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\begin{document}
\begin{titlepage}
\begin{flushleft}
       \hfill                      {\tt hep-th/0304051}\\
       \hfill                       FIT HE - 03-02 \\
       \hfill                       KYUSHU-HET 65 \\
\end{flushleft}
\vspace*{3mm}
\begin{center}
{\bf\LARGE Dilaton coupled brane-world and field trapping \\ }
%\vspace*{5mm}
\vspace*{12mm}
{\large Kazuo Ghoroku\footnote[2]{\tt gouroku@dontaku.fit.ac.jp},
Motoi Tachibana\footnote[3]{\tt motoi@postman.riken.go.jp} and
Nobuhiro Uekusa\footnote[4]{\tt uekusa@higgs.phys.kyushu-u.ac.jp}}\\
\vspace*{2mm}

\vspace*{2mm}

\vspace*{4mm}
{\large ${}^{\dagger}$Fukuoka Institute of Technology, Wajiro, 
Higashi-ku}\\
{\large Fukuoka 811-0295, Japan\\}
\vspace*{4mm}
{\large ${}^{\ddagger}$Theoretical Physics Laboratory, RIKEN, 2-1 Hirosawa,}\\
{\large  Wako, Saitama 351-0198, Japan\\}
\vspace*{4mm}
{\large ${}^{\ddagger}$Department of Physics, Kyushu University, Hakozaki,
Higashi-ku}\\
{\large Fukuoka 812-8581, Japan\\}
\vspace*{10mm}
\end{center}

\begin{abstract}
We give brane-world solutions with cosmological constant $\lambda$
by introducing the dilaton in 5d bulk,
and we have examined the localization of graviton, gauge bosons and 
dilaton. 
For the solutions given here, we find that both graviton and gauge bosons 
can be trapped for any sign, positive and negative, and wide range of 
$\lambda$ due to the dilaton. 
On the other hand, the bulk dilaton can not be trapped as a stable 
particle on a 
brane of $\lambda\geq 0$. This implies a possible resolution of
the moduli problem and the equivalence principle in superstring theory
since the dilaton can not be localized on the brane
as a partner of the graviton.
\end{abstract}
\end{titlepage}

\section{Introduction}

The idea of the brane-world given in \cite{RS1,RS2} is strongly
motivated by the recent development in the superstring/M 
theory \cite{HW,MGW}. 
In string theory however, the dilaton is necessarily
included as a scalar partner
of the graviton and plays many important dynamical roles. For example,
the bulk configuration of the dilaton would give an insight for the 
running coupling 
constant of the boundary 4d Yang-Mills theory in an appropriate 10d background 
manifold. It would therefore be interesting to see the role of the dilaton 
in the brane-world and to examine whether the dilaton
could live also in our 4d world after the warped compactification 
and how it behaves. When it survives on the brane 
as a massless scalar partner of the graviton as
in 10d, the equivalence principle, which will be observed in the free-fall
experiments, would be remained as a problem to be resolved as in
\cite{DPV}. While it could appear
as a 4d massive scalar, its mass ($m_{\phi}$) is very small since it
could not exceed the 4d cosmological constant $\lambda$. Then we can 
estimate its magnitude as $m_{\phi}\sim \sqrt{\lambda}$, and the 
coupling to the light particles like photons could be estimated
as $1/M_{\rm pl}$. This implies a very long life time 
$\tau\sim M_{\rm pl}^2/m_{\phi}^3$ of the massive 
dilaton, and this fact gives a serious difficulty known as the moduli 
problem
\cite{RaTh,RT,DV}. In this case, we could however consider this massive 
particle as a candidate of the dark matter \cite{DG,Di}.

It would be meaningful to study the problem of localization of the dilaton
in order to approach to the above issues from the viewpoint of the 
brane-world. Although the localization of
the graviton has been extensively studied \cite{RS2,KR,Mie,KT,bre,BBG},
much attention has not been given to scalar field \cite{GN1,GY}, especially
for the dilaton.
However, it would be important to see the situation of the dilaton 
as well as the graviton in the context of the warped compactification
within the superstring theory. 


%%%%%%%%%%%%
In addition to the dilaton and graviton, we also examine the localization 
of gauge bosons which are the important force of our 4d world. 
Up to now, the localization of the graviton has been
assured in a wide range of brane-world solutions except
for AdS${}_4$ brane \cite{KR,Mie}.
As for the gauge bosons, its localization has been examined
in some models \cite{DS,DSG,DFKK,DSL,Oda,KT,GN2,GU}. 
To trap gauge bosons, it was necessary
to introduce positive $\lambda$ or the bulk-dilaton field. 
The latter case might be related to the massive vector model \cite{GN2} by a 
special form of brane-vector coupling \cite{MT}. This point is an 
another
reason to examine the brane-world with bulk-dilaton.
One more interest to study dilaton is in the proposal of the
self-tuning model \cite{KSS,ADKS,CE} 
in obtaining the Poincare invariant brane. Brief comments
related to this are given here.

\vspace{.2cm}
We study these issues in terms of a simple class of
brane-world solutions with non-trivial dilaton, which are essentially
equivalent to the one given in \cite{KLO}. However, the use and extension
of these solutions are different.
The dilaton ($\phi$) is identified here with a scalar 
field which couples exponentially to the brane
like $e^{\gamma\phi}$ and has the bulk
potential of the exponential form. 
We concentrate on the localization of the fields mentioned above.
Since the solutions given here have the same form of warp factor with the 
one, which are
given before \cite{bre} for the case without the dilaton, then many
problems of the localization would be easily understood. 

\vspace{.3cm}
Several new aspects caused by dilaton are given here.
The localization of both the graviton and 
gauge bosons is realized for any brane of positive, negative and zero
$\lambda$, i.e., dS$_4$, AdS$_4$ and Minkowski brane. 
When dilaton is absent,
the gauge bosons are localized only for $\lambda>0$, dS$_4$ brane \cite{GU}, 
but this
is not a necessary condition when we consider dilaton. 
The essential point needed for the localization of gauge bosons
would be in the deformation of the warp factor from its exact AdS form. 
And the necessary 
deformation can be obtained either by the positive $\lambda$ or by the 
dilaton.
In any case, the conformal symmetry in the bulk is 
broken, so we could say that
this breaking of the conformal symmetry would be inevitable for
the localization of gauge bosons. 

As for the graviton, it can be localized even if $\lambda<0$ when
dilaton is included. This point is important since the general coordinate
invariance is not necessarily broken for AdS$_4$ brane in the case
when dilaton is considered.

As for the localization of dilaton, it 
depends on the strength of its coupling to the brane.
When the coupling is weak, the dilaton is trapped as a tachyon or a 
massless
scalar. The latter case is realized for $\lambda<0$, however our universe
is expected as $\lambda\geq 0$. In this sense,
the brane of weak coupling will be unfavorable.
For the case of strong coupling, dilaton can not be
trapped and it could live only in the bulk. 
Then we can not observe the dilaton
as a particle on the brane. However, dilaton plays a role to trap
other bulk fields, and dilaton itself lives outside of the 
brane.
This situation is favorable to evade the problems stated above related
to the dilaton in the string theory. 
In this sense, we can say that the superstring theory can 
survive as the basic theory in the brane-world story.

\vspace{.2cm}
In Section 2, we set our model with dilaton, and brane solutions
are shown. We comment on the relation between these new solutions and the 
one given before without dilaton. In Section 3, the localization of 
graviton, gauge bosons and dilaton are examined.
In section 4,
the relation between the dilaton model and massive vector model is 
discussed.
In the final section, summary and discussions are given.


\section{Dilaton coupled brane solution}


We begin with the following action with dilaton ($\phi$),
\bea
   S_g=\int d^4\!xdy\sqrt{-g}
   \left\{{1\over 2\kappa^2}(R-2\Lambda) 
    -{1\over 2}(\partial\phi)^2-V(\phi)\right\}  
     -\tau\int d^4\!x\sqrt{-\det g_{\mu\nu}}\ e^{\gamma\phi} \ ,
                                                     \label{acg}
\eea
where the parameters are the five-dimensional gravitational constant 
$\kappa^2$, 
bulk cosmological constant $\Lambda$, 
brane tension $\tau$ and 
dilaton-brane coupling $\gamma$.
The brane is set at $y=0$ by imposing $y\rightarrow -y$ symmetry on 
the above action.
And the form of the potential $V$ is not specified at this stage.
The equations of motion derived from the action (\ref{acg}) are given as
\bea
  &&  G_{MN}=
      \kappa^2\left\{
    \partial_M\phi\partial_N\phi-g_{MN}
    \left({1\over 2}(\partial\phi)^2+{\Lambda\over \kappa^2}+V\right)
    -g_{\mu\nu}\delta_M^{\mu}\delta_N^{\nu}
            \tau e^{\gamma\phi}\delta(y)\right\} \ , \label{eins}
\\
  &&\qquad\qquad
      {1\over\sqrt{-g}}\partial_M\left\{\sqrt{-g}g^{MN}\partial_N\phi\right\}
      ={\partial V\over \partial\phi}
        +{\sqrt{-\det g_{\mu\nu}}\over\sqrt{-g}}
           \tau\gamma e^{\gamma\phi}\delta(y) \ .   \label{dila}
\eea
Here we solve these under the following ansatz for metric, 
\bea
   ds^2=g_{MN}dx^M dx^N=A^2(y)(-dt^2+a^2(t)\gamma_{ij}dx^i dx^j)
         +dy^2 \ ,                                 \label{fmet}
\eea
where $\gamma_{ij}=(1+k\delta_{mn}x^m x^n/4)^{-2}\delta_{ij}$.
As long as we do not mention, $k=0$.
While $a(t)$ is solved for each $k$, for example
$a_0=e^{H_0 t}$, $H_0=\sqrt{\lambda}$ for $k=0$.
If we take as $\phi=\phi(y)$, 
Einstein equations (\ref{eins}) and dilaton equation (\ref{dila}) are 
written as
\bea
   &&{A''\over A}+\left({A'\over A}\right)^2
     -{\lambda \over A^2}
  =-{\kappa^2\over 3}
       \left({1\over 2}(\phi')^2 +{\Lambda\over \kappa^2}+V(\phi)\right)
     -{\kappa^2\tau\over 3}e^{\gamma\phi}\delta(y) \ ,             
\label{tteq}
\\
    &&\qquad\qquad
      \left({A'\over A}\right)^2-{\lambda \over A^2} 
    ={\kappa^2\over 6}
     \left({1\over 2}(\phi')^2 -{\Lambda\over \kappa^2}-V(\phi)\right) \ ,
                                                  \label{yyeq}
\\
    &&\qquad\qquad
          \phi''+4\phi'{A'\over A}
        ={\partial V\over \partial \phi}
         +\tau\gamma e^{\gamma\phi}\delta(y) \ ,  \label{deq}
\eea
where $'=d/dy$.
Let us assume the following form for $\phi$,
\bea
    \phi'=\alpha A^{n}\theta(y)\ ,   \label{ans}
\eea
Eqs.(\ref{yyeq}) and (\ref{deq}) are solved as
\bea
    &&\qquad
      \left({A'\over A}\right)^2-{\lambda \over A^2} 
    =-{\kappa^2\over 6}
     \left({2\alpha^2\over n}A^{2n} + V_0\right) -{\Lambda\over 6}\ ,
                                                  \label{yyeq1}
\\
    &&\qquad
     V= {n+4\over 2n}({\alpha A^n})^2+V_0, \qquad 
           2\alpha =\gamma\tau e^{\gamma\phi_0}\ ,
          \label{deq1}
\eea
where $V_0$ is a constant, and we take the boundary conditions $A(0)=1$ and 
$\phi(0)=\phi_0$.
From eq.(\ref{yyeq1}), we can see that the parameters $V_0$ and $\alpha$
can be absorbed into $\Lambda$ and $\lambda$ when $n=-1$. We can solve for 
other
values of $n$, but we consider this simple solution in the following 
analysis
since it is included in the
previously obtained one for the case without dilaton by replacing the 
modified parameters. We should however notice here
that this solution of $n=-1$ corresponds to the one which has 
already been found by P.~Kanti %\textit
{et al.}\cite{KLO}. Another ansatz which could lead to the solutions
similar to our previous ones are also possible, but they are a little bit
complicated. So we concentrate here on the solution of $n=-1$ with the above
ansatz (\ref{ans}).

\vspace{.5cm}
Then the potential is given as
\bea
   V= -{3\over 2}\biggl({\alpha\over A}\biggr)^2+V_0,    \label{poten}
\eea
and the equations (\ref{tteq}) and (\ref{yyeq}) can be written as
\bea 
    &&
   {A''\over A}+\left({A'\over A}\right)^2
     -{\tilde{\lambda}\over A^2}+{\tilde{\Lambda}\over 3}
    =-{\kappa^2\tau\over 3}e^{\gamma\phi_0}\delta(y) \ .        
\label{tteq3}
\\   &&\qquad
     \left({A'\over A}\right)^2+{\tilde{\Lambda}\over 6}
    ={\tilde{\lambda}\over A^2} \ ,        \label{yyeq2}
\\
    &&
       \tilde{\Lambda}=\Lambda+\kappa^2V_0 \ , \quad
      \tilde{\lambda}=\lambda+{\kappa^2\alpha^2\over 3} \ . \label{lam}
\eea
As stated above,
the equations (\ref{tteq3}) and (\ref{yyeq2}) are the same form with the 
one
obtained before \cite{bre} for the case where dilaton is suppressed. 
The dilaton modifies superficially the brane and bulk cosmological 
constants
($\lambda, \Lambda$) to $(\tilde{\lambda}, \tilde{\Lambda})$ as shown 
above.
So the
solutions are obtained by replacing the parameters ($\lambda, \Lambda$) 
in the previously given solutions by the new one, 
$(\tilde{\lambda}, \tilde{\Lambda})$. 
Then the various analyses can be
performed in a parallel way.
For example,
the solutions $A(y)$ and $\phi(y)$ and the potential $V(\phi)$ are given as
follows.

\vspace{.4cm}
\noindent For $\tilde{\lambda}>0,\ \tilde{\Lambda}=0$,
\bea
   &&\qquad\qquad
    A(y)=1-\sqrt{\tilde{\lambda}}|y|\equiv 1-{|y|\over y_H} \ 
,\label{singsol}
\\
   &&\phi(y)=-{\alpha\over\sqrt{\tilde{\lambda}}}
           \ln(1-\sqrt{\tilde{\lambda}}|y|)+\phi_0 \ ,\quad
   V(\phi)=-{3\alpha^2\over 2}e^{{2\sqrt{\tilde{\lambda}}\over\alpha}
                   (\phi-\phi_0)}
       +V_0 \ .                     \label{scalarp1}
\eea
For $\tilde{\lambda}>0,\ \tilde{\Lambda}<0$,
\beq
    %&&\qquad\qquad\qquad\qquad
       A(y)={\sqrt{\tilde{\lambda}}\over \mu}\sinh\left[\mu(y_H-|y|)\right] 
\ ,
\eeq
\beq
     \phi(y)={\alpha\over \sqrt{\tilde{\lambda}}}
           \ln\left({\coth\left[\mu(y_H-|y|)/2\right]\over
                     \coth(\mu y_H/2)}\right)+\phi_0 \ ,
\eeq
\beq
     V(\phi)=-{3\alpha^2\over 2}{\mu^2\over \tilde{\lambda}}
             % \sinh^2\left\{{\sqrt{\tilde{\lambda}}\over \alpha}(\phi-\phi_0)
              \sinh^2\left\{\sqrt{\tilde{\lambda}}(\phi-\phi_0)/ \alpha
             +\chi\right\}
              +V_0 \ ,            \label{scalarp2}
\eeq
\beq
     \mu=\sqrt{-\tilde{\Lambda}/6} \ ,\quad
      \sinh(\mu y_H)=\mu/\sqrt{\tilde{\lambda}} \ ,\quad
     e^{\chi}= \coth(\mu y_H/2) \ .
\eeq
For $\tilde{\lambda}>0,\ \tilde{\Lambda}>0$,
\beq
   %% &&%\qquad\qquad\qquad\qquad
    A(y)={\sqrt{\tilde{\lambda}}\over \mu_d}\sin\left[\mu_d(y_H-|y|)\right] 
\ ,
%\\
  % &&
     \phi(y)={\alpha\over \sqrt{\tilde{\lambda}}}
           \ln\left({\cot\left[\mu_d(y_H-|y|)/2\right]\over
                    \cot(\mu_d y_H/2)}\right)+\phi_0\ ,\quad
\eeq
\beq
     V(\phi)=-{3\alpha^2\over 2}{\mu_d^2\over \tilde{\lambda}}
            \cosh^2\left\{\sqrt{\tilde{\lambda}}(\phi-\phi_0)/\alpha
            +\chi_d\right\}
              +V_0 \ ,                     \label{scalarp3}
\eeq
\beq
   %&&\qquad
     \mu_d=\sqrt{\tilde{\Lambda}/6} \ ,\quad
      \sin(\mu_d y_H)=\mu_d/\sqrt{\tilde{\lambda}} \ ,\quad 
       e^{\chi_d}=\cot(\mu_d y_H/2) \ .
\eeq
For any above solution $A(y)$,
there exists horizon whose position is represented by $y=y_H$
and which has curvature singularity 
since 5$D$ Ricci scalar is written as 
\beq
    R=-4\left(2{A''\over A}+3\biggl({A'\over A}\biggr)^2-3{\lambda\over A^2}\right) .
\eeq
So we interpret the extra dimension is in a finite region,
$0\le y\le y_H$. 
%%%%%%%
Hereafter we set $\phi_0=0$ for the sake of brevity, or we interpret
$\tau$ as $\tau e^{\gamma\phi_0}$ as long as we do not mention it especially.
And we obtain from eq.(\ref{tteq3}) for the above solutions
\bea
      \tilde{\lambda}={\tilde{\Lambda}\over 6}+
              {\kappa^4\tau^2\over 36} \ .        \label{lambda}
\eea
This is the same form of the relation given before by $\Lambda$ and 
$\lambda$ in the case
of the theory without dilaton.
%%%%%%%%%%

At the end of this section, we like to comment on the self-tuning 
solutions.
When $\Lambda=V=0$, the solution called as self-tuning has been found
\cite{KSS,ADKS,CE}, in which
the brane of $\lambda=0$ is always obtained by choosing an appropriate 
value
of the constant part of $\phi$. It is given as a constant of the 
integration
of the equation of motion, then it is not a parameter of the theory. In our 
case, both the $\Lambda$ and $V$ are not zero, then it would be impossible 
to
find such a self-tuning solution in terms of the dilaton. However, it would 
be
possible to obtain such a solution when other fields are added to the 
action.

The easy way is to introduce the four form fields. It has no dynamical 
degree of freedom in $5D$, but it could generate a dynamical cosmological
constant \cite{DN}. This could change the original bulk cosmological 
constant
to any value which we want, 
then we can find a self-tuning of $\lambda=0$ by this dynamical
constant. It is seen as follows in our model.

Using the four form field strength $F_{M_1\cdots M_5}$, its action is
given as
\bea
     S_f=-\int d^4xdy\ {1\over 2\times 5!}
      \sqrt{-g}F_{M_1\cdots M_5}F^{M_1\cdots M_5} \ .\label{acf}
\eea

Then the energy momentum tensor and the equation of motion of this field 
given below are added to our system,
 \beq
     T_{MN}^{F}= 
   {\kappa^2\over 4!}F_{MP_1\cdots P_4}F_N^{\quad P_1\cdots P_4}
   -{\kappa^2\over 2\times 5!}F_{P_1\cdots P_5}F^{P_1\cdots P_5}g_{MN} \ ,
\eeq
\beq
      \partial_{M_1}\left(\sqrt{-g}F^{M_1\cdots M_5}\right)=0 \ .\label{44}
\eeq 
As known previously, we obtain the solution,
\beq
    F^{y\mu_1\cdots\mu_4}
       =\Lambda_f{1\over \sqrt{-g}}\epsilon^{\mu_1\cdots\mu_4} \ ,
\eeq
where $\epsilon^{\mu_1\cdots\mu_4}$ is normalized as $\pm 1$.
Then since $F_{P_1\cdots P_5}F^{P_1\cdots P_5}=-120\Lambda_f^2$,
the $\Lambda_f$ shifts the bulk cosmological constant $\Lambda$ as
\beq
 \Lambda\to \Lambda-{\kappa^2\over 2}\Lambda_f^2.
\eeq
The $\Lambda_f$ is the integral constant and is not a parameter of theory.
Therefore we claim this to be a self-tuning model. 


\vspace{.5cm}           




\section{Localization}

Here we consider the localization of the bulk fields in terms of the
linearized field equations for the graviton, dilaton and gauge bosons
around the background configuration obtained above.

\subsection{Graviton} 

Under the metric (\ref{fmet}), it is convenient to take the gravitational 
fluctuation $h_{ij}$ as follows,
\beq
 ds^2= A(y)^2(-dt^2+a^2(t)[\gamma_{ij}(x^i)+h_{ij}(x^{\mu},y)]dx^{i}dx^{j})
           +dy^2  \,. \label{metricape}
\eeq
We are interested in the localization of the traceless transverse
component, which represents the graviton on the brane.
It is projected out by the conditions, $h_i^i=0$ and
$\nabla_i h^{ij}=0$, where $\nabla_i$ denotes the covariant derivative
with respect to the three-metric $\gamma_{ij}$ which is used to raise
and lower the three-indices $i,j$. The transverse and traceless part
is denoted by $h$ hereafter for simplicity. The linearized field equation
for $h$ is obtained in the same form with the one given previously for
the case without dilaton,
\beq
 \nabla^2_5h=0,  \label{scalar}
\eeq
in terms of
the five dimensional covariant derivative $\nabla^2_5=\nabla_M\nabla^M$.
The above equation is given for $k=0$, and
this is equivalent to the field equation of a five dimensional 
free massless scalar. For $k=\pm 1$, the term proportion to this appears,
but it is not essential here and is abbreviated.

Then we arrive at the conclusion that the graviton can be localized on
the brane if $\tilde{\lambda}\geq 0$. The new point is that the graviton
can be trapped even if $\lambda <0$ differently from the case without
dilaton. When the dialton is not considered, massive spin-2 field is 
trapped
on the brane of negative $\lambda$. Then the general coordinate invariance
is broken for AdS$_4$ brane from the viewpoint of the brane world scenario.
But this is not always true as seen above.

For scalar fields, we can obtain similar results given for the case
without dilaton by replacing the parameters $(\Lambda,\lambda)$ by
$(\tilde{\Lambda},\tilde{\lambda})$. The discussions are abbreviated here.

\subsection{Dilaton} 

Here we discuss the dilaton according to its linearized equation
around the background solution given here. The equation is obtained
by denoting as $\phi=\bar{\phi}+\delta\phi$, where $\bar{\phi}$
represents the classical solution for $\phi$.
The bulk ``mass'' term of the dilaton is obtained from 
the potential, $V(\bar{\phi}+\delta\phi)$ given above. 
Expanding it around the classical solutions,
we get
\beq
 V=\bar{V}+{1\over 2}m_{\phi}^2(\delta{\phi})^2,
\eeq
\beq
   m_{\phi}^2=-{6\tilde{\lambda}\over A^2}+{\tilde{\Lambda}\over 2},
\eeq
where $\bar{V}=V(\bar{\phi})$. 
The explicit forms of $m_{\phi}^2$ for each solutions
are written as
%%%%%%%%%%%%%%%%%%%%%%%%%
%\beq
% m_{\phi}^2=3\mu^2(2{\tilde{\lambda}\over \mu^2A^2}+1),
%\eeq
%for $\tilde{\Lambda}<0$ and $\tilde{\lambda}>0$,
%\beq
% m_{\phi}^2=3\mu_d^2(2{\tilde{\lambda}\over \mu_d^2A^2}-1),
%\eeq
%for $\tilde{\Lambda}>0$ and $\tilde{\lambda}>0$, and
%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
   &&\qquad 
     m_{\phi}^2=-6\tilde{\lambda}
      e^{{2\sqrt{\tilde{\lambda}}\over\alpha}\bar{\phi}} \ ,
     \qquad\qquad\qquad\  \textrm{for } \tilde{\Lambda}=0,\ 
\tilde{\lambda}>0 ,
\\
  && m_{\phi}^2=-3\mu^2\cosh\left[\right.2
        ({\sqrt{\tilde{\lambda}}\over \alpha}\bar{\phi}+\chi)\left.\right] 
\ ,
     \qquad \textrm{for } \tilde{\Lambda}<0, \ \tilde{\lambda}>0 ,
\\
  && m_{\phi}^2=-3\mu_d^2\cosh\left[\right.2
       ({\sqrt{\tilde{\lambda}}\over \alpha}\bar{\phi}+\chi_d)\left.\right] 
\ ,
     \qquad \textrm{for } \tilde{\Lambda}>0,\ \tilde{\lambda}>0 .
\eea 
%%%%%%%%%%%%%%%%%%%%%%
%and 
%\beq
% m_{\phi}^2=3\mu^2({1\over 2}\sin({2H\bar{\phi}\over \alpha})+1),
%\eeq
%for $\tilde{\Lambda}<0$ and $\tilde{\lambda}<0$. Here $\bar{\phi}$
%denotes the classical solution for $\phi$. 
%%%%%%%%%%%%%%%%%%%%%%%%
They are all not constant
but the functions of $y$. In this sense, we can not see the mass of
the dilaton in a normal form. On the other hand, we can see that they are all
negative in all region of $y$. So the brane obtained here seems to be
unstable for the dilaton fluctuation. However, we 
should notice the other mass 
term coming from the brane, and the total mass term is given as
\beq
   m_{\phi}^2=-{6\tilde{\lambda}\over A^2}+{\tilde{\Lambda}\over 2}
             +\tau\gamma^2\delta(y).  \label{dilaton-mass}
\eeq
We can see that the instability stated above might
be cured by the third term as shown below.

\vspace{.5cm}
Then the linearized equation of the dilaton fluctuation
$\delta\phi$ is given as,
\beq
   ({\sq}_5-m_{\phi}^2)\delta\phi=0  \,, \label{mscalar}
\eeq
where $\sq_5$ denotes the bulk laplacian, and $m_{\phi}^2$ is given 
above by (\ref{dilaton-mass}).
Then, $\delta\phi$ is decomposed as follows
in terms of the four-dimensional continuous mass eigenstates:
\beq
 \delta\phi=\int\! dm \ \varphi_m(t,x^i)\,\Phi(m,y) \, , \label{eigenex}
\eeq
where the 4d mass $m$ is defined by
\beq
  -\sq_4\varphi=\ddot{\varphi}_m+3{\dot{a}_0\over a_0}\dot{\varphi}_m
           +{-\partial_i^2\over a_0^2}\varphi_m=-m^2\varphi_m , 
\label{masseig}
\eeq
and $\dot{}=d{}/dt$. In order to see the localization, we do not
need the solution of this equation, and
we need only the explicit form
of $\Phi(m,y)$, and its equation is given as 
\beq
  {\Phi}''+4{A'\over A}{\Phi}'
           +{\tilde{m}^2\over A^2}\Phi
          =(M^2+\tau\gamma^2\delta(y))\Phi , \label{warp2}
\eeq
\beq
  \tilde{m}^2=m^2+6\tilde{\lambda} \, ,\qquad 
       M^2={\tilde{\Lambda}\over 2} \, ,
\eeq
where ${}'=d{}/dy$. The equation (\ref{warp2}) is equivalent to the
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
wave equation of a scalar with its mass $M$, and it is rewritten 
%%%%%%% Massive scalar
into the form of one-dimensional 
Schr\"{o}dinger-like equation with the eigenvalue $\tilde{m}^2$,
\beq
 [-\partial_z^2+V(z)]u(z)=\tilde{m}^2 u(z) , \ \label{warp3}
\eeq
where we introduced $u(z)$ and $z$ defined as 
$\phi=A^{-3/2}u(z)$ and $\partial z/\partial y=\pm A^{-1}$.
The potential $V(z)$ in this equation is determined by $A(y)$,
\beq
 V(z)={9\over 4}(A')^2+{3\over 2}AA''+A^2 M^2
                  +\tau\gamma^2\delta(y).
\eeq 
Since $A$ is singular at $y=0$ ( at $z=z_0$), two $\delta$-functions
are included. Then $u$ must satisfy the following boundary condition
%%%%%%%
%%%%%%%%%%% boundary condition
at $z=z_0$,
\beq
 u'(z_0)=-({\kappa^2\tau\over 4}-{\tau\gamma^2\over 2})u(z_0).  
                   \label{boundzero1}
\eeq
And the eigenvalue $\tilde{m}^2$ of the bound state is given as the 
solution
of this equation (\ref{boundzero1}).
%%%%%%%%%%%

\vspace{.2cm}
Before solving this equation, we can say the followings without an
explicit analysis.
In any case of the above solutions for $\tilde{\lambda}>0$, the bound
state is obtained for $\tilde{m}^2\leq 9\tilde{\lambda}/4$ \cite{GY}. 
This implies
\beq
   m^2\leq -{15\over 4}\tilde{\lambda}.
\eeq
Then the stable brane would be obtained only for $\tilde{\lambda}=0$ and
$m=0$ dilaton. However, this brane is AdS${}_4$ one since $\lambda<0$ in 
this
case. Then we can say that it is impossible to obtain a realistic and 
stable
brane solution when the dilaton is trapped on the brane. The remaining 
such a solution is the one in which the dilaton is not trapped on the 
brane.
Such a solution can be obtained when the coupling between the dilaton and 
the
brane, $\gamma$, or $\tilde{\Lambda}$ is large enough. While the 
superstring
theory implies $\tilde{\Lambda}<0$ or AdS${}_5$ bulk which is also acquired
to realize the Newton's law on the brane. So we are restricted to the 
solution of large $\gamma$ and the dilaton is not localized on the brane.

\vspace{.3cm}
An example of such a solution is given for $\tilde{\lambda}=0.1$ and 
$\mu=1$.
This setting of the parameters corresponds to the bulk AdS and small 
$\lambda$,
and this brane world would be the most probable one.
Solving
the boundary condition (\ref{boundzero1}) in this case, we see that
the weak coupling region is in the range 
$(\gamma/\kappa)^2 < 0.34$.
The eigenvalues of $\tilde{m}^2$ are shown in the Fig.1. From
this figure, we also see the negative $\tilde{m}^2$ for 
$(\gamma/\kappa)^2<0.26$ and
positive one for $0.26< (\gamma/\kappa)^2< 0.34$. In both regions, however, 
the 
value of $m^2(=\tilde{m}^2-6\tilde{\lambda})$ is negative since 
$\tilde{m}^2\leq
9\tilde{\lambda}/4$ as stated above and it is also assured from the figure. 
Then the dilaton is trapped as a tachyon in this region.
%%%%%%%%%%%%%%% Fig %%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\voffset=15cm
\includegraphics[width=9cm,height=7cm]{dilaton2.eps} 
%  \includegraphics[width=8cm,height=7cm]{zerola2}
\caption{The curve shows the value of $\tilde{m}^2$, which is the 
solution of equation 
(\ref{boundzero1}) and scaled by $9\tilde{\lambda}/4$ in the figure,
for $\mu=1$ and $\tilde{\lambda}=0.1$. And $C_d=1-2(\gamma/\kappa)^2$.
\label{mxigraph}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%% Fig %%%%%%%%%%%%%%

While in the strong coupling region, $(\gamma/\kappa)^2>0.34$, we can not 
get
any real solution of the boundary condition (\ref{boundzero1}). In this 
region,
we find instead only the KK modes and there is no particle pole on the 
brane.
Then the dilaton is essentially living in the bulk in this case.
%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Gauge field}

Here
we consider a case where the dilaton couples to gauge fields
with the following action
\bea 
   &&  S=S_g+S_{\textrm{\scriptsize gauge}} \ ,   \label{action}
\\
   &&  S_{\textrm{\scriptsize gauge}}=
           \int d^4x \int dy \sqrt{-g}
             \left(-{1\over 4}e^{4\zeta\phi}g^{MN}g^{PQ}F_{MP}F_{NQ}\right) 
\ ,
                                                    \label{actiond}
\eea
where $\zeta$ denotes the dilaton-gauge field coupling.
When the dilaton is decoupled from the gauge fields, then $\zeta=0$ and
the gauge fields can be trapped for $\tilde{\lambda}>0$ in this case. 
This is easily 
understood from the previous analysis \cite{GU} given for the case 
without dilaton. In fact, the warp 
factor and the equation of the gauge-boson fluctuation have the same form 
with the one given in \cite{GU} if ($\tilde{\lambda}$, $\tilde{\Lambda}$) 
were replaced by ($\lambda$, $\Lambda$). 


\vspace{.2cm}
%%%%%%%%%%%%%%%%%%%%
For $\zeta\neq 0$,
we expand the fields in terms of the four-dimensional mass eigenstates,
$A_M=\int dm a_M(t,x^i,m)\phi_M(y,m)$, and 
the equation of motion for the spatial transverse
component $A_i^T$
is derived as
\bea
  &&  (\partial_y^2
   +\{2{A'\over A}+4\zeta\phi'\}\partial_y
   +{m^2\over A^2})\phi_i^T           
    =0 \ ,                                 \label{atrans}
\\
  &&\qquad
     (-\partial_t^2-{\dot{a}_0\over a_0}\partial_t+{\partial_j^2\over 
a_0^2}
     )a_i^T=m^2 a_i^T \ .              
\eea
By introducing $u(z)$ and $z$ defined as 
$\phi_i^T=A^{-1/2}u(z)$ and 
$\partial z/\partial y=\pm A^{-1}$, Eq.(\ref{atrans}) can be written as 
\beq
 [-\partial_z^2-4\xi\partial_z+V(z)]u(z)=m^2 u(z) , \ \label{warp31}
\eeq
where $\xi=\zeta\alpha$ and the potential $V(z)$ is given as
\beq
 V(z)={1\over 4}(A')^2+{1\over 2}AA''+2\xi A' .  \label{pott}
\eeq 
%
%\vspace{.5cm}
It is difficult to get an analytic solution
of the equation (\ref{warp31}) for non-zero $m$
except for some special cases, which
are given in two examples below. Before showing them, we show that
the zero mode is trapped on the brane for all solutions given above.
In other words, the gauge bosons can be trapped for those solutions
when the dilaton couples to the gauge bosons. It is proved as follows.

\vspace{.3cm}
For $\zeta=0$, the eigenvalue equation is written as
\beq
 [-\partial_z^2+V_0(z)]u(z)=m^2 u(z) , \ \label{warp4}
\eeq
\beq
 V_0(z)={1\over 4}(A')^2+{1\over 2}AA'' ,
\eeq 
For the zero-mode $u_0(z)$ ($m=0$), Eq.(\ref{warp4}) is written as, 
\beq
 [-\partial_z^2+V_0(z)]u_0(z)=0 , \ \label{warp40}
\eeq
and its normalizable solution is obtained as
\beq
  u_0(z)=c_0 X^{1/4}                 \label{solution}
\eeq
where $c_0$ is a constant. And $X={\rm sinh}^{-2}(\sqrt{\tilde{\lambda}}z)$
($X={\rm cosh}^{-2}(\sqrt{\tilde{\lambda}}z)$) for ${\tilde{\lambda}}>0$
and ${\tilde{\Lambda}}<0$ (${\tilde{\Lambda}}>0$). 

\vspace{.3cm}
It is easily seen that the above solution $u_0(z)$ also satisfies the zero-eigenvalue equation of $\zeta\neq 0$,
\beq
 [-\partial_z^2-4\xi\partial_z+V(z)]u_0(z)=0 , \ \label{warp310}
\eeq
where $\xi=\zeta\alpha$ and the potential $V(z)$ is given as
\beq
 V(z)=V_0(z)+2\xi A' .  \label{pott01}
\eeq 
The proof of (\ref{warp310}) is shown as follows.
By substituting the solution (\ref{solution}) into (\ref{warp310}),
we find
\beq
 [-\partial_z^2-4\xi\partial_z+V(z)]u_0(z)=2\xi(-2\partial_z+A')u_0(z)
 , \ \label{warp311}
\eeq
and the right hand side vanishes when the explicit forms of $u_0(z)$
and $A'(z)$ for each solution given above are used. Then the
zero-mode solutions are not affected by the dilaton coupling.

\vspace{.3cm}
On the other hand, 
the normalizability condition is modified by the dilaton coupling.
When dilaton couples to the gauge bosons, the condition of normalizability 
of zero mode is given as
\beq
  \int_{z_0}^{\infty}dz e^{4\xi z}u_0(z)^2 < \infty , \label{normali}
\eeq
From this and the explicit form of $u_0(z)$, 
we find for both solutions the following common condition,
\beq
  4\xi < \sqrt{\tilde{\lambda}} . \label{const1}
\eeq

\vspace{.5cm}
As a result, we can say 
that the gauge bosons can be trapped for the case of $\zeta\neq 0$ whenever the constraint (\ref{const1})
is satisfied.
In the followings, we can see this relation through the explicit solutions
for the soluble cases for general $m$.

%%%%%%%%%%%%%%%%%%%%%%%% New sol %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%% First example %%%%%%%%%%%%%%%%%%%
\vspace{.5cm}
As a first example, we consider the case of $\tilde{\lambda}>0$ and
$\tilde{\Lambda}=0$. This is given as a limit of $\tilde{\Lambda}\to 0$
from either solution of $\tilde{\Lambda}>0$ or $\tilde{\Lambda}<0$. 
In this case, we can solve for $u(z)$ of non-zero $m$, but
the setting of
$\tilde{\Lambda}=0$ would not be realistic when we respect the Newton's law
to be observed in our world. Although the graviton is trapped on the brane,
its KK modes would be overwhelming \cite{GNY}. 
So the analysis of this case is 
performed from the theoretical interest.

As above the same form of equation
(\ref{warp31}) and (\ref{pott}) are obtained. And the explicit form
of the potential is given as
\beq
 V(z)={\sqrt{\tilde{\lambda}}\over 4}(\sqrt{\tilde{\lambda}}-8\xi)
            -\sqrt{\tilde{\lambda}}\delta(z-z_0) .  \label{pott0L}
\eeq 
Then $u(z)$ is solved as,
\bea
 u(z)=c_1e^{k_+z}+c_2e^{k_-z},  \qquad 
         k_{\pm}=-2\xi \pm\sqrt{(2\xi-\sqrt{\tilde{\lambda}}/2)^2-m^2}
\label{soldsL}
\eea
where $c_1$ and $c_2$ are constants of integration. The normalizable 
solution is obtained by choosing $k_-$ and the boundary condition,
(\ref{boundzero}), can be written as
\beq
 -2\xi+\sqrt{\tilde{\lambda}}/2=\sqrt{(2\xi-\sqrt{\tilde{\lambda}}/2)^2-m^2}\ .
                           \label{abovc} 
\eeq
Then we obtain $m=0$ and $\xi<\sqrt{\tilde{\lambda}}/4$, which is nothing 
but the one given above, (\ref{const1}). This indicates that the result given
above for the zero-mode is satisfied also 
in the limit of $\tilde{\Lambda}\to 0$.

%%%%%%%%%%%%%%%%%%%%%%%% Second example %%%%%%%%%%%%%%%
\vspace{.3cm}
As a second soluble example, we 
consider another limit, $\tilde{\lambda}=0$. However,
$\lambda$ is zero or negative in this limit as seen from the relation 
$\tilde{\lambda}=\lambda+{\kappa^2\alpha^2\over 3}$ (see (\ref{lam})), so
the brane is not a realistic also in this case 
when we respect the present observation
of small but positive $\lambda$. 
However it is meaningful from a theoretical viewpoint to study this case. 

In this case,
the warp factor $A(y)$ takes the simple Randall-Sundrum form, and
the potential is given as
\beq
 V(z)={3\over 4z^2}-{2\xi\over z}-\mu\delta(|z|-z_0) .  \label{pott0}
\eeq 
Then $u(z)$ is solved as,
\bea
 u(z)=e^{-2z(\xi+d/2)}z^{3/2}\{c_1~{}_1F_1(b_1,b_2;2zd)
+c_2 U(b_1,b_2;2zd)\}, 
        \label{solds}
\eea
where $c_1$ and $c_2$ are constants of integration and
\beq
    d=\sqrt{4\xi^2-m^2},\qquad b_1={3\over 2}-{\xi\over d},
          \qquad b_2=3     \label{para1}
\eeq
Here ${}_1F_1(b_1,b_2;X)$ denotes the Kummer's hypergeometric function,
and the confluent hypergeometric function denoted by
$U(b_1,b_2;X)$ is another independent solution of the same differential
equation.
It follows from this solution that $u(z)$ oscillates 
with $z$ when $m^2>4\xi^2$, 
where the continuum KK modes appear. 

\vspace{.5cm}
Here we concentrate on the bound state which is restricted to the region
of $m^2<4\xi^2$. From the normalization condition, 
which is obtained by replacing $u_0(z)$ in (\ref{normali}) by $u(z)$,
\beq
  \int~e^{4\xi z}u^2(z)dz <\infty,
\eeq
the solution for the localized state is obtained
by setting $c_1=0$ since 
$$e^{2\xi z}~e^{-2z(\xi+d/2)}z^{3/2}{}_1F_1(b_1,b_2;2zd)\to e^{2|\xi| 
z}/\sqrt{z}$$ for 
$z\to \infty$. Then $u(z)$ is written as,
\beq
 u(z)=c_2e^{-2z(\xi+d/2)}z^{3/2}U(b_1,b_2;2zd).
 \label{solb}
\eeq

At the next step, we consider the boundary condition at $z=z_0$ (or $y=0$), which
is given as follows by taking into account of the 
$\delta$-function potential in (\ref{pott0}),
\beq
{du(z_0)\over dz}=-{1\over 2}({\kappa^2\tau\over 6})u(z_0).  
\label{boundzero}
\eeq
This condition can be written as
\beq
  {1\over 2}+{\mu-\xi\over d}={U(b_1,b_2+1;2d/\mu)\over U(b_1,b_2;2d/\mu)},
   \label{bound03}
\eeq
where we used $z_0=1/\mu$. This equation provides the value of $m$ when
$\xi$ and $\mu$ are given. Our interest is in the trapped gauge bosons, so
we see this equation for $m=0$. For $\xi>0$, the above equation is
written as
\beq
  {2\over x}={U(1,4;x)\over U(1,3;x)},
   \label{bound00}
\eeq
where $x=4|\xi|/\mu$. While the right hand side is obtained as 
\footnote[4]{
The relations (\ref{bound02}) and (\ref{bound04}) are easily obtained 
from the identities, $U(a,a+1;x)=x^{-a}$ and 
$U'(a,c;x)=U(a,c;x)-U(a,c+1;x)$}
\beq
  {U(1,4;x)\over U(1,3;x)}={2\over x}+{x\over 1+x},
   \label{bound02}
\eeq
and we can see (\ref{bound00}) is not satisfied for $x>0$. 
From this we can say that there is no solution of $m=0$ for $\xi>0$.

For $\xi<0$, equation (\ref{bound03}) is written as
\beq
  1+{2\over x}={U(2,4;x)\over U(2,3;x)}.
   \label{bound04}
\eeq
This equation is the identity, so we find always the solution $m=0$
for $\xi< 0$. As a result, we can say that the gauge bosons can be
trapped in the region, $\xi< 0$,
for $\tilde{\lambda}=0$. This result is again consistent with the
relation $4\xi<\sqrt{\tilde{\lambda}}$, (\ref{const1}).
In order to see clearly the above result, the numerical estimation
of the value of $m$ as the solutions of (\ref{boundzero}) is shown
in the Fig.\ref{mxigraph2} as a function of $\xi$ for fixed $\mu$, $\mu=1$.

%%%%%%%%%%%%%%% Fig %%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\voffset=15cm
  \includegraphics[width=9cm,height=7cm]{zerol3.eps} 
%  \includegraphics[width=8cm,height=7cm]{zerola2}
\caption{The value $m^2$ (vertical-axis) of the solution of equation 
(\ref{boundzero}) versus $\xi$ (horizontal-axis) for $\mu=1$.
\label{mxigraph2}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%% Fig %%%%%%%%%%%%%%

\vspace{.2cm}
%%%%%%%%%%%%%%%%%
For the case of $\tilde{\lambda}<0$,
the graviton can not be trapped as stated above, then this brane 
is also unrealistic. While 
it would be worthwhile from the theoretical viewpoint
to study this case, but we like to discuss in the future.

\vspace{.4cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Finally at this sub-section, we comment on another interesting point
stated in the introduction for the case of nonzero $\zeta$.
%As stated above, the case of nonzero $\zeta$ has another point of view.  
%It has been indicated that
The dilaton-coupled action (\ref{action}) could be
related to the action described in terms of massive vector field.
According to \cite{MT}, where the case of zero effective cosmological constant 
$\lambda$ is considered,
we define the massive vector field as
$\tilde{A}_M \equiv \epsilon A_M$ and $\epsilon\equiv e^{2\zeta\phi}$.  
Then the mass-like term for the component $\tilde{A}_{\mu}$
is given by
\beq
    -\left({A'\over A}{\epsilon'\over \epsilon}+
           {1\over 2} \left( {\epsilon'\over \epsilon}\right)'
                                  +\frac{1}{2}\left({\epsilon'\over 
\epsilon}\right)^2\right)
            \sqrt{-g}g^{\mu\nu}\tilde{A}_{\mu}\tilde{A}_{\nu} \ ,
\label{gaugemass}
\eeq
where the delta function term is also included.
For the brane solutions given above,
the ``mass'' is dependent on $y$ in general, 
then
it can not be identified as the standard mass. 
This 
would imply that we need special brane solution to rewrite
the theory to the massive gauge theory with a special coupling to the 
brane.
On this point, we discuss in the next section.


%%%%%%%%%%%%%%%%%%%%%%%%%%55
\section{Relation to the massive theory}

In this section, let us discuss some relation between the system of bulk 
gauge boson 
coupled 
to the dilaton and that of massive vector field.
The standard way to obtain massive vector theory
is to apply Higgs mechanism, then
gauge variant scalars have the vacuum expectation values  
and consequently gauge symmetry is spontaneously broken.
Since the dilaton considered here is however a gauge singlet,
we need an additional consideration when the original gauge symmetry
is broken.
To investigate this,
we consider the following action lifting the factor $\epsilon^2$
in front of $F_{MN}F^{MN}$ in the action (\ref{actiond})
to the more general factor $f^2$ which depends on dilaton
and is a function of $x^M$ in general,
\begin{equation}
    S=-\int d^5x \sqrt{-g}f^2 F_{MN}F^{MN} \ , \label{fgau}
\end{equation} 
where a canonical normalization constant is omitted, also in following. 
Now we assume that by the field redefinition 
\begin{equation}
    \tilde{A}_M=f(x^N)A_M \ ,   
\end{equation}
the gauge invariant action (\ref{fgau}) can be rewritten 
as a massive vector form, 
\begin{equation}
   S_{\rm M}=-\int d^5 x \sqrt{-g}(\tilde{F}_{MN}\tilde{F}^{MN}
                            +M^2 \tilde{A}_M\tilde{A}^M ) \ ,\label{fgaum}
\end{equation}
where $\tilde{F}_{MN}=F_{MN}(\tilde{A}_P)$. This action
is obviously gauge non-invariant with respect to $\tilde{A}_M$. Further,
the original gauge invariance with respect to $A_M$ 
is also lost in $S_{\rm M}$. This is seen by a direct 
%%%%%%%%%%%%%%%%%%%% direct gt %%%%%%%%%%%%%%%%%%%
infinitesimal gauge transformation, $\delta A_M=\partial_M\omega$ 
in (\ref{fgaum}). Then the variation of $S_{\rm M}$ is derived up to 
total derivative as
\begin{eqnarray}
     \delta S_{\rm M}=-\int d^5 x
                \left\{-4\partial_P(\sqrt{-g}g^{MP}g^{NQ}
                          \left[(\partial_Mf)(\partial_N\omega)
                            -(\partial_Nf)(\partial_M\omega)\right])
                        \tilde{A}_Q\right.
\nonumber
\\                \left.
                    +2\sqrt{-g}M^2f(\partial_M\omega)\tilde{A}^M\right\} \ 
.
                                         \label{fgamv}
\end{eqnarray}
In the above variation, (\ref{fgamv}), we can see
the second derivative on $\omega$ yields with different combination 
on the index in the variation of the kinetic term.
Hence there does not exist $f$ such that $\delta S_{\rm M}$  vanishes  
for an arbitrary parameter $\omega$.

%%%%%%%%%%%%%%%%%%%%% Dropped term %%%%%%%%
This implies that some part of (\ref{fgau}) is dropped in (\ref{fgaum}).
In fact, we find a term like
\beq
  g^{MP}g^{NQ}(\partial_QA_P)A_N\partial_Mf
\eeq
in rewriting into the form of (\ref{fgaum}), and this term
can not be reformed into the mass-like term. One way to remove this 
term is to impose a gauge condition. This is performed by imposing $A_y=0$
for $f=f(y)$. Actually
the mass term given
in the previous section by (\ref{gaugemass}) can be obtained
under the gauge condition
$A_y=0$ as in \cite{MT}. It would be possible to get another form of mass-like term
by using another gauge condition, but some gauge condition is necessary
to get the form of $S_{\rm M}$. In other words, the difference
of two actions, (\ref{fgau}) - (\ref{fgaum}), 
necessarily includes quadratic terms
of $A_M$.
In this sense, the original gauge invariance
is lost in $S_{\rm M}$. It would be a challenging problem to find a way
to get this kind of action without losing the original gauge invariance.

\vspace{.2cm}
Furthermore, it depends on the background solutions 
whether dilatonic action can be transformed to the action with a constant 
mass after this rewriting with an appropriate gauge condition.
We give here explicit examples of such a transformation
for the solutions given in section 2 with the $A_y=0$ gauge.

As shown in (\ref{gaugemass}) in this case,
the mass-like term for the component $\tilde{A}_{\mu}$ is given by
\beq
-2\zeta\biggl(\frac{A'}{A}\phi'+\frac{1}{2}\phi''+\zeta(\phi')^2\biggr)
\sqrt{-g}g^{\mu\nu}\tilde{A}_{\mu}\tilde{A}_{\nu},
\label{gaugemass2}
\eeq
where $A' \equiv {dA\over dy}$ and $\zeta$ denotes the dilaton-gauge field 
coupling. 
Here we find a constant mass only for one case, i.e. 
$\tilde{\lambda}>0$ and $\tilde{\Lambda}=0$.
Plugging the solution of the Einstein equations for $\tilde{\lambda}>0$ and 
$\tilde{\Lambda}=0$ (eqs.(15) and (16)) into (\ref{gaugemass2}), 
we have the following form
\beq
-2\zeta\alpha\Biggl(\frac{-\sqrt{\tilde{\lambda}}+2\zeta\alpha}{2(1-\sqrt{\tilde{\lambda}}|y|)^2}
+\delta(y)\Biggr)\sqrt{-g}g^{\mu\nu}\tilde{A}_{\mu}\tilde{A}_{\nu}\label{gaugemass3}
\eeq
Obviously the bulk gauge boson mass could be constant if 
$\tilde{\lambda}= 0$.
In this case, $\lambda=-\frac{\kappa^2 \alpha^2}{3}\leq 0$ and we obtain the
gauge field with the bulk mass as well as the boundary one as
%%%%%%%%%%%%%%%%%%%%%%%%%%% Solution 1 %%%%%%%%%%% 
the following form
$$
  -{1\over 2}\Biggl(M^2+c_1\delta(y)\Biggr)
     \sqrt{-g}g^{\mu\nu}\tilde{A}_{\mu}\tilde{A}_{\nu}  , \label{gaugem2}
$$
\beq
 M^2=(-2\zeta\alpha)^2, \quad c_1=4\zeta\alpha=-2M,
\eeq
where the mass should be positive, then $M=-2\zeta\alpha >0$.
For this solution, we can assure the gauge boson trapping as follows.
When we take the parameter $c$ as follows \cite{GN2},
\beq
 c=-2\mu(\sqrt{1+{M^2\over \mu^2}}-1),  \label{cval}
\eeq
where $\mu=\sqrt{-\tilde{\Lambda}/6}$. Then, taking the limit $\mu\to 0$, 
we find
\beq
 c\to -2M.  \label{cval2}
\eeq
Then it is obvious that the gauge bosons are trapped in this case according to
our previous model \cite{GN2}.


In this sense, this is an example of the conjecture given in \cite{MT}
for the correspondence of massive vector and the dilaton-coupled gauge theory.
However, we must notice in this case that there is no tension, $\tau=0$,
then the brane is characterized by the position of the localized vector
mass. Then, for $\lambda<0$,
this localization is seen in the 5d space with the following metric,
\bea
   ds^2=g_{MN}dx^M dx^N=(-dt^2+a^2(t)\gamma_{ij}dx^i dx^j)+dy^2 \ ,                                 \label{fmet2}
\eea
$$ a(t)={1\over H}\sin (Ht), \quad H=\sqrt{-\lambda}, \quad k=-1.$$

%%%%%%%%%%%%%%%%%%%%%%%%%%% Solution 2 %%%%%%%%%%% 

\vspace{.3cm}
As another possibility, we may take
$\tilde{\lambda}=(2\zeta \alpha)^2$, where the first term of 
(\ref{gaugemass3}) vanishes, then $M=0$.
In this case, only the boundary mass term which is proportional to the 
delta function remains.
%%%%%%%%%%%%%%%%%%%%%%%%%%
Then the gauge bosons are not trapped in this case since the 
relation (\ref{cval}) is not satisfied any more. 

\vspace{.2cm}
The above two examples are consistent with the criterion for the trapping
of gauge bosons given in the previous section, $4\xi<\sqrt{\tilde{\lambda}}$.
The first example satisfies this, but the second does not. 
From these, it might be said that 
the correspondence between the system of gauge boson coupled to dilaton and 
that of 
gauge field with some constant masses is possible in our model 
when 
some appropriate
choice of the parameter set and a gauge condition are
achieved. So the situation is rather similar 
to the case
of classical solution with nontrivial profiles such as solitons. In those 
cases, the mass
of the fluctuation modes around classical configuration, in general, 
depends on the
coordinates and we have to solve the Schrodinger-like equation for the 
fluctuation modes
to determine their masses.

%%%%%%%%%%%%%%%%%%% End %%%%%%%%%%%%%%%%%%%


\section{Summary and Discussions}


We have examined the brane-world solutions by taking into account of the 
dilaton,
and investigated the localization of the bulk fields in the 
dilaton-graviton background. Owing to a specific ansatz (\ref{ans}), 
the solutions are obtained in a way where
only the brane and bulk cosmological constant  
$(\lambda,\Lambda)$ are modified to $(\tilde{\lambda},\tilde{\Lambda})$
without changing the equations of motion compared to the case without
dilaton field. The new point is the appearance of the dynamical bulk 
scalar.

Then the solutions are classified by these new parameters 
$(\tilde{\lambda},\tilde{\Lambda})$. For $\tilde{\lambda}>0$, the graviton
and the gauge bosons are trapped according to the previous analysis. 
The new feature is seen from $\tilde{\lambda}=\lambda+\kappa^2\alpha^2/3$.
When dilaton is decoupled, the graviton is trapped only for $\lambda\geq 
0$.
However, it is trapped even if $\lambda<0$ when the dilaton couples and
$\tilde{\lambda}>0$ were satisfied. This is always realizable since 
$\kappa^2\alpha^2/3>0$. This point is important in the sense that we can 
see
the graviton also in AdS$_4$ brane. The breaking of the general coordinate
invariance is expected in AdS$_4$ brane before, but this is evaded by 
including the dilaton in such a way to satisfy $\tilde{\lambda}>0$. 
We should however notice that this breaking would be seen
in AdS$_4$ brane when $\lambda$ is small enough and $\tilde{\lambda}<0$. 

As for the gauge bosons, the new analyses are given here by including
the coupling ($\zeta$) of the dilaton to gauge bosons. We find that
the gauge bosons are trapped for small enough
coupling $4\xi(=4\zeta\alpha)<\sqrt{\tilde{\lambda}}$ (see (\ref{const1})).
This condition is further assured by studying two concrete examples, where
solutions for any 4d mass-state could be obtained.

When we consider the smallness of $\lambda$ and 
$\tilde{\lambda}=\lambda+(\kappa\alpha)^2/3$, the above condition is
satisfied if $\zeta$ is the same order of the 5d gravitational coupling
$\kappa$ or smaller than it. As a limit of this constraint, we can consider
the case of $\zeta=0$, the decoupling limit. Our results given here
are consistent with the one given in the previous analysis for
$\zeta=0$ where the trapping of the gauge bosons is observed for
small $\lambda$.

One more point to be noticed is the relation between the dilaton coupled
gauge theory and the massive vector theory. Within our simple model,
we have shown that
the former theory can be rewritten to the latter by choosing an appropriate
solution for the brane world and a gauge condition for 5d gauge symmetry.
Furthermore, the criterion of the gauge-boson trapping stated above is
assured by showing two simple examples of the rewriting of the theory.


\vspace{.3cm}
Finally we comment on the localization of the dilaton which 
should be considered as an important dynamical field as well as graviton
from the string theory viewpoint. We find that the situation of the trapping of
this field depends heavily on the coupling ( denoted by $\gamma$)
of the dilaton to the brane. This parameter $\gamma$ is independent of the
parameter $\zeta$ stated above. For weak
coupling (small $\gamma$), the dilaton
can be trapped as a tachyon or a massless scalar.
The latter case is realized for $\tilde{\lambda}=0$ (or $\lambda\leq 0$).
While for strong coupling, there is no trapping state of the dilaton and
it lives in the bulk. It is observed only as continuum KK modes on the 
brane, but its particle mode is absent.

When we respect the observation of a small positive $\lambda$ in our 
present world, the brane with localized massless dilaton would be rejected
as a candidate of our world within our model. The tachyonic dilaton is
also unfavourable from the stability of the brane. 
Then the remaining solution is the brane
which does not include dilaton. However, the massless dilaton is 
usually expected as 
a scalar partner of the graviton in the superstring theory. This is true
in the 10d theory, but our analysis implies that the dilaton is absent
in our 4d world after a dimensional compactification. 
From the viewpoint of superstring theory, this seems to be
a difficulty at a glance, but it
could give a possible resolution for the moduli problem and the equivalence
principle when the massless dilaton is rejected to exist in our 4d world.

On the other hand, its non-trivial configuration would be
related to the problem of the AdS/CFT correspondence. Although
the brane considered here is coupled with the gravity, it has been
observed that the propagator
of the graviton on the brane seems to receive quantum corrections from 
CFT living on the same brane \cite{DL,Gidd}. This correction is
seen in the 
Newton potential between two massive objects, and the same 
correction is obtained
from the 5d tree propagator of the graviton in the AdS bulk.
This AdS/CFT correspondence has been 
assured for the brane solution with a trivial configuration for the 
dilaton,
which is consistent with the conformal invariance of the field theory on 
the boundary. For the case of the non-trivial dilaton, we 
expect a non-conformal the field theory on the boundary.
So it would be interesting problem to study the same problem
in our model with non-trivial dilaton configurations.
A report 
on these points would be given in a future paper.

\vspace{.3cm}
\section*{Acknowledgments}
This work has been supported in part by the Grants-in-Aid for
Scientific Research (13135223)
of the Ministry of Education, Science, Sports, and Culture of Japan.
M.T. was supported by Special Postdoctral Research
Program in RIKEN.

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