370 lines
16 KiB
TeX
370 lines
16 KiB
TeX
\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)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|