Vacuum expectation value profiles of the bulk scalar field in the generalized
Randall-Sundrum model
A<EMAIL_ADDRESS>M. Moazzenb, A. Farokhtabara
aDepartment of Physics, Faculty of Basic Sciences,
University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran.
bYoung Researchers and Elite Club, Bojnourd Branch,
Islamic Azad University, Bojnurd, Iran.

###### Abstract

In the generalized Randall-Sundrum warped braneworld model the cosmological
constant induced on the visible brane can be positive or negative. In this
paper we investigate profiles of vacuum expectation value of the bulk scalar
field under general Dirichlet and Neumann boundary conditions in the
generalized warped braneworld model. We show that the VEV profiles generally
depends on the value of the brane cosmological constant. We find that the VEV
profiles of the bulk scalar field for a visible brane with negative
cosmological constant and positive tension is quite distinct from those of
Randall-Sundrum model. In addition we show that the VEV profiles for a visible
brane with large positive cosmological constant are also different from those
of the Randall-Sundrum model.
We also verify that Goldberger and Wise mechanism can work under non-zero
Dirichlet boundary conditions in the generalized Randall-Sundrum model.
PACS: 11.25.Mj, 04.50.-h
Key Words: Higher-dimensional gravity, Bulk scalar field, Warped
compactification model

##  1 Introduction

Extra dimension is an important subject in the realm of theoretical physics
that provides many creative ways to solve some problem in physics such as
hierarchy problem. Unlike the model of Arkani-Hamed, et. al.$[1]$, Randall and
Sundrum (RS) proposed an alternative scenario $[2]$ to solve the hierarchy
problem that does not require large extra dimensions. They assumed an
exponential function of the compactification radius called a warp factor in a
$5-$dimensional anti-de sitter space-time compactified on a $S^{1}/Z_{2}$
orbifold. Two 3-branes are located at the orbifold fixed point $\phi=\pi$
(visible brane) and $\phi=0$ (hidden brane). Due to the non-factorizable
geometry of metric all fundamental scalar masses are subject to an exponential
suppression on the visible brane.
In the original RS model the cosmological constant induced on the visible
brane is zero and the brane tension is negative. This model has been
generalized such that the cosmological constant on the brane as well as brane
tension can be positive or negative $[3]$. In the RS model the size of extra
dimension, $r_{c}$, is not determined by the dynamic of the model. For this
scenario to be relevant, it is necessary to find a mechanism for generating a
potential to stabilize the value of $r_{c}$. This mechanism was proposed by
Goldberger and Wise (GW) $[4]$ so that the dynamic of a five dimensional bulk
scalar field in such model could stabilize the size of extra dimension. In the
GW mechanism the potential for the radion that sets the size of the fifth
dimension is generated by a bulk scalar with quartic interactions localized on
the two $3-$branes. The minimum of this potential yields a compactification
scale that solves the hierarchy problem without fine tuning of parameters. The
back reaction of the bulk scalar field was neglected in the original GW
mechanism but it was studied in $[5]$ later. Some studies about this mechanism
can be found in $[6-8]$.
Recently, GW mechanism of radius stabilization in the brane-world model with
non zero brane cosmological constant has been considered $[9]$. It was shown
that for a generalized RS model, the modulus stabilization condition
explicitly depends on the brane cosmological constant. Furthermore Haba, et.
al. $[10]$ have analyzed profiles of vacuum expectation value (VEV) of the
bulk scalar field under the general boundary conditions (BCs) on the RS warped
compactification. They have investigated GW mechanism in several setups with
the general BCs of the bulk scalar field. Also they showed that $SU(2)_{R}$
triplet Higgs in the bulk left-right symmetric model with custodial symmetry
can be identified the Goldberger-Wise scalar.
The motivation for the present study is to study the vev profile of the scalar
field, including a brane cosmological constant and with different combinations
of Dirichlet and Neumann boundary conditions at the two branes. We also want
to know if this scalar can stabilize the size of the warped extra-dimension?
We note that he present accelerating phase of the universe is due to the
presence of small positive cosmological constant with a tiny value of
$10^{-124}$ in Planck unit. Hence it is desirable to consider the effect of
non-zero cosmological constant on the brane.
This paper is organized as follows. In section $2$, we briefly summarize the
braneworld model with non zero brane cosmological constant. In section $3$ we
study the behavior of the bulk scalar field under four BCs on the general RS
warped model in the case without brane localized potential and in the next
section we analyze profiles of VEV of the bulk scalar field in the case with
brane localized scalar potential. Also we investigate the GW mechanism under
the BCs in the generalized braneworld scenario. Finally in section $5$ we
conclude with the summary of our results.

##  2 The generalized Randall-Sundrum model

One of the many different possibilities that can be explored to solve the
hierarchy problem is Randall-Sundrum (RS) model $[2]$. The RS model proposes
that spacetime is described by a $5D$ Anti-de Sitter (AdS) metric. Some good
reviews on this subject can be found in, e.g., Refs. $[11-13]$. In the
Randall-Sundrum model the visible brane tension is negative and the
cosmological constant on the visible brane assumed to be zero. It has been
shown in $[3]$ that one can indeed generalize the model with nonzero
cosmological constant on the brane and still can have Planck to TeV scale
warping from the resulting warp factor. In this section we want to study the
warped braneworld model with non zero brane cosmological constant briefly.
Other studies for this model have been reported in $[14-16]$.
The action of a bulk scalar field, $\Phi$, on the warp braneworld model that
suggested by Randall and Sundrum is

$S=\int
d^{5}x\sqrt{-G}[-G^{MN}(\partial_{M}\Phi^{{\dagger}})(\partial_{N}\Phi)-V].$
(1)

Where $x^{M}=(x^{\mu},y)$, $\mu=0,1,2,3$. The general form of the warped
metric for a five dimensional space-time is given by

$G_{MN}dx^{M}dx^{N}=e^{-2{A(y)}}g_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$ (2)

where $g_{\mu\nu}$ stands for four dimensional curved brane. The brane tension
$k$ is defined by

$k\equiv\pm\sqrt{\frac{-\Lambda}{6M_{5}^{3}}}=\frac{V_{UV}}{6M_{5}^{3}}=-\frac{V_{IR}}{6M_{5}^{3}},$
(3)

where $M_{5}$ is the Planck mass in five-dimensions. We assume that the scalar
field $\Phi$ is a function of the extra dimension $y$ only that can be defined
as $[10]$

$\Phi=\frac{\Phi_{R}+i\Phi_{I}}{\sqrt{2}},$ (4)

and for simplicity, we take the bulk potential as

$V=m^{2}|\Phi|^{2}=\frac{m^{2}}{2}(\Phi_{R}^{2}+\Phi_{I}^{2}).$ (5)

A scalar mass on the visible brane gets warped through the warp factor
$e^{-A(kr\pi)}=10^{-n}$ where the warp factor index, $n$, is set to $16$ to
achieve the desired warping from Planck to TeV scale. The magnitude of the
induced cosmological constant on the brane in the generalized RS model is non-
vanishing and is given by $10^{-\aleph}$ in planck unit. For the negative
value of the cosmological constant, $\Omega<0$, on the visible brane which
leads to AdS-Schwarzschild space-times the solution of the warp factor can be
written as $[9]$

$e^{-A(y)}=\omega\cosh(\ln\frac{\omega}{c_{1}}+ky),$ (6)

where $\omega^{2}\equiv\frac{-\Omega}{3k^{2}}\geq 0$ and
$c_{1}=1+\sqrt{1-\omega^{2}}$. It was shown that real solution for the warp
factor exists if and only if $\omega^{2}\leq 10^{-2n}$. This leads to an upper
bound for the magnitude of the cosmological constant as $\aleph=2n$. When
$\aleph=\aleph_{min}=2n$, the visible brane tension is zero. For AdS brane,
there are degenerate solutions of $kr\pi$ whose values will depend on
$\omega^{2}$ and $n$. The brane tension is negative for some of these
solutions and is positive for others. Next, for AdS brane we want to
investigate the solution of the equation of motion which is obtained from the
action given by $eq.(1)$. As it has been discussed in $[3]$, to resolve the
gauge hierarchy problem without introducing any intermediate scale, the brane
cosmological constant should be tuned to a very small value therefore in this
case the warp factor in $eq.(6)$ can be given by $[9]$

$e^{-4A(y)}=e^{-4ky}+\omega^{2}e^{-2ky}.$ (7)

We use background field method to study behaviors of the bulk scalar field of
the generalized Randall-Sundrum model. In this method one separates the field
into classical and quantum fluctuation parts. The configuration of the
classical field obeys an equation of motion $[10]$.
With the above warp factor equation of motion for the classical field can be
written down

$\partial_{y}^{2}\Phi_{X}-(4k-2k\omega^{2}e^{2ky})\partial_{y}\Phi_{X}-m^{2}\Phi_{X}=0,$
(8)

where $X$ stands for $R$ and $I$. We solve the above equation and obtain the
solution for the scalar field as

$\Phi_{X}(z)=Az^{\nu+2}+Bz^{-(\nu-2)}+\alpha Az^{\nu+4}+\beta Bz^{-(\nu-4)},$
(9)

where

$\alpha=\frac{-\omega^{2}}{2}(\frac{\nu+2}{\nu+1}),\quad\quad\quad\beta=\frac{-\omega^{2}}{2}(\frac{2-\nu}{1-\nu}),$
(10)

and $\nu=\sqrt{4+\frac{m^{2}}{k^{2}}}$ and $z=e^{ky}$. In the above solution
$A$ and $B$ are arbitrary constant which evaluated by using the appropriate
boundary conditions at the locations of the brane.
For the positive induced brane cosmological constant $\Omega>0$ which
corresponds to dS-Schwarzschild space-time the warp factor turns out to be

$e^{-A(y)}=\omega\sinh(\ln\frac{c_{2}}{\omega}-ky),$ (11)

where $\omega^{2}\equiv\frac{\Omega}{3k^{2}}\geq 0$ and
$c_{2}=1+\sqrt{1+\omega^{2}}$. In this case there is no bound on the value of
$\omega^{2}$, and the positive cosmological constant can be of arbitrary
magnitude. For dS brane, the brane tension is negative for the entire range of
values of the positive cosmological constant. Also for a small $\omega$ the
warp factor can be written down

$e^{-4A(y)}=e^{-4ky}-\omega^{2}e^{-2ky}.$ (12)

By using the above warp factor the solution of the equation of motion for the
dS brane is

$\Phi_{X}(z)=Az^{\nu+2}+Bz^{-(\nu-2)}-\alpha Az^{\nu+4}-\beta Bz^{-(\nu-4)}.$
(13)

It is obvious that by taking $\alpha\rightarrow-\alpha$ and
$\beta\rightarrow-\beta$ we can obtain the above solution from AdS one. Notice
that from now on we will use $\omega^{2}$ to represent induced brane
cosmological constant. With this generalized RS warped model we now
investigate the profile of the bulk scalar field under boundary conditions in
the next section.

##  3  VEV profiles in the absence of brane localized potential

In this section, we study the VEV profiles of the bulk scalar field in a case
without brane localized scalar potential in the generalized Randall-Sundrum
model. By utilizing the $eq.(2)$, the action can be rewritten as

$S=\int
d^{4}x\int_{0}^{L}dye^{-4A(y)}[-e^{2A(y)}|\partial_{\mu}\Phi|^{2}-|\partial_{y}\Phi|^{2}-V].$
(14)

The above action was defined on a line segment as $0\leq y\leq L$. If we write
the bulk scalar as $eq.(4)$ then the variation of the action is given by

$\delta S=\int
d^{4}x\int_{0}^{L}dye^{-4A(y)}[\delta\Phi_{X}(\rho\Phi_{X}-\frac{\partial
V}{\partial\Phi_{X}})+\delta(y)\delta\Phi_{X}(+\partial_{y}\Phi_{X})+\delta(y-L)\delta\Phi_{X}(-\partial_{y}\Phi_{X})].$
(15)

Where $\rho=e^{2A(y)}\Box+e^{4A(y)}\partial_{y}e^{-4A(y)}\partial_{y}$. The
VEV profile is obtained by the action principal $\delta S=0$, that is

$(\rho\Phi_{X}-\frac{\partial V}{\partial\Phi_{X}})=0.$ (16)

Notice that the above equation of motion has been solved in the pervious
section for dS and AdS brane in $eq.(9)$ and $eq.(13)$ respectively. The
boundary conditions at $y=0$ and $L$ reads either Dirichlet

$\delta\Phi_{X}|_{y=0,L}=0,$ (17)

or Neumann

$\pm\partial_{y}\Phi_{X}|_{y=0,L}=0.$ (18)

Where plus and minus signs are for $y=0$ and $y=L$ respectively. As mentioned
in $[10]$ we can have four choices of combination of Dirichlet and Neumann
boundary conditions showed by $(D,D)$, $(D,N)$, $(N,D)$ and $(N,N)$ at $y=0$
and $L$. Here by using the solution of the equation of motion that given by
$eq.(9)$ and $eq.(13)$ we want to verify the profile of the bulk scalar field
under the four boundary conditions in the generalized Randall- Sundrum model
with non-zero brane cosmological constant.

###  3.1 $(D,D)$ case

We discuss a case in which both boundary conditions on the $y=0$ and $y=L$
branes are the Dirichlet type boundary conditions. The most general form of
the Dirichlet BC is $\delta\Phi|_{z=\xi}=0$ and

$\upsilon(1)=\upsilon_{1},\quad\quad\quad\upsilon(z_{L})=\upsilon_{2},$ (19)

where $\xi$ is taken as $1$ and $z_{L}=e^{kL}$. For AdS brane these boundary
conditions can be rewritten by $eq.(9)$ as

$A(1+\alpha)+B(1+\beta)=\upsilon_{1}\quad\quad\quad Az_{L}^{\nu+2}(1+\alpha
z_{L}^{2})+Bz_{L}^{(-\nu+2)}(1+\beta z_{L}^{2})=\upsilon_{2}.$ (20)

The above equations lead to

$A=-\frac{\upsilon_{1}z_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2})-\upsilon_{2}(1+\beta)}{z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})(1+\beta)-z_{L}^{(-\nu+2)}(1+\beta z_{L}^{2})(1+\alpha)},$ (21)

and

$B=\frac{\upsilon_{1}z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})-\upsilon_{2}(1+\alpha)}{z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})(1+\beta)-z_{L}^{(-\nu+2)}(1+\beta z_{L}^{2})(1+\alpha)}.$ (22)

Therefore, under $(D,D)$ boundary conditions, in the generalized RS model with
negative brane cosmological constant the VEV profile is

$\upsilon(z)=-\frac{\upsilon_{1}z_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2})-\upsilon_{2}(1+\beta)}{z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})(1+\beta)-z_{L}^{(-\nu+2)}(1+\beta z_{L}^{2})(1+\alpha)}z^{\nu+2}$
$+\frac{\upsilon_{1}z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})-\upsilon_{2}(1+\alpha)}{z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})(1+\beta)-z_{L}^{(-\nu+2)}(1+\beta z_{L}^{2})(1+\alpha)}z^{-\nu+2}$
$-\alpha(\frac{\upsilon_{1}z_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2})-\upsilon_{2}(1+\beta)}{z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})(1+\beta)-z_{L}^{(-\nu+2)}(1+\beta z_{L}^{2})(1+\alpha)})z^{\nu+4}$
$+\beta(\frac{\upsilon_{1}z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})-\upsilon_{2}(1+\alpha)}{z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})(1+\beta)-z_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2})(1+\alpha)})z^{-\nu+4},$ (23)

notice that for $\omega=0$ (i.e., the RS case) which corresponds to $\alpha=0$
and $\beta=0$ above equations lead to the results that proposed by Haba $[10]$
for $(D,D)$ BCs.

Figure 1: The VEV profile under $(D,D)$ BCs as a function of extra dimension
for $\aleph=34$ in the absence of brane potential with $\Omega<0$ and a
positive tension visible brane. We take
$\upsilon\equiv\upsilon_{0,1,2,L}\equiv M^{\frac{3}{2}}_{pl}$, $k\equiv
M_{pl}$, $\nu=2.1$, $M_{pl}=2.4\times 10^{18}GeV$ .

In figure $1$ the VEV profile for $\aleph=34$ is shown for the $AdS$ brane. In
this figure we have assumed the brane tension of visible brane to be positive
(see $eq.(66)$ for detail). we find that the pattern of localization is
different from that of the $RS$ case. It is seen that the position of the IR
brane shifts to the larger value of $z_{L}$ so the drastic change of the VEV
profile occurs later than RS case. For this figure we have taken
$\upsilon\equiv\upsilon_{0,1,2,L}\equiv M^{\frac{3}{2}}_{pl}$, $k\equiv
M_{pl}$, $\nu=2.1$, $M_{pl}=2.4\times 10^{18}GeV$ .
For dS brane ,$\alpha\rightarrow-\alpha$ and $\beta\rightarrow-\beta$
therefore $A$ and $B$ can be written as

$A=-\frac{\upsilon_{1}z_{L}^{(-\nu+2)}(1-\beta
z_{L}^{2})-\upsilon_{2}(1-\beta)}{z_{L}^{(\nu+2)}(1-\alpha
z_{L}^{2})(1-\beta)-z_{L}^{(-\nu+2)}(1-\beta z_{L}^{2})(1-\alpha)},$ (24)

and

$B=\frac{\upsilon_{1}z_{L}^{(\nu+2)}(1-\alpha
z_{L}^{2})-\upsilon_{2}(1-\alpha)}{z_{L}^{(\nu+2)}(1-\alpha
z_{L}^{2})(1-\beta)-z_{L}^{(-\nu+2)}(1-\beta z_{L}^{2})(1-\alpha)}.$ (25)

By utilizing the above form of $A$ and $B$ in $eq.(13)$ the VEV profile of
bulk scalar field is obtained for dS brane. We investigate this VEV profile
for large and small values of $\omega$. We find that for the small values of
positive brane cosmological constant the VEV profile has tiny difference with
respect to the RS case.

Figure 2: The dependence of VEV profile under $(D,D)$ BCs for large positive
brane cosmological constant with $\aleph=20$ (dotted line), $\aleph=28$
(dashed line) and $\aleph=\infty$ (solid line i.e., RS case) . We take the
same values for other parameter as ones in Figure $1$ .

In figure.$2$ we show the results for the large positive brane cosmological
constant, namely $\aleph=20$ (dotted line) and $\aleph=28$ (dashed line). For
the sake of comparison we also include $\aleph=\infty$ (solid line i.e., RS
case). It is observed that the profile localizes near the IR brane. But the
location of the IR brane is different in comparison to the RS case (see
$eq.(69)$ of the appendix.
In $(D,N)$ case, the boundary conditions can be given by

$\upsilon(1)=\upsilon_{1},\quad\quad\quad\partial_{z}\upsilon(z)|_{z=z_{L}}=0.$
(26)

Then, for AdS brane they are written down by

$A(1+\alpha)+B(1+\beta)=\upsilon_{1},$ (27)

and

$Az_{L}^{\nu+2}(v+2+\alpha(\nu+4)z_{L}^{2})-Bz_{L}^{-(\nu-2)}(\nu-2+\beta(\nu-4)z_{L}^{2})=0.$
(28)

These lead to

$A=\frac{\upsilon_{1}z_{L}^{(-\nu+2)}[(\nu-2)+\beta
z_{L}^{2}(\nu-4)]}{z_{L}^{(-\nu+2)}(1+\alpha)[(\nu-2)+\beta
z_{L}^{2}(\nu-4)]+z_{L}^{(\nu+2)}(1+\beta)[(\nu+2)+\alpha z_{L}^{2}(\nu+4)]},$
(29)

and

$B=\frac{\upsilon_{1}z_{L}^{(\nu+2)}[(\nu+2)+\alpha
z_{L}^{2}(\nu+4)]}{z_{L}^{(-\nu+2)}(1+\alpha)[(\nu-2)+\beta
z_{L}^{2}(\nu-4)]+z_{L}^{(\nu+2)}(1+\beta)[(\nu+2)+\alpha z_{L}^{2}(\nu+4)]}.$
(30)

By substituting the above form of $A$ and $B$ in $eq.(9)$ the VEV profile in
$(D,N)$ case of boundary conditions is obtained.

###  3.2 $(N,D)$ case

Next, we consider the $(N,D)$ case. These boundary conditions can be described
by

$\partial_{z}\upsilon(z)|_{z=1}=0,\quad\quad\quad\upsilon(z_{L})=\upsilon_{2}.$
(31)

For negative brane cosmological constant, they are

$A(\nu+2+\alpha(\nu+4))-B(\nu-2+\beta(\nu-4))=0,$ (32)

and

$Az_{L}^{\nu+2}(1+\alpha z_{L}^{2})+Bz_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2})=\upsilon_{2}.$ (33)

These lead to

$A=\frac{\upsilon_{2}[(\nu-2)+\beta(\nu-4)]}{z_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2})[(\nu+2)+\alpha(\nu+4)]+z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})[(\nu-2)+\beta(\nu-4)]},$ (34)

and

$B=\frac{\upsilon_{2}[(\nu+2)+\alpha(\nu+4)]}{z_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2})[(\nu+2)+\alpha(\nu+4)]+z_{L}^{(\nu+2)}(1+\alpha
z_{L}^{2})[(\nu-2)+\beta(\nu-4)]}.$ (35)

Like pervious subsection we substitute the above form of $A$ and $B$ in
$eq.(9)$ to obtain the VEV profile in $(N,D)$ case of boundary conditions.

###  3.3 $(N,N)$ case

Finally, we investigate the $(N,N)$ case. These boundary conditions are

$\partial_{z}\upsilon(z)|_{z=1}=0,\quad\quad\quad\partial_{z}\upsilon(z)|_{z=z_{L}}=0.$
(36)

Then, for AdS brane they are written down by

$A(\nu+2+\alpha(\nu+4))-B(\nu-2+\beta(\nu-4))=0,$ (37)

and

$Az_{L}^{\nu+2}(\nu+2+\alpha(\nu+4)z_{L}^{2})-Bz_{L}^{-(\nu-2)}(\nu-2+\beta(\nu-4)z_{L}^{2})=0.$
(38)

We find that like RS model there is no solution to satisfy the above boundary
conditions except for a trivial one, $(A,B)=(0,0)$ which is not of physical
interest.
In this section we studied the VEV profile of a bulk scalar field under four
boundary conditions on a generalized warped brane-world model in the absence
of brane localized potential. We found that the VEV profiles depend on the
vanishingly small brane cosmological constant.

##  4  VEV profiles in the presence of brane localized potentials

In this section, we formulate the VEV profiles of the bulk scalar field in a
case with brane localized scalar potentials in the generalized Randall-Sundrum
model. The action of a bulk field in the presence of brane potentials reads as

$S=\int
d^{4}x\int_{0}^{L}dye^{-4A(y)}[-e^{2A(y)}|\partial_{\mu}\Phi|^{2}-|\partial_{y}\Phi|^{2}-V-\delta(y)V_{0}-\delta(y-L)V_{L}].$
(39)

The variation of the action is

$\delta S=\int
d^{4}x\int_{0}^{L}dye^{-4A(y)}[\delta\Phi_{X}(\rho\Phi_{X}-\frac{\partial
V}{\partial\Phi_{X}})$
$+\delta(y)\delta\Phi_{X}(+\partial_{y}\Phi_{X}-\frac{\partial
V_{0}}{\partial\Phi_{X}})+\delta(y-L)\delta\Phi_{X}(-\partial_{y}\Phi_{X}-\frac{\partial
V_{L}}{\partial\Phi_{X}})].$ (40)

From above equation the Dirichlet boundary condition is the same as $eq.(17)$
while the Neumann boundary condition should be modified as

$\pm\partial_{y}\Phi_{X}-\frac{\partial
V_{\eta}}{\partial\Phi_{X}}|_{y=\eta=0,L}=0.$ (41)

In this paper we seek to generalize the Randall-Sundrum model and moreover we
want to investigate the GW mechanism. Therefore we use the brane potentials as
$[2,4,10]$

$V_{\eta}=\lambda_{\eta}(|\Phi|^{2}-\frac{v_{\eta}^{2}}{2})^{2}=\frac{\lambda_{\eta}}{4}(\Phi_{R}^{2}+\Phi_{I}^{2}-v_{\eta}^{2})^{2},$
(42)

where $\lambda_{0}$ and $\lambda_{L}$ stand for the brane tension and
$\upsilon_{0}$ and $\upsilon_{L}$ are the VEV of the bulk scalar field on the
UV and IR brane respectively. By utilizing the above scalar potentials we
study the VEV profile in the case with brane localized potential in the
generalized RS model. We notice that like the RS model, the $(D,D)$ boundary
conditions in this case is the same as one for $(D,D)$ case with brane
potentials.
For some other choices of the brane potentials see $[8]$.

###  4.1 $(D,N)$ case

Here, we consider the $(D,N)$ case that can be given by

$\upsilon(1)=\upsilon_{1}\quad\quad\quad\partial_{z}\upsilon(z)|_{z=z_{L}}+\frac{\partial
V_{L}}{\partial\Phi}|_{z=z_{L}}=0.$ (43)

For AdS brane they are written down by

$A(1+\alpha)+B(1+\beta)=\upsilon_{1},$ (44)

and

$k[Az_{L}^{\nu+2}(\nu+2+\alpha(\nu+4)z_{L}^{2})-Bz_{L}^{-(\nu-2)}(\nu-2+\beta(\nu-4)z_{L}^{2})]+$
$\lambda_{L}(Az_{L}^{\nu+2}(1+\alpha z_{L}^{2})+Bz_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2}))[(Az_{L}^{\nu+2}(1+\alpha z_{L}^{2})+Bz_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2}))^{2}-\upsilon_{L}^{2}]=0.$ (45)

We should remember that when the coupling in the boundary potential is
infinite, the Neumann boundary conditions turn to Dirichlet one if we choose
$\upsilon_{L}=\upsilon_{2}$.
When the boundary quartic coupling is finite, the numerical calculation
indicates that $A\ll B\simeq z_{L}^{\nu-2}\upsilon_{L}(1-\beta z_{L}^{2})$ as
a solution of $eq.(44)$ and $eq.(45)$. For the small boundary quartic
coupling, $A$ and $B$ can be approximated by

$A\simeq\chi B[\frac{\nu-2+\beta z_{L}^{2}(\nu-4)}{\nu+2+\alpha
z_{L}^{2}(\nu+4)}]z_{L}^{-2\nu}$ (46)

and

$B\simeq\sqrt{\upsilon_{L}^{2}+\frac{k(1-\chi)[\nu-2+\beta
z_{L}^{2}(\nu-4)]}{\lambda_{L}(1+\beta
z_{L}^{2})}}\frac{z_{L}^{\nu-2}}{1+\beta z_{L}^{2}},$ (47)

where $\chi$ is determined by $eq.(44)$ and numerical calculation shows that
$\chi\simeq O(1)$ and $\chi=1$ for $\lambda_{L}=0$. Also, $\chi$ is limited by

$\chi\leq 1+\frac{\lambda_{L}(1+\beta
z_{L}^{2})\upsilon_{L}^{2}}{k[\nu-2+\beta z_{L}^{2}(\nu-4)]}.$ (48)

Figure 3:  The VEV profile for negative induced brane cosmological constant
with positive brane tension for $\aleph=33$ (dotted line), $\aleph=34$ (dashed
line) and $\aleph=\infty$ (solid line i.e., RS case) in the $(D,N)$ case for
$\widetilde{\lambda}_{L}=100$. The values of other parameters of the model are
the same as the ones in Figure $1$ .

In Fig. $3$ we show the dependence of VEV profile for negative induced brane
cosmological constant with positive brane tension with
$\widetilde{\lambda}_{L}=100$.

###  4.2 $(N,D)$ case

For the $(N,D)$ case, these boundary conditions given by

$\partial_{z}\upsilon(z)|_{z=1}-\frac{\partial
V_{0}}{\partial\Phi}|_{z=1}=0,\quad\quad\quad\partial_{z}\upsilon(z)|_{z=z_{L}}=0.$
(49)

Then, for negative brane cosmological constant they are written down by

$k[A(\nu+2+\alpha(\nu+4))-B(\nu-2+\beta(\nu-4))]$
$-\lambda_{0}(A(1+\alpha)+B(1+\beta))[(A(1+\alpha)+B(1+\beta))^{2}-\upsilon_{0}^{2}]=0,$
(50)

and

$Az_{L}^{\nu+2}(1+\alpha z_{L}^{2})+Bz_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2})=\upsilon_{2}.$ (51)

Numerical calculation indicates that $A\sim
B\sim\frac{\upsilon_{2}}{z_{L}^{2}(1+\alpha z_{L}^{2})}$ and for small
$\lambda_{0}$, $\widetilde{\lambda}_{0}\leq O(0.01)$, $A$ and $B$ are
approximated by

$A\simeq(1-\frac{\Gamma_{1}}{\Gamma_{1}+\Gamma_{2}z_{L}^{2\nu}})\frac{\upsilon_{2}z_{L}^{-(\nu+2)}}{1+\alpha
z_{L}^{2}},$ (52)

and

$B\simeq(\frac{\Gamma_{1}}{\Gamma_{1}+\Gamma_{2}z_{L}^{2\nu}})\frac{\upsilon_{2}z_{L}^{\nu-2}}{1+\beta
z_{L}^{2}}.$ (53)

Where

$\Gamma_{1}=(1+\beta
z_{L}^{2})[k(\nu+2+\alpha(\nu+4))+\lambda_{0}\upsilon_{0}^{2}(1+\alpha)],$
(54)

and

$\Gamma_{2}=(1+\alpha
z_{L}^{2})[k(\nu-2+\beta(\nu-4))-\lambda_{0}\upsilon_{0}^{2}(1+\beta)].$ (55)

Figure 4:  The VEV profile for large positive brane cosmological constant with
$\aleph=20$ (dotted line), $\aleph=28$ (dashed line) and $\aleph=\infty$
(solid line i.e., RS case)in the $(N,D)$ case for
$\widetilde{\lambda}_{0}=100$. The values of other parameters of the model are
the same as the ones in Figure $1$ .

Figure $4$ shows the VEV profile for large positive brane cosmological
constant, when $\widetilde{\lambda}_{0}=100$.

###  4.3 $(N,N)$ case

The $(N,N)$ case of boundary conditions for AdS brane are

$\partial_{z}\upsilon(z)|_{z=1}-\frac{\partial
V_{0}}{\partial\Phi}|_{z=1}=0,\quad\quad\quad\partial_{z}\upsilon(z)|_{z=L}+\frac{\partial
V_{L}}{\partial\Phi}|_{z=L}=0.$ (56)

Then, the boundary conditions can be written down as

$k[A(\nu+2+\alpha(\nu+4))-B(\nu-2+\beta(\nu-4)]$
$-\lambda_{0}(A(1+\alpha)+B(1+\beta))[(A(1+\alpha)+B(1+\beta))^{2}-\upsilon_{0}^{2}]=0,$
(57)

and

$k[Az_{L}^{\nu+2}(\nu+2+\alpha(\nu+4)z_{L}^{2})-Bz_{L}^{-(\nu-2)}(\nu-2+\beta(\nu-4)z_{L}^{2})]+$
$\lambda_{L}(Az_{L}^{\nu+2}(1+\alpha z_{L}^{2})+Bz_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2}))[(Az_{L}^{\nu+2}(1+\alpha z_{L}^{2})+Bz_{L}^{(-\nu+2)}(1+\beta
z_{L}^{2}))^{2}-\upsilon_{L}^{2}]=0.$ (58)

Figure 5: The VEV profile for negative induced brane cosmological constant
with positive brane tension for $\aleph=33$ (dotted line), $\aleph=34$ (dashed
line) and $\aleph=\infty$ (solid line i.e., RS case) in the $(N,N)$ case for
$\widetilde{\lambda}_{0}=100,\widetilde{\lambda}_{L}=100$. The values of other
parameters of the model are the same as the ones in Figure $1$ .

In the limit of very small boundary couplings, $\widetilde{\lambda}_{0}\ll
O(0.01)$ and $\widetilde{\lambda}_{L}\ll O(0.1)$, $A$ and $B$ are zero but for
large boundary quartic couplings which is considered in GW mechanism we have

$A_{GW}\simeq-\upsilon_{0}\frac{(1+\beta
z_{L}^{2})}{\Delta_{1}}z_{L}^{-2\nu}+\upsilon_{L}\frac{(1+\beta)}{\Delta_{1}}z_{L}^{-(\nu+2)},$
(59)

and

$B_{GW}\simeq\upsilon_{0}\frac{(1+\alpha
z_{L}^{2})}{\Delta_{1}}(1+\frac{\Delta_{2}}{\Delta_{1}}z_{L}^{-2\nu})-\upsilon_{L}\frac{(1+\alpha)}{\Delta_{1}}z_{L}^{-(\nu+2)}.$
(60)

Where

$\Delta_{1}=(1+\beta)(1+\alpha z_{L}^{2}),$ (61)

and

$\Delta_{2}=(1+\alpha)(1+\beta z_{L}^{2}).$ (62)

Here we numerically analyze the VEV profile for finite boundary couplings. The
result is shown in Figure $5$ for large values of $\widetilde{\lambda}_{0}$
and $\widetilde{\lambda}_{L}$. At the end of this section we reconsider the
realizations of the GW mechanism under general boundary conditions in the
generalized RS model. In the GW mechanism the effective potential for the
modulus can be generated by the total action integral of a bulk scalar field
with quartic interactions localized in two $3-$branes. The bulk scalar field
acts an important role to stabilize the radion which yields a compactification
scale in terms of VEVs profile of the scalar field at two branes. The VEV of
the bulk scalar can be obtained by solving the equation of motion.
Recently, this mechanism was studied in the generalized warped brane models
with a nonzero brane cosmological constant by $[9]$. They have been obtained
the modulus stabilization condition both for positive and negative values of
the brane cosmological constant which increases (decreases) from RS value with
increasing $\omega^{2}$.
we have investigated the unknown coefficients A and B of the VEV profile of
the bulk scalar field in the generalized RS model by imposing boundary
conditions which are given in the sections $3$ and $4$. It is well known that
the $(N,N)$ type boundary conditions given by $eq.(59)$ and $(60)$ should be
taken in the GW mechanism. The situation in the generalized warped model is
similar to RS warped model as discussed in Ref. $[10]$. When the boundary
quartic couplings $\lambda_{0}$ and $\lambda_{L}$ are large the $(N,N)$ BCs
which imposed in the GW mechanism are equivalent to $(D,D)$ ones given in
$(20)$ if we take $\upsilon_{1}=\upsilon_{0}$ and $\upsilon_{2}=\upsilon_{L}$,
so by this assumption in the limit of large boundary couplings, A and B given
by $eq.(21)$ and $eq.(22)$ are $A\simeq A_{GW}$ and $B\simeq B_{GW}$,
therefore the GW mechanism can be realized in the $(D,D)$ case with
appropriate VEV profile.
In the generalized warped model as mentioned in Ref. $[10]$, when the brane
localized potential is considered, GW mechanism can also work in $(N,D)$ and
$(D,N)$ BCs with large boundary quartic couplings. The GW mechanism can not
work when one of the boundary coupling $\lambda_{0}$ or $\lambda_{L}$ becomes
small in $(N,N)$ case.

##  5 Conclusions

In the RS model a negative tension visible brane is utilized to describe our
Universe. However, Such negative tension branes are unstable. Hence in this
work we considered a generalized RS warped model.
In this paper we investigated a bulk scalar field under Dirichlet and Neumann
boundary conditions in a generalized RS warped model with a non zero
cosmological constant on the brane. First we have obtained the profiles of VEV
of the bulk scalar in the absence of brane localized potential for the
negative and positive brane cosmological constant. We find that:
$(1)$ For an AdS visible brane with negative brane tension the results are
similar to RS case.
$(2)$ However for an AdS visible brane with positive brane tension the results
are different from the RS case. In this case the VEV profile undergo sudden
changes near the IR brane. Moreover we see that the IR brane is displaced from
it’s RS location. The coefficients $A$ and $B$ are functions of the length of
the stabilized modulus. Hence the change of the length of the stabilized
modulus modifies the shape of the VEV in comparison to the RS case.
$(3)$ For a dS visible brane the results for small values of $\omega^{2}$ are
not different from the RS case. But for larger values of $\omega^{2}$ we have
departures from the RS result.
Next we have considered the VEV profiles under general boundary conditions in
the presence of brane localized potential for non zero brane cosmological
constant. We find that to see drastic changes of the profile near the IR brane
for an AdS visible brane with positive brane tension the value of
$\widetilde{\lambda}_{L}\sim O(1)$ in the RS case but the value of
$\widetilde{\lambda}_{L}\sim O(10)$ in the present case.
We have also verified GW mechanism in the generalized RS model with Dirichlet
and Neumann Boundary conditions. Analogous to RS model $[10]$ the GW mechanism
can work under non zero Dirichlet BCs with appropriate VEVs in the generalized
RS model. It will be interesting to use these results to study the Dirichlet
Higgs as radion stabilizer in the generalized warped campactification.
Our analysis of consistent boundary conditions is similar to that of Csaki,
et. al. $[17]$. It is possible to extend this work by studying the behavior of
the bulk scalar field on the warped five dimension by utilizing the background
field method, separating the field into classical and quantum fluctuation
parts and to investigate the phenomenological consequences of the profile of
the scalar field $[10]$.
There are quantum corrections, from Casimir effect and induced gravity on the
brane. One requires that the brane world metric be a solution of the quantum-
corrected Einstein equations. For the RS model this issue has been addressed
in Refs. $[18,19]$. From $eq.(12)$ we see that in the generalized RS model
$A(y)=k|y|$ plus extra small term due to the curved branes. In the RS model
the sum of brane tensions of two branes are zero. However in the generalized
RS model this sum is not zero $[3]$. Hence the discussion of the issue of
self-consistency is different from that of the RS model. However it is
reasonable to expect that GRS be self consistent for some specific values of
the parameters of the model .
The analysis presented in this work we have ignored the backreaction of the
scalar field on the spacetime geometry. The stress tensor for the scalar field
in the limit of large modulus can be obtained $[4]$ . For the $(N,N)$ case
with large boundary coupling,stress tensor for the scalar field can we find
that if $v^{2}_{0}\Delta^{2}_{1}/M_{5}^{3}<<1$,
$v^{2}_{L}\Delta^{2}_{1}/M_{5}^{3}<<1$ and $m^{2}/k^{2}<<1$, then one can
neglect the stress tensor for the scalar field in comparison to the stress
tensor induced by the bulk cosmological constant. Furtheremore the parameters
of the bulk and brane potential are constrained by these inequalities.
In this work our choices for the bulk and the brane potentials were those of
Ref. $[2]$. Other choices for the bulk potential and brane potentials are
possible. The potential for the $GW$ scalar $\Phi$ in the bulk $[20]$ has the
general form of $V(\Phi)=m^{2}\Phi^{2}+C_{3}\Phi^{3}+C_{4}\Phi^{4}+......$ .
It would be interesting to study the VEV profiles of the bulk scalar field in
such generalized warped models.
In a previous work we studied the case of a scalar field non-minimally coupled
to the Ricci curvature scalar $[21]$. The case of massless, conformally
coupled scalar field are self-conistent after the inclusion of quantum
corrections. Hence it will be of interest to consider massless, conformally
coupled scalar field in the generalized RS framework.

Appendix : The variation of $kr\pi$ and the brane tensions

In the $RS$ $[2]$ case the brane tension of the visible brane is negative and
to solve the hierarchy problem the value of $kr\pi=36.84$.
The situation is different in the generalized $Randall-Sundrum$ model.
This subject is discussed in Ref. $[3]$. Here we provide a concise
presentation.

The AdS case

If we denote $\varsigma=kr\pi$ then for the case where the cosmological
constant of the brane is negative we have $[3]$

$e^{-\varsigma}=\frac{10^{-n}}{c_{1}}[1\pm\sqrt{1+\omega^{2}10^{2n}}],$ (63)

with $c_{1}=1+\sqrt{1-\omega^{2}}$. Let $\varsigma_{1}$ and $\varsigma_{2}$
corresponds to the plus and minus sign respectively. The $\varsigma_{1}$ case
is not much different from the $RS$ case. But the $\varsigma_{2}$ is quite
distinct from the $RS$ case. In the limit $\aleph-2n\gg 1$

$\varsigma_{1}\simeq
nln10+\frac{1}{4}10^{-(\aleph-2n)}\qquad\varsigma_{2}\simeq(\aleph-n)ln10+ln4,$
(64)

the value of $\varsigma_{2}$ can be as large as $250$.
Next we will show below in this situation the brane tension of the visible
brane is positive. As far as the stability of the brane is concern this is a
desirable feature. To compute the brane tension $U_{vis}$ of the AdS case we
rewrite the brane tension of the visible brane of Ref. $[3]$ as

$U_{vis}=12M^{3}k[\frac{1-(c_{1}e^{-\xi}\omega^{-1})^{2}}{1+(c_{1}e^{-\xi}\omega^{-1})^{2}}].$
(65)

Now inserting $eq.(63)$ in $eq.(65)$ and with $\omega^{2}=10^{-\aleph}$, the
correct form of brane tension of the visible brane is

$U_{vis}=(12M^{3}k)\frac{1-10^{\aleph-2n}(1\pm\sqrt{1-10^{-(\aleph-2n)}})^{2}}{1+10^{\aleph-2n}(1\pm\sqrt{1-10^{-(\aleph-2n)}})^{2}}.$
(66)

And finally from this result we can compute the brane tension in the limit
where $\aleph-2n\gg 1$. For the plus sign case the brane tension for the
visible brane is negative and it is given

$U_{vis-1}\simeq-12M^{3}k.$ (67)

Which is in agreement with the result of Ref. $[3]$.
But for the minus sign case the brane tension for the visible brane is
positive and it is given

$U_{vis-2}\simeq 12M^{3}k.$ (68)

Which differs from the result of Ref. $[3]$ by a factor $3$.

The dS case

When the cosmological constant of the brane is positive we have

$e^{-\varsigma}=\frac{10^{-n}}{c_{2}}[1+\sqrt{1+\omega^{2}10^{2n}}],$ (69)

with $c_{2}=1+\sqrt{1+\omega^{2}}$. It turns out that the brane tension of the
visible brane is negative for the entire range of the positive values of
$\Omega$. However the difference with the $RS$ case is due to the value of
$\varsigma$. For instance for $\aleph=20$ we obtain $\varsigma=23.7$.

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