Self-tuning of the cosmological constant \\ and Einstein Gravity on the brane \footnote{\uppercase{T}alk presented by \uppercase{B. Kyae} at {\uppercase{SUSY} 2003: \uppercase{S}upersymmetry in the \uppercase{D}esert}\/, held at the \uppercase{U}niversity of \uppercase{A}rizona, \uppercase{T}ucson, \uppercase{AZ}, \uppercase{J}une 5-10, 2003. \uppercase{T}o appear in the \uppercase{P}roceedings. %\uppercase{T}he talk was based on the work in collaboration with %\uppercase{J. E. Kim} and \uppercase{Q. Shafi}. }
JIHN E. Kim
We show that a self-tuning mechanism of the cosmological constant could work in 5D non-compact space-time. %with a $Z_2$ symmetry in the presence of a massless scalar field. The standard model matter fields live only on the 4D brane. The change of vacuum energy on the brane % %by, for instance, electroweak and QCD phase transitions, just gives rise to dynamical shifts of the profiles of the background metric and a bulk scalar field in the extra dimension, keeping 4D space-time flat. % % without any fine-tuning. %To avoid naked singularities in the bulk, the brane cosmological %constant should be negative. % To obtain the Newtonian potential on the brane at long distances, we introduce an additional brane-localized 4D Einstein-Hilbert term. % %which is constructed with a special form of the metric. % %Without %assuming a large hierarchy between 5D and 4D gravity scales, % %The Newtonian potential successfully arises on the brane %at long distances.
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\begin{document}
\title{Self-tuning of the cosmological constant
\\
and Einstein Gravity on the brane 
\footnote{\uppercase{T}alk presented 
by \uppercase{B. Kyae} 
at {\it \uppercase{SUSY} 2003: 
\uppercase{S}upersymmetry in the \uppercase{D}esert}\/,
held at the \uppercase{U}niversity of \uppercase{A}rizona,
\uppercase{T}ucson, \uppercase{AZ}, \uppercase{J}une 5-10, 2003.
\uppercase{T}o appear in the \uppercase{P}roceedings. 
%\uppercase{T}he talk was based on the work in collaboration with 
%\uppercase{J. E. Kim} and \uppercase{Q. Shafi}.
}}
%%  PLEASE USE THE ABOVE FOOTNOTE FOR ALL ARXIV POSTINGS,
%%   (SUBSTITUTING SPEAKER NAME, OF COURSE;  NO SPEAKER NAME NEEDED
%%      FOR SINGLE-AUTHOR CONTRIBUTIONS).
%%  NOT NECESSARY TO HAVE FOOTNOTE FOR VERSIONS THAT ARE SENT TO
%%  SUSY 2003 CONFERENCE FOR PUBLICATION


\author{JIHN E. Kim}

\address{School of Physics, Seoul National University \\
Seoul, 151-747 Korea
%%\\
%%E-mail: jekim@phyp.snu.ac.kr
}
%
\author{BUMSEOK KYAE,~~~Qaisar Shafi} 

\address{Bartol Research Institute, University of Delaware,\\
Newark, DE 19716 USA
%\\
%E-mail: bkyae@bartol.udel.edu, shafi@bartol.udel.edu
}


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\maketitle

\abstracts{
We show that a self-tuning mechanism of the
cosmological constant could work in 5D non-compact space-time.  
%with a $Z_2$ symmetry in the presence of a massless scalar field.
The standard model matter fields live only on the
4D brane. The change of vacuum energy on the brane  
%
%by, for instance, electroweak and QCD phase transitions, 
just gives rise to dynamical shifts of the
profiles of the background metric and a bulk scalar field in the extra
dimension, keeping 4D space-time flat. 
%
% without any fine-tuning. 
%To avoid naked singularities in the bulk, the brane cosmological
%constant should be negative. 
%
To obtain the Newtonian potential on the brane at long distances, 
we introduce an additional
brane-localized 4D Einstein-Hilbert term.   
%
%which is constructed with a special form of the metric.     
%
%Without
%assuming a large hierarchy between 5D and 4D gravity scales, 
%
%The Newtonian potential successfully arises on the brane
%at long distances.
}

\section{Introduction and Summary} 

The problem of the cosmological constant or the absolute scale of
the potential energy has been considered as the most
serious mass hierarchy problem in modern particle
physics. In recent years, however, there has been a
hope to understand the vanishing cosmological constant in extra
dimensional field theories~\cite{original,kkl1,kachru}.
%
%In four dimensional (4D) space-time, the vacuum energy $V_0$ ($=3M_P^2R$)
%determines the curvature of our space-time. The flat 4D space-time is
%possible only for $V_0=0$. Thus, in 4D one must fine-tune the
%vacuum energy to zero to obtain a flat space-time.  However, it is
%more involved in higher dimensional field theories.
%
In the Randall-Sundrum-II (RS-II) type five dimensional (5D) theories
with a 3-brane embedded in a {\it non-compact} 5D space,
the 5D gravity sector contains a 5D cosmological constant $\Lambda_b$ and
a brane vacuum energy $\Lambda_1$.
For a flat four dimensional (4D) subspace, a fine-tuning relation
between $\Lambda_b$ and $\Lambda_1$
is inevitable in the RS-II model. 
However, by introducing a massless scalar field in the bulk,
it was shown that a self-tuning of the cosmological constant
is possible in the RS-II type setup~\cite{kkl1,original}.
To guarantee a massless scalar field, the scalar can be regarded  
as the dual field of a 5D 3 form tensor field $A_{MNP}$.  
%
%the four-form field
%strength $H_{MNPQ}$ has been very useful~\cite{kkl1,townsend}.

%In the initial attempts of RS-II type models, one restricted to a
%finite value for the integration of the warp factor
%through the extra dimension,
%so that a finite effective 4D Planck mass is obtained
%even in non-compact space.
%
%In the model with the conventional scalar kinetic term in the RS-II setup,
%the effective 4D Planck mass turned out to be divergent
%(or unwanted naked singularities at some points
%in the bulk arise)~\cite{nilles,kkl2}.    
%
%an exotic kinetic term $+1/H^2$ was employed~\cite{kkl1}. However, 
%
To obtain $-\frac{1}{r}$ gravity potential,   
one might make the extra space compact by employing two branes.
However, the introduction of two branes accompanies
a fine-tuning relation between two brane cosmological
constants~\cite{nilles}. 
%
%if the intervel length between two branes
%must be somehow stabilized~\cite{nilles2}.
%
In this paper,
we employ only one brane, and  
introduce an additional brane-localized
4D Einstein-Hilbert term.    
As shown in Refs.~\cite{dvali,newDGP},
if one restricts matter fields only on the brane,
the $-\frac{1}{r}$ potential could be obtained   
even with an infinite volume extra dimension
by introducing a brane-localized gravity term.
%
%The above observation of Ref.~\cite{newDGP,dvali} that
%the introduction of a brane-localized gravity term
%$\delta(y)M_4^2\sqrt{|\bar{g}_4|}\bar{R}_4$
%can give a $-\frac1r$ potential 
%triggered our reconsideration on the model with $-H^2$
%as a kinetic energy term of the three-form anti-symmetric tensor field
%$A_{MNP}$ under the RS-II setup.
%
%While Ref.~\cite{dvalicc} tried to explain the vacuum energy
%$(0.003\ {\rm eV})^4$ with a very small bulk gravity mass parameter,
%in this paper we try to search for the possibility
%that the self-tuning idea works in the model with the standard kinetic
%term $-H^2$ and the brane gravity term
%$\delta(y)M_4^2\sqrt{|\bar{g}_4|}\bar{R}_4$.
%Especially, we are interested in a relatively large bulk gravity
%mass parameter case.
%
%The explicit verification of the self-tuning solution in this paper
%is composed of several ingredients.
%They will be summarized in the conclusion.
%
%In principle, the case with $\Lambda_b<0$ as well as with $\Lambda_b=0$ is
%workable.
%
In this paper, we will discuss only the $\Lambda_b=0$ case.  
%
%because this case gives an analytic solution of the (linearized) graviton. 
%
\section{Self-tuning of the cosmological constant}  
We consider the following action in 5D space-time $(x^\mu,y)$,
\begin{eqnarray}
S&=&\int d^4xdy\bigg[\sqrt{|g_5|}\bigg(\frac{M_5^3}{2}R_5
-\frac{1}{2}\partial_M\phi\partial^M\phi+L_m^b\bigg)
\nonumber \\
&&~~~~~+\delta(y)\sqrt{|\bar{g}_4|}\bigg(\frac{M_4^2}{2}\bar{R}_4
+L_m^1-\Lambda_1\bigg) \bigg]~,~~\label{action}
\end{eqnarray}
%
%where $g_5\equiv{\rm Det}g_{MN}$, $M,N=0,1,2,3,5$,
%and $\bar{g}_4={\rm Det}\bar{g}_{\mu\nu}$, $\mu,\nu=0,1,2,3$.
%
%$\Lambda_1$ indicates the brane cosmological constant.
where $\bar{R}_4$ is the 4D Einstein-Hilbert term 
$\bar{g}^{\mu\nu}\bar{R}^\rho_{\mu\rho\nu}$ 
constructed with a metric different from
the bulk metric $g_{MN}$:  
%\begin{eqnarray}
$\bar{g}_{\mu\nu}(x,y)=(1+\frac{\bar{\alpha}}{M_4^2}\partial_y^2)
[g_{\mu\nu}(x,y)\times\Omega^{-2}(x,y)]$   
%\end{eqnarray}
with a dimensionless coupling $\bar{\alpha}$.   
%$\Omega(x,z)$ can not be removed from the action
%by redefining the metric~\cite{newDGP}.
%
%As will be discussed later, the brane metric $\bar{g}_{\mu\nu}$
%could be more generalized to
%$\hat{g}_{\mu\nu}\equiv\bar{g}_{\mu\nu}+\frac{\bar{\alpha}}{M_4^2}
%\partial_y^2\bar{g}_{\mu\nu}$ with a dimensionless coupling $\bar{\alpha}$.
%
At the linearized level, the $z$ derivative terms coming from
$\delta(y)M_4^2\sqrt{|\bar{g}_4|}\bar{R}_4$ maintain
the 4D gauge symmetry required for the spin-2 field~\cite{newDGP}.
A 4D general coordinate transformation still can be defined as
$\partial_z^2\bar{g}'_{\mu\nu}=\frac{\partial x^\rho}{\partial x^{'\mu}}
\frac{\partial x^\sigma}{\partial x^{'\nu}}\partial_z^2\bar{g}_{\rho\sigma}$
with $\partial_z^2(\frac{\partial x^\rho}{\partial x^{'\mu}})|_{z=0}=0$.
%
The ordinary standard model (SM) matter fields are assumed
to live only on the brane.  
The presence of brane terms with $\delta(y)$
in Eq.~(\ref{action}) explicitly breaks the 5D general covariance
into the 4D one at $y=0$. 
The introduction of $\delta(y)M_4^2\sqrt{|\bar{g}_4|}\bar{R}_4$
does not spoil any given symmetry~\cite{dvali,newDGP}.
We suppose that $M_4$ is comparable to $M_5$.
%
%$\phi$ is a 5D massless scalar field.
%To protect the massless scalar from radiative large mass generation,
%it is desirable to regard $\phi$
%as the dual field of a 5D 3 form tensor field $A_{MNP}$. 
%
%
% and $L_m^1$ denotes the Lagrangian
%describing SM matter dynamics. 
%
We introduce a $Z_2$ symmetry in
the extra space, and assign even (odd) parities to $g_{\mu\nu}$,
$g_{55}$ ($g_{\mu 5}$, $\phi$) under $y\leftrightarrow -y$.

The 4D flat background solutions turn out to be~\cite{original}   
\begin{eqnarray}
&&ds^2=\beta^2(y)~\eta_{\mu\nu}dx^\mu dx^\nu+dy^2 ~, 
\label{az1}\\
%
%&&~~~~~~~~~~~~\partial_y\phi= \frac{\sqrt{2A}}{\beta^4}~,
%~~\partial_\mu\phi=0 ~, \label{az2} 
&&\beta(y)=\bigg[4a|y|+c\bigg]^{1/4}~, ~~\Omega(y)=\beta(y) ~, 
\\
&&\phi(y)=\frac{\sqrt{12M_5^3}}{4}~{\rm ln}\bigg[
\frac{4a}{c}|y|+1\bigg]\times{\rm sgn}(y)\label{phi3} ~, 
\label{sol3}
\end{eqnarray}
where 
%$\eta_{\mu\nu}={\rm diag.}(-1,1,1,1)$, and 
%
$a$ ($>0$) is a dimensionful arbitrary constant.  
${\rm sgn}(y)$ is defined as $+1$ ($-1$) for $y>0$ ($y<0$). 
%
%$\Omega(y)$ can be normalized to unity at $y=0$
%by rescaling $M_4$.
%
%Eq.~(\ref{az2}) trivially satisfies the scalar field equation
%with the metric ansatz Eq.~(\ref{az1}),
%\begin{eqnarray}
%\partial_M\bigg(\sqrt{-g_5}~\partial^M\phi\bigg)=0.
%\end{eqnarray}
%With the metric ansatz (\ref{az1}), the 5D Einstein equation leads
%to~\cite{kkl2}
%\begin{eqnarray} \label{backeq1}
%3\bigg(\frac{\ddot{\beta}}{\beta}\bigg)
%+3\bigg(\frac{\dot{\beta}}{\beta}\bigg)^2
%&=&-\frac{\Lambda_b}{M_5^3}-\delta(y)\frac{\Lambda_1}{M_5^3}
%-\frac{A}{M_5^3}~\frac{1}{\beta^8} ~, \\
%6\bigg(\frac{\dot{\beta}}{\beta}\bigg)^2
%&=&-\frac{\Lambda_b}{M_5^3}+\frac{A}{M_5^3}~\frac{1}{\beta^8} ~,
%\label{backeq2}
%\end{eqnarray}
%where the dots denote differentiations with respect to $y$, and
%\begin{eqnarray} \label{para}
%
%\lambda_b\equiv\sqrt{\frac{|\Lambda_b|}{6M_5^3}}~~,
%~~~
%Note that the brane-localized Einstein-Hilbert term does not affect
%the solutions.     
%
%Depending on the sign of $\Lambda_b$, we have three classes of
%bulk solutions~\cite{kkl2}
%\begin{eqnarray}
%&&\Lambda_b>0 : \ \
%\beta(y)=\bigg(\frac{a}{\lambda_b}\bigg)^{1/4}\bigg[ {\rm sin}
%(4\lambda_b|y|+c_+)\bigg]^{1/4}
%~~~~~~~ \label{sol1}\\
%&&\Lambda_b<0 :\ \
%\beta(y)=\bigg(\frac{a}{\lambda_b}\bigg)^{1/4}\bigg[{\rm sinh}
%(4\lambda_b|y|+c_-)\bigg]^{1/4} ~~~~~\label{sol2}\\
%
%&&
%\Lambda_b=0 :\ \ \beta(y)=\bigg[4a|y|+c\bigg]^{1/4}
%~~~~~~~~~~~~~~~~~ \label{sol3}
%\end{eqnarray}
%where ${\rm sgn}(y)$ is defined as $+1$ ($-1$) for $y>0$ ($y<0$). 
%
The integration constant $c$ should be
determined by the boundary conditions: 
\begin{eqnarray} \label{self1}
%c_-=-{\rm coth}^{-1}\bigg(\frac{\lambda_1}{\lambda_b}\bigg)~>0 ~~~
%&&{\rm for}~~\Lambda_b<0 ~,
%\\  \label{self2}
%
c=-\bigg(\frac{6M_5^3a}{\Lambda_1}\bigg)~>0 ~.    
%
%~~~~~~~~~~~~&&{\rm
%for}~~\Lambda_b=0 ~,
\end{eqnarray}
To avoid naked
singularities at some points in the bulk, we should take only {\it positive}
values of $c$. 
Hence, the brane vacuum energy is required to be negative. 
In the boundary condition Eq.~(\ref{self1}), we note that $\Lambda_1$ is
never completely fixed by the Lagrangian parameters. Any arbitrary negative
value of $\Lambda_1$ allows a 4D flat space-time solution, by
adjusting $c$ and $a$ such that the boundary conditions at the brane
are fulfilled. Even if the electroweak and QCD phase transitions
change the brane vacuum energy $\Lambda_1\rightarrow\Lambda_1'$, a flat
4D space-time could be maintained by proper shifts
$c\rightarrow c'$ and/or $a\rightarrow a'$. It means that
the profiles of the metric and the scalar field in the extra
dimension are changed. Since they are dynamical fields, it is
always possible. 
We note that our self-tuning mechanism is not associated
with any parameter sensitive to low energy physics. 
As will be shown, the 4D Newtonian constant
is given by $M_4$ rather than $M_5$ in our model.

According to Ref.~\cite{hawking},
the flat universe is quantum mechanically most probable in 4D space-time,
even though universe with a non-zero 4D curvature
is also classically allowed.
Hence, the flat universe can always be {\it dynamically} chosen
through the self-tuning mechanism.

%
%From Eq.~(\ref{az2}), and with the solutions Eqs.~(\ref{sol2})
%and (\ref{sol3}), one can easily find the background solution of $\phi_0(y)$,
%\begin{eqnarray}
%&&\Lambda_b<0 :\ \  \label{sol<0}
%\phi_0(y)=\frac{\sqrt{12M_5^3}}{4}~{\rm ln}\bigg[
%\frac{{\rm tanh}\bigg(\frac{4\lambda_b|y|+c_-}{2}\bigg)}
%{{\rm tanh}\frac{c_-}{2}}\bigg]\times{\rm sgn}(y)
%\label{phi2} \\  \label{sol=0}
%&&
%\Lambda_b=0 :\ \ \phi_0(y)=\frac{\sqrt{12M_5^3}}{4}~{\rm ln}\bigg[
%\frac{4a}{c}|y|+1\bigg]\times{\rm sgn}(y)\label{phi3} ~,
%\end{eqnarray}
%where ${\rm sgn}(y)$ is defined as $+1$ ($-1$) for $y>0$ ($y<0$).
%Note that $\phi$ has the odd parity under $y\leftrightarrow -y$.
%Since it is a continuous function at $y=0$,
%$\partial_y\phi$ does not induce the delta function at $y=0$, which is
%consistent with Eq.~(\ref{az2}).
%
%With the 5D gravity action, the gravity potential $-\frac{1}{r^2}$
%is naively expected in non-compact 5D space-time.
%In Refs.~\cite{kkl2,kkl3,cho,nilles},
%
%The simplest way to obtain $-\frac{1}{r}$ gravity potential is 
%to make the extra dimension compact by employing two branes. 
%However, the introduction of two branes accompanies 
%a fine-tuning relation between two brane cosmological 
%constants~\cite{kkl2,nilles}, 
%if the intervel length between two branes 
%must be somehow stabilized~\cite{nilles2}. 
%In this paper, instead of employing two branes,
%we introduce just one brane with an additional brane-localized 
%Einstein-Hilbert term in 5D non-compact space-time, and
%examine how the gravity potential on the brane is given in this setup.  
%
%
%to get a realistic
%$-\frac{1}{r}$ gravity potential.
%It is possible because the introduction of a
%brane-localized Einstein-Hilbert term provides
%a desirable 4D gravity interaction between the brane matter embedded
%in non-compact extra dimension~\cite{dvali,newDGP,shifman}.
%To see it explicitly, we need to consider metric perturbations
%near the background solutions Eqs.~(\ref{sol3}) and (\ref{sol2}), and
%examine how the graviton propagator is given in this setup.

\section{Metric perturbation and Conclusion}

%The relevant perturbed metric near the above background solutions is
%\begin{eqnarray}
%ds^2=\frac{1}{K^2(z)}\bigg(1+F\bigg)
%\bigg[\bigg(\eta_{\mu\nu}+h_{\mu\nu}\bigg)dx^\mu dx^\nu
%+\bigg(1+h_{55}\bigg)dz^2\bigg]~, \label{ds2} 
%\end{eqnarray}
%where $|z|=\int^z_0 dy/\beta(y)
%=\frac{1}{3a}[(4a|y|+c)^{3/4}-c^{3/4}]$, 
%$K(z)=\frac{1}{\beta(y(z))}=(3a|z|+c^{3/4})^{-1/3}$.
%$F(x,z)$ ($<<1$) denotes perturbation of the warp factor,
%while $h_{\mu\nu}(x,z)$ is interpreted as the 4D graviton.
%%
%%$f(x)$ could be interpreted as the brane bending mode~\cite{gt}.
%%
%Small energy-momentum tensors by brane and bulk matter
%cause such small metric perturbations.
%Since the brane metric $\bar{g}_{\mu\nu}$ can be relatively different
%from the bulk metric $g_{\mu\nu}$ by $\Omega^2$
%($\approx [1+\omega(x,z)]/K^2$),
%we can always redefine $h_{\mu\nu}$ (or $\eta^{\mu\nu}h_{\mu\nu}\equiv h$)
%and $F$ such that on the brane
%$\bar{g}_{\mu\nu}\approx \eta_{\mu\nu}+\bar{h}_{\mu\nu}$,
%while in the bulk $g_{\mu\nu}\approx (\eta_{\mu\nu}+\bar{F}\eta_{\mu\nu}
%+\bar{h}_{\mu\nu})/K^2$
%with $\bar{h}_{\mu\nu}=h_{\mu\nu}+(F-\omega)\eta_{\mu\nu}$
%(or $\bar{h}=h+4(F-\omega)$) and $\bar{F}=\omega$~\cite{newDGP}.
%From now on we will drop the ``bar''s.
%%
%%Since $M_4^2\sqrt{|\bar{g}_4|}{R}_4$ will turn out to be dominant
%%in gravity interaction on the brane,
%%it is convenient to redefine the metric such that
%%$\omega$ appears only in the bulk side for proprer interpretation of
%%gravity interaction on the brane.
%%
%We define $G$ such that $g_{55}\approx (1+G)/K^2$ (i.e. $G=F+h_{55}$).
%
%\begin{eqnarray}
%(\mu\nu)&:&  \label{mn}
%~-\bigg(\partial_\mu\partial_\nu-\eta_{\mu\nu}\nabla_4^2\bigg)\bigg[
%F+\frac{1}{2}G\bigg]+\frac{3}{2}\eta_{\mu\nu}\partial_z^2F
%\nonumber \\
%&&~
%+3\eta_{\mu\nu}\bigg[\bigg(\frac{K''}{K}-2\frac{K^{'2}}{K^2}\bigg)G
%-\frac{3K'}{2K}\partial_zF+\frac{K'}{2K}\partial_zG\bigg]
%\nonumber \\
%&&~-\frac{1}{2}\bigg(\nabla_5^2-3\frac{K'}{K}\partial_z\bigg)
%\bigg[h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h\bigg]
%+\frac{1}{4}\eta_{\mu\nu}\bigg[\partial_z^2h-3\frac{K'}{K}\partial_zh\bigg]
%%
%%+\frac{1}{4}\eta_{\mu\nu} \bigg(\nabla_5^2+\partial_z^2
%%-6\frac{K'}{K}\partial_z\bigg)h
%%
% \\
%&&~-\frac{M_4^2}{2M_5^3}\delta(z)K\nabla_4^2\bigg[
%h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h\bigg]
%\nonumber \\
%&&
%=\frac{\eta_{\mu\nu}}{2M_5^3}\bigg[(\phi_0')^2
%+\frac{\Lambda_1}{K}\delta(z)\bigg]G
%-\frac{\eta_{\mu\nu}}{M_5^3}\phi_0'\varphi'
%+\frac{1}{M_5^3}T^b_{\mu\nu}+\frac{1}{M_5^3}\delta(z)KT^1_{\mu\nu}
%%
%%+\bigg(\partial_\mu\partial_\nu f(x)-\eta_{\mu\nu}\nabla^2_4f(x)\bigg)
%~,
%~~~~~\nonumber \\
%(\mu 5)&:&  \label{m5}
%-\frac{3}{2}\partial_\mu\partial_z\bigg[F+\frac{1}{6}h\bigg]
%-\frac{3K'}{2K}\partial_\mu G
%=\frac{1}{M_5^3}\phi_0'\partial_\mu\varphi+\frac{1}{M_5^3}T^b_{\mu5} ~,
%\\
%(55)&:&   \label{55}
%\frac{3}{2}\nabla_4^2\bigg[F+\frac{1}{6}h\bigg]-6\frac{K'}{K}\partial_z\bigg[
%F+\frac{1}{4}h\bigg]
%=
%%\bigg[6\frac{K^{'2}}{K^2}-\frac{(\phi')^2}{2M_5^3}\bigg]h_{55}
%-\frac{\Lambda_b}{M_5^3}\frac{G}{K^2}
%+\frac{1}{M_5^3}\phi_0'\varphi'+\frac{1}{M_5^3}T^b_{55}~,~
%%-\frac{3}{2}\nabla_4^2f(x) ~,
%\end{eqnarray}
%where $'$ denotes the derivative with respect to $z$, and
%$\nabla_5^2$ and $\nabla_4^2$ indicate
%$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu} +\partial_z^2$ and
%$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$, respectively.
%
%$\varphi(x,z)$ denotes the linearly perturbed bulk scalar
%near the background solution $\phi_0(z)$ given
%in Eq.(\ref{sol<0}) or (\ref{sol=0}).
%Thus, the odd parity should be assigned to it.
%
%Small perturbations by dynamical brane and bulk matter are responsible
%for the brane and bulk energy-momentum tensors
%in Eqs.~(\ref{mn})--(\ref{55}).
%
%When we derived Eqs.~(\ref{mn})--(\ref{55}),
%we utilized the background Einstein equations
%Eqs.~(\ref{backeq1}) and (\ref{backeq2}), which are
%translated in the $z$ coordinate into
%\begin{eqnarray} \label{backeq1'}
%K^2\bigg[\frac{K''}{K}-2\frac{K^{'2}}{K^2}\bigg]&=&\frac{1}{3M_5^3}\bigg[
%\Lambda_b+\delta(z)K\Lambda_1+AK^8\bigg] ~,  \\
%K^2\bigg(\frac{K'}{K}\bigg)^2&=&-\frac{1}{6M_5^3}\bigg[\Lambda_b-AK^8\bigg] ~.
%\label{backeq2'}
%\end{eqnarray}
%
%We found that the solutions of $\varphi$, $G$, $F$ are
%\begin{eqnarray} \label{solphi}
%&&\varphi(x,z)=\sqrt{\frac{M_5^3}{48}}~\bigg[h(x,z)-h^0(x)\bigg]
%\times{\rm sgn}(z) ~,  \\  \label{sol55}
%&&G(x,z)=\frac{1}{3}\bigg[h(x,z)-h^0(x)\bigg] ~, \\
%&&F(x,z)=-\frac{1}{6}\bigg[h(x,z)-f(x)\bigg] ~,  \label{solF}
%\end{eqnarray}
%
%
%One can check Eqs.~(\ref{solphi}), (\ref{sol55}), (\ref{solF}) fulfill
%Eqs.~(\ref{m5}) and (\ref{55}) using the relations
%\begin{eqnarray} \label{rel1}
%&&~~~~~\phi_0'=\sqrt{2A}~K^3 ~,
%\\ \label{rel2}
%&&\frac{K'}{K}{\rm sgn}(z)=-\sqrt{\frac{A}{6M_5^3}}~K^3 ~,
%\end{eqnarray}
%which came from Eqs.~(\ref{az2}) and (\ref{backeq2'}), respectively.
%Note that $\varphi$ in Eq.~(\ref{solphi}) is continuous at $z=0$, and so
%$\partial_z\varphi\propto{\rm sgn}(z)\partial_zh$.
%
%
%By Eqs.~(\ref{solphi}), (\ref{sol55}), (\ref{solF}),
%and Eqs.~(\ref{backeq1'}), (\ref{rel1}), (\ref{rel2}),

The brane-localized gravity term does not affect
the self-tuning mechanism, but it is essential 
in gravity interaction on the brane.     
Let us consider metric perturbation near the background solutions.   
For convenience of the further analysis,
we change the $(x,y)$ coordinate into the conformal coordinate $(x,z)$.
With the gauge choices $\partial^\mu
(h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h)=h_{\mu 5}=0$,  
the $(\mu\nu)$ component of the linearized Einstein equation turns out 
to be~\cite{original} 
\begin{eqnarray}
&&-\frac{1}{2}\bigg[\bigg(\nabla_5^2-3\frac{K'}{K}\partial_z\bigg)
%\bigg[h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h\bigg]
%\nonumber \\
%&&-
+\frac{M_4^2}{M_5^3}\delta(z)K\nabla_4^2\bigg(1+\frac{\alpha}{M_4^2}
\partial_z^2\bigg)\bigg]\bigg[h_{\mu\nu}
-\frac{1}{2}\eta_{\mu\nu}h\bigg] ~~~~
\nonumber \\
&&~~~~~=\frac{1}{6}\bigg(
\partial_\mu\partial_\nu-\eta_{\mu\nu}\nabla_4^2\bigg)\bigg[f-h^0\bigg]
+\frac{1}{M_5^3}T^b_{\mu\nu}+\frac{1}{M_5^3}\delta(z)KT^1_{\mu\nu} ~,
\label{simplemn}
\end{eqnarray}
where $K(z)=\frac{1}{\beta(y(z))}=(3a|z|+c^{3/4})^{-1/3}$ with  
$|z|=\int^z_0 dy/\beta(y)
=\frac{1}{3a}[(4a|y|+c)^{3/4}-c^{3/4}]$.   
$'$ denotes the derivative with respect to $z$, and
$\nabla_5^2$ and $\nabla_4^2$ indicate
$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu} +\partial_z^2$ and
$\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$, respectively.      
$h$ is defined as $\eta^{\mu\nu}h_{\mu\nu}$, and $h^0(x)\equiv h(x,z=0)$.   
%
%The singular terms proportional to $\delta(z)K$ ($=\delta(y)$)
%in the left hand side are
%originated from the brane-localized Einstein-Hilbert term in the action
%Eq.~(\ref{action}).
%
The SM matter fields only contribute to $T^1_{\mu\nu}$.
For simplicity, we assume that another massless bulk scalar field
independent of $z$ 
makes a dominant contribution to $T^b_{MN}$. 
%
%Thus they satisfy $T^b\equiv \eta^{\mu\nu}T^b_{\mu\nu}=2T^b_{55}$.
%
$f(x)$ satisfies $\nabla_4^2f(x)=\frac{2}{M_5^3}T^b(x)$. 
Indeed, gravity interaction on the brane is insensitive to $T^b_{MN}$
in our model~\cite{newDGP}.
%
%where $\delta(z)G(x,z)=0$ was used.
%The trace of Eq.~(\ref{simplemn}) gives
%\begin{eqnarray}\label{traceeq}
%\bigg(\nabla_5^2-3\frac{K'}{K}\partial_z\bigg)\bigg[h(x,z)-h^0(x)\bigg]
%+\frac{M_4^2}{M_5^3}\delta(z)K\nabla_4^2h^0(x)
%=\frac{2}{M_5^3}\delta(z)KT^1(x) ~,
%\end{eqnarray}
%
%Here we used Eq.~(\ref{f}) and $T^b=2T_{55}$.
%
In Eq.~(\ref{simplemn}) we note that at $z=0$, 
the equation of motion for the trace mode is
$\nabla_4^2h^0(x)=\frac{2}{M_4^2}T^1(x)$, whereas at $z\neq 0$, 
%dynamics of $H(x,z)\equiv h(x,z)-h^0(x)$ is described with 
%
it is $[\nabla_5^2-3\frac{K'}{K}\partial_z](h-h^0)=0$.
%
%Using Eq.~(\ref{traceeq}) and
%${\rm sgn}(z)\delta(z)~(=\frac{1}{4}\partial_z[({\rm sgn}(z))^2])~=0$
%together with Eqs.~(\ref{solphi}), (\ref{sol55}) and (\ref{solF}),
%one can easily check that the linearized scalar equation
%for $\varphi$~\cite{kkl3},
%\begin{eqnarray}
%\bigg(\nabla_5^2-3\frac{K'}{K}\partial_z\bigg)\varphi
%+\frac{\phi'}{2}\bigg(4F'+h'-G'\bigg)=0
%\end{eqnarray}
%is trivially satisfied.
%
%After some algebra, Eq.~(\ref{simplemn}) takes the following form,
%\begin{eqnarray} \label{maineq}
%&&~~~~~~~~\bigg(\nabla_5^2-3\frac{K'}{K}\partial_z\bigg)h_{\mu\nu}
%+\frac{M_4^2}{M_5^3}\delta(z)K\nabla_4^2h_{\mu\nu}
%\\
%&&=-\frac{2}{M_5^3}\bigg(S_{\mu\nu}
%-\frac{1}{2}\eta_{\mu\nu}S\bigg)
%-\frac{2}{M_5^3}\delta(z)K\bigg(
%T^1_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}T^1\bigg) ~,
%\nonumber
%\end{eqnarray}
%
%Here we see that the graviton has the same tensor structure
%as that in the Eistein gravity theory.
%
%$z$ and $K(z)$ are calculated to give
%$|z|=\int^y_0\frac{dy}{(4a|y|+c)^{1/4}}
%=\frac{1}{3a}\bigg[(4a|y|+c)^{3/4}-c^{3/4}\bigg]$, 
%$K(z)=\frac{1}{\beta(y(z))}=(3a|z|+c^{3/4})^{-1/3}$ ~.    
%
%\begin{eqnarray} \label{eom}
%&&~~~~~~~~\bigg[k^2+\partial_z^2
%+\frac{3a{\rm sgn}(z)}{(3a|z|+c^{3/4})}\partial_z
%+\frac{M_4^2}{M_5^3}\delta(z)c^{-1/4}k^2\bigg]\tilde{h}_{\mu\nu}(k,z)
%\\
%&&=-\frac{2}{M_5^3}\bigg(\tilde{S}_{\mu\nu}(k)
%-\frac{1}{2}\eta_{\mu\nu}\tilde{S}(k)\bigg)
%-\frac{2c^{-1/4}}{M_5^3}\delta(z)\bigg(\tilde{T}^{1}_{\mu\nu}(k)
%-\frac{1}{2}\eta_{\mu\nu}\tilde{T}^1_{\mu\nu}(k)\bigg) ~,
%\nonumber
%\end{eqnarray}
%
%The traceless and trace parts of the bulk solution are given by
%\begin{eqnarray} \label{bulksolless}
%\tilde{h}^T_{\mu\nu}(k,z)=-\frac{2\tilde{S}^T_{\mu\nu}(k)}{M_5^3k^2}
%+p_{\mu\nu}(k)
%\bigg(J_0[k(|z|+c^{3/4}/3a)]+u(k)Y_0[k(|z|+c^{3/4}/3a)]\bigg),~
%\\ \label{bulksoltrace}
%\tilde{H}(k,z)\equiv \tilde{h}(k,z)-\tilde{h}^0(k)=
%q(k)\bigg(J_0[k(|z|+c^{3/4}/3a)]+v(k)Y_0[k(|z|+c^{3/4}/3a)]\bigg).~
%\end{eqnarray}
%
%In Eq.~(\ref{bulksolless})
%$\tilde{h}^T_{\mu\nu}$, $\tilde{S}_{\mu\nu}^T$ denote
%the traceless parts of $\tilde{h}_{\mu\nu}$, $\tilde{S}_{\mu\nu}$, i.e.
%$\tilde{h}^T_{\mu\nu}\equiv \tilde{h}_{\mu\nu}
%-\frac{1}{4}\eta_{\mu\nu}\tilde{h}$ and
%$\tilde{S}^T_{\mu\nu}\equiv\tilde{S}_{\mu\nu}
%-\frac{1}{4}\eta_{\mu\nu}\tilde{S}$.
%Similarly we define $\tilde{T}^{1T}_{\mu\nu}\equiv\tilde{T}^1_{\mu\nu}
%-\frac{1}{4}\eta_{\mu\nu}\tilde{T}$.
%In Eq.~(\ref{bulksoltrace}) we used Eq.~(\ref{f}).
%$p_{\mu\nu}(k)$, $q(k)$ and $u(k)$, $v(k)$ should be determined
%by boundary conditions.
%
%
%In Ref.~\cite{newDGP}, the higher derivative correction
%to the $(\mu\nu)$ component of the linearized gravity equation,
%\begin{eqnarray} \label{new}
%-\frac{\alpha}{2M_5^3}\delta(z)K\nabla_4^2\partial_z^2\bigg[
%h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h\bigg]
%\end{eqnarray}
%
%They could appear in the linearized equation derived from the Lagrangian
%with a more extended brane-localized term like
%$\delta(z)KM_4^2\sqrt{|\hat{g}_4|}\hat{R}_4[\hat{g}_{\mu\nu}]$,
%which is constructed with
%$\hat{g}_{\mu\nu}\equiv\bar{g}_{\mu\nu}+\frac{\alpha}{M_4^2}
%\partial_z^2\bar{g}_{\mu\nu}$.
%
%After some algebra, in 4D momentum space $(p,z)$,  
%where $k\equiv\sqrt{-p^\mu p_\mu}$ ($\ge 0$).
%
The bulk solution of Eq.~(\ref{simplemn}) in 4D momentum space $(p,z)$ 
is given by Bessel functions of order zero~\cite{original}.  

The terms with the coupling $\alpha$ in Eq.~(\ref{simplemn}) 
require to satisfy a non-trivial boundary condition at $z=0$: 
%only smooth graviton wave functions near the brane are allowed,
%
$\partial_z\tilde{h}_{\mu\nu}|_{z=0^+}
=\partial_z\tilde{h}_{\mu\nu}|_{z=0^-}=0$~\cite{newDGP}.
Otherwise highly singular terms proportional to $\delta^2(z)$ arise, 
which can not be matched to the right handed side of Eq.~(\ref{simplemn}).  
%
%As shown in Ref.~\cite{newDGP}, the boundary condition
%by Eq.~(\ref{new}) makes
%the effect of $\sqrt{|g_5|}M_5^3R_5$ suppressed on the brane.
%Hence, $M_5$ can be arbitrarily large.
%
%$\delta(z)KM_4^2\sqrt{|\hat{g}_4|}\hat{R}_4$ would
%ensure $-\frac{1}{r}$ gravity potential on the brane at low energy.
%In this paper, we also consider Eq.~(\ref{new}) and impose
%the consequent boundary condition, 
%by which $u(k)$ and $v(k)$ are determined to be
%\begin{eqnarray} \label{uv}
%u(k)=v(k)=-\frac{J_1[kc^{3/4}/3a]}{Y_1[kc^{3/4}/3a]}  ~.
%\end{eqnarray}
%
Then, since $\partial^2_zh_{\mu\nu}$, $\partial^2_zh$ do not
induce any delta function,  
we have only to compare the coefficients of the delta functions
appearing in Eq.~(\ref{simplemn}) to fulfill the boundary condition at $z=0$.  
%
%
%In fact, there does not exist the solution of $\tilde{H}(k,z)$
%($\equiv\tilde{h}(k,z)-\tilde{h}^0(k)$) at $z\neq 0$,
%because only the vanishing $q(k)$ is consistent
%with the boundary conditions.
%However, $h^0(x)$ still can propagate
%only in the longitudinal direction of the brane.
%At $z=0$, as discussed in Eq.~(\ref{traceeq}),
%\begin{eqnarray} \label{soltrace}
%\tilde{h}^0(k)=\frac{2\tilde{T}^1(k)}{M_4^2k^2} ~.
%\end{eqnarray}
%%
%%On the other hand, the boundary condition for the bulk solution
%%of $\tilde{h}^T_{\mu\nu}$ just gives
%%\begin{eqnarray} \label{lessbdy}
%%p_{\mu\nu}(k)\bigg(J_0[kc^{3/4}/3a]+v(k)Y_0[kc^{3/4}/3a]\bigg)
%%=-\frac{2}{k^2}\bigg[\frac{\tilde{T}^{1T}_{\mu\nu}(k)}{M_4^2}
%%-\frac{\tilde{S}^T_{\mu\nu}(k)}{M_5^3}\bigg]\bigg[
%%1+O\bigg(\frac{\alpha k^2}{M_4^2}\bigg)\bigg],
%%\end{eqnarray}
%
%On the other hand, from Eqs.~(\ref{eom}) and (\ref{new}),
%the solution of $\tilde{h}^T_{\mu\nu}$ at $z=0$ is given by
%\begin{eqnarray} \label{solless}
%\tilde{h}^T_{\mu\nu}(k,z=0)=-\frac{2}{M_4^2k^2}
%\bigg[\tilde{T}^{1T}_{\mu\nu}(k)+O\bigg(
%\frac{\alpha k^2\tilde{T}^{1T}_{\mu\nu}}{M_4^2},
%\frac{\alpha k^2\tilde{S}^T_{\mu\nu}}{M_5^3}\bigg)\bigg] ~.
%\end{eqnarray}
%
We found the graviton solution at $z=0$ is given by~\cite{original}   
\begin{eqnarray} \label{final}
\tilde{h}_{\mu\nu}(p,z=0)=\frac{2}{M_4^2p^2}\bigg[
\tilde{T}^{1}_{\mu\nu}(p)-\frac{1}{2}\eta_{\mu\nu}\tilde{T}^1(p)\bigg]
+O\bigg(\frac{\alpha\tilde{T}^{1T}_{\mu\nu}}{M_4^4},
\frac{\alpha \tilde{S}^T_{\mu\nu}}{M_5^3M_4^2}\bigg) ~,  
\end{eqnarray}
where $\tilde{T}^1\equiv \eta^{\mu\nu}\tilde{T}^1_{\mu\nu}$, and   
$\tilde{S}_{\mu\nu}\equiv -\frac{M_5^3}{6}(p_\mu p_\nu
-\eta_{\mu\nu}p^2)[\tilde{f}-\tilde{h}^0]+\tilde{T}^b_{\mu\nu}$.   
The unspecified part of order $\alpha$ are negligible at low energies.  
Hence, at low energies 
$\tilde{h}_{\mu\nu}$ in Eq.~(\ref{final})
reproduces the same result that the 4D Einstein gravity theory predicts 
with $G_N\equiv 1/(8\pi M_4^2)$.    
%
%Thus, with $T^1_{00}(x)=\rho(x)>>T^1_{ii}(x)$ ($i=1,2,3$),
%the non-relativistic gravity potential on the brane is 
%given by~\cite{dvali}
%\begin{eqnarray}
%V(r)=-\int dt\int\frac{d^4p}{(2\pi)^4}e^{ipx}
%\frac{1}{2}\tilde{h}_{00}(p,z=0)
%%\nonumber \\
%\approx -G_N\int d^3\vec{r}'
%\frac{\rho(\vec{r}')}{|\vec{r}-\vec{r}'|} ~,  
%\end{eqnarray}
%where $r\equiv \sqrt{(\vec{x}\cdot \vec{x})}$ and $G_N\equiv 1/(8\pi M_4^2)$.
%Hence, the Newtonian $-\frac{1}{r}$ potential is guaranteed at low energies.

%\section{Conclusion}
In conclusion, as we have shown, 4D flat solution is always possible
independent of the magnitude of 4D brane cosmological constant
in 5D non-compact space-time, through dynamics 
by the 5D gravity and a massless scalar.   
Since the SM fields are localized on the brane,
4D vacuum energy by them and its variation do not destroy the 4D flatness.
%To obtain the observed 4D gravity in non-compact extra space,
An additional brane-localized gravity term 
successfully induces $-\frac{1}{r}$ potential on the brane.
%
%\section*{Acknowledgments}
%The work is partially supported
%by DOE under contract number DE-FG02-91ER40626.
%
%This work is supported in part by the BK21 program of Ministry of
%Education(JEK), the KOSEF Sundo Grant(JEK), Korea Research
%Foundation Grant No. KRF-PBRG-2002-070-C00022(JEK), and by DOE
%under contract number DE-FG02-91ER40626(BK,QS).
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{thebibliography}{0}

\bibitem{original} J.~E.~Kim, B.~Kyae and Q.~Shafi,
 %``Brane gravity, massless bulk scalar and self-tuning solution of the
%cosmological constant problem,''
arXiv:hep-th/0305239. 

\bibitem{kkl1} J.~E.~Kim, B.~Kyae and H.~M.~Lee,
%``A model for self-tuning the cosmological constant,''
Phys.\ Rev.\ Lett.\  {\bf 86}, 4223 (2001).

%\bibitem{weinberg} S.~Weinberg,
%%``The Cosmological Constant Problem,''
%Rev.\ Mod.\ Phys.\  {\bf 61}, 1 (1989).
%
\bibitem{kachru} N.~Arkani-Hamed, S.~Dimopoulos, N.~Kaloper and R.~Sundrum,
%``A small cosmological constant from a large extra dimension,''
Phys.\ Lett.\ B {\bf 480}, 193 (2000);  
%[arXiv:hep-th/0001197];
%
S.~Kachru, M.~B.~Schulz and E.~Silverstein,
%``Self-tuning flat domain walls in 5d gravity and string theory,''
Phys.\ Rev.\ D {\bf 62}, 045021 (2000).  
%[arXiv:hep-th/0001206].  
%
%P.~Binetruy, C.~Charmousis, S.~C.~Davis and J.~F.~Dufaux,
%%``Avoidance of naked singularities in dilatonic brane world scenarios
%%with a Gauss-Bonnet term,''
%Phys.\ Lett.\ B {\bf 544}, 183 (2002)
%[arXiv:hep-th/0206089].

\bibitem{nilles} S.~Forste, Z.~Lalak, S.~Lavignac and H.~P.~Nilles,
%``A comment on self-tuning and vanishing cosmological constant 
%in the  brane world,''
Phys.\ Lett.\ B {\bf 481}, 360 (2000); 
%[arXiv:hep-th/0002164];
%
%``The cosmological constant problem from a brane-world perspective,''
JHEP {\bf 0009}, 034 (2000).  
%[arXiv:hep-th/0006139].
%
%\bibitem{kkl1} J.~E.~Kim, B.~Kyae and H.~M.~Lee,
%``A model for self-tuning the cosmological constant,''
%Phys.\ Rev.\ Lett.\  {\bf 86}, 4223 (2001).  
%[arXiv:hep-th/0011118];   
%J.~E.~Kim, B.~Kyae and H.~M.~Lee,
%``Self-tuning solution of the cosmological constant problem
%with  antisymmetric tensor field,''
%Nucl.\ Phys.\ B {\bf 613}, 306 (2001); 
%[arXiv:hep-th/0101027];
%J.~E.~Kim, B.~Kyae and H.~M.~Lee,
%``Wave function of the radion in the brane background
%with a massless  scalar field and a self-tuning problem,''
%Phys.\ Rev.\ D {\bf 66}, 106004 (2002). 
%[arXiv:hep-th/0110103].
%  
%
%K.~S.~Choi, J.~E.~Kim and H.~M.~Lee,
%%``Dual description of brane world cosmological constant with H(MNPQ),''
%J.\ Korean Phys.\ Soc.\  {\bf 40}, 207 (2002)
%[arXiv:hep-th/0201055].
%
%\bibitem{rs2} L.~Randall and R.~Sundrum,
%%``An alternative to compactification,''
%Phys.\ Rev.\ Lett.\  {\bf 83}, 4690 (1999)
%[arXiv:hep-th/9906064].
%
%
%\bibitem{townsend} The 4-form field strength was used earlier by
%A.~Aurilia, H.~Nicolai and P.~K.~Townsend,
%%``Hidden Constants: The Theta Parameter Of QCD And The Cosmological Constant
%%Of N=8 Supergravity,''
%Nucl.\ Phys.\ B {\bf 176}, 509 (1980);
%E. Witten, {\it Fermion quantum numbers in Kaluza-Klein theory},
%{\it Proc. Shelter Island II Conference}
%(Shelter Island, 1983), ed. R. Jackiw, N. N. Khuri, S. Weinberg, and
%E. Witten (MIT Press, 1985), p.369;
%S.~W.~Hawking,
%%``The Cosmological Constant Is Probably Zero,''
%Phys.\ Lett.\ B {\bf 134}, 403 (1984).
%More general functions of $H^2$ with the self-tuning property
%without the brane gravity have
%been considered in J.~E.~Kim and H.~M.~Lee,
%%``More self-tuning solutions with H(MNPQ),''
%JHEP {\bf 0209}, 052 (2002)
%[arXiv:hep-th/0207260].
%

\bibitem{dvali} G.~R.~Dvali, G.~Gabadadze and M.~Porrati,
%``4D gravity on a brane in 5D Minkowski space,''
Phys.\ Lett.\ B {\bf 485}, 208 (2000).   
%[arXiv:hep-th/0005016];
%
%G.~R.~Dvali and G.~Gabadadze,
%``Gravity on a brane in infinite-volume extra space,''
%Phys.\ Rev.\ D {\bf 63}, 065007 (2001); 
%[arXiv:hep-th/0008054];
%
%C.~Deffayet, G.~R.~Dvali, G.~Gabadadze and A.~Lue,
%``Braneworld flattening by a cosmological constant,''
%Phys.\ Rev.\ D {\bf 64}, 104002 (2001). 
%[arXiv:hep-th/0104201].
%See also
%U.~Ellwanger,
%%``Blown-up p-branes and the cosmological constant,''
%JCAP {\bf 0311}, 013 (2003)
%[arXiv:hep-th/0304057].

%
%\bibitem{dvalicc}
%G.~Dvali, G.~Gabadadze and M.~Shifman,
%%``Diluting cosmological constant in infinite volume extra dimensions,''
%Phys.\ Rev.\ D {\bf 67}, 044020 (2003)
%[arXiv:hep-th/0202174];
%N.~Arkani-Hamed, S.~Dimopoulos, G.~Dvali and G.~Gabadadze,
%%``Non-local modification of gravity and the cosmological constant problem,''
%arXiv:hep-th/0209227;
%O.~Corradini, A.~Iglesias and Z.~Kakushadze,
%%``Diluting solutions of the cosmological constant problem,''
%Mod.\ Phys.\ Lett.\ A {\bf 18}, 1343 (2003)
%[arXiv:hep-th/0305164].
%
\bibitem{newDGP} B.~Kyae,
%``Einstein gravity on a brane in 5D non-compact flat spacetime:
%DGP model revisited,''
arXiv:hep-th/0312161.

%
%\bibitem{ks} B.~Kyae and Q.~Shafi,
%%``GUT scale and leptogenesis from 5D inflation,''
%Phys.\ Lett.\ B {\bf 556}, 97 (2003)
%[arXiv:hep-ph/0211059];
%B.~Kyae and Q.~Shafi,
%%``Inflation and leptogenesis with five dimensional SO(10),''
%to appear in Phys.\ Rev.\ D [arXiv:hep-ph/0212331].
%
%
%\bibitem{nilles2} See H.~P.~Nilles, A.~Papazoglou and G.~Tasinato,
%``Selftuning and its footprints,''
%Nucl.\ Phys.\ B {\bf 677}, 405 (2004). 
%[arXiv:hep-th/0309042].
%
\bibitem{hawking} 
E.~Baum,
%``Zero Cosmological Constant From Minimum Action,''
Phys.\ Lett.\ B {\bf 133}, 185 (1983);
S.~W.~Hawking,
%``The Cosmological Constant Is Probably Zero,''
Phys.\ Lett.\ B {\bf 134}, 403 (1984).
%
%
%\bibitem{cho} I.~Y.~Cho,
%%``Brane world in a texture,''
%Phys.\ Rev.\ D {\bf 67}, 065014 (2003)
%[arXiv:hep-th/0210279].
%
%
%
%\bibitem{shifman} See also G.~Gabadadze and M.~Shifman,
%%``Softly Massive Gravity,''
%arXiv:hep-th/0312289;
%C.~Middleton and G.~Siopsis,
%%``The Schwarzschild solution in the DGP model,''
%arXiv:hep-th/0311070;
%M.~A.~Luty, M.~Porrati and R.~Rattazzi,
%%``Strong interactions and stability in the DGP model,''
%JHEP {\bf 0309}, 029 (2003)
%[arXiv:hep-th/0303116].
%
\end{thebibliography}

\end{document}



hep-ph0402111.extracted_bib
\begin{thebibliography}{0}

\bibitem{original} J.~E.~Kim, B.~Kyae and Q.~Shafi,
 %``Brane gravity, massless bulk scalar and self-tuning solution of the
%cosmological constant problem,''
arXiv:hep-th/0305239. 

\bibitem{kkl1} J.~E.~Kim, B.~Kyae and H.~M.~Lee,
%``A model for self-tuning the cosmological constant,''
Phys.\ Rev.\ Lett.\  {\bf 86}, 4223 (2001).

%\bibitem{weinberg} S.~Weinberg,
%%``The Cosmological Constant Problem,''
%Rev.\ Mod.\ Phys.\  {\bf 61}, 1 (1989).
%
\bibitem{kachru} N.~Arkani-Hamed, S.~Dimopoulos, N.~Kaloper and R.~Sundrum,
%``A small cosmological constant from a large extra dimension,''
Phys.\ Lett.\ B {\bf 480}, 193 (2000);  
%[arXiv:hep-th/0001197];
%
S.~Kachru, M.~B.~Schulz and E.~Silverstein,
%``Self-tuning flat domain walls in 5d gravity and string theory,''
Phys.\ Rev.\ D {\bf 62}, 045021 (2000).  
%[arXiv:hep-th/0001206].  
%
%P.~Binetruy, C.~Charmousis, S.~C.~Davis and J.~F.~Dufaux,
%%``Avoidance of naked singularities in dilatonic brane world scenarios
%%with a Gauss-Bonnet term,''
%Phys.\ Lett.\ B {\bf 544}, 183 (2002)
%[arXiv:hep-th/0206089].

\bibitem{nilles} S.~Forste, Z.~Lalak, S.~Lavignac and H.~P.~Nilles,
%``A comment on self-tuning and vanishing cosmological constant 
%in the  brane world,''
Phys.\ Lett.\ B {\bf 481}, 360 (2000); 
%[arXiv:hep-th/0002164];
%
%``The cosmological constant problem from a brane-world perspective,''
JHEP {\bf 0009}, 034 (2000).  
%[arXiv:hep-th/0006139].
%
%\bibitem{kkl1} J.~E.~Kim, B.~Kyae and H.~M.~Lee,
%``A model for self-tuning the cosmological constant,''
%Phys.\ Rev.\ Lett.\  {\bf 86}, 4223 (2001).  
%[arXiv:hep-th/0011118];   
%J.~E.~Kim, B.~Kyae and H.~M.~Lee,
%``Self-tuning solution of the cosmological constant problem
%with  antisymmetric tensor field,''
%Nucl.\ Phys.\ B {\bf 613}, 306 (2001); 
%[arXiv:hep-th/0101027];
%J.~E.~Kim, B.~Kyae and H.~M.~Lee,
%``Wave function of the radion in the brane background
%with a massless  scalar field and a self-tuning problem,''
%Phys.\ Rev.\ D {\bf 66}, 106004 (2002). 
%[arXiv:hep-th/0110103].
%  
%
%K.~S.~Choi, J.~E.~Kim and H.~M.~Lee,
%%``Dual description of brane world cosmological constant with H(MNPQ),''
%J.\ Korean Phys.\ Soc.\  {\bf 40}, 207 (2002)
%[arXiv:hep-th/0201055].
%
%\bibitem{rs2} L.~Randall and R.~Sundrum,
%%``An alternative to compactification,''
%Phys.\ Rev.\ Lett.\  {\bf 83}, 4690 (1999)
%[arXiv:hep-th/9906064].
%
%
%\bibitem{townsend} The 4-form field strength was used earlier by
%A.~Aurilia, H.~Nicolai and P.~K.~Townsend,
%%``Hidden Constants: The Theta Parameter Of QCD And The Cosmological Constant
%%Of N=8 Supergravity,''
%Nucl.\ Phys.\ B {\bf 176}, 509 (1980);
%E. Witten, {\it Fermion quantum numbers in Kaluza-Klein theory},
%{\it Proc. Shelter Island II Conference}
%(Shelter Island, 1983), ed. R. Jackiw, N. N. Khuri, S. Weinberg, and
%E. Witten (MIT Press, 1985), p.369;
%S.~W.~Hawking,
%%``The Cosmological Constant Is Probably Zero,''
%Phys.\ Lett.\ B {\bf 134}, 403 (1984).
%More general functions of $H^2$ with the self-tuning property
%without the brane gravity have
%been considered in J.~E.~Kim and H.~M.~Lee,
%%``More self-tuning solutions with H(MNPQ),''
%JHEP {\bf 0209}, 052 (2002)
%[arXiv:hep-th/0207260].
%

\bibitem{dvali} G.~R.~Dvali, G.~Gabadadze and M.~Porrati,
%``4D gravity on a brane in 5D Minkowski space,''
Phys.\ Lett.\ B {\bf 485}, 208 (2000).   
%[arXiv:hep-th/0005016];
%
%G.~R.~Dvali and G.~Gabadadze,
%``Gravity on a brane in infinite-volume extra space,''
%Phys.\ Rev.\ D {\bf 63}, 065007 (2001); 
%[arXiv:hep-th/0008054];
%
%C.~Deffayet, G.~R.~Dvali, G.~Gabadadze and A.~Lue,
%``Braneworld flattening by a cosmological constant,''
%Phys.\ Rev.\ D {\bf 64}, 104002 (2001). 
%[arXiv:hep-th/0104201].
%See also
%U.~Ellwanger,
%%``Blown-up p-branes and the cosmological constant,''
%JCAP {\bf 0311}, 013 (2003)
%[arXiv:hep-th/0304057].

%
%\bibitem{dvalicc}
%G.~Dvali, G.~Gabadadze and M.~Shifman,
%%``Diluting cosmological constant in infinite volume extra dimensions,''
%Phys.\ Rev.\ D {\bf 67}, 044020 (2003)
%[arXiv:hep-th/0202174];
%N.~Arkani-Hamed, S.~Dimopoulos, G.~Dvali and G.~Gabadadze,
%%``Non-local modification of gravity and the cosmological constant problem,''
%arXiv:hep-th/0209227;
%O.~Corradini, A.~Iglesias and Z.~Kakushadze,
%%``Diluting solutions of the cosmological constant problem,''
%Mod.\ Phys.\ Lett.\ A {\bf 18}, 1343 (2003)
%[arXiv:hep-th/0305164].
%
\bibitem{newDGP} B.~Kyae,
%``Einstein gravity on a brane in 5D non-compact flat spacetime:
%DGP model revisited,''
arXiv:hep-th/0312161.

%
%\bibitem{ks} B.~Kyae and Q.~Shafi,
%%``GUT scale and leptogenesis from 5D inflation,''
%Phys.\ Lett.\ B {\bf 556}, 97 (2003)
%[arXiv:hep-ph/0211059];
%B.~Kyae and Q.~Shafi,
%%``Inflation and leptogenesis with five dimensional SO(10),''
%to appear in Phys.\ Rev.\ D [arXiv:hep-ph/0212331].
%
%
%\bibitem{nilles2} See H.~P.~Nilles, A.~Papazoglou and G.~Tasinato,
%``Selftuning and its footprints,''
%Nucl.\ Phys.\ B {\bf 677}, 405 (2004). 
%[arXiv:hep-th/0309042].
%
\bibitem{hawking} 
E.~Baum,
%``Zero Cosmological Constant From Minimum Action,''
Phys.\ Lett.\ B {\bf 133}, 185 (1983);
S.~W.~Hawking,
%``The Cosmological Constant Is Probably Zero,''
Phys.\ Lett.\ B {\bf 134}, 403 (1984).
%
%
%\bibitem{cho} I.~Y.~Cho,
%%``Brane world in a texture,''
%Phys.\ Rev.\ D {\bf 67}, 065014 (2003)
%[arXiv:hep-th/0210279].
%
%
%
%\bibitem{shifman} See also G.~Gabadadze and M.~Shifman,
%%``Softly Massive Gravity,''
%arXiv:hep-th/0312289;
%C.~Middleton and G.~Siopsis,
%%``The Schwarzschild solution in the DGP model,''
%arXiv:hep-th/0311070;
%M.~A.~Luty, M.~Porrati and R.~Rattazzi,
%%``Strong interactions and stability in the DGP model,''
%JHEP {\bf 0309}, 029 (2003)
%[arXiv:hep-th/0303116].
%
\end{thebibliography}