CP-Violation in Kaluza-Klein and Randall-Sundrum Theories
S. Ichinose \address {Laboratory of Physics, School of Food and Nutritional Sciences,
The Kaluza-Klein theory and Randall-Sundrum theory are examined comparatively, with focus on the five dimensional (Dirac) fermion and the dimensional reduction to four dimensions. They are treated in the Cartan formalism. The chiral property, localization, anomaly phenomena are examined. The electric and magnetic dipole moment terms naturally appear. The order estimation of the couplings is done. This is a possible origin of the CP-violation. \vspace{1pc}
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\title{
CP-Violation in Kaluza-Klein and Randall-Sundrum Theories
}

\author{S. Ichinose
   \address
{Laboratory of Physics, 
School of Food and Nutritional Sciences, \\
University of Shizuoka,
Yada 52-1, Shizuoka 422-8526, Japan 
}%
        \thanks{E-mail address:\ ichinose@u-shizuoka-ken.ac.jp}
        }
       
\begin{document}

\begin{abstract}
The Kaluza-Klein theory and Randall-Sundrum
theory are examined comparatively, with focus
on the five dimensional (Dirac) fermion
and the dimensional reduction to four dimensions.
They are treated in the Cartan formalism. 
The chiral property, localization, anomaly phenomena
are examined. The electric and magnetic dipole
moment terms naturally appear. The order estimation
of the couplings is done. This is a possible
origin of the CP-violation.
\vspace{1pc}
\end{abstract}

% typeset front matter (including abstract)
\maketitle


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%  Sec.1  Intro  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction and Conclusion}
If our present research direction 
of the string and D-brane is right, 
the unification of various forces should be, 
effectively at some scale, explained by some higher-dimensional
theory. 
Then it is quite sure that
the real world of 4 dimensions should be some approximation
of the higher dimensional one. It is realized by 
the procedure called {\it dimensional 
reduction}. There are two 
representative and contrastive
approaches, that is, the Kaluza-Klein and the Randall-Sundrum
theories. 
In the former case, the reduction is achieved by
the {\it compactification} of the extra space, while
in the latter one it is done by the {\it localization}
of the configuration along the extra dimension(s). 
%At present 
%it is fair to say that we do not have any strong reason
%to believe that one is better than the other. 
%In fact many literatures on both theories
%have been produced and are now being produced. 
The two approaches look to have
both advantages and disadvantages in the phenomenological 
application. Here we treat them in a comparative way and examine
their 4 dimensional(D) reduction properties. 

We will present the higher dimensional approach
to the CP-violation mechanism. As was stressed by
Thirring for the KK model\cite{Thirr72}, the CP-violation naturally
occurs also in the RS model.
%Both give the origin of the CP-violation. 

%In the original papers by 
%Randall and Sundrum\cite{RS9905,RS9906}, 
%the distinction between the two models is stressed 
%mainly in the mass-hierarchy relations. 
%%In KK model, ****.Whereas the DW model, ***. 
%The origin of the CP-violation has been discussed
%so long time. (***Thirring 72,'tHooft 76, Weinberg ?,
%***) 
%The fundamental importance of the discrete symmetry
%is well known both theoretically and experimentally.
%We pursue the origin in the higher dimensional
%models. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%    SEC. 2  Kaluza-Klein Theory     %%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fermions in Kaluza-Klein Theory}
Let us first review the 5D Kaluza-Klein theory.
%based on the Cartan formalism\cite{Thirr72}. 
%(For a good textbook, see \cite{Naka91}.) 
This serves as the preparation for the same treatment of 
the Randall-Sundrum theory in the next section.
The 5D space-time manifold is described by the 4D coordinates
$x^a$ ($a=0,1,2,3$) and an extra coordinate $y$. We also use
the notation ($X^m$)=($x^a, y$), ($m=0,1,2,3,5$). 
With the general 5D metric $\gh_{mn}$:\ 
%*** KK1%%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$ ds^2=\gh_{mn}(X)dX^mdX^n $
%\com
%\label{KK1}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
, we assume the $S^1$ compactification condition for the
extra space:\ 
%*** KK1b%%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$ \gh_{mn}(x,y)=\gh_{mn}(x,y+{2\pi}/{\mu}) $
%\com
%\label{KK1b}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
, where $\mu^{-1}$ is the {\it radius} of the extra space circle.
We specify the form of the metric as
%*** KK1c%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
ds^2
=g_{ab}(x)dx^adx^b+\e^{\si(x)}(dy-fA_a(x)dx^a)^2
\com
\label{KK1c}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $g_{ab}(x), A_a(x)$ and $\si(x)$ are 
the 4D metric, the U(1) gauge field and the dilaton field
respectively. 
$f$ is a coupling constant. 
This specification is based on the
following additional assumptions:\ 
1.
$y$ is a space coordinate;\ 
2.
The geometry is invariant under
the U(1) symmetry, 
$y\ra y+\La(x),\ A_a(x)\ra A_a(x)+\frac{1}{f}\pl_a\La$.
%\item[3.]
%When the gauge field $A_a(x)$ switches off ( or $f=0$),
%the 4D world \{$x^a$\} and the extra one \{$y$\} are
%decomposable (separate worlds). 
We take $\si(x)=0$ in (\ref{KK1c}) for simplicity.

We take the Cartan formalism to
compute the geometric quantities\cite{Thirr72}. 
%(For a good textbook, see \cite{Naka91}.) 
The basis \{$\thhat^\m$\} 
($\m= 0,1,2,3,5$ are the local Lorentz (tangent) frame indices)
of the cotangent manifold($T_p^*M$)
and the f\"{u}nf-bein $\ehat^\m_{~m}$
are obtained as
%*** KK5 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\thhat^\al=\theta^\al=e^\al_{~a}dx^a,\ \thhat^5=dy-fA_adx^a,
A_\al\equiv e_\al^{~a}A_a,\nn
(\ehat^\m_{~m})=\left(
\begin{array}{cc}
e^\al_{~a} & 0 \\
-fA_a   & 1
\end{array}
                 \right),\ 
(\ehat_\m^{~m})=\left(
\begin{array}{cc}
e_\al^{~a} & 0 \\
fA_\al   & 1
\end{array}
                 \right),
\label{KK5}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\theta^\al$ and $e^\al_{~a}$ ($\al=0,1,2,3$)
are the 4D part of $\ehat^\m_{~m}$ and $\thhat^\m$
respectively. 
%the f\"{u}nf-bein
The first Cartan's structure equation gives
the connection 1-form $\omhat^\m_{~\n}$,
%*** KK9 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\omhat^5_{~5}=0,\ \omhat^5_{~\al}=-\frac{f}{2}F_\ab\theta^\be,
\nn
\omhat^\al_{~\be}=\om^\al_{~\be}
-\frac{f}{2}F^\al_{~\be}\thhat^5,\ 
F_\ab\equiv e_\al^{~a}e_\be^{~b}F_{ab}
\com
\label{KK9}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\om^\al_{~\be}$ is the 4D part.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% SEC. 3  Fermions in Kaluza-Klein Theory  %%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Fermions in Kaluza-Klein Theory}
The 5D Dirac equation is generally given by
%(see \cite{Naka91})
%*** ferKK1 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\left\{
\ga^\m\ehat_\m^{~m}\frac{\pl}{\pl X^m}
+\frac{1}{8}(\omhat^\si)_\mn\ga_\si [\ga^\m,\ga^\n]-\mhat
\right\}\psihat=0
.
\label{ferKK1}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The spin connection above $(\omhat^\si)_\mn$ is defined by
$\omhat^\mu_{~\nu}=(\omhat_\la)^\mu_{~\nu}\thhat^\la$.
The 5D Dirac matrix $\ga^\m$ satisfies
$\{\ga^\m,\ga^\n\}=2\etahat^\mn,
\etahat^\mn=\mbox{diag}(-1,1,1,1,1)$.
For simplicity we switch off the 4D gravity: 
$e^\al_{~a}\ra\del^\al_a\ ,\ \om^\al_{~\be}\ra 0$. 
The parameter $\mhat$ is the 5D fermion mass.
In the present case, using (\ref{KK5}) and (\ref{KK9}),
the eq.(\ref{ferKK1}) says
%*** ferKK2 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\left\{
\ga^a(\pl_a+fA_a\pl_5)+\ga^5\pl_5\right.\nn
\left.
+\frac{f}{16}F_{ab}\ga^5[\ga^a,\ga^b]-\mhat
\right\}\psihat=0
\pr
\label{ferKK2}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Corresponding to a {\it massive} mode in the 4D reduction
%(\ref{KK2}), 

We consider the following form of the fermion:\ 
%*** ferKK3 %%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$ \psihat(x,y)=\e^{i(\phi\gago+\m y)}\psi(x) $
%\pr
%\label{ferKK3}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
. Here we regard the fermion as a KK-{\it massive} mode.
The angle parameter $\phi$ is chosen as
%*** ferKK4 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
(i\gago\m+\mhat)\e^{2i\phi\gago}
=\sqrt{{\mhat}^2+\m^2}\equiv M,\ 
\tan 2\phi=-\frac{\m}{\mhat}
.
\label{ferKK4}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Then (\ref{ferKK2}) reduces to
%*** ferKK5 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\left\{
\ga^a(\pl_a+ieA_a)-M
+\right.\nn
\left.
\frac{1}{16M}(\mhat\frac{e}{\m}-ie\gago)F_{ab}\ga^5[\ga^a,\ga^b]
\right\}\psi=0
\com
\label{ferKK5}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $e\equiv f\m$ is the electric coupling constant.
We notice, in this expression, 
the electric and  magnetic moments appear.
%*** ferKK6 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{1}{16}\frac{\mhat}{M\m}eF_{ab}\psibar\ga^5[\ga^a,\ga^b]\psi,\ 
-\frac{i}{16}\frac{1}{M}eF_{ab}\psibar[\ga^a,\ga^b]\psi
.
\label{ferKK6}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The first term violate the CP symmetry. 
Note that the second term usually appear, in the 4D QED, 
as the {\it quantum} effects. 

Let us do the order estimation. From the reduction
of $
\int\sqrt{-\gh}\Lcal^{grav}=\frac{1}{G_5}\int d^5X\sqrt{-\gh}\Rhat
$,
%*** ferKK7 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{1}{G_5}\frac{2\pi}{\m}\int d^4x 
\sqrt{-g}(R-\frac{f^2}{4}F^\ab F_\ab+\cdots)
,
\label{ferKK7}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
we know
%*** ferKK8 %%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$ {G_5\m}\sim {G},\ 
{f^2}/{G_5\m}\sim 1 $
%\com
%\label{ferKK8}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
, where $G$ is the gravitational constant. 
This gives $f\sim \sqrt{G}=10^{-19}\mbox{GeV}^{-1}$.
On the other hand, we know $e=\m f\sim 10^{-1}$.
Hence we obtain $\m\sim 10^{-1}f^{-1}\sim 10^{18}\mbox{GeV}$.
Assuming $\mhat\sim\m$, we can estimate 
the electric and magnetic couplings as
%*** ferKK9 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{\mhat}{M\m}e\sim
\frac{e}{M}\sim 10^{-19}\mbox{GeV}^{-1}\sim 10^{-32}\mbox{e cm}
\com
\label{ferKK9}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
which is {\it far below} the experimental upper bound
of the neutron electric dipole moment $6.3\times 10^{-26}$e cm.
%should be compared with the observed (or expected) ones:\ 
%$10^3$ GeV$^{-1}$ for the electron, 
%$1$ GeV$^{-1}$ for the proton,
%$10^{-2}$ GeV$^{-1}$ for the top quark.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% SEC. 4  Fermions in Randall-Sundrum Geometry %%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Fermions in Randall-Sundrum Geometry}
We consider the following 5D space-time geometry\cite{RS9905}.
%In terms of the 4D coordinates $x^a$ and the extra one
%$y$,  the metric is taken as follows. 
%(We also use the notation $(X^m)=(x^a,y)$. ) 
%*** RS1 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
ds^2=\e^{-2\si(y)}\eta_{ab}dx^adx^b+dy^2
=\gh_{mn}dX^mdX^n,\nn
-\infty<y<+\infty\ ,\ -\infty<x^a<+\infty
,
\label{RS1}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\si(y)$ is regarded as a scale factor field.
$(\eta_{ab})=\mbox{diag}(-1,1,1,1)$.
When the geometry is AdS$_5$, $\si(y)=c|y|, c>0$. 
Such a situation, in the present case, 
occurs in the asymptotic region
of the extra space $y\ra\pm\infty$. 
The 1-form $\thhat^\m$, 
the f\"{u}nf-bein $\ehat^\m_{~m}$ and
the connection 1-form $\omhat^\m_{~\n}$
are given by
%*** RS3 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\thhat^\al=\e^{-\si}\eta^\al_{~a}dx^a\com\q
\thhat^5=dy
\com\nn
(\ehat^\m_{~m})=\left(
\begin{array}{cc}
e^{-\si}\eta^\al_{~a} & 0 \\
0   & 1
\end{array}
                 \right),\ 
(\ehat_\m^{~m})=\left(
\begin{array}{cc}
e^\si\eta_\al^{~a} & 0 \\
0   & 1
\end{array}
                 \right)
                 ,\nn
\omhat^5_{~5}=0\com\q 
\omhat^\al_{~5}=-\omhat_5^{~\al}=-\si'\theta^\al\com\nn
\omhat^5_{~\al}=-\omhat_\al^{~5}=\si'\theta_\al\com\q
\omhat^\al_{~\be}=0\com
\label{RS3}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\si'=\frac{d\si}{dy}$.

%2nd Cartan, Riemann curvature ***********
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% SEC. 5  Fermions in Randall-Sundrum Theory  %%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Fermions in Randall-Sundrum Theory}
As the fermion model, 
the (5D) Dirac theory (\ref{ferKK1}), $\sqrt{-\gh}\Lcal^{Dirac}$, 
is not accepted physically 
because the fermion configuration does not localize (around
$y=0$) in the extra space.
%To see it, let us examine the zero mode
%%($\mhat=0$) 
%solution of (\ref{ferRS6}). 
%*** ferRS9 %%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
%(2\si'-\pl_y)\xi_0=(2\si'-\pl_y)\eta_0=0\com\nn
%\xi_0=A\e^{2\si(y)}\com\q \eta_0=B\e^{2\si(y)}
%\com
%\label{ferRS9}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%where $A,B$ are integration constants. Since $\si(y)$
%is the even function ($\si(y)=\si(-y)$),
%the oddness of $\xi_0(y)$ requires $A=0$. 
%Hence $\xi_0(y)$ vanishes. From the requirement that 
%the 5D geometry is AdS$_5$ in the asymptotic region 
%$|y|\ra\infty$,
%and the use of the Einstein equation, we can deduce
%$\si(y)\sim +\om |y|, \om>0$ in the region.
%Hence the zero mode 
%$\psihat_0(x,y)=\psi^{(0)}_R(x)\eta_0(y)$
%is neither normalized nor localized. 
%%The situation is similar to the massive modes.****
Nature requires the {\it Yukawa interaction} 
between the 5D fermion and the 5D Higgs\cite{BG99}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      SEC. 6  Fermion-Higgs System             %%%%%%%%%%%%%%%%%%
%%%%              in the Randall-Sundrum Geometry  %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Fermion-Higgs System in the Randall-Sundrum Geometry}
Hence we examine the fermion behavior
under the Yukawa coupling with the Higgs scalar:\ 
%*** yuka1 %%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$
\sqrt{-\gh}\Lcal=\sqrt{-\gh}(\Lcal^{Dirac}+\Lcal^Y),\ 
\Lcal^Y=ig_Y{\bar \psihat}\psihat\Phi
,
$
%\label{yuka1}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\Phi$ is the 5D(bulk) Higgs scalar field
and $g_Y$ is the Yukawa coupling. This reduces to
%*** yuka5 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
i\e^{\si}\{
\ga^a\pl_a+2\e^{-\si}(\si'-\half\pl_y)\gago-g_Y\e^{-\si}\Phi
                           \}\psihat=0
.
\label{yuka5}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A solution of the right chirality zero mode:\ 
%*** yuka6 %%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$
\psihat(x,y)=\psi^0_R(x)\eta(y),\ 
\gago\psi^0_R=+\psi^0_R,\ 
\ga^a\pl_a\psi^0_R=0
$
%\label{yuka6}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
, is obtained by
%*** yuka7 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\pl_y\eta=(2\si'-g_Y\Phi(y))\eta
\pr
\label{yuka7}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the far region, the solution behaves as
%*** yuka8 %%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$ \eta(y)=\mbox{const}\times \e^{-(g_Y\vz-2\om)|y|} $
%\com
%\label{yuka8}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
, which shows the {\it exponentially damping} for the large
yukawa coupling $g_Y>2\om/\vz$. 
($\om$ and $\vz$ is the asymptotic values of $\si'(y)$
and $\Phi(y)$\cite{SI0003}.
)
This is called
{\it massless chiral fermion localization}. In the near region, 
%*** yuka9 %%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$ \eta(y)=\mbox{const}\times \e^{-\frac{k}{2}(g_Y\vz-2\om)y^2} $
%\com
%\label{yuka9}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
, which shows the {\it Gaussian damping}. 
($1/k$ is the {\it thickness} parameter of the wall.
)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%      SEC. 7  5D QED                           %%%%%%%%%%%%%%%%%%
%%%%              in the Randall-Sundrum Geometry  %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Five Dimensional QED in the Randall-Sundrum Geometry}
Now we analyze the 5D QED, 
$\Lcal^{QED}=-e{\bar \psihat}
\ga^\m\ehat_\m^{~m}\psihat A_m$, 
with the Yukawa interaction in RS geometry:\ 
%*** qed1 %%%%%%%%%%%%%%%%%%%%
%\begin{eqnarray}
$ \sqrt{-\gh}(\Lcal^{Dirac}+\Lcal^{EM}+\Lcal^{QED}+\Lcal^Y), 
\Lcal^{EM}=-\fourth \gh^{mn}\gh^{kl}F_{mk}F_{nl} $
%=\sqrt{-\gh}\left[ i{\bar \psihat}
%\left\{
%\ga^\m\ehat_\m^{~m}(\pl_m+ieA_m)\right.\right.\nn
%\left.\left. +\frac{1}{8}(\omhat^\si)_\mn\ga_\si [\ga^\m,\ga^\n]
%\right\}\psihat+ig_Y{\bar \psihat}\psihat\Phi
%                \right]\\
%\com
%\label{qed1}
%\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
.
We assume, as in the previous paragraph, $\gh^{mn}$ and $\Phi$ are
the brane background obtained from the gravitational and Higgs systems:
$\sqrt{-\gh}(\Lcal^{grav}+\Lcal^S)$.

Let us examine the {\it bulk quantum} effect. It induces the 5D 
effective action $S_{eff}$ which reduces to the 4D action
in the {\it thin wall limit}. 
From the diagram of Fig.1(left), we expect
%*** qed3 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{\del S^{(1)}_{eff}}{\del A^\mu(X)}\equiv
<J_\m>\sim e^2g_Y\ep_{\mn\ls\tau}\Phi F^{\nu\la}F^{\si\tau}
\pr
\label{qed3}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% Fig.2  %%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
%\vspace{3pt}
\includegraphics[height=5pc]{anomaly}\hspace{30pt} 
\includegraphics[height=5pc]{magnetic}
\caption{The left diagram induces the effective action
$S^{(1)}_{eff}$, while the right one induces $S^{(2)}_{eff}$.}
\label{fig:anomaly}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Then the effective action is integrated as
%*** qed4 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
S^{(1)}_{eff}\sim
e^2g_Y\int d^5X\ep_{\mn\ls\tau}\Phi A^\m F^{\nu\la}F^{\si\tau}
\pr
\label{qed4}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the {\it thin wall limit} we may approximate as
$\Phi=\Phi(y)\sim \vz\ep(y)$ where $\ep(y)$ is the step function.
Under the U(1) gauge transformation $\del A^\m=\pl^\m\La$, 
$S^{(1)}_{eff}$ changes as
%*** qed5 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\del_\La S^{(1)}_{eff}\sim
e^2g_Y\vz\int d^5X\ep_{\mn\ls\tau}\ep(y) \pl^\m\La
F^{\nu\la}F^{\si\tau}\nn
=-e^2g_Y\vz\int d^4x
\La(x) F^{\ab}{\tilde F}_{\ab}
\com
\label{qed5}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where ${\tilde F}_{\ab}\equiv \ep_{\ab\ga\del}F^{\ga\del}$. 
In the above we assume that the boundary term vanishes. 
Callan and Harvey interpreted this result as 
the "anomaly flow" between
the boundary (our 4D world) and the bulk\cite{CH85}. 

Through the analysis of 
the induced action in the bulk,
we can see the "dual" aspect of the 4D QED. 

Another interesting bulk quantum effect is given by Fig.1(right). 
The induced effective action $S^{(2)}_{eff}$ is expected
to satisfy
%*** qed6 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{\del S^{(2)}_{eff}}{\del F^\mn}\equiv
<J_\mn>\sim e^2g_Y\ep_{\mn\ls\tau}\pl^\la\Phi 
{\bar \psihat}\Si^{\si\tau}\psihat
\pr
\label{qed6}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Then $S^{(2)}_{eff}$ is obtained as, in the thin wall limit, 
%*** qed7 %%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
S^{(2)}_{eff}
\sim e^2g_Y\ep_{\mn\ls\tau}\int d^5X\pl^\la\Phi F^\mn
{\bar \psihat}\Si^{\si\tau}\psihat\nn
= e^2g_Y\vz\ep_{\ab\gd}\int d^4x F^\ab
{\bar \psi}\si^{\gd}\psi
\pr
\label{qed7}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This term is the "dual" of the magnetic moment term of
(\ref{ferKK6}). It is a CP-violating term. The coupling
depends on the vacuum expectation value of $\Phi$. 


%\vspace{2cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%% reference %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\bibitem{Thirr72}  %*Thirr72*
W. Thirring, Acta Physica Austriaca, Suppl.IX, 256-271(1972),
Springer-Verlag,
"Fivedimensional Theories and CP-Violation"
%\bibitem{Naka91}  %*Naka91*
%M. Nakahara, "Geometry Topology and Physics", 
%Adam Hilger, Bristol, England, 1990.
\bibitem{RS9905} %*RS9905*
L.Randall and R.Sundrum,
\PRL {\bf 83}(1999)3370,4690
%,hep-ph/9905221
%,MIT-CTP-2860,
%"A Large Mass Hierarchy from a Small Extra Dimension"
%}
%\bibitem{RS9906} %*RS9906*
%L.Randall and R.Sundrum, {
%\PRL {\bf 83}(1999)4690
%,hep-th/9906064
%,MIT-CTP-2874,
%"An Alternative to Compactification"
%
%
\bibitem{BG99} %*BG99*
B.Bajc and G.Gabadadze, Phys.Lett.{\bf B474}(2000)282
\bibitem{SI0003}   %*SI0003*
S.Ichinose,Class.Quant.Grav.{\bf 18}(2001)421,5239%,hep-th/0003275 
%Univ.of Shizuoka preprint,US-00-01, 
%"A Solution of the Randall-Sundrum Model and
%the Mass Hierarchy Problem"
%\bibitem{SI0107}   %*SI0107*
%S.Ichinose,{
%Class.Quant.Grav.{\bf 18}(2001)5239%,hep-th/0107254
%Univ.of Shizuoka preprint,US-01-03,hep-th/0107254, 
%"Some Properties of Domain Wall Solution in the Randall-Sundrum Model"
\bibitem{CH85} %*CH85*
C.G.Callan and J.A.Harvey,{\NP{\bf B250}(1985)427}
%\bibitem{CHNOY99} *CHNOY99*
%S.Chang, J.Hisano, H.Nakano, N.Okada and M.Yamaguchi,
%Phys.Rev.{\bf D62}(2000)084025, hep-ph/9912498,
%''Bulk Standard Model in the Randall-Sundrum Background''
\end{thebibliography}

\end{document}

hep-ph0203037.extracted_bib
\begin{thebibliography}{99}
\bibitem{Thirr72}  %*Thirr72*
W. Thirring, Acta Physica Austriaca, Suppl.IX, 256-271(1972),
Springer-Verlag,
"Fivedimensional Theories and CP-Violation"
%\bibitem{Naka91}  %*Naka91*
%M. Nakahara, "Geometry Topology and Physics", 
%Adam Hilger, Bristol, England, 1990.
\bibitem{RS9905} %*RS9905*
L.Randall and R.Sundrum,
\PRL {\bf 83}(1999)3370,4690
%,hep-ph/9905221
%,MIT-CTP-2860,
%"A Large Mass Hierarchy from a Small Extra Dimension"
%}
%\bibitem{RS9906} %*RS9906*
%L.Randall and R.Sundrum, {
%\PRL {\bf 83}(1999)4690
%,hep-th/9906064
%,MIT-CTP-2874,
%"An Alternative to Compactification"
%
%
\bibitem{BG99} %*BG99*
B.Bajc and G.Gabadadze, Phys.Lett.{\bf B474}(2000)282
\bibitem{SI0003}   %*SI0003*
S.Ichinose,Class.Quant.Grav.{\bf 18}(2001)421,5239%,hep-th/0003275 
%Univ.of Shizuoka preprint,US-00-01, 
%"A Solution of the Randall-Sundrum Model and
%the Mass Hierarchy Problem"
%\bibitem{SI0107}   %*SI0107*
%S.Ichinose,{
%Class.Quant.Grav.{\bf 18}(2001)5239%,hep-th/0107254
%Univ.of Shizuoka preprint,US-01-03,hep-th/0107254, 
%"Some Properties of Domain Wall Solution in the Randall-Sundrum Model"
\bibitem{CH85} %*CH85*
C.G.Callan and J.A.Harvey,{\NP{\bf B250}(1985)427}
%\bibitem{CHNOY99} *CHNOY99*
%S.Chang, J.Hisano, H.Nakano, N.Okada and M.Yamaguchi,
%Phys.Rev.{\bf D62}(2000)084025, hep-ph/9912498,
%''Bulk Standard Model in the Randall-Sundrum Background''
\end{thebibliography}