\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)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
