\LARGE Effective de Sitter brane\\ via running radius
%\vspace{.7cm} Nobuhiro Uekusa
\vspace{.3cm} We present the solution for effective couplings on a running de Sitter brane. A renormalization group formalism is developed for linearized gravity in a two de Sitter brane model. It is shown that the effective tension %has radius dependence %%%%%%% v3 %in contrast remains a constant similarly to the zero cosmological constant case. We also discuss the suppression of higher derivative terms. %In a two brane model with a positive cosmological constant, %we find the solution %for the effective coupling on the running brane for linearized gravity %using a drivative expansion. %Then the effective tension has the radius dependence %in contrast to the zero cosmological constant case. %We also discuss the validity of the linearized approximation of our analysis.
crun2v3.tex
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\newcommand{\indi}{\nu_1\cdots\nu_{p+1}}
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\
\vspace{.3cm}
\preprint{KYUSHU-HET-68}
%\begin{flushleft}
%%       \hfill                      {\tt hep-th/03*****}\\
%       \hfill                       KYUSHU-HET 6* \\
%\end{flushleft}

%\vspace*{3mm}

\title{\LARGE Effective de Sitter brane\\ via running radius }
%\vspace{1cm}

\author{%\large \vspace{.7cm}
Nobuhiro Uekusa}
\email[]{uekusa@higgs.phys.kyushu-u.ac.jp}
\affiliation{%\large \vspace{.3cm}
Department of Physics, Kyushu University, Hakozaki, 
Higashi-ku, Fukuoka 812-8581, Japan\\ \ \\  }

%\begin{center}
%{\bf\LARGE Effective de Sitter brane\\ via running radius \\ }
%\vspace*{12mm}
%{\large Nobuhiro Uekusa\footnote[2]{\texttt{uekusa@higgs.phys.kyushu-u.ac.jp}}}\\
%\vspace*{3mm}
%\vspace*{3mm}
%\vspace*{4mm}
%{\large %${}^{\ddagger}$
%Department of Physics, Kyushu University, Hakozaki, 
%Higashi-ku, Fukuoka 812-8581, Japan}\\
%\vspace*{10mm}
%\end{center}


%\date{\today}



\begin{abstract}


\vspace{.3cm}


We present the solution for effective couplings on a 
running de Sitter brane.
A renormalization group formalism 
is developed for linearized gravity in a two de Sitter brane model.
It is shown that the effective tension 
%has radius dependence   %%%%%%% v3
%in contrast 
remains a constant similarly
to the zero cosmological constant case.
We also discuss the suppression of higher derivative terms. 



%In a two brane model with a positive cosmological constant,
%we find the solution  
%for the effective coupling on the running brane for linearized gravity
%using a drivative expansion. 
%Then the effective tension has the radius dependence 
%in contrast to the zero cosmological constant case.
%We also discuss the validity of the linearized approximation of our analysis. 



\end{abstract}

%\pacs{04.50.+h, 11.15.-q, 04.62.+v, 11.10.Kk}

\maketitle 

%\newpage

\section{Introduction}


The holographic renormalization group 
plays an important role 
to gain a deep understanding of effective field theory.
This is motivated by the AdS/CFT conjecture
where
the size of an extra dimension is identified with the renormalization 
group scale in dual field theory \cite{Mal,GKP,Wit}. 
In particular, 
it becomes a useful tool when the calculation of field theory
is complicated.
Much attention has been attracted by the applications to
phenomenologically interesting brane world context 
such as the Randall-Sundrum (RS) model \cite{RS,Ver,BK,DL}.


A renormalization group formalism has been developed
for theories which have a fixed position of visible brane  
but a variable position of  hidden brane \cite{LMS,LR}.
It simplifies the calculation of
running couplings for localized operators  
generated by quantum corrections 
\cite{GGH,DGP}.
%\cite{DGP,GR,KTT}. 
These analyses were applied to the case
where cosmological constants on branes are zero. %\cite{RS1,RS2}.
Since the observation favors a positive cosmological constant \cite{WMAP}
and 4d Newton potential 
on de Sitter (dS) branes has been confirmed \cite{GNY},
we are interested in the case where 
cosmological constants on branes are nonzero.
The purpose of the present paper is to apply 
the renormalization group formalism 
to a model with two dS branes.
%We should note that the bulk background is not AdS space 
%in order to construct dS branes 
%and that conformal symmetry
%would be broken in dual field theory even if it exists.
%The analysis we will perform is thus 
%a simple extension to non-AdS space in classical gravity side.


%The idea that the observed world is a brane embedded in 
%the higher dimensinal spacetime has been attractive such as 
%the Randall-Sundrum scenario, where
%4-dimensional Newtonian gravity is reproduced by warped geometry
%in bulk AdS space \cite{RS1,RS2}.
%It also has been proposed that
%4d Newtonian gravity is originated from a 4d localized Einstein term
%which may be generated by quantum corrections \cite{DGP,GR,KTT}.
%These have extended the application of the AdS/CFT conjecture
%as the holographic renormalization group where
%the size of extra dimension is identified with the renormalization 
%group scale in dual field theory \cite{Ver,BK,DL},
%involving that bulk couplings give rise to renormalization group 
%running of brane couplings \cite{GGH}.

%for classical linearized gravity,
%where the cosmological constant on branes is zero.
%The purpose of the present paper is to apply 
%this formalism for the effective theory
%to two de Sitter branes model
%since the observation favors a positive cosmological constant \cite{WMAP}
%and 4d Newton potential on dS branes has been confirmed \cite{GNY}.
%We should note that the bulk background is not AdS space 
%in order to construct dS branes 
%and conformal symmetry
%would be broken in dual field theory if and even if it exists.
%The analysis we will perform is thus 
%a simple extension to non-AdS space in classical gravity side.


\vspace{.3cm}
In this paper, 
we vary the distance 
between two dS branes located at fixed points of 
the orbifold S$^1$/Z$_2$, 
while we keep the same physics on the visible brane.
The requirement that bulk classical field equations do not change
leads to 
%the running of hidden brane couplings. %%% v3
the variation of the coefficients of hidden brane operators.
We give the exact solution for the 
%running couplings  %%%% v3
coefficients 
of localized quadratic graviton
using a derivative expansion, %.
%%%% v3
where the corresponding
differential equation has the smooth zero cosmological constant limit.
%or a expansion by Fourier-transformed momentum $q^2$.
In particular, 
we find out the radius independent tension of the effective hidden brane. 
%%%%%%%% v3 independent
%although it is constant for %%%% v3
This is similar to the case of 
the RS branes \cite{LR}. 
We also confirm that  higher derivative terms 
in the effective hidden brane action are suppressed 
compared to a localized Einstein-Hilbert term 
against a small variation of the radius.
%near the original position of the hidden brane. 
%up to ${\cal O}(q^6)$.



The paper is organized as follows.
In Section \ref{s2}, we present the solution for couplings
on effective dS brane.
In Section \ref{s3}, we investigate the suppression of higher derivative terms.
Summary and discussions are given in the final section.


\section{\label{s2}Couplings on effective de Sitter brane}


We begin with the gravitational action,
\bea
  &&\qquad 
      S=S_{\textrm{\scriptsize bulk}}
        +S_{\textrm{\scriptsize hid}}
        +S_{\textrm{\scriptsize vis}} \ ,\label{act}
\\
  &&   S_{\textrm{\scriptsize bulk}}={1\over 2\kappa^2}\int d^4x dy\sqrt{-g} 
      (R-2\Lambda)  \ ,
\\
  &&\quad
    S_{\textrm{\scriptsize hid}}=
      -\tau_h\int d^4x \sqrt{-g_h}  \ ,    \label{hid}
\\
  &&\quad  
    S_{\textrm{\scriptsize vis}}= -\tau_v\int d^4x \sqrt{-g_v} \ ,
\eea
where $\Lambda(<0)$, $\kappa$, $\tau_h$ and $\tau_v$ denote
the bulk cosmological constant, 
the five-dimensional gravitational coupling constant,
the tensions of the hidden and visible branes, respectively.
The hidden and visible branes are located at $y=y_h$ and $y=y_v$
which correspond to two fixed points of the orbifold 
S$^1$/Z$_2$.
The determinants of the metrics in the  
bulk and on the hidden and visible branes
are denoted as  
$g$, $g_h$ and $g_v$, respectively.
%The subscripts $h$ and $v$ denote the quantities with respect to the 
%hidden and visible branes which correspond to two fixed planes under 
%the orbifolding S${}^1$/Z${}_2$.


For the action (\ref{act}),
the background solution with positive cosmological constants on
the hidden and visible branes is given by\footnote{The 
localization of various bulk fields
on a brane with a positive cosmological constant
has been investigated for 
the background (\ref{ba}) \cite{GNY,BGO,ue}.}
\beq
    ds^2=A^2(y)(-dt^2+a^2(t)\delta_{ij}dx^i dx^j)+dy^2 \ ,\label{ba}
\eeq
\beq
     A(y)={\sqrt{\lambda}\over\mu}\sinh\left[\mu(y_H-|y|)\right] \ , \label{wa}
\eeq
where $\mu=\sqrt{-\Lambda/6}$. 
The constant of integration $y_H$ is determined by 
the boundary condition $A(y_h)=1$.
%and $\sinh\left[\mu (y_H-y_h)\right]=\mu/\sqrt{\lambda}$.
%$\gamma_{ij}=\delta_{ij}/(1+kx^kx^l\delta_{kl}/4)$.
The feature of the background (\ref{ba}) is summerized as follows.
The cosmological constants on the visible and hidden branes are 
written equally as
\beq
    \lambda=\left({\dot{a}\over a}\right)^2 \ .  \label{la4}
\eeq
The terminology ``dS brane ''
means a brane with positive cosmological constant $\lambda$.
As a consistency condition,
the cosmological constants on these branes
are related to
the parameters in the action (\ref{act}),
\beq
    \lambda={\Lambda\over 6}+{\kappa^4\tau_v^2\over 36} \ .
\eeq
In addition, the brane tensions are constrained as
\beq
    \tau_h=-\tau_v \ .
\eeq
%
%
As seen from the warp factor (\ref{wa}),
the location at $y=y_H$ is interpreted as a horizon. 
In the following, the region of the fifth dimension is 
taken as $y_h\leq y \leq y_v<y_H$.





\vspace{.5cm}

We consider the gravitational fluctuation around the background (\ref{ba}),
\beq
    ds^2=A^2(y)(-dt^2
    +a^2(t)\left[\delta_{ij}+h_{ij}(x^{\mu},y)\right]dx^i dx^j)+dy^2 \ ,
                                              \label{flu}
\eeq
where $x^{\mu}=(t,x^i)$.
%where 
%$\nabla_i$ denotes the covariant derivative with respect to the metric
%$\gamma_{ij}$ which is used to raise and lower the three-indices $i,j$,
%%%%%% v3 added
This is convenient for the present calculations although
in the flat case, the fluctuation is often written as 
$A^2\eta_{\mu\nu}+h_{\mu\nu}$ instead of
$A^2(\eta_{\mu\nu}+h_{\mu\nu})$.
Under the gauge conditions, 
$h_i^i=0$ and $\partial_i h^{ij}=0$,
the field equation derived from the equations (\ref{act}) and (\ref{flu}) 
is given
by
%\beq           %%%%%%%%%%%%%%%%%%%%%% eom with curvature index k %%%%
%   \left(\nabla_M^2-{2k\over A^2 a^2} -2\kappa^2\tau_h\delta(y-y_h)
%                +2\kappa^2\tau_v\delta(y-y_v)\right)h_{ij}   =0 \ ,
%\eeq     
%\beq    %%%%% v3
%    \left(\nabla_M^2 -2\kappa^2\tau_h\delta(y-y_h)
%                +2\kappa^2\tau_v\delta(y-y_v)\right)h_{ij}   =0 \ ,
%                                           \label{buleq}
%\eeq
\beq 
    \nabla_M^2 h_{ij}   =0 \ ,
                                           \label{buleq}
\eeq
where $\nabla_M$ denotes the five-dimensional covariant derivative.
In the equation (\ref{buleq}), we work with a system of 
only gravity
and the distance between the two branes is not dynamical.
We assume the proper radius is chosen 
so as to fit a possible explanation of the hierarchy  
as in the RS scenario \cite{RS}.
%We will not refer to this anymore since 
%we are interested in the derivation of
%the effective theory equivalent to the original theory, Eq.(\ref{act}).   
Thus the fluctuation in the bulk is completely determined by 
the equation (\ref{buleq}).


\vspace{.5cm}

%Here %%% v3 
According to \cite{LMS,LR},
we shorten the distance between the two branes
without varying the bulk field equation of the equation (\ref{buleq})
with fixed $y_v$.
Then effective local terms can be induced on the running hidden brane.
%This situation is speculated  intuitively via the AdS/CFT correspondence
%since 
%the radius is related to  the cutoff scale in field theory side
%and the loop corrections in field theory with boundary 
%yield localized terms. 
We would like to derive the radius dependence of
these local terms. 
To investigate it,
we introduce the effective hidden brane $S_l$ located at $y=l$ $(\geq y_h)$. 
%where $y_h<l <y_v$.
In other words, 
by replacing $S_{\textrm{\scriptsize hid}}$
 in the action (\ref{act})
with $S_l$,
we obtain the effective theory in $l\leq y\leq y_v$,
\beq
   S_{\textrm{\scriptsize eff}}
    =S_{\textrm{\scriptsize bulk}}
     +S_l 
     +S_{\textrm{\scriptsize vis}} \ .   \label{eff}
\eeq
As the fluctuation $h_{ij}$ is small,
the quadratic action
of the effective hidden brane at $y=l$ is given
by  
%%%%%%% v3 h^2 -> (A^2 h)^2
\beq 
    S_l=-\tau_h\int d^4x
    \,{1\over 4}\, \alpha(x,l)\sqrt{-g_l}\,(A^2h)^2 \ , \label{aact}
\eeq
where 
$g_l$ denotes the determinant of the metric on the effective hidden 
brane.
%which corresponds to the quadratic expansion of 
%the covariant form $\int \tau_v\sqrt{-g}(\alpha^{(0)}+\alpha^{(2)}R+\cdots)$. 
The $\alpha(x,l)$ stands for a series of couplings multiplied
by derivative operators. 
In four-dimensional Fourier transforms 
$\alpha(x,l)\rightarrow \alpha(q,l)$, it is represented as 
\beq
   \alpha(q,l)=\sum_{j=0,1,\cdots} \alpha^{(2j)}(l)(q/A)^{2j} \ , \label{der}
\eeq
where momentum $q$ is defined as a small dimensionless quantity
divided by some large mass scale.
In the equations (\ref{aact}) and (\ref{der}), 
we omitted the explicit tensor structure and $a(t)$-dependence
since we are particularly interested in the position dependence of the coupling
$\alpha$.



\vspace{.3cm}

The linearized field equation derived from
the effective action $S_{\textrm{\scriptsize eff}}$ is given by\footnote{
In the flat case,
the factor $(3\alpha-4)$ of the equation (3.7) in \cite{LR}
can be found in the equation for the rescaled fluctuation $\hat{h}=A^2 h$,
where
$\partial_y^2 h=4\mu\delta(y-l)\hat{h}/A^2+(\textrm{no delta functions})$
and $\kappa^2\tau_h=6\mu$ are used.
}
%\beq     %%% v3
%     \left(A^{-2}\nabla_{\mu}^2%-{2k\over A^2}
%    +\partial_y^2+4{(\partial_y A)\over A}\partial_y
%    -2\kappa^2\tau_h \alpha\delta(y-l)
%    +2\kappa^2\tau_v\delta(y-y_v)
%   \right)h =0  \ .\label{fieq}
%\eeq
\beq
     \left(A^{-2}\nabla_{\mu}^2%-{2k\over A^2}
    +\partial_y^2+4{(\partial_y A)\over A}\partial_y
    -2\kappa^2\tau_h (\alpha-1)\delta(y-l)
   \right)h =0  \ .\label{fieq}
\eeq
%where ${}'=\partial/\partial y$.
From the equation (\ref{fieq}), the boundary condition at $y=l$ is obtained as 
\beq
      \partial_y h(y)|_{y=l}=\kappa^2 \tau_h (\alpha-1) h(l) \ . 
\eeq
Combining this with the field equation (\ref{fieq})
leads to the 
%flow %% v3
differential equation for $\alpha$,
%%% v3 alpha -> alpha-1
\beq 
    {\partial\over \partial l}\alpha(l)=-\kappa^2\tau_h(\alpha-1)^2
        -4{A'\over A}(\alpha-1)+{q^2\over \kappa^2 \tau_h A^2} \ .  \label{flowal}
\eeq
where $A=A(l)$ and ${}'=\partial/\partial l$.
When we take the $\lambda\rightarrow 0$ limit 
$A=e^{-\mu l}\equiv 1/(\mu z)$, the equation (\ref{flowal})
reduces to
\beq
   z{\partial\over \partial z}\alpha
   =-6(\alpha-1)^2+4(\alpha-1)+{(qz)^2\over 6} \ ,
\eeq
where we used $6\mu=\kappa^2\tau_h$.
This agrees with the equation in the flat case derived in \cite{LR}.


We can find the exact solution for 
%this equation, %%% v3
the equation (\ref{flowal}),
%the coupling $\alpha$ on the effective
%hidden brane
%from Eq.(\ref{flowal}),
%%% v3 lhs 1+
\beq
    \alpha=1+{2\mu\over\kappa^2\tau_h} X\sqrt{1+X}
 \left[
   {2\over X}-
    {{\beta_{+}\beta_{-} \over 3}
         F(1+\beta_{+},1+\beta_{-},4;-X)
       -B G_{22}^{20}\left(-X \left|
       {\begin{array}{cc}
        {}_{-\beta_{+}} & {}_{-\beta_{-}}\\
        {}_{-3}&  {}_{0}   \\
      \end{array}}\right.\right)
      \over 
      F(\beta_{+},\beta_{-},3;-X)
      +B G_{22}^{20}\left(-X \left|
         \begin{array}{cc}
        {}_{1-\beta_{+}} & {}_{1-\beta_{-}}\\
       {}_{-2}&  {}_{0}   \\
      \end{array}\right.\right)}\right] \ , \label{alpha}
\eeq
\beq
    X={1\over \sinh^2\left[\mu(y_H-l)\right]} \ ,\qquad
  \beta_{\pm}={1\over 4} \left(5\pm 3\sqrt{1+{4q^2\over 9\lambda}}\right) \ ,
    %\delta={1\over 4} \left(5+3\sqrt{1+{4q^2\over 9\lambda}}\right) \ ,\quad
    %\beta={1\over 4} \left(5-3\sqrt{1+{4q^2\over 9\lambda}}\right) \ ,\quad
    %\gamma=3 \ ,
\eeq
%%%%%%%%%%%%%%%%%
%\beq
%    \alpha={2\mu\over T}X\sqrt{1+X}\left[
%   {2\over X}-
%    {{\delta\beta \over \gamma}
%      {}_2F_1(\bar{\delta},\bar{\beta},\bar{\gamma};-X)
%      -B G_{22}^{20}\left(-X \left|
%       \begin{array}{cc}
%        \bar{a}_1 & \bar{a}_2\\
%        \bar{b}_1&   0   \\
%      \end{array}\right.\right)
%      \over 
%     {}_2F_1(\delta,\beta,\gamma;-X)
%      +B G_{22}^{20}\left(-X \left|
%       \begin{array}{cc}
%        a_1 & a_2\\
%        b_1&   0   \\
%      \end{array}\right.\right)}\right] \ ,   \label{alpha}
%\eeq
%\bea
%  &&  \alpha={2\mu\over k}X\sqrt{1+X}\tilde{\alpha}  \ ,
%\\
%  &&\tilde{\alpha}=
%   {2\over X}-
%    {{\delta\beta \over \gamma}
%       {}_2F_1(\bar{\delta},\bar{\beta},\bar{\gamma};-X)
%      -B G_{22}^{20}\left(-X \left|
%       \begin{array}{cc}
%        \bar{a_1} & \bar{a_2}\\
%        \bar{b_1}&   0   \\
%      \end{array}\right.\right)
%      \over 
%     {}_2F_1(\delta,\beta,\gamma;-X)
%      +B G_{22}^{20}\left(-X \left|
%       \begin{array}{cc}
%        a_1 & a_2\\
%       b_1&   0   \\
%      \end{array}\right.\right)} \ ,
%\eea
%\beq
%    X={1\over \sinh^2\left[\mu(y_H-l)\right]} \ ,
%\eeq
%where 
%\bea
%  &&  \delta={1\over 4} \left(5+3\sqrt{1+{4q^2\over 9\lambda}}\right) \ ,\quad
%    \beta={1\over 4} \left(5-3\sqrt{1+{4q^2\over 9\lambda}}\right) \ ,\quad
%    \gamma=3 \ ,
%\\
% &&  \bar{\delta}={1\over 4} \left(9+3\sqrt{1+{4q^2\over 9\lambda}}\right) \ ,
%         \quad
%    \bar{\beta}={1\over 4} \left(9-3\sqrt{1+{4q^2\over 9\lambda}}\right) \ ,
%        \quad
%    \bar{\gamma}=4 \ ,\quad
%\eea
%and
%\bea
%   &&a_1={1\over 4}\left(-1+3\sqrt{1+{4q^2\over 9\lambda}}\right)  ,\ \
%     a_2={1\over 4}\left(-1-3\sqrt{1+{4q^2\over 9\lambda}}\right)  ,\ \
%     b_1=-2  \ ,  
%\\
%   &&\bar{a}_1={1\over 4}\left(-5+3\sqrt{1+{4q^2\over 9\lambda}}\right)  ,
%       \ \
%     \bar{a}_2={1\over 4}\left(-5-3\sqrt{1+{4q^2\over 9\lambda}}\right)  ,
%       \ \
%     \bar{b}_1=-3 \ .\quad\quad
%\eea
%%%%%%%%%%%%%%%%%%%%%%%%
where
%%% v3 added
we introduced the new postion coordinate $X$
such that smaller $X$ is closer to the original hidden brane,
while $X\rightarrow \infty$ corresponds to the horizon.
The functions
$F(\delta,\beta,\gamma;-X)$ and 
$G_{22}^{20}\left(-X \left|
         \begin{array}{cc}
        {}_{\delta} & {}_{\beta}\\
        {}_{\gamma}&   {}_{0}   \\
      \end{array}\right.\right)$
stand for the Gauss's hypergeometric function and 
Meijer's $G$-function, respectively.
The constant of integration $B$ is determined by 
the boundary condition $\alpha(y_h)=1$ from the equation (\ref{buleq}) 
and (\ref{fieq}),
%%%%%%% v3 signature ... factor 1/2
\beq
   B=-{
      F(\beta_{+},\beta_{-},3;-{\lambda\over \mu^2})
    -{\beta_{+}\beta_{-}\over 3}{\lambda\over 2\mu^2}
       F(1+\beta_{+},1+\beta_{-},4;-{\lambda\over \mu^2})
    \over
          G_{22}^{20}\left(-{\lambda\over \mu^2}\left|
           \begin{array}{cc}
        {}_{1-\beta_{+}}& {}_{1-\beta_{-}} \\
         {}_{-2}& {}_{0}\\
        \end{array}
      \right.\right)
    +{\lambda\over 2\mu^2}G_{22}^{20}\left(-{\lambda\over \mu^2}\left|
         \begin{array}{cc}
         {}_{-\beta_{+}}& {}_{-\beta_{-}} \\
         {}_{-3}  & {}_{0}\\
        \end{array}
      \right.\right)} \ .
\eeq
%\beq  v3
%   B=-{
%      F(\beta_{+},\beta_{-},3;-{\lambda\over \mu^2})
%    +{\beta_{+}\beta_{-}\over 3}{\lambda\over \mu^2}
%       F(1+\beta_{+},1+\beta_{-},4;-{\lambda\over \mu^2})
%    \over
%          G_{22}^{20}\left(-{\lambda\over \mu^2}\left|
%           \begin{array}{cc}
%        {}_{1-\beta_{+}}& {}_{1-\beta_{-}} \\
%         {}_{-2}& {}_{0}\\
%        \end{array}
%      \right.\right)
%    -{\lambda\over \mu^2}G_{22}^{20}\left(-{\lambda\over \mu^2}\left|
%         \begin{array}{cc}
%         {}_{-\beta_{+}}& {}_{-\beta_{-}} \\
%         {}_{-3}  & {}_{0}\\
%        \end{array}
%      \right.\right)} \ .
%\eeq  <- v3
%\beq
%   B=-{\left({T\over 2\sqrt{\lambda+\mu^2}} 
%                            -2\right)
%      {}_2F_1(\delta,\beta,\gamma;-{\lambda\over \mu^2})
%    +{\delta\beta\over\gamma}{\lambda\over \mu^2}
%       {}_2F_1(\bar{\delta},\bar{\beta},\bar{\gamma};-{\lambda\over \mu^2})
%    \over
%    \left({T\over 2\sqrt{\lambda+\mu^2}} 
%                            -2\right)
%      G_{22}^{20}\left(-{\lambda\over \mu^2}\left|
%         \begin{array}{cc}
%        a_1& a_2 \\
%          b_1& 0\\
%        \end{array}
%      \right.\right)
%    -{\lambda\over \mu^2}G_{22}^{20}\left(-{\lambda\over \mu^2}\left|
%         \begin{array}{cc}
%        \bar{a}_1& \bar{a}_2 \\
%         \bar{b}_1& 0\\
%        \end{array}
%      \right.\right)}
%\eeq
%it was not able to find the $(q/A)^2$-expantion of the solution (\ref{alpha})
%analytically.
Fig.\ref{alq} shows
the behavior of our solution (\ref{alpha}) for the coupling 
$\alpha$
against the position $X$ as well as the derivative operator $(q/A)^2$. 
The coupling $\alpha|_{q^2=0}$
%varies with  %%% v3
is independent of
the position and the slope
$\partial_{(q^2/A^2)}\alpha|_{q^2=0}$ is nonzero
except at $X(l)=X(y_h)\equiv X_h$.
%$X\neq X_{\textrm{\scriptsize min}}$.
This means that the tension of the effective hidden brane,
$\tau_h\alpha^{(0)}$, is 
%not  %%% v3 
constant and
that the localized curvature term appears on the effective hidden brane.
%%% v3 dep -> indep
The radius independent behavior of the effective hidden brane tension
is 
%remarkably different from %%% v3
similar to
the case of RS branes. 
%So the effective brane approaching to visible brane  
%leads to the larger effective tension and curvature scalar term
%on the brane as expected.
%While, the effective tension on the brane at $y=y_v$ is identified as
%effective cosmological constant
%by involving the visible brane tension.
%For the background given above 
%which has a positive cosmological constant, 
%the cosmological constant on the brane, 
%$\tau_v(\alpha(q^2=0)-1)$, remain nonzero
%differently from the case of RS brane as we can see in Fig.\ref{alq}.   

\vspace{.3cm}
%%%%%%%%%%%%%%% Fig %%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\voffset=15cm
  \includegraphics[width=11cm,height=8cm]{modblwh2.eps}
\caption{$X$- and $(q/A)^2$-dependences of the coupling $\alpha$,
where
$\kappa^2\tau_h=1.2$, 
$\mu=\sqrt{390}\times 10^{-2}$, $\lambda=10^{-3}$ and $X_h=1/39$.
\label{alq}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%% Fig %%%%%%%%%%%%%%


It is useful to take some limits  
in order to study properties of the solution (\ref{alpha}).
We look at the coupling $\alpha$ in 
two limits, $q^2=0$ and large $X$.
%We investigate the behavior of the couplings $\alpha^{(2j)}$, $j>0$ 
%near the original hidden brane position
%in next section.


For $q^2=0$, 
%%%% v3
%the solution (\ref{alpha}) reduces to
%\beq
%   \alpha^{(0)}={3\mu\over \kappa^2\tau_h}
%      X^2
%       \left[ 
%    (X-2)\sqrt{1+X}+{3\lambda^2\over \kappa^2\tau_h\mu^3}
%     -\left({\lambda\over \mu^2}-2\right)\sqrt{1+{\lambda\over \mu^2}}
%        \right]^{-1} \ .    \label{azero}
%\eeq
%From this, 
%we can calculate the slope of $\alpha^{(0)}$ at $X=X_h$,
%\beq
%    \left.{\partial\over \partial X}\alpha^{(0)}\right|_{X=X_h}
%     =-{\mu^2\over \lambda} \ .              \label{slop}
%\eeq
%The equation (\ref{slop}) implies an abrupt decrease of
%the  effective hidden brane tension against 
%a small variation of the radius 
%as shown in Fig.\ref{alq}.
%   v3 <-
%
%Near the original hidden brane,
%the second term of denominator in Eq.(\ref{azero}) dominates 
%since $X\rightarrow \lambda\mu^{-2}e^{2\mu l}$ for small $\lambda$.
%This leads to $\alpha^{(0)}<1$ as shown in Fig.\ref{alq}.
%
%
%%% v3
%If we take the limit where cosmological constants 
%on the two branes, $\lambda$ are zero,
%one might worry about a divergence in the equation (\ref{slop}).
%However when we return to the equation (\ref{azero}),
%we can check that
%the constant coupling is recovered     
%as $\alpha^{(0)}\rightarrow 1$, where $X\rightarrow \lambda/\mu^2$ 
%for $\lambda\rightarrow 0$.
%%% v3 <-
%From (\ref{alpha}),
%$\alpha\rightarrow 1$ for $\Lambda\rightarrow 0$.
%This cancels out the tension on the visible brane. 
%Therefore when the bulk contribution vanishes,
%the cosmological constant on the brane becomes zero.  
%%% v3
%The appearance of radius dependence of $\alpha$ in the dS case
%is also read off the radius dependence of $A'/A$ term in 
%the equation (\ref{flowal}). 
%%% v3 <-
the equation (\ref{flowal}) reduces to
\beq 
    {\partial\over \partial l}\alpha^{(0)}(l)
   =-\kappa^2\tau_h(\alpha^{(0)}-1)^2
        -4{A'\over A}(\alpha^{(0)}-1)
  \ .  
\eeq
The general solution for this equation is given by
\beq
   \alpha^{(0)}=1+
    {3\mu\over \kappa^2\tau_h}
   {X^2\over (X-2)\sqrt{1+X}+C} \ ,
\eeq
where $C$ is a constant of integration.
By imposing the boundary condition 
$\alpha^{(0)}(y_h)=1$, we obtain the solution,
\beq
   \alpha^{(0)}=1 \ .
\eeq
Thus the value of the hidden brane tension $\tau_h$
is preserved in the variation of the radius. 

\vspace{.3cm}
For a large $X$ limit,  
the coupling $\alpha$ is given by
\beq 
    \alpha \simeq {4\mu\over \kappa^2\tau_h}\sqrt{X}
       (1-{\cal F}(q^2)) \ ,  \label{alx}
\eeq
where ${\cal F}(q^2)$ denotes the coefficients dependent on $q^2$.
%%% v3 
%\beq
%     \alpha
%       \simeq{3\mu\over 2\kappa^2\tau_h}\sqrt{X}
%       \left(1+\sqrt{1+{4\over 9\mu^2 X}
%                        \left(q\over A\right)^2}\right) \ .
%                                                 \label{alx}
%\eeq v3 <-
%\nonumber \\
%   &&\quad
%         ={2\mu\over T}{1\over \sqrt{1+X}}\left[
%        \left({3\over 2}X+2\right)+{1\over 6\mu^2}\left(q\over A\right)^2
%       -{1\over 27\mu^4X}\left({q\over A}\right)^4
%        +{\cal O}\left(\left({q\over A}\right)^6\right)\right] \ .
%\eea
%%% v3
%By a derivative expansion of the equation (\ref{alx}),
%we obtain the following couplings for large $X$, 
%\beq
%   \alpha^{(0)}={3\mu\over \kappa^2\tau_h}\sqrt{X} \ ,
%  \quad
%    \alpha^{(2j)}={3\mu\over 2\kappa^2\tau_h}(-1)^{j+1}
%              \left({2\over 9\mu^2 X}\right)^j \sqrt{X} \ ,  \label{ahi}
%\eeq
%where $j>0$.
Since ${\cal F}$ depends on $q^2$ only through the form
$\sqrt{1+4q^2/(9\lambda)}=\sqrt{1+4q^2/(9\mu^2X A^2)}$,
we can identify   
a ratio of higher derivative terms,
% in (\ref{alx})
%%% v3
%is directly proportional to $q^2$
%but inversely proportional to $\lambda$,
%\beq
%    \alpha^{(2j+2)}\left({q\over A}\right)^{2j+2}\left/
%    \alpha^{(2j)}\left({q\over A}\right)^{2j}     \right.
%    =-{2\over 9} {q^2\over \lambda} \ ,\qquad\textrm{for } j>0 \ .
%\eeq
\beq  
    \alpha^{(2j+2)}\left({q\over A}\right)^{2j+2}\left/
    \alpha^{(2j)}\left({q\over A}\right)^{2j}     \right.
    \propto
     -{q^2\over \mu^2 X A^2}=-{q^2\over \lambda}\ ,\qquad\textrm{for } j>0\ .
\eeq 
This result seems to imply 
that for tiny $\lambda$, the contributions of
higher derivative terms in
the action (\ref{aact})
can exceed the Einstein-Hilbert term (i.e. the $q^2$ term)
far away from the original position of the hidden brane.
%even if $q^2$ is small. 
However before
concluding the breakdown of linearized approximation,
one must require to take KK-modes into account 
since %zero-mode and 
KK-modes are localized 
near %the original hidden and 
the visible brane
%, respectively 
\cite{GKR}.  
We leave this problem in future.
In next section, we will study the contributions
of higher derivative terms  at $l\sim y_h$.




\section{\label{s3}The suppression of higher derivative terms}

We would like to examine the radius dependence of 
$\alpha$
at each order of derivative expansion.
We need to check particularly
whether the contributions of higher derivative terms are negligible. 
Since it is difficult to calculate
directly a derivative expansion of our solution (\ref{alpha}),
%To investigate this,
we return to the 
%flow  %%% v3
differential
equation (\ref{flowal}). 


%So far we have employed the linearized approximation (\ref{aact}).
%If higher derivative terms are comparable with 
%lower derivative terms,
%this approximation is not valid.
%
%Thus we need to confirm that 
%the contributions of the higher derivative terms are small. 

Expanding the %flow %%% v3
equation (\ref{flowal}) about $(q/A)^2$,
we obtain a set of the %flow 
equations as
\bea
   &&\alpha^{(2)}{}'+2{A'\over A}\alpha^{(2)}+
      -{1\over \kappa^2\tau_h}=0 \ ,
\nonumber\\
   &&\alpha^{(4)}{}'
     +\kappa^2\tau_h \alpha^{(2)}{}^2
    =0 \ ,
\nonumber\\
   && \cdots \ ,
\nonumber\\     \label{set}
\eea
%\bea
%    &&\alpha^{(0)}{}'+4{A'\over A}\alpha^{(0)}
%    +\kappa^2\tau_h\alpha^{(0)}{}^2=0\ ,
%\nonumber\\
%   &&\alpha^{(2)}{}'+2{A'\over A}\alpha^{(2)}+
%        2\kappa^2\tau_h\alpha^{(0)}\alpha^{(2)}-{1\over \kappa^2\tau_h}=0 \ ,
%\nonumber\\
%   &&\alpha^{(4)}{}'
%     +\kappa^2\tau_h(\alpha^{(2)}{}^2+2\alpha^{(0)}\alpha^{(4)})=0 \ ,
%\nonumber\\
%   && \cdots \ ,
%\nonumber\\     \label{set}
%\eea
where %${}'=\partial/\partial l$ and 
the ellipsis corresponds to
the equations for $(q/A)^{2n}$, $n>2$.
The %flow  v3
equations in %the equation 
(\ref{set}) are numerically solved.  
%
%%%%%%%%%%%%%%% Fig %%%%%%%%%%%%%%
\begin{figure}[htbp]
\begin{center}
\voffset=15cm
  \includegraphics[width=8cm,height=6cm]{modex2.eps}
\caption{$\alpha^{(2n)}$-$X$ graph for $n\leq 2$,
where
$\kappa^2\tau_h=1.2$, $\mu=\sqrt{390}\times 10^{-2}$, $\lambda=10^{-3}$
and $X_h=1/39$.
\label{alqq}}
\end{center}
\end{figure}
%%%%%%%%%%%%%%% Fig %%%%%%%%%%%%%%
%
The position dependences of the couplings
$\alpha^{(0)}$, $\alpha^{(2)}$ and $\alpha^{(4)}$ 
are shown in Fig.\ref{alqq}.
%%% v3 added 
For small $X$,
higher derivative terms are 
%at most %%% v3 
of the order of lower derivative terms
up to ${\cal O}((q/A)^2)$.
Therefore as long as we consider 
%the neighborhood of 
%the original hidden brane position, 
a small variation of the radius,
where $A\sim 1$ is satisfied,
%it is clear that
the contributions of higher derivative terms are suppressed by 
powers of $q^2$.
%and the linearized approximation is useful 





\section{Summary and discussions}


In this paper, we have found the solution for  
effective couplings
on the running dS brane for linearized gravity.
%%% v3
The corresponding differential equation has
the smooth $\lambda\rightarrow 0$ limit.
The solution 
%has radius dependence  v3
is a constant
at zero-th order 
(corresponding to the effective tension)
of a derivative expansion 
%in contrast  v3
similarly
to the case of a RS brane model. 
% v3
%When we starts shortening the radius,
%the effective tension quickly varies
%since the variation of the tension
%is inversely  proportional to the cosmological constant 
%on the effective hidden brane.
%This radius dependence in the dS case  
%is originated from the characteristic form of the dS warp factor.
In addition, 
an Einstein-Hilbert term and higher derivative terms appear
in the effective hidden brane action with radius dependence.
We have %also
confirmed up to ${\cal O}(q^6)$ 
that higher derivative terms 
are suppressed against a small variation of the radius.
Far away from the original position of the hidden brane,
the KK-mode contribution becomes significant.
%
%Without the consideration of KK-modes,
%the contributions of higher derivative terms 
%exceed 
%the ones of lower derivative terms
%away from the original position.
%
%
As for KK-modes,
it would be important to investigate not only the contributions
of higher derivative terms
but  
the spectrum of the KK gravitons 
which might be modified near the visible brane \cite{DHR}.
%The modified spectrum and
%their couplings to fields on the brane 
%would need to be studied for a dS model where we have analyzed here.

%\vspace{.3cm}
%As in \cite{LR},
%we can find the fixed point 
%for $\alpha$ from Eq.(\ref{flowal}).
%The fixed point for $\alpha^{(0)}$ in the dS case
%has the radius dependence  
%by the same reason as the one for $\alpha^{(0)}$ itself.


\vspace{.3cm}

We finally note that
the bulk background in our model is not AdS space 
in order to construct dS branes.
Beyond the AdS/CFT conjecture,
it remains to be examined whether some dual field theory exists.
This would require explicit calculations of quantum corrections
in field theory.
If dual theory  exists, 
it is worthwhile to examine whether and how conformal symmetry is broken. 
%since 
%conformal symmetry is probably not preserved.   


 

\section*{Acknowledgement}

The author thanks K.~Yoshioka and K.~Harada for reading the manuscript and discussions.
He is also grateful to K.~Ghoroku and K.~Inoue for discussions.

%%%%%%%%%%%%%%%%%%%%%%%
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hep-th0307107.extracted_bib
\begin{thebibliography}{99} 
\bibitem{Mal} J.~Maldacena,
     Adv. Theor. Math. Phys. {\bf 2}, 231 (1998), 
     Int. J. Theor. Phys. {\bf 38}, 1113 (1999).
%THE LARGE N LIMIT OF SUPERCONFORMAL FIELD THEORIES AND SUPERGRAVITY. 
%By Juan M. Maldacena (Harvard U.). HUTP-98-A097, Nov 1997. 19pp. 
%Published in Adv.Theor.Math.Phys.2:231-252,1998, Int.J.Theor.Phys.38:1113-1133,1999 Also in *Buenos Aires 1998,
%Trends in theoretical physics 2* 51-63 
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\bibitem{GKP} 
        S.S.~Gubser, I.R.~Klebanov and A.M.~Polyakov, 
     Phys. Lett. {\bf B} 428, 105 (1998). 
%GAUGE THEORY CORRELATORS FROM NONCRITICAL STRING THEORY.
%By S.S. Gubser, Igor R. Klebanov, Alexander M. Polyakov (Princeton U.). PUPT-1767, Feb 1998. 14pp. 
%Published in Phys.Lett.B428:105-114,1998 
%e-Print Archive: hep-th/9802109 
\bibitem{Wit}          
         E.~Witten,
   Adv. Theor. Math. Phys. {\bf 2}, 253 (1998). 
%ANTI-DE SITTER SPACE AND HOLOGRAPHY.
%By Edward Witten (Princeton, Inst. Advanced Study). IASSNS-HEP-98-15, Feb 1998. 39pp. 
%Published in Adv.Theor.Math.Phys.2:253-291,1998 
%e-Print Archive: hep-th/9802150         
%\bibitem{NO} 
%            S.~Nojiri and S.D.~Odintsov        %%%%%%%
%          J. High Energy Phys {\bf 0112}, 033 (2001).
%QUANTUM COSMOLOGY, INFLATIONARY BRANE WORLD CREATION AND DS / CFT
%CORRESPONDENCE.
%By Shin'ichi Nojiri (Japan, Natl. Defence Academy), Sergei D. Odintsov (Tomsk State U. & Guanajuato U.). Jul 2001. 14pp. 
%Published in JHEP 0112:033,2001 
%e-Print Archive: hep-th/0107134 
\bibitem{RS} L.~Randall and R.~Sundrum, 
                Phys. Rev. Lett. {\bf 83}, 3370 (1999);
                Phys. Rev. Lett. {\bf 83}, 4690 (1999).
\bibitem{Ver} H.~Verlinde,           
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\bibitem{BK} 
          V.~Balasubramanian and P.~Kraus,
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\bibitem{DL} 
           S.~Nojiri, S.D.~Odintsov and S.~Zerbini
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%QUANTUM (IN)STABILITY OF DILATONIC ADS BACKGROUNDS AND HOLOGRAPHIC
%RENORMALIZATION GROUP WITH GRAVITY.
%By Shin'ichi Nojiri (Japan, Natl. Defence Academy), Sergei D. Odintsov (Tomsk Pedagogical Inst.), Sergio Zerbini (Trento U. &
%INFN, Trento & INFN, Turin). NDA-FP-71, Jan 2000. 19pp. 
%Published in Phys.Rev.D62:064006,2000 
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         M.J.~Duff and J.T.~Liu
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        Class. Quant. Grav. {\bf 18}, 3207-3214, (2001);\\
         S.~Nojiri and S.D.~Odintsov 
            Phys. Lett. B {\bf 484}, 119 (2000). 
%BRANE WORLD INFLATION INDUCED BY QUANTUM EFFECTS.
%By Shin'ichi Nojiri (Japan, Natl. Defence Academy), Sergei D. Odintsov (Guanajuato U.). NDA-FP-73, Apr 2000. 10pp. 
%Published in Phys.Lett.B484:119-123,2000 
%e-Print Archive: hep-th/0004097
\bibitem{LMS} A.~Lewandowski, M.J.~May and R.~Sundrum,
       %  {\texttt hep-th/0209050},
        Phys. Rev. D {\bf 67}, 024036 (2003). 
\bibitem{LR} A.~Lewandowski and M.~Redi,
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\bibitem{GGH} H.~Georgi, A.K.~Grant and G.~Hailu,
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\bibitem{DGP} G.~Dvali, G.~Gabadadze and M.~Porrati,
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%\bibitem{GR} I.~Giannakis and H.~Ren,
%        %  {\texttt hep-th/0111127}.
%      Phys. Lett. B {\bf 528}, 133 (2002).
%\bibitem{KTT} E.~Kiritsis, N.~Tetradis and T.N.~Tomaras,
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%       J. High Energy Phys {\bf 0203}, 019 (2002). 
\bibitem{WMAP} C.L.~Bennett, M.~Halpern, G.~Hinshaw, N.~Jarosik, A.~Kogut, 
              M.~Limon, S.S.~Meyer, L.~Page, D.N.~Spergel, G.S.~Tucker, 
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              E.~Komatsu, M.R.~Nolta, N.~Odegard, H.V.~Peiris, L.~Verde 
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\bibitem{GNY}  A.~Kehagias and K.~Tamvakis,
           Class. Quant. Grav. {\bf 19}, L185 (2002);\\
      S.~Nojiri and  S.D.~Odintsov,
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     K.~Ghoroku, A.~Nakamura and M.~Yahiro,
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\bibitem{BGO} I.~Brevik, K.~Ghoroku, S.D.~Odintsov and M.~Yahiro,
         Phys. Rev. D {\bf 66}, 064016 (2002);\\ 
%LOCALIZATION OF GRAVITY ON BRANE EMBEDDED IN ADS(5) AND DS(5).
%By Iver Brevik (Norwegian U. Sci. Tech.), Kazuo Ghoroku (Fukuoka Inst. Tech.), Sergei D. Odintsov (Tomsk State U.),
%Masanobu Yahiro (Ryukyus U.). FIT-HE-02-01, Apr 2002. 19pp. 
%Published in Phys.Rev.D66:064016,2002 
%e-Print Archive: hep-th/0204066
            M.~Ito
               {\texttt hep-th/0204113};\\ 
% LOCALIZED GRAVITY ON DE SITTER BRANE IN FIVE-DIMENSIONS.
%By Masato Ito (Nagoya U.). DPNU-02-08, Apr 2002. 8pp. 
%e-Print Archive: hep-th/0204113 
          P.~Singh and N.~Dadhich,
                 {\texttt hep-th/0208080}. 
%LOCALIZED GRAVITY ON FRW BRANES.
%By Parampreet Singh, Naresh Dadhich (IUCAA, Pune). Aug 2002. 10pp. 
%e-Print Archive: hep-th/0208080 
\bibitem{ue}
     K.~Ghoroku and N.~Uekusa 
     {\texttt hep-th/0212102};\\
%GAUGE SYMMETRY AND COSMOLOGICAL CONSTANT ON A BRANE.
%By Kazuo Ghoroku (Fukuoka Inst. Tech.), Nobuhiro Uekusa (Kyushu U.). FIT-HE-02-07, KYUSHU-HET-63, Dec 2002.
%4pp. 
%e-Print Archive: hep-th/0212102 
  K.~Ghoroku, M.~Tachibana and N.~Uekusa,
    {\texttt hep-th/0304051}. 
%DILATON COUPLED BRANE WORLD AND FIELD TRAPPING.
%By Kazuo Ghoroku (Fukuoka Inst. Tech.), Motoi Tachibana (Wako, RIKEN), Nobuhiro Uekusa (Kyushu U.).
%FIT-HE-03-02, KYUSHU-HET-65, Apr 2003. 19pp. 
%e-Print Archive: hep-th/0304051 
\bibitem{GKR} S.T.~Giddings, E.~Katz and L.~Randall,
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{thebibliography}