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thesis/chapters/chap2_mu_e_conv.tex

@@ -9,7 +9,7 @@ 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^-} 
+  \binom{\nu_\tau}{\tau^-}. 
 \end{equation*}
 
 Each lepton is assigned a lepton flavour quantum number, $L_e$, $L_\mu$,
@@ -24,7 +24,7 @@ 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
+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
@@ -40,15 +40,25 @@ are violated by one unit or more, is forbidden:
   \end{aligned}
   \label{eq:mueg}
 \end{equation} 
+
+However, it is observed that neutrinos do change flavour in the so-called
+neutrino oscillations where a neutrino of a certain lepton flavour
+can be measured to have a different flavour as it travels in space-time. The
+phenomenon has been confirmed in many experiments with solar neutrinos,
+atmospheric neutrinos, reactor neutrinos and beam neutrinos. The observation
+of neutrino oscillations means that the lepton flavour is not strictly
+conserved and neutrinos are massive. The massive neutrinos allow lepton
+flavour violation in the charged leptons, but at an unmeasurably small level
+as described in \cref{sec:lepton_flavour_violation}.
 %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
+ %massless neutrinos are assumed, lepton 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.
+ %lepton violation (LFV) in charged lepton processes.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Muon and its decays in the Standard Model}
@@ -111,15 +121,15 @@ or with an associated $e^+ e^-$ pair:
   \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,
+The dominant process, \micheldecay is commonly called the 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
+decay. Experimental results from extensive measurements of the Michel parameters
 are consistent with the predictions of the V-A
 theory~\cite{Michel.1950,FetscherGerber.etal.1986,BeringerArguin.etal.2012}.
 
@@ -127,9 +137,9 @@ 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
+from the 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
+the electron and photon spectra in this decay channel. Like the case of the
 Michel decay, experiments results on the branching ratio and the parameter are
 in agreement with the SM's predictions~\cite{BeringerArguin.etal.2012}.
 
@@ -221,8 +231,8 @@ 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
+SM. Many theoretical 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}
@@ -257,15 +267,15 @@ significantly larger CLFV
 %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
+experimental limits on these two decay modes are set respectively by the MEG
+experiment~\cite{Adam.etal.2013} and the SINDRUM-II
 experiment~\cite{Bertl.etal.2006}:
 \begin{equation}
-  \mathcal{B}(\mu^+ \rightarrow e^+ \gamma) < 5.7 \times 10^{-13}
+  \mathcal{B}(\mu^+ \rightarrow e^+ \gamma) < 5.7 \times 10^{-13}\,
 \end{equation}
-, and:
+and:
 \begin{equation}
-  \mathcal{B} (\mu^- + Au \rightarrow e^- +Au) < 7\times 10^{-13}
+  \mathcal{B} (\mu^- + Au \rightarrow e^- +Au) < 7\times 10^{-13}\.
 \end{equation}
 
 %\hl{TODO: mueg and muec relations, Lagrangian \ldots}
@@ -278,25 +288,26 @@ experiment~\cite{Bertl.etal.2006}:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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
+The conversion of a captured negative muon in a muonic atom 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
+When a negatively charged muon is stopped in a material, it is quickly captured
+by an atom 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: $\mu^- p \rightarrow \nu_\mu n$
+  \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. 
+expected, but has never been observed: 
 \begin{equation}
-  \mu^{-} + N(A,Z) \rightarrow e^{-} + N(A,Z)
+  \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
@@ -322,8 +333,8 @@ The quantity measured in searches for \mueconv is the ratio between the rate of
   \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. 
+%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}