard Model
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(SM) where they are mediated by $\gamma$ and $Z^0$ bosons, but arise at loop
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level via weak charged currents mediated by the $W^{\pm}$ boson. Because flavor
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violation requires mixing between generations, CLFV exactly vanishes in the SM
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with massless neutrinos. Even in the framework of the SM with massive neutrinos
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and their mixing, branching ratio of CLFV is still very small - for example, in
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case of \mueg~\cite{marciano}:
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\begin{equation}
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\mathcal{B}(\mu^{+} \rightarrow e^{+}\gamma) \simeq
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10^{-54} \left( \frac{sin^{2}2\theta_{13}}{0.15}\right)
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\end{equation}
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This is an unobservably tiny
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branching ratio so that any experimental evidence of CLFV would be a clear
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sign of new physics beyond the SM.
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One of the most prominent CLFV processes is
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a process of coherent muon-to-electron conversion ($\mu
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- e$ conversion) in the field of a nucleus: \muecaz. When muons are stopped in
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a target, they are quickly
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captured by atoms ($~10^{-10}$ s) and cascade down to the 1S orbitals. There,
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they can undergo:
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(a) ordinary decay, (b) weak capture, $\mu^- p \rightarrow \nu_\mu n$, or (c)
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$\mu - e$ conversion, \muec. The last of these reactions is a CLFV process
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where lepton flavor numbers, $L_\mu$ and $L_e$, are violated by one unit.
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The $\mu - e $ conversion is attractive both from theoretical and experimental
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points of view. Many extensions of the SM predict that it would has sizeable
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branching ratio~\cite{altman}. One possible supersymmetric contribution to the
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$\mu - e$ conversion is shown in Fig.~\ref{fig:susy_contr}. Experimentally, the
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simplicity and distinctive signal, a mono-energetic electron of energy $E_{e}$:
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$
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E_{e} = m_{\mu} - B_{\mu}(Z, A) - R(A) \simeq \textrm{105 MeV},
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$
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where $m_\mu$ is the muon mass, $B_\mu(Z, A)$ is the muonic atom binding
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energy, and $R(A)$ is the nuclear recoil energy, allow experimental searches
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without accidentals and thus in extremely high rates. As a result, one of the
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best upper limits of CLFV searches comes from a search for $\mu - e$ conversion
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in muonic gold done by the SINDRUM--II collaboration:
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\sindrumlimit~\cite{sindrumii}.
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\begin{figure}[tbh]
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\centering
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\includegraphics[width=\textwidth]{figs/susy_contr}
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\caption{Possible SUSY contributions to the CLFV processes \mueg
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(left) and \muec (right).}
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\label{fig:susy_contr}
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\end{figure}
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%\section{Motivation}
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%\label{sec:motivation}
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\subsection{COMET experiment}
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At the Japan Proton Accelerator Research Complex (J-PARC), an experiment to
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search for \muec~conversion, which is called COMET (COherent Muon to Electron
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Transition), has been proposed~\cite{comet07}. The experiment received Stage--1
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approval in
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2009. Utilising a proton beam of 56 kW (8 GeV $\times$ 7 $\mu$A) from the
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J-PARC main ring, the COMET aims for a single event sensitivity of $3 \times
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10^{-17}$, which is 10000 times better than the current best
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limit at SINDRUM--II. As of April 2013, the COMET collaboration has 117
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members in 27 institutes from 12 countries.
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The COMET experiment is designed to be carried out at the Hadron
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Experimental Facility using a bunched proton beam that is
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slowly-extracted from the J-PARC main ring. The experimental set-up consists of
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a dedicated proton beam line, a muon beam transport section, and a detector
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section. The muon beam section is composed of superconducting magnets: pion
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capture solenoid and a pion/muon transport solenoid. The
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detector section has a multi-layered muon stopping target, an electron
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transport beam line for $\mu - e$ conversion signals,
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followed by detector systems.
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The COMET collaboration has adopted a staging approach with two
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phases~\cite{comet12}. COMET Phase--I is scheduled to
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have an engineering run in 2016, followed by a physics run in 2017. Phase--I
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should achieve a sensitivity
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of $3 \times 10^{-15}$, 100 times better than that of SINDRUM--II; while
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Phase--II will reach a sensitivity of $2.6 \times 10^{-17}$, which is
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competitive with the Mu2e project at Fermilab~\cite{mu2e08}.
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A schematic layout of the COMET experiment with its two phases is
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shown in Fig.~\ref{fig:comet_phase1}, and a schedule for two phases is shown in
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Fig.~\ref{fig:sched}.
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\begin{figure}[tbh]
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\centering
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\includegraphics[width=\textwidth]{figs/comet_phase1}
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\caption{Schematic layout of the COMET experiment with two phases: Phase--I
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(left) and Phase--II (right).}
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\label{fig:comet_phase1}
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\end{figure}
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\begin{figure}[tbh]
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\centering
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\includegraphics[width=0.8\textwidth]{figs/sched}
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\caption{The anticipated schedule of the COMET experiment.}
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\label{fig:sched}
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\end{figure}
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COMET Phase--I has two major goals:
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\begin{itemize}
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\item Background study for the COMET Phase--II by using the actual COMET beam
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line constructed at Phase--I,
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\item Search for $\mu-e$ conversion with a single event sensitivity of $3
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\times 10^{-15}$.
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\end{itemize}
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In order to realize the goals, COMET Phase--I proposes to have two systems of
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detector. A straw tube detector and an electromagnetic calorimeter will be used
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for the background study. For the $\mu-e$ conversion search, a cylindrical
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drift chamber (CDC) will be built.
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\subsection{Proton emission issue}
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We, as a jointed force between Mu2e and COMET, would like to measure rates and
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energy spectrum of charged particle emission after nuclear muon capture on
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aluminum. The rates and spectra of charged particle emission, in particular
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protons, is very important to optimize the detector configuration both for the
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Mu2e and COMET Phase-I experiments.
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\noindent The tracking chambers of COMET Phase-I and Mu2e are designed to be
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measure charged particles of their momenta greater than 70 MeV/$c$ and 53
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MeV/$c$ respectively. In that momentum ranges, it turns out that single hit
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rates of the tracking chambers would be dominated by protons after nuclear muon
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capture.
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The second source of the hit rate will be electrons from muon decays in orbit
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(DIO). In order to limit the single hit rate of the tracking chamber to an
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acceptable level, both experiments are considering to place proton absorbers in
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front of the tracking chambers to reduce proton hit rates. However, the proton
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absorber would deteriorate the reconstructed momentum resolution of electrons
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at birth. And similarly the rate of proton emission is important to determine
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thickness of the muon stopping target made of aluminum. Therefore it is
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important to know the rate so that the detector system can be optimized in
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terms of both hit rate and momentum resolution.
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\noindent Unfortunately the yield, energy spectrum and composition of the
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charged particles emitted in muon capture on Al and Ti have not been measured
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in the relevant energy range for COMET Phase-I and Mu2e.
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Figure~\ref{fg:silicon-proton} shows the spectrum of charged particle emission
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from muons being stopped and captured in a silicon detector \cite{sobo68}. The
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peak below 1.4 MeV is from the recoiling heavy ions, mainly $^{27}$Al, when no
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charged particles were emitted. Hungerford~\cite{hung34} fitted the silicon
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spectrum in Fig.~\ref{fg:silicon-proton} with an empirical function given by
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%
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\begin{equation} p(T) = A(1-{T_{th} \over T})^{\alpha} e^{-(T/T_0)}
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\label{eq:protons} \end{equation}
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%
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where $T$ is the kinetic energy and the fitted parameters are $A=0.105$
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MeV$^{-1}$, $T_{th}$ = 1.4 MeV, $\alpha$=1.328 and $T_0$ = 3.1 MeV. The
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spectrum is normalized to 0.1 per muon capture. Some other results in the past
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experiments are summarized in Table~\ref{tb:proton}.
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%\begin{figure}[htb]
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%\centering
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%\includegraphics[width=0.7\textwidth]{figs/si-proton.pdf}
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%\caption{Charged particle spectrum from muons stopping and being captured in
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%a silicon detector~\cite{sobo68}.}
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%\label{fg:silicon-proton}
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%\end{figure}
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\begin{table}[htb]
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\centering \caption{Probabilities in unites of $10^{-3}$ per
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muon capture for inclusive proton emission calculated by Lifshitz and
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Singer~\cite{lifshitz80}.
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The numbers in crescent parenthesis are estimates for the total inclusive
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rate derived from the measured exclusive channels by the use of the
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approximate regularity, such as $(\mu, \nu p):(\mu, \nu p n):(\mu, \nu p
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2n):(\mu, \nu p 3n) = 1:6:4:4$.}
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\label{tb:proton}
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\vskip 3mm
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\begin{tabularx}{\textwidth}{ccccX}
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\toprule
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Target nucleus & Calculation & Experiment & Estimate & Comments \\
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\midrule
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%$_{10}$Ne & & $200\pm 40$ & & \\
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$^{27}_{13}$Al & 40 & $>28 \pm 4$ & (70) & 7.5 for $T>40$ MeV \\
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$^{28}_{14}$Si & 144 & $150\pm30$ & & 3.1 and 0.34 $d$ for $T>18$ MeV \\
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$^{31}_{15}$P & 35 & $>61\pm6$ & (91) & \\
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$^{46}_{22}$Ti & & & & \\
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$^{51}_{23}$V & 25 & $>20\pm1.8$ & (32) & \\
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\bottomrule
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\end{tabularx}
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\end{table}
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\noindent The limited information available at present makes it difficult to
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draw quantitative conclusive detector design. From Table~\ref{tb:proton}, the
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yield for Al can be taken from experiment to be $>$3\% for $T>40$ MeV, or from
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theory to be 4\%, or estimated based on the ratio of exclusive channels from
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other nuclei to be 7\%, or speculated to be as high as Si
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%or Ne
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, namely 15-20\%. The
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energy spectrum can only be inferred from the Si data or from
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Ref.~\cite{bala67}. At this moment, for both COMET Phase-I and Mu2e, this
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analytical spectrum has been used to estimate proton emission. And also the $p,
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d, \alpha$ composition is not known. The Ti proton yield can only be estimated
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from V to be around 3\%.
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\noindent It might be worth to present how proton emission affects a single
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rate of the tracking chambers. As an example for COMET Phase-I, single rates
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of the tracking chamber (cylindrical drift chamber) have been simulated based
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on the spectrum given in Eq.(\ref{eq:protons}). To reduce protons entering the
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tracking chamber, in addition to the inner wall of the drift chamber (of 400
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$\mu$m) a cylindrical proton absorber of different thickness is located in
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front of the tracking chamber. Monte Carlo simulations were done with three
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different thickness of proton degrader, namely 0~mm, 5~mm, and 7.5~mm.
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%Figure~\ref{fig:protongenerated} shows a proton momentum spectrum generated
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(larger than 50 MeV/$c$) in the simulation study, and regions in red show
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protons reaching the first layer. The results are summarized in
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Table~\ref{tb:protonhits}, where the proton emission rate of 0.15 per muon
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capture is assumed. If we assume the number of muons stopped in the
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muon-stopping target is $5.8 \times 10^{9}$/s, the number of muon capture on
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aluminum is about $3.5 \times 10^{9}$/s since the fraction of muon capture in
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aluminum is $f_{cap}=0.61$. Therefore the total number of hits in all the cells
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in the first layer is estimated to be 530 kHz (1.3 MHz) for the case of a
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proton degrader of 5 mm (0 mm) thickness. This example present the importance
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to understand the proton emission, rate and spectrum, from nuclear muon capture
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on aluminum for COMET Phase-I and Mu2e.
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%
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\begin{table}[htb]
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\begin{center}
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\caption{Total numbers of hits in the first
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layer by protons emitted from muon capture for different trigger counter
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thickness. 100 k proton events were generated for COMET Phase-I. 15 \%
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protons per muon capture is assumed.}
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\label{tb:protonhits}
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\vspace{5mm}
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\begin{tabular}{lccc}
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\toprule
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Proton degrader thickness & 0 mm & 5 mm& 7.5 mm\\
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\midrule
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% number of 1 hit events & 2467 & 87 & 28 \cr\hline number of 2 hit events &
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% 73 & 8 & 1 \cr\hline number of 3 hit events & 9 & 0 & 0 \cr\hline\hline
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% number of 4 hit events & 1 & 0 & 0 \cr\hline\hline
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Hits & 2644 & 103 & 30 \cr
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Hits per proton emission & 2.6 \% & 0.1 \% & 0.03 \% \cr
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Hits per muon capture & $3.9\times10^{-3}$ & $1.5\times10^{-4}$ & $4.5\times10^{-5}$ \cr
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\bottomrule
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\end{tabular}
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\end{center}
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\end{table}
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\subsection{Any physics implication??}
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% section _mu_e_conversion (end)
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