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\begin{document}
\begin{titlepage}
\begin{flushright}
BA-TH/01-415\\ CERN-TH/2001-149 \\ hep-th/0106019
\end{flushright}
%\vspace*{3mm}
\begin{center}
\huge {\bf Localization of Scalar Fluctuations \\ in a Dilatonic
Brane-World Scenario} \vspace*{1cm}
\large{ V. Bozza\footnote{valboz@sa.infn.it}${}^{(a,b)}$, M.
Gasperini\footnote{gasperini@ba.infn.it}${}^{(c,d)}$ and G.
Veneziano\footnote{venezia@nxth04.cern.ch}${}^{(e,f)}$}
\bigskip
\normalsize ${}^{(a)}$ {\sl Dipartimento di Fisica ``E. R.
Caianiello", Universit\`a di Salerno, \\ Via S. Allende, 84081
Baronissi (SA), Italy\\ \vspace{0.2cm} ${}^{(b)}$ INFN, Sezione di
Napoli, Gruppo Collegato di Salerno, Salerno, Italy} \\
\vspace{0.2cm} ${}^{(c)}$ {\sl Dipartimento di Fisica,
Universit\`a di Bari, \\ Via G. Amendola 173, 70126 Bari, Italy\\
\vspace{0.2cm} ${}^{(d)}$ INFN, Sezione di Bari, Bari, Italy} \\
\vspace{0.2cm} ${}^{(e)}$ {\sl Theoretical Physics Division, CERN,
CH-1211 Geneva 23, Switzerland \\ \vspace{0.2cm} ${}^{(f)}$
Laboratoire de Physique Th\'eorique, Universit\'e Paris Sud, 91405
Orsay, France}
%\vspace{0.3cm}
\vspace*{5mm}
\begin{abstract}
We derive and solve the full set of scalar perturbation equations
for a class of $Z_2$-symmetric five-dimensional geometries
generated by a bulk cosmological constant and by a 3-brane
non-minimally coupled to a bulk dilaton field. The massless scalar
modes, like their tensor analogues, are localized on the brane,
and provide long-range four-dimensional dilatonic interactions,
which are generically present even when matter on the brane
carries no dilatonic charge. The shorter-range corrections induced
by the continuum of massive scalar modes are always present: they
persist even in the case of a trivial dilaton background (the
standard Randall--Sundrum configuration) and vanishing dilatonic
charges.
\end{abstract}
\end{center}
%\vspace{5mm}
\begin{center}
To appear in {\bf Nucl. Phys. B}
\end{center}
\vspace*{5mm}
\end{titlepage}
%\newpage
\section{Introduction}
\setcounter{equation}{0}
\def\theequation{\thesection.\arabic{equation}}
The possibility, first discovered in the context of Horava--Witten
(HW) heterotic M-theory \cite{1,2}, that our Universe could lie
on a hypersurface (a ``brane") embedded in some higher-dimensional
``bulk" space-time --the so-called brane-world scenario-- has
recently attracted considerable attention. In the original HW
paper, our world is a 9-brane sitting at one of the boundaries of
an eleven-dimensional bulk, while in a large class of M-theory
models \cite{4,5}, in which six dimensions are compactified in a
more traditional Kaluza--Klein (KK) way, one can envisage
constructing fully consistent four-dimensional brane-world
scenarios with, effectively, a five-dimensional bulk.
Unfortunately, finding brane configurations which are consistent
with all stringy constraints has proved to be a very hard, if not
impossible, task.
At a more phenomenological level, i.e. when problems with
quantization of the higher-dimensional gravity theory are ignored,
one can consider the dimensions orthogonal to the 3-brane as either
compact and large \cite{9,6,7}, or as having infinite proper
size \cite{3,8,9a}. In the latter (so-called Randall--Sundrum (RS)) case,
the bulk geometry is bent
by an appropriate ``warp-factor", providing a crucial difference
with respect to the ``old" KK scenario, where the bulk geometry is
simply the direct product of an ``internal" and an ``external"
manifold. In particular, unlike the KK models, RS models are able
to reproduce the four-dimensional Newton law at large distances
on the brane by {\it dynamically} binding the massless gravitons
to it \cite{9a}.
As a consequence of the non-factorized structure of the metric in
RS-type scenarios, previous approaches to the study of metric
fluctuations in higher-dimensional backgrounds
\cite{10}--\cite{15}, based on the isometries of a factorizable
geometry, cannot be applied directly to the brane-world scenario.
A new gauge-invariant formalism is required, like the one
developed in \cite{16}. The classical and quantum analysis of
metric fluctuations is of primary importance for understanding the
possible localization of massless modes on the brane, as well as
the nature of the corrections to the long-range interaction due
to the continuum of massive modes that are typically living in
the bulk. Until now, the study of this problem has been mainly
focused on the structure of tensor (i.e. transverse-traceless)
perturbations of the bulk geometry (see \cite{17} for a general
discussion).
In all string/M-theory models, however, the graviton enjoys the
company of (perturbatively) massless scalar partners (the dilaton,
compactification moduli, etc.). These typically induce
long-range interactions of gravitational strength \cite{TV}, and
are therefore dangerous in view of the existing experimental
tests (see for instance \cite{tests}). The standard way to
solve this problem is to assume that these scalars get a
SUSY-breaking non-perturbative mass (for alternatives see e.g.
\cite{DP}). However, one may ask whether RS-type scenarios can
offer an alternative solution to (or an alleviation of) this
problem, e.g. by {\it not} confining scalar fluctuations on the
brane. If this were the case, scalar interactions on the brane
would be suppressed, or possibly become short-ranged, and
brane-world scenarios would naturally solve one of the most
serious potential problems with higher-dimensional or stringy
extensions of the Standard Model and General Relativity. This
particular aspect of brane-world scenarios has never been
completely addressed, to the best of our knowledge, in spite of
many studies recently appeared in the literature, and devoted to
the perturbations of a brane-world background
\cite{18}--\cite{26}.
The main purpose of this paper is to directly address, and to
discuss in detail, the possible localization (or de-localization) of scalar
fluctuations in a typical brane-world scenario of the RS type, taking
into account the possible existence of scalar sources and scalar fields
living in the bulk. To this purpose, we will take a phenomenological
approach, considering a scalar-tensor model of gravity in which the
scalar fluctuations of the metric are also non-trivially coupled to the
scalar fluctuations of the matter sources (unlike tensor fluctuations,
i.e. gravitons, which are decoupled from matter). We shall
consider, in particular,
a non-compact, $Z_2$-symmetric, five-dimensional background,
generated by a positive tension 3-brane and by a bulk dilaton field
coupled to the brane and to the (negative) bulk energy density.
We shall restrict ourselves to the gravi-dilaton solutions discussed in
\cite{27,31a} (hereafter called CLP backgrounds, for short), which
generalize the AdS$_5$ RS scenario \cite{9a} in the presence of a bulk
scalar field, and which are already known to guarantee the localization
of tensor metric fluctuations \cite{27,31a} and of bulk fields of various
spin \cite{31b} (the localization of scalar and higher spin fields
in the AdS$_5$ background was previously discussed in \cite{31c}).
By extending the analysis of the perturbations, and by using an
appropriate gauge-invariant formalism \cite{16}, we find that the
same class of CLP backgrounds that localize massless tensor
perturbations on the four-dimensional brane also localize
massless scalar perturbations (i.e. the dilatonic interaction).
However, the short-range corrections to the scalar interactions,
due to the massive modes propagating in the extra-dimension, are
in general different from the higher-dimensional corrections
affecting the pure tensor part of the gravitational interaction.
We also find that, in general, scalar metric fluctuations exhibit
a non-trivial, ``self-sustained" spectrum of solutions, even for a
trivial dilaton background. This implies that long- and
short-range gravitational interactions on the brane are
effectively of the scalar-tensor type, in agreement with previous
results \cite{27a}, and are therefore subject to strong
phenomenological constraints.
The paper is organized as follows. In Section 2 we present the
action and the classical equations of motion for a 3-brane
non-minimally coupled to a five-dimensional gravi-dilaton
background, and we retrieve the whole class of CLP solutions
\cite{27}. In Section 3 we give the full set of scalar
perturbation equations in the so-called ``generalized longitudinal
gauge" \cite{16}, and we find the four independent canonical
variables diagonalizing them.
In Section 4 we discuss the localization of
massless modes, and determine the class of backgrounds admitting
long-range dilatonic interactions confined on the brane. In
Section 5 we present the general spectrum of solutions for the
massive modes that propagate throughout the bulk, and determine
the relative magnitude of their amplitudes. In Section 6 we
evaluate, in the weak field limit, the leading-order corrections
to the effective gravitational potential generated by a static
source with a point-like mass and dilatonic charge, confined on
the brane. The main results of this paper are finally summarized
in Section 7.
\section{Background equations}
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\def\theequation{\thesection.\arabic{equation}}
We shall consider a five-dimensional scalar-tensor background
$\{g_{AB}, \phi\}$, possibly arising from the bosonic sector of a
dimensionally reduced string/supergravity theory, and
non-trivially coupled to a negative cosmological constant
$\Lambda$ and to a 3-brane of positive tension $T_3$: \bea
S=S_{\rm bulk}+ S_{\rm brane}&=&{M_5^3}\int d^5 x \sqrt{|g|}
\left( -R+\frac{1}{2} g^{AB} \partial_A \phi
\partial_B \phi -2 \Lambda e^{\alpha_1 \phi}
\right) \nonumber\\ &-& \frac{T_3}{2} \int d ^4 \xi
\sqrt{|\gamma|}\left[ \gamma^{\alpha \beta} \partial_\alpha X^A
\partial_\beta
X^B g_{AB}
e^{\alpha_2 \phi} -2 \right]. \label{21} \eea Here $M_5$ is the
fundamental mass scale of the five-dimensional bulk space-time,
and the parameters $\alpha_1, \alpha_2$ control the coupling of
the bulk dilaton to $\Lambda$ and to the brane. Particular values
of these parameters may simply correspond to the rescaling of the
minimally coupled $\sigma$-model action in the canonical Einstein
frame \cite{28}, but here we take a phenomenological approach,
allowing in general for non-minimal
couplings (a single exponential potential for the dilaton
can also be derived from the dimensional reduction of a suitable
higher-dimensional model \cite{27}). The brane action (see
for instance \cite{29}) is parametrized by the coordinates
$X^A(\xi)$ describing the embedding of the brane in the bulk
manifold, and by the auxiliary metric tensor $\gamma_{\alpha
\beta}(\xi)$ defined on the four-dimensional world-volume of the
brane, spanned by the coordinates $\xi^\alpha$. Consequently,
$\pa_{\a} X^A$ is a short-cut notation for $\partial
X^A(\xi)/\partial \xi^\alpha$.
Conventions: Greek indices run from $0$ to $3$, capital Latin
indices from $0$ to $4$, lower-case Latin indices from $1$ to $3$.
For the bulk coordinates we use the notation $x^A= (t,x^i,z)$. The
metric signature is $(+,-,-,-,-)$, and the curvature tensor is
defined by ${R_{MNA}}^B=\partial_M {\Gamma_{NA}}^B+{\Gamma_{MP}}^B
{\Gamma_{NA}}^P - (M \longleftrightarrow N)$,
$R_{NA}={R_{MNA}}^M$.
The variation of the action with respect to $g_{AB}$, $\phi$,
$X^A$ and $\ga_{\a\b}$ gives, respectively, the Einstein equations
(in units such that $M_5^3=1$):
\begin{eqnarray}
&{G^A}_B & =\frac{1}{2}\left(
\partial^A \phi \partial_B \phi -
\frac{1}{2}\delta^A_{B}\partial_C \phi
\partial^C \phi \right) + \Lambda e^{\alpha_1 \phi}\delta^A_B+
\nonumber \\ &&+\frac{T_3}{2} \frac{1}{ \sqrt{|g|}} g_{BC}\int d
^4 \xi \sqrt{|\gamma|} \delta^5(x-X) \gamma^{\alpha \beta}
\partial_\alpha X^A \partial_\beta
X^C e^{\alpha_2 \phi}, \label{Eq Einstein}
\end{eqnarray}
the dilaton equation:
\begin{equation}
\nabla_M \nabla^M \phi +2\alpha_1 \Lambda e^{\alpha_1\phi}
+\frac{\alpha_2 T_3}{2}\frac{1}{\sqrt{|g|}} \int d ^4 \xi
\sqrt{|\gamma|} \delta^5(x-X) \gamma^{\alpha \beta}
\partial_\alpha X^A
\partial_\beta X^B g_{AB} e^{\alpha_2 \phi}
=0, \label{Eq dilaton}
\end{equation}
the equation governing the evolution of the brane in the bulk
space-time:
\begin{equation} \partial_\alpha
\left(\sqrt{|\gamma|}\gamma^{\alpha\beta} \partial_\beta X^B
g_{AB} e^{\alpha_2 \phi}\right)=\frac{1}{2}
\sqrt{|\gamma|}\gamma^{\alpha \beta} \partial_\alpha X^B
\partial_\beta X^C \left.
\pa_A \left(g_{BC} e^{\alpha_2 \phi} \right) \right|_{x=X(\xi)} ,
\label{Eq brane}
\end{equation}
and the induced metric on the brane:
\begin{equation}
\gamma_{\alpha \beta} = \partial_\alpha X^A \partial_\beta X^B
g_{AB} e^{\alpha_2 \phi} \label{Eq gamma}.
\end{equation}
We now specialize these equations to the case of a conformally
flat bulk metric, with a warp factor $a$ and a dilaton $\phi$,
both of which only depend on the fifth coordinate $z$. Also, we
shall look for $Z_2$-symmetric solutions, describing a flat brane
rigidly located at $z=0$, and we set \beq g_{AB}= a^2(z)\eta_{AB},
~~~~~~ \phi = \phi(z), ~~~~~~ X^A = \delta^A_\mu\xi^\mu.
\label{26} \eeq where $\eta_{AB}$ is the five-dimensional
Minkowski metric. The induced metric thus reduces to
\begin{equation}
\gamma_{\alpha \beta} = \delta^A_\alpha \delta^B_\beta g_{AB}~
e^{\alpha_2 \phi},
\end{equation}
while the brane equations (\ref{Eq brane}) are identically
satisfied
thanks to the $Z_2$ symmetry.
The dynamical equations are obtained from the dilaton equation
(\ref{Eq dilaton}), which becomes
\begin{equation}
3\frac{a'}{a}\phi'+\phi''-2\alpha_1 \Lambda a^2 e^{\alpha_1
\phi}-2\alpha_2 T_3 a e^{2\alpha_2 \phi} \delta(z)=0 , \label{Eq
reduced dilaton}
\end{equation}
and from the $(\a,\b)$ and $(4,4)$ components of the Einstein
equations (\ref{Eq Einstein}), which give, respectively, \bea &&
-3\frac{a''}{a} =\frac{\phi'^2}{4}+\Lambda a^2 e^{\alpha_1
\phi}+\frac{T_3}{2}a e^{2\alpha_2 \phi} \delta(z), \label{Eq
G00}\\ && -6\frac{a'^2}{a^2} = -\frac{\phi'^2}{4}+\Lambda a^2
e^{\alpha_1 \phi}
\label{Eq G44}
\eea (a prime denotes differentiation with respect to $z$). The
last three equations are not independent: the dilaton equation,
for instance, can be obtained by differentiating eqs. (\ref{Eq
G00}), (\ref{Eq G44}), as a consequence of the Bianchi
identities.
No general solution is known for arbitrary values of
$\alpha_1,\alpha_2, T_3$ and $\La$. However, if we fine-tune these
parameters by choosing: \beq \alpha_1= 4 \alpha_2 \;,~~~ T_3 = 8
\sqrt{\La/\Delta}\;, ~~~ \alpha_1^2=\Delta+\frac{8}{3}\; ,
\label{alpha rel}
\end{equation}
where the last equation defines $\Delta$, we recover the
four-dimensional sector of a known, one-parameter family of exact
domain wall solutions \cite{30}, which can be written in an
explicitly $Z_2$-symmetric form. For $\Delta = -2$ the solution
is: \beq a(z) = e^{-\frac{k|z|}{3}}, ~~~~~~
\phi(z)=\sqrt{\frac{2}{3}} k |z|,~~~~~~ k^2=-2 \Lambda,\nonumber
\\ \label{213} \eeq otherwise ($\Delta \not= -2$): \beq a(z)
=\left( 1+k |z| \right)^{\frac{2}{3(\Delta+2)}}, ~~~~~~
e^{\phi(z)}= \left( 1+k |z|
\right)^{-\frac{2\alpha_1}{\Delta+2}},~~~~~~
k^2=\frac{(\Delta+2)^2 \Lambda}{\Delta}.
\label{214}
\eeq
We shall choose $k>0$ so that the $z$ coordinate, transverse to
the brane, may run from $-\infty$ to $+\infty$ (the proper size of
the transverse dimension is finite, however, unless $\Da=-8/3$).
In that case, the solution corresponds to a brane of positive
tension, $T_3>0$, provided $\Da \leq -2$. This range of $\Da$
guarantees a positive tension and also avoids the presence of
naked singularities \cite{27}. On the other hand, the reality of
$\a_1$ requires $\Da \geq -8/3$. In the rest of this paper we
shall thus assume \beq k>0, ~~~~~~~~~~~~~~~~~~~-{8\over 3} \leq
\Da \leq -2. \label{215} \eeq We may note that in the limit $\Da
= -8/3$ the dilaton decouples and becomes trivial, and the
solution reduces to the well studied pure AdS$_5$ background ,
originally introduced to localize gravity on a 3-brane \cite{9a}.
In the following section we will obtain the canonical equations
governing the evolution of scalar (metric + dilaton) fluctuations
around the above CLP background solutions.
\section{Scalar perturbations}
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We now perturb to first order the full set of bulk equations
(\ref{Eq Einstein})--(\ref{Eq gamma}), keeping the position of the
brane fixed, $\da X^A=0$. It is known, indeed, that the brane
location can be consistently assumed to remain unchanged, to the
relevant linear order, when perturbing a background of the type we
are considering \cite{16} (see \cite{21} for a study that
includes, instead, a non-vanishing deformation of the brane). We
thus set \beq \delta g_{AB}=h_{AB}, ~~~~~~ \delta g^{AB}=-h^{AB},
~~~~~~ \delta\phi=\chi , ~~~~~~ \da X^A=0, \label{Pert dil} \eeq
where the indices of the perturbed fields are raised and lowered
by the unperturbed metric, and the background fluctuations
$h_{AB}, \chi$ are assumed to be inhomogeneous.
The perturbation of the background equations (\ref{Eq
Einstein})--(\ref{Eq gamma}) gives, respectively, the linearized
equations for the Einstein tensor:
\begin{eqnarray}
&& \delta {G^A}_B =\frac{1}{2} \left(-h^{AC}\partial_C \phi
\partial_B \phi+\partial^A \phi \partial_B \chi+ \partial^A \chi
\partial_B \phi \right)-\frac{1}{4} \delta^A_B \left( 2\partial ^M
\partial_M \chi -h^{MN} \partial_M \phi \partial_N \phi \right)
\nonumber \\ && + \Lambda \alpha_1 \chi e^{\alpha_1 \phi}
\delta^A_B+\frac{T_3}{2}\frac{1}{\sqrt{|g|}} \int d ^4 \xi
\sqrt{|\gamma|} \delta^5(x-X) \gamma^{\alpha \beta}
\partial_\alpha X^A \partial_\beta
X^C e^{\alpha_2 \phi} \nonumber
\\ &&
\times \left[ h_{BC}+ g_{BC} \left( -\frac{1}{2} g^{MN} h_{MN}
+\frac{1}{2} \gamma^{\mu \nu} \delta \gamma_{\mu \nu}+ \alpha_2
\chi \right) \right] \nonumber
\\ &&
+\frac{T_3}{2}\frac{1}{\sqrt{|g|}}g_{BC} \int d ^4 \xi
\sqrt{|\gamma|} \delta^5(x-X)
\partial_\alpha X^A \partial_\beta
X^C e^{\alpha_2 \phi}\delta \gamma^{\alpha\beta}, \label{32}
\end{eqnarray}
for the dilaton:
\begin{eqnarray}
&&\nabla_M \nabla^M \chi -h^{MN} \nabla_M \nabla_N \phi-g^{MN}
\delta {\Gamma_{MN}^B} \partial_B \phi+ 2 \alpha_1^2 \Lambda \chi
e^{\alpha_1 \phi} \nonumber \\ && +\frac{\alpha_2
T_3}{2}\frac{1}{\sqrt{|g|}} \int d ^4 \xi \sqrt{|\gamma|}
\delta^5(x-X) \gamma^{\alpha \beta}
\partial_\alpha X^A \partial_\beta
X^B g_{AB} e^{\alpha_2 \phi} \nonumber\\ && \times \left(
-\frac{1}{2} g^{MN} h_{MN} +\frac{1}{2} \gamma^{\mu \nu} \delta
\gamma_{\mu \nu}+ \alpha_2 \chi \right)\nonumber
\\ &&
+\frac{\alpha_2 T_3}{2}\frac{1}{\sqrt{|g|}} \int d ^4 \xi
\sqrt{|\gamma|} \delta^5(x-X)
\partial_\alpha X^A \partial_\beta
X^B e^{\alpha_2 \phi}\left( g_{AB}\delta
\gamma^{\alpha\beta}+h_{AB} \gamma^{\alpha\beta} \right)=0,
\end{eqnarray}
for the brane:
\begin{eqnarray}
&&\partial_\alpha \left\{ \sqrt{|\gamma|}
\partial_\beta X^B e^{\alpha_2 \phi} \left[ \delta
\gamma^{\alpha\beta}
g_{AB} +\gamma^{\alpha\beta}
h_{AB}+ \gamma^{\alpha\beta}
g_{AB} \left(\alpha_2 \chi +\frac{1}{2} \gamma^{\mu\nu} \delta
\gamma_{\mu\nu}
\right) \right] \right\}= \nonumber \\
&&=\frac{1}{2}
\sqrt{|\gamma|} \partial_\alpha X^B
\partial_\beta X^C \left[ \left( \delta\gamma^{\alpha \beta}
+\frac{1}{2} \gamma^{\alpha \beta} \gamma^{\mu \nu}\delta
\gamma_{\mu \nu} \right)\left. \frac{\partial}{\partial x^A}
\right|_{x=X(\xi)} \left( g_{BC} e^{\alpha_2 \phi} \right) \right.
\nonumber \\ && \left.+ \gamma^{\alpha \beta}\left. \pa_A \left(
h_{BC} e^{\alpha_2 \phi} +\alpha_2 \chi g_{BC} e^{\alpha_2
\phi}\right) \right|_{x=X(\xi)} \right],
\end{eqnarray}
and for the induced metric:
\begin{equation}
\delta \gamma_{\alpha \beta} = \partial_\alpha X^A \partial_\beta
X^B e^{\alpha_2 \phi} \left( h_{AB} + \alpha_2 \chi g_{AB}
\right). \label{35}
\end{equation}
Here all geometrical quantities, such as the perturbed connection
$\da \Ga_{AB}\,^C$, the perturbed scalar curvature $\da R=
-h^{AB}R_{AB}+ g^{AB}\da R_{AB}$, and so on, are computed to first
order in $h_{AB}$.
By expanding around the background (\ref{26}), it is now easy to
study the propagation of the spin-2 physical degrees of freedom
on the brane, represented by the transverse and traceless
perturbations $\overline{h}_{ij}$, \beq h_{AB}=a^2\delta^i_A
\delta^j_B \overline{h}_{ij}, ~~~~~~~~~~~ {\overline{h}^i}_i=0,
~~~~~~~~~~~~ \nabla^i \overline{h}_{ij}=0. \label{36} \eeq In the
linear approximation, the tensor fluctuations $\overline{h}_{ij}$
are decoupled from the scalar and matter fluctuations. We can
consistently set $\chi=0$, and find that the dilaton and the
brane equations are trivially satisfied; in addition, the
right-hand side of the Einstein equations (\ref{32}) is
identically vanishing, and the linearized Ricci tensor leads to
the well known covariant wave equation for
gravitons:\begin{equation} \Box_5 \overline{h}_{ij}\equiv \left(
{\pa^2\over \pa t^2}- {\pa^2\over \pa x^{i2}} -{\pa^2\over \pa
z^2}-{3 a'\over a} {\pa\over \pa z}\right)\overline{h}_{ij}=0,
\label{Eq libera}
\end{equation}
where $\Box_5 \equiv \nabla_M \nabla^M$ is the five-dimensional
covariant d'Alembert operator, describing free propagation in the
warped bulk geometry.
In this paper (also in preparation of future cosmological
applications) we are primarily interested in the scalar
fluctuations of the bulk metric, which are coupled to the dilaton
fluctuations. Thus, we shall keep $\da \phi =\chi \not=0$, and we
shall expand around the background (\ref{26}) in the so-called
``generalized longitudinal gauge" \cite{16}, which extends the
longitudinal gauge of standard cosmology \cite{31} to the
brane-world scenario. As discussed in \cite{16}, in five
dimensions there are four independent degrees of freedom for the
scalar metric fluctuations: in the generalized longitudinal gauge
they are described by the four variables $\{ \varphi, \psi,
\Gamma, W\}$, defined by \bea && h_{00}= 2 \varphi a^2,
~~~~~~~~~~~~~~~ h_{ij}=2\psi a^2 \da_{ij}, \nonumber\\ && h_{44}=
2 \Ga a^2, ~~~~~~~~~~~~~~~ h_{04}=-W a^2. \label{38} \eea
(Off-diagonal metric fluctuations have been taken into account
also in a recent study of linearized gravity in a brane-world
background \cite{32}: in that case, however, there are no scalar
sources in the bulk, and no long-range scalar interactions).
By inserting the explicit form of the background (\ref{26}) and of
the metric fluctuations (\ref{38}) into the perturbed equations
(\ref{32})--(\ref{35}), we obtain the full set of constraints and
dynamical equations governing the linearized evolution of the five
scalar variables $\{ \varphi, \psi, \Gamma, W,\chi\}$. Let us give
them in components, starting from the Einstein equations, and
using eq. (\ref{35}) for the perturbations of the induced metric.
Equation $(0,0)$ gives:
\begin{eqnarray}
&&2\nabla^2 \psi+ 3\psi'' +\nabla^2\Gamma +9\frac{a'}{a}\psi'
-3\frac{a'}{a}\Gamma' -\frac{\phi'}{2}\chi' -6\frac{a''}{a}\Gamma
-\frac{{\phi'}^2}{2}\Gamma \nonumber \\ &&-a^2 e^{\alpha_1 \phi}
\Lambda \alpha_1 \chi - \frac{1}{2} a e^{2\alpha_2\phi} T_3 \left(
\Gamma+ 2\alpha_2 \chi \right) \delta(z)=0. \label{Eq00}
\end{eqnarray}
Equation $(i,i)$ gives:
\begin{eqnarray}
&& -\nabla^2 \varphi -\varphi'' -2\ddot \psi +\nabla^2 \psi
+2\psi'' -\ddot \Gamma +\nabla^2 \Gamma \nonumber \\ &&
-3\frac{a'}{a} \varphi' -\dot W' -3\frac{a'}{a}\dot W +6
\frac{a'}{a} \psi' -3\frac{a'}{a}\Gamma' -\frac{\phi'}{2}\chi'
-6\frac{a''}{a}\Gamma-\frac{{\phi'}^2}{2}\Gamma \nonumber \\ &&
-a^2 e^{\alpha_1 \phi} \Lambda \alpha_1 \chi - \frac{1}{2} a
e^{2\alpha_2\phi} T_3 \left( \Gamma+ 2\alpha_2 \chi \right)
\delta(z)=0. \label{Eqii}
\end{eqnarray}
Equation $(i,j)$, with $i \neq j$, gives:
\begin{equation}
\partial_i\partial_j \left( \varphi -\psi -\Gamma \right)=0.
\label{Eqij}
\end{equation}
Equation $(4,4)$ gives:
\begin{eqnarray}
&& -\nabla^2 \varphi -3\ddot\psi +2\nabla^2\psi
-3\frac{a'}{a}\varphi' -3\frac{a'}{a}\dot W +9\frac{a'}{a} \psi'
+\frac{{\phi'}}{2}\chi' \nonumber \\ &&-12\frac{a'^2}{a^2}
\Gamma+\frac{{\phi'}^2}{2}\Gamma-a^2 e^{\alpha_1 \phi} \Lambda
\alpha_1 \chi =0. \label{Eq44}
\end{eqnarray}
Equation $(i,0)$ gives:
\begin{equation}
\partial_i \left(\frac{W'}{2} +\frac{3}{2}\frac{a'}{a}W +2 \dot \psi +\dot
\Gamma \right) =0. \label{Eq0i}
\end{equation}
Equation $(4,0)$ gives:
\begin{equation}
\frac{1}{2}\nabla^2 W -3\dot \psi' +3\frac{a'}{a} \dot \Gamma
+\frac{\phi'}{2} \dot \chi=0. \label{Eq40}
\end{equation}
Equation $(4,i)$ gives:
\begin{equation}
\partial_i \left( \frac{\dot W}{2} +\varphi' -2\psi'
+3\frac{a'}{a} \Gamma +\frac{\phi'}{2} \chi\right)=0. \label{Eq4i}
\end{equation}
The dilaton equation gives:
\begin{eqnarray}
&& \Box_5 \chi - \phi' \varphi' -\phi' \dot W +3 \phi' \psi'
-\phi' \Gamma' -6 \frac{a'\phi'}{a}\Gamma -2\phi''\Gamma
\nonumber \\ && +2a^2 e^{\alpha_1 \phi} \Lambda \alpha_1^2 \chi +2
a e^{2\alpha_2\phi} T_3 \alpha_2 \left( \Gamma+ 2\alpha_2 \chi
\right) \delta(z)=0 \label{Eq prechi}
\end{eqnarray}
(the dots denote differentiation with respect to Minkowski time
on the brane). Finally, the brane perturbation equation gives a
constraint at $z=0$ which is always satisfied because of the $Z_2$
symmetry. A similar set of equations was already derived in
\cite{16}. We have no contributions from the time derivatives of
the background, since our background is static.
In the absence of bulk sources with anisotropic stresses, we can
now eliminate $\varphi$ from eq. ($i \not=j$), thus reducing to
four scalar degrees of freedom, by setting:
\begin{equation}
\varphi=\psi+\Gamma. \label{318}
\end{equation}
As a consequence, we immediately find that the variable $W$
decouples from the other fluctuations. The combination of the time
derivative of eq. ($4,i$) with the $z$-derivative of eq. ($i,0$)
and with eq. ($4,0$) leads in fact to the equation
\begin{equation}
\Box_5 W=3\left( \frac{a''}{a}-\frac{a'^2}{a^2} \right)W.
\label{319}
\end{equation}
Thus $W$ is decoupled but, because of the non-trivial
self-interactions, it does not freely propagate in the background
geometry like the graviton, eq. (\ref{Eq libera}).
In order to discuss the dynamics of the remaining variables $\psi,
\Gamma$ and $\chi$, it is now convenient to recombine their
differential equations in an explicitly covariant way, to obtain a
canonical evolution equation. To this aim, we can use eq.
(\ref{318}) and eq. ($4,i$) to eliminate $\varphi$ and $\dot W$ in
eqs. ($i,i$), ($4,4$) and in the dilaton equation (\ref{Eq
prechi}). We then combine the simplified version of eqs. ($4,4$),
($i,i$) with eq. ($0,0$), and obtain the following system of
coupled equations, where the source terms
depend only on $\Gamma$ and $\chi$:
\begin{eqnarray}
\Box_5 \psi &=&-2\frac{a''}{a}\Gamma+2\frac{a'^2}{a^2}\Gamma
+\frac{a'\phi'}{a}\chi -\frac{2}{3}a^2 e^{\alpha_1 \phi} \Lambda
\alpha_1 \chi \nonumber\\ &- &\frac{1}{6} a e^{2\alpha_2\phi} T_3
\left( \Gamma+ 2\alpha_2 \chi \right) \delta(z), \label{Eq psi} \\
\Box_5 \Gamma&= &-2\frac{a''}{a}\Gamma
+8\frac{a'^2}{a^2}\Gamma -\phi'^2\Gamma +\phi''\chi
+\frac{a'\phi'}{a}\chi -\frac{2}{3}a^2 e^{\alpha_1 \phi} \Lambda
\alpha_1 \chi \nonumber\\ &-& \frac{2}{3} a e^{2\alpha_2\phi} T_3
\left( \Gamma+ 2\alpha_2 \chi \right) \delta(z), \label{Eq Gamma}
\\ \Box_5 \chi &=&2\phi''\Gamma -\phi'^2\chi -2a^2 e^{\alpha_1
\phi} \Lambda \alpha_1^2 \chi \nonumber\\ &-& 2 a
e^{2\alpha_2\phi} T_3 \alpha_2 \left( \Gamma+ 2\alpha_2 \chi
\right) \delta(z). \label{Eq chi}
\end{eqnarray}
This system can be diagonalized by introducing the fields \beq
\omega_1=2\psi+ \Gamma, ~~~~~ \omega_2=6\alpha_2 \Gamma +\chi,
~~~~~ \omega_3=\Gamma- 2\alpha_2 \chi, \label{323}
\eeq
relations
that can be inverted as:
\begin{eqnarray}
&&\psi=\frac{\omega_1}{2}-\frac{2\alpha_2 \omega_2
+\omega_3}{2\left(1+12\alpha_2^2\right)}, ~~~~~~~
\Gamma=\frac{2\alpha_2 \omega_2
+\omega_3}{\left(1+12\alpha_2^2\right)}\nonumber\\ &&
\chi=\frac{\omega_2-6\alpha_2
\omega_3}{\left(1+12\alpha_2^2\right)}. \label{324}
\end{eqnarray}
In terms of these new variables, the perturbation equations
(\ref{Eq psi})--(\ref{Eq chi}) reduce to
\begin{eqnarray}
\Box_5 \omega_1&=&0, \label{325}\\
\Box_5 \omega_2&=&0 , \label{326}\\
\Box_5 \omega_3&=&
\left[-\frac{\phi''}{2\alpha_2}
-\phi'^2-\frac{a'\phi'}{2\alpha_2a} +
\left(\frac{\alpha_1}{3\alpha_2}-2\alpha_1^2 \right)a^2
e^{\alpha_1 \phi} \Lambda\right]\om_3 \nonumber\\ &+&
\left[\left( \frac{2}{3}-4\alpha_2^2
\right) a e^{2\alpha_2\phi} T \delta (z)\right]\omega_3.
\label{327}
\end{eqnarray}
Together with eq. (\ref{319}), and the constraints
(\ref{Eq0i})--(\ref{Eq4i}), the above decoupled equations
describe the complete evolution of the scalar (metric + dilaton)
fluctuations in the CLP brane-world background (\ref{213}),
(\ref{214}). Two variables ($\om_1, \om_2$) are (covariantly) free
on the background like the graviton, while the other two variables
($\om_3, W$) have non-trivial self-interactions.
In all cases, we can introduce the corresponding ``canonical
variables" $\ha W$, $\ha \om_i$ ($i=1,2,3$), which have
canonically normalized kinetic terms \cite{31} in the action,
simply by absorbing the geometric warp factor as follows:
\begin{equation}
W= \ha W a^{-3/2}, ~~~~~~~~~~~~~~~ \om_i= \ha \om_i a^{-3/2}.
\label{328}
\end{equation}
The variables $\ha W$, $\ha \om_i$ are required for a correct
normalization of the scalar perturbations to a quantum fluctuation
spectrum, as they satisfy canonical Poisson (or commutation)
brackets. When the general solution is written as a superposition
of free, factorized plane-waves modes on the brane, \beq \ha W=
\Psi_w(z) e^{-ip_\mu x^\mu}, ~~~~~~~~~~~ \ha \om_i= \Psi_i(z)
e^{-ip_\mu x^\mu}, \label{329} \eeq they define the inner product
of states with measure $dz$ \cite{17}, as in conventional
one-dimensional quantum mechanics, $\int dz |\Psi(z)|^2 $.
Such a product is required for an appropriate definition of
normalizable solutions.
It should be stressed that in this paper we are primarily interested in
the quantum fluctuations of the metric, and of the other background
fields, around their mean values determined by the classical solution.
For a consistent quantization of such perturbations we have thus to
impose appropriate boundary conditions, in order to eliminate the
non-normalizable solutions (blowing up in $z$). In the case of classical
perturbations, the existence of solutions that are initially small enough
to justify the linear approximation, and that blow up at late times,
would signal an instability of the background. In this paper we have not
analyzed this problem in full generality. It looks however that,
whenever we have looked at tachyonic modes that grow at late times,
we have found that they also blow up at large $z$ (see Section 5),
making the linear analysis of this paper inadequate for drawing any
strong conclusion.
In the case of quantum fluctuations, discussed in this paper,
the allowed mass spectrum of $m^2=\eta^{\mu\nu} p_\mu p_\nu$, for
the scalar fluctuations on the brane, can then be obtained by
solving an eigenvalue problem in the Hilbert space $L^2(R)$ for
the canonical variables $\Psi_w,\Psi_i$, satisfying a
Schr\"odinger-like equation in $z$, which is obtained from the
equations (\ref{319}), (\ref{325})--(\ref{327}) for $W$ and
$\om_i$, and which can be written in the conventional form as:
\beq \Psi_w''+\left(m^2- {\xi_w''\over \xi_w}\right)\Psi_w=0,
~~~~~~~~~~~~ \Psi_i''+\left(m^2- {\xi_i''\over
\xi_i}\right)\Psi_i=0. \label{330} \eeq Here, by analogy with
cosmological perturbation theory \cite{31}, we have introduced
four ``pump fields" $\xi_w,\xi_i$, defined as follows:\bea &&
\xi_w=a^{\b_w}, ~~~~~~~~~~~~ \xi_i=a^{\b_i}, \nonumber\\ &&
\b_w=-{3\over 2}, ~~~~~~~~~ \b_1=\b_2 ={3\over 2}, ~~~~~~~~~
\b_3=-{1\over 2}(1+3 \a_1^2)=-{3\over 2}(\Da+3). \label{332} \eea
The effective potential generated by the derivatives of the pump
fields depends on $\b_w,\b_i$, and contains in general a smooth
part, peaked at $z=0$, plus a positive or negative $\da$-function
contribution at the origin. We may have, in principle, not only
volcano-like potentials, which correspond to the free covariant
d'Alembert equation with $\b=3/2$ \cite{27} (and which are known
to localize gravity \cite{9a,17}), but also potentials that are
positive everywhere and admit no bound states. The possible
localization of scalar interactions on the 3-brane, for the given
background and perturbation equations, will be discussed in the
next section.
\section{Localization of the massless modes}
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The general solutions of the canonical perturbation equations
(\ref{330}) are labelled by the mass eigenvalue $m$, by their
parity with respect to $z$-reflections, and by the parameters
$\b_w,\b_i$, which depend on the type of perturbation. The
massless scalar modes corresponding to bound states of the
effective potential \cite{9a,17,27} will describe long-range
scalar interactions confined to the four-dimensional brane.
Concerning $z$-reflections, we shall follow the perturbative
formalism developed in \cite{16} consistently, restricting
ourselves to a perturbed background that is still $Z_2$-symmetric,
namely to $Z_2$-odd solutions for $W$, and to $Z_2$-even solutions
for $\om_i$. To describe a bound state, we shall further restrict
such solutions to those with a normalizable canonical variable
w.r.t. the measure $dz$, namely to $\Psi(z)~ \in ~ L^2(R)$. Among
the acceptable solutions, we shall finally select those satisfying
the constraints (\ref{Eq0i})--(\ref{Eq4i}) derived in the previous
section. The above set of conditions will determine the class of
brane-world backgrounds allowing the four-dimensional localization
of long-range scalar interactions.
For $m=0$ the exact solution of the canonical equations
(\ref{330}), for a generic pump field $\xi(z)$, can be separated
into an even and an odd part, $\Psi^+$ and $\Psi^-$, as follows:
\beq \Psi_0^+(z)=c_+\xi(z), ~~~~~~~~~~~ \Psi_0^-(z)= c_-\xi(z)\int
dz' \xi^{-2}(z'), \label{41} \eeq where $c_+$ and $c_-$ are
integration constants. In order to parametrize the solutions for
different values of $\Da$ and $\b$ (and also in view of
subsequent applications to the massive mode solutions), it is now
convenient to introduce the two indices $\nu$ and $\nu_0$, defined
by \beq
\nu_0=\frac{\Delta}{2(\Delta+2)}, ~~~~~~~~~~~~
\nu=\frac{1}{2}-\frac{2\beta}{3(\Delta+2)}. \label{42} \eeq For
the three possible values of $\beta_w, \b_i$ (see eq.
(\ref{332})), they are related by
\begin{eqnarray}
&&\nu=1-\nu_0, \hspace{2.25cm}{\b=\b_w=-{3\over 2}} , \nonumber\\
&&\nu=\nu_0, \hspace{2.85cm}{\b=\b_{1,2}={3\over 2}}, \nonumber\\
&&\nu=2-\nu_0, \hspace{2.25cm}{\b=\b_3=-{3\over 2}(\Da+3) }.
\label{43}
\end{eqnarray}
We recall that, in the class of backgrounds we are considering,
the parameter $\Da$ is constrained in the range $-8/3\leq \Da \leq
-2$ (because of the conditions of positive tension, absence of
naked singularities and reality of the dilaton couplings). As a
consequence, the index $\nu_0$ can range from $2$ to $+\infty$.
Let us first discuss the limiting case $\nu=\nu_0=\infty$,
corresponding to $\Da=-2$. For a generic pump field, with
exponent $\beta$, the massless solutions are: \beq
\Psi_{0,\infty}^+(z) = c^+_{0,\infty}e^{-\frac{\beta}{3} k|z|},
~~~~~~~~~~ \Psi_{0,\infty}^-(z)= c^-_{0,\infty}{\rm sgn}\{z\}
\left(e^{\frac{\beta}{3} k|z|}-e^{-\frac{\beta}{3} k|z|}\right).
\label{44} \eeq In our background $k>0$, so that the fluctuations
$W$ and $\om_3$, both with $\b=-3/2$, are not normalizable. The
even solutions of the free d'Alembert equation, with $\beta=3/2$,
are instead normalizable, so we have acceptable solutions for
$\omega_1$ and $\omega_2$. However, from the constraint ($i,0$) of
eq. (\ref{Eq0i}), $\om_1$ is forced to vanish when $W=0$, unless
the fluctuations are static, $\dot \om_1=0$. All the other
constraints (using also the background equations) are instead
identically satisfied by $\om_2$.
We can also check, by using the inverted expressions (\ref{324}),
that the perturbation equation (0,0), as well as all the original
perturbed set of evolution equations, are identically satisfied by the
above solutions.
It follows that, for $\Da=-2$,
there are two independent massless modes localized on the brane:
one, $\omega_2$, is propagating and the other, $\om_1$, is
static.
The same is true for the case $\Da<-2$, i.e. finite $\nu_0$. In
that case
the massless solutions can be written in the form
\begin{eqnarray}
&& \Psi_{0,\nu}^+(z)= c^+_{0,\nu} \left(1+k|z|
\right)^{\frac{1}{2}-\nu },\nonumber
\\ &&
\Psi_{0,\nu}^-(z)= c^-_{0,\nu} {\rm sgn}\{z\}\left[ \left(1+k|z|
\right)^{\frac{1}{2}+\nu } -\left(1+k|z| \right)^{\frac{1}{2}-\nu
}\right]. \label{45}
\end{eqnarray}
The solutions for $W$ and $\om_3$ always correspond to $\nu<0$,
and are not normalizable. The even d'Alembert modes, with
$\nu=\nu_0$, are normalizable for $\nu>1$, i.e. $\Da >-4$, so that
$\om_1$ and $\om_2$ are always acceptable in our class of
backgrounds, for which $\Da \geq -8/3$. Again, however, the
constraint (\ref{Eq0i}) implies $\om_1=0$, unless $\dot \om_1=0$,
while the other constraints are identically satisfied by $\om_2$,
so we are left with non-trivial massless solutions only for a
propagating fluctuation, $\om_2$, and for a static one, $\om_1$.
Again, we can check that all the perturbed evolution equations
(for instance, eq. (0,0)), are satisfied by the above solution.
We may thus conclude that all backgrounds of the class defined by
the conditions (\ref{215}) localize on the brane not only the
massless spin-2 degrees of freedom \cite{27} (long-range tensor
interactions), but also one propagating massless scalar degree of
freedom ($\om_2$), corresponding to a long-range scalar
interaction generated by the dilaton field. In the longitudinal
gauge, the canonical representation $\om_2$ of such a scalar
interaction is associated not only with the dilaton fluctuation
$\chi$, but also with the four-dimensional ``Bardeen potential"
$\psi$, and with the ``breathing mode" $\Ga$ of the dimension
orthogonal to the brane (see eqs. (\ref{324})). In addition, we
have a second independent massless degree of freedom localized on
the brane ($\om_1$), which is not propagating ($\dot \om_1=0$),
but is essential to reproduce the standard long-range
gravitational interaction in the static limit, as we shall discuss
in Section 6.
The localization of the scalar interactions does not impose any
further constraints on the background, besides those of eq.
(\ref{215}). Also, in the limiting case of a pure AdS$_5$
solution ($\Da=-8/3$, $\nu_0=2$, $\a_1=0=\a_2$) the dilaton
background disappears, and the dilaton fluctuation $\chi=\om_2$
decouples from the others. The only (static) contribution to the
scalar sector of metric fluctuations comes from $\om_1$, which
generates the long-range Newton potential $\varphi=\psi$ on the
brane (see Section 6). However, even in the case $\Da=-8/3$, the
propagating dilaton flucutuation $\om_2$ is non-vanishing, and
remains there to describe a (possibly dangerous) long-range scalar
interaction. By contrast, no propagating massless scalar modes
appear in the ``pure-gravity" models without bulk scalar fields in
the action, as the AdS$_5$ brane-world scenario discussed in
\cite{9a}.
\section{The massive mode spectrum}
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The massive part of the spectrum of the canonical equations
(\ref{330}) is not localized on the brane; it may induce
higher-dimensional (and short-range) corrections to the long-range
scalar forces (just as in the case of the pure tensor part of the
gravitational interaction \cite{9a,17,27}). Such corrections thus
provide physical effects from the fifth
dimension on our brane.
In order to determine the massive modes able to survive the
constraints of Section 3, and to evaluate the corresponding
short-range corrections, we shall first present the exact
solutions of the massive canonical equations (\ref{330}),
distinguishing, as before, the even and odd parts under
$z$-reflections. For $m\not=0$, $\Da=-2$ (i.e. $\nu_0=\infty$),
and a generic pump field with power $\b$, such solutions can be
written as follows
\begin{eqnarray}
&& \Psi^+_{m,\infty}(z)=\frac{1}{\sqrt{\pi m q }} \left( q \cos
q|z| -\frac{\beta }{3} k \sin q|z| \right),\nonumber \\
&&
\Psi^-_{m,\infty}(z)=-\sqrt{\frac{m}{\pi q } }\sin q z ,
\label{51} \eea where \beq q=
\left(m^2-\frac{k^2}{4}\right)^{1/2}. \label{52} \eeq For
$\Da<-2$, i.e. finite $\nu_0$, the solution can be written as a
combination of first- and second-kind Bessel functions $J_\nu$
and $Y_\nu$ \cite{33}, of index $\nu$ given by eq. (\ref{42}):
\begin{eqnarray}
&& \Psi^+_{m,\nu}(z)=c_{m,\nu-1} \sqrt{1+k|z|} \left[ Y_{\nu-
1}\left(\frac{m}{k} \right) J_{\nu }\left(y \right)- J_{\nu
-1}\left(\frac{m}{k} \right) Y_{\nu }\left(y \right)\right] ,
\nonumber\\ &&\Psi^-_{m,\nu}(z)=c_{m,\nu} ~{\rm
sgn}\{z\}\sqrt{1+k|z|} \left[ Y_{\nu}\left(\frac{m}{k} \right)
J_{\nu }\left(y \right)- J_{\nu }\left(\frac{m}{k} \right) Y_{\nu
}\left(y\right)\right] , \label{53} \eea where \beq c_{m,\nu}=
\sqrt{\frac{m}{2k}} \left[ J_{\nu}^2 \left(\frac{m}{k} \right) +
Y_{\nu }^2\left(\frac{m}{k} \right) \right]^{-\frac{1}{2}},
~~~~~~~~y= \frac{m}{k}(1+k|z|). \label{54} \eeq Note that in the
above equations we have adopted the $\da$-function normalization
of the continuum modes, as for plane waves in one-dimensional
quantum mechanics. As a consequence, $\Psi_m$ is dimensionless
(unlike in \cite{27}, where a different normalization is adopted).
It is important to note that modes with
negative squared mass (tachyons) are not
included in the spectrum, as they would not correspond to a
normalizable canonical variable ($\Psi$ would blow up in $z$).
This can be regarded as a direct check of the stability of the
given class of CLP backgrounds against quantum scalar
perturbations, since tachyonic modes would also blow up in time, and
would destroy the assumed homogeneity of the four-dimensional brane.
Another consequence of the normalization condition is the mass gap
($m^2>k^2/4$, see eq. (\ref{52})) between the localized massless
mode and the massive corrections, in the limiting background with
$\nu_0=\infty$ (already noticed in \cite{27} for the case of pure
tensor interactions).
We shall now impose the constraints (\ref{Eq0i})--(\ref{Eq4i})
following from the perturbed background equations. It is
convenient to introduce the four amplitudes $A_i, A_w$, defined by
\begin{eqnarray}
&& \omega_1=\frac{a^{-3/2} A_1}{c_{m,\nu_0-1}}
\Psi_{m,\nu_0}^+(z)e^{-ip_\mu x^\mu} , ~~~~~~~~~~~~~
\omega_2=\frac{a^{-3/2} A_2}{c_{m,\nu_0-1}}
\Psi_{m,\nu_0}^+(z)e^{-ip_\mu x^\mu}\nonumber \\
&& \omega_3=\frac{a^{-3/2} A_3}{c_{m,1-\nu_0}}
\Psi_{m,2-\nu_0}^+(z)e^{-ip_\mu x^\mu}, ~~~~~~~ W=\frac{a^{-3/2}
A_w}{c_{m,1-\nu_0}} \Psi_{m,1-\nu_0}^-(z)e^{-ip_\mu x^\mu},
\end{eqnarray}
where $\Psi^{\pm}$ are the above normalized solutions. Plugging
this ansatz into the constraint equations
(\ref{Eq0i})--(\ref{Eq4i}), we find that, in contrast with the
massless case, none of the four scalar fluctuations is forced to
vanish. However, only two amplitudes are independent. By taking,
for instance, $\om_2$ and $\om_3$ as independent variables, we
can indeed express $A_1$ and $A_w$ in terms of $A_2$ and $A_3$,
for all values of $\nu_0$, in such a way that all the constraints
are identically satisfied. For a generic mode of mass $m$ and
momentum $p$, we find, in particular, \bea &&
A_1=\frac{m^2}{9m^2+6p^2}\frac{\sqrt{2\nu_0-1}}{\nu_0-1}\left(
\sqrt{3\left(\nu_0-2 \right)} A_2 +3 \sqrt{2\nu_0-1} A_3 \right),
\nonumber\\ && A_w=\frac{2 i m \sqrt{m^2+p^2}}
{9m^2+6p^2}\frac{\sqrt{2\nu_0-1}}{\nu_0-1}\left(
\sqrt{3\left(\nu_0-2 \right)} A_2 +3 \sqrt{2\nu_0-1} A_3 \right),
\label{57} \eea which also hold when $\nu_0 \rightarrow\infty$.
Note that a single combination of $A_2$ and $A_3$ determines both
$A_1$ and $A_w$. We can check that the above constrained solutions
satisfy eq. (0,0) and, more generally, the complete set of perturbed
evolution equations.
For such backgrounds we thus have four types of
higher-dimensional contributions to the scalar interactions on the
brane, arising from the massive spectrum of $\om_i$ and $W$. The
exchange of such massive modes generates corrections to the
four-dimensional scalar forces. The corrections are in general
different from those of tensor interactions, arising from the
massive spectrum of a variable satisfying the free d'Alembert
equation, like $\om_2$. In the weak field limit, however, the
leading-order contributions to the non-relativistic potential
generated by a static scalar sources have the same qualitative
behaviour as in the case of tensor interactions, as will be
illustrated in the next section.
\section{Static limit and leading-order corrections}
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To make contact with previous works, and for future
phenomenological applications, we shall now compute the effective
scalar-tensor interaction induced on the brane, in the weak field
limit, by a static and point-like source of mass $M$ and dilatonic
charge $Q$.
In our longitudinal gauge (\ref{36}), (\ref{38}), in which the
decomposition of the metric fluctuations is based on the $O(3)$
symmetry of the spatial hypersurfaces of the brane, the energy
density of a point-like particle only contributes to the scalar
part of the perturbed matter stress tensor (with $T_{00}$ as the
only non-vanishing component), and provides a $\delta$-function
source to the $(0,0)$ scalar perturbation equation (\ref{Eq00}).
Similarly, the charge $Q$ acts as a point-like source in the
dilaton perturbation equation (\ref{Eq prechi}).
As a consequence, we obtain three $\delta$-function sources in
the equations for the three $\om_i$ fluctuations, $S_i
\da^3(x-x')\da(z)$, with three scalar charges $S_i$, which are
``mixtures" of $M$ and $Q$, while no source term is obtained in
the static limit for the $W$ fluctuation (in agreement with the
fact that $W=0$, in the static limit, according to the constraint
$(i,0)$). The effective sources $S_i$ for the massless and
massive $\om_i$ fluctuations are defined by the combination of
eqs. $(4,4), (i,i), (0,0)$, and by the dilaton equation (\ref{Eq
prechi}), as follows: \beq S_1= M, ~~~~~~~~ S_2=Q+2\a_2 M,
~~~~~~~~ S_3= {M\over 3} -2 \a_2 Q. \label{61} \eeq
The exact static solutions of eqs. (\ref{325})--(\ref{327}), with
the above point-like sources, can be easily obtained using the
static limit of the retarded Green function evaluated on the brane
($z=0$), i.e. \beq \om_i(\nu,x,x')= -S_i G_i (\nu,x,x'),
\label{62} \eeq where \beq G_i (\nu,\vec x, \vec x', z=z'=0)= \int
{d^3p\over (2 \pi)^3} e^{i \vec p \cdot (\vec x -\vec
x')}\left\{{\left[\psi^+_{0,\nu}(0)\right]^2\over p^2}
+\int_{m_0}^\infty dm {\left[\psi^+_{m,\nu}(0)\right]^2\over
p^2+m^2} \right\}
\label{63}
\eeq (see also \cite{27,27a,34}). Here $\nu=\nu_0$ for $i= 1, 2$,
while $\nu=2-\nu_0$ for $i = 3$, and $\psi^+_{0,\nu}(z),
\psi^+_{m,\nu}(z)$ (including the case $\nu =\infty$) are the
exact solutions (\ref{44}), (\ref{45}), (\ref{51}), (\ref{53}),
obtained in the previous sections. The first term in the integrand
corresponds to the long-range forces generated by the massless
modes, the second term to the ``short-range" corrections due to
the massive modes, and $m_0$ is the lower bound for the massive
spectrum ($m_0=k/2$ if $\nu_0=\infty$, while $m_o=0$ if
$\nu_0<\infty$, see eqs. (\ref{51}), (\ref{52})).
We should note that in the $\om_1, \om_2$ case we have to include
both the massless and massive contributions, while in the $\om_3$
case only the massive ones survive (indeed, we recall
that for $m=0$ the $\om_3$
modes are not normalizable, and that the massless sector of
$\om_1$, in the static limit, is not eliminated by the constraint
$(i,0)$, see Section 4). We also note that the amplitude of the
massless solutions (\ref{44}), (\ref{45}) has to be fixed by the
correct normalization, i.e. \beq c^+_{0,\infty}=
\left(k\b/3\right)^{1/2}, ~~~~~~~~~~~~ c^+_{0,\nu}=\left[k
(\nu-1)\right]^{1/2}. \label{64} \eeq
Let us start with $\om_1$, for which $\nu=\nu_0$, and with the
limiting background $\Da=-2$, i.e. $\nu=\infty$. By setting
$\b=3/2$, and using eqs. (\ref{44}) and (\ref{51}) for
$\psi^+_{0,\infty}$ and $\psi^+_{m,\infty}$, respectively, we
obtain \beq \om_1(\nu=\infty) =-S_1 {k\over 8\pi r}
\left[1+{2\over k\pi} \int_{k/2}^\infty {dm\over m}
\left(m^2-{k^2\over 4}\right)^{1/2} e^{-mr}\right], \label{65}
\eeq where $r= |\vec x-\vec x'|$. The same integral had already
been obtained in \cite{27} when discussing the localization of
tensor fluctuations, and the associated leading-order corrections
(in the large-distance limit) are known to be of the form
$e^{-kr/2} (kr)^{-3/2}$.
In the case $\Da<-2$, i.e. $\nu=\nu_o <\infty$, we shall use
instead eqs. (\ref{45}) and (\ref{53}) for $\psi^+_{0,\nu_0}$ and
$\psi^+_{m,\nu_0}$, respectively. Also, we shall estimate the
contribution of the massive modes by expanding the Bessel
functions in the small-$m$ regime (which is relevant at the long
distances typical of the weak field limit). The small argument
limit \cite{33} of $J_\nu$, $Y_\nu$ then gives \beq
\left[\psi^+_{m,\nu_0}(0)\right]^2 ={1\over \Ga^2 (\nu_0-1)}
\left(m\over 2k\right)^{2\nu_0-3}, \label{66} \eeq and we are led
to \beq \om_1(\nu<\infty) =-S_1 {k(\nu_0-1)\over 4\pi r} \left[1+
{\Ga (\nu_0-1/2)\over \sqrt \pi \Ga (\nu_0)}\left(1\over
kr\right)^{2\nu_0-2} \right] \label{67} \eeq (here $\Ga$ obviously
represents the Euler function, not to be confused with the
$g_{44}$ component of the metric fluctuations).
Exactly the same results (\ref{65}), (\ref{67}) are obtained for
$\om_2$, which satisfies the same free d'Alembert equation as
$\om_1$, with the only difference that $S_1$ has to be replaced by
$S_2$.
Let us then compute $\om_3$, for which $\nu=2-\nu_0$, and which
has only the massive mode contribution to the Green function. For
$\Da=-2$, i.e $\nu=\infty$, we shall use eq. (\ref{51}) for
$\psi^+_{m,\infty}$. At $z=0$ the solution however is
$\b$-independent, and we obtain the (massive part) of the
result already given in eq. (\ref{65}) (with $S_1$ replaced by
$S_3$).
In the case $\Da<-2$, i.e $\nu<\infty$, we shall use
$\psi^+_{m,2-\nu_0}$ from eq. (\ref{53}). In the large-distance
(small-$m$) regime, however, the contribution at $z=0$ to the
Green function is exactly the same as that of eq. (\ref{66}), so
that we obtain \beq \om_3(\nu<\infty) =-S_3 {k(\nu_0-1)\over 4\pi
r} {\Ga (\nu_0-1/2)\over \sqrt \pi \Ga (\nu_0)}\left(1\over
kr\right)^{2\nu_0-2} . \label{68} \eeq
We are now ready to estimate, in the static limit, the dilaton and
metric fluctuations $\chi, \Ga, \psi, \varphi$. We shall
explicitly consider the case $\nu_0<\infty$, for an easy
comparison with previous results relative to an AdS$_5$
background, for which $\nu_0=2$. Defining \beq
A_{\nu_0}={k(\nu_0-1)\over 4\pi}, ~~~~~~~~~~~~ B_{\nu_0}={\Ga
(\nu_0-1/2)\over \sqrt \pi \Ga (\nu_0)}, \label{69} \eeq we first
rewrite the $\om_i$ solutions in compact form as: \bea &&
\om_1=-{S_1 A_{\nu_0}\over r} \left[1+ B_{\nu_0}\left(1\over
kr\right)^{2\nu_0-2}\right], \nonumber\\ && \om_2=-{S_2
A_{\nu_0}\over r} \left[1+ B_{\nu_0}\left(1\over
kr\right)^{2\nu_0-2}\right], \nonumber\\ && \om_3=-{S_3
A_{\nu_0}\over r} B_{\nu_0}\left(1\over kr\right)^{2\nu_0-2} .
\label{610} \eea We may note, as a check of our previous
computations, that the relative amplitude of the massive
corrections, in the weak, static limit, is controlled by the
correspondig scalar charges, which satisfy the relation (from eq.
(\ref{61})): \beq S_3+2 \a_2S_2= \left({1\over 3} +
4\a_2^2\right)S_1. \label{611} \eeq By eliminating $\a_2$ in
favour of $\Da$ according to eq. (\ref{alpha rel}), and $\Da$ in
terms of $\nu_0$, according to eq. (\ref{42}), we obtain \beq
S_1={1\over \nu_0-1} \left[ {1\over 3} \sqrt{3
(\nu_0-2)(2\nu_0-1)}S_2 +(2 \nu_0-1) S_3\right], \label{612} \eeq
which exactly reproduces, in the static limit $m^2+p^2 \ra 0$, the
general relation (\ref{57}) between the massive amplitudes.
It is convenient, at this point, to explicitly introduce the
four-dimensional gravitational constant $M_p^{-2}$, by noting that
\beq A_{\nu_0}={k(\nu_0-1)\over 4\pi}={1\over 4\pi}
\left[\psi^+_{0,\nu_0}(0)\right]^2={1\over 4\pi} \left[\int dz~
a^3(z)\right]^{-1}. \label{613} \eeq The above normalization
integral, when expressed in terms of a new bulk coordinate $y$,
with $dy= a(z)dz$, represents the warped extra-dimensional volume
that controls the ratio between the four- and five-dimensional
gravitational constants \cite{8,9a}, i.e. \beq M_p^2= M_5^3 \int
dy ~a^2(y)= M_5^3 \int dz~a^3(z). \label{614} \eeq It follows
that, in units such that $M_5^3=1$, \beq A_{\nu_0}=(4\pi
M_p^2)^{-1}=2G, \label{615} \eeq where $G$ is the Newton coupling
constant. The scalar and dilaton fluctuations (\ref{324}), using
the static solutions (\ref{610}), can then be written in the
form: \bea && \varphi =-{GM\over r}\left[1+ {2 \a_2\over 1+12
\a_2^2} \left({Q\over M} + 2 \a_2\right) + {4\over 3}B_{\nu_0}
\left(1\over kr\right)^{2\nu_0-2}\right], \nonumber\\ && \psi
=-{GM\over r}\left[1-{2 \a_2\over 1+12 \a_2^2} \left({Q\over M} +
2 \a_2\right) + {2\over 3}B_{\nu_0} \left(1\over
kr\right)^{2\nu_0-2}\right], \nonumber\\ && \Ga =-{GM\over
r}\left[{4 \a_2\over 1+12 \a_2^2} \left({Q\over M} + 2 \a_2\right)
+ {2\over 3}B_{\nu_0} \left(1\over kr\right)^{2\nu_0-2}\right],
\nonumber\\ && \chi =-{GQ\over r}\left[{2 \over 1+12 \a_2^2}
\left(1+ 2\a_2 {M\over Q}\right) + {2}B_{\nu_0} \left(1\over
kr\right)^{2\nu_0-2}\right]. \label{616} \eea It should be noted
that the short-range corrections induced by the massive scalar
modes have the same qualitative behaviour as in the tensor case,
discussed in \cite{27}, in spite of the fact that the massive
scalar modes have different spectra. The dimensional decoupling
(i.e., the suppression of the higher-dimensional corrections) is
thus effective for all scales of distance $r$ such that $kr \gg
1$, where $k=(M_5^3/M_p^2)(\nu_0-1)^{-1}$ is the mass scale
relating the five- and four-dimensional gravitational constants,
in our class of backgrounds.
The limiting case $\Da=-8/3$, i.e. $\a_2=0$ and $\nu_0=2$,
corresponds to a pure AdS$_5$ background, if there are no scalar
charges on the brane. In that case $\om_2$ exactly corresponds to
the dilaton fluctuation $\chi$ (see eq. (\ref{323})), and can be
consistently set to zero (toghether with the dilaton background)
if we want to match, in particular, the ``standard" brane-world
configuration originally considered by Randall and Sundrum
\cite{9a}. In this limit, $B_{2}=1/2$, and we exactly recover
previous results for the effective gravitational interaction on
the brane \cite{27a}, i.e. \beq \varphi= -{GM\over r} \left(1+
{2\over 3 k^2r^2}\right), ~~~~~~ \psi= -{GM\over r} \left(1+
{1\over 3 k^2r^2}\right). \label{617} \eeq
The massless-mode truncation reproduces in this case the static,
weak field limit of linearized general relativity (note, however,
that for $Q \ne 0$ there is no way to get rid of the long-range
scalar interactions). The massive tower of scalar fluctuations,
however, induces deviations from Einstein gravity already in the
static limit (as noted in \cite{27a}), and is the source of a
short-range force due to the ``breathing" of the fifth dimension,
\beq \Ga= - {GM\over 3 r}{1\over (kr)^2}, \label{618} \eeq even in
the absence of bulk scalar fields, and of scalar charges for the
matter on the brane.
In a more general gravi-dilaton background ($\Da \not= -8/3$,
$\a_2\not=0$), the static expansion (\ref{616}) describes an
effective scalar-tensor interaction on the brane, which is
potentially dangerous for the brane-world scenario, as it contains
not only short-range corrections, but also long-range scalar
deviations from general relativity (and, possibly, violations of
the Einstein equivalence principle), even in the interaction of
ordinary masses, i.e. for $Q=0$ (similar results have been
recently obtained also in the context of a multibrane scenario
\cite{39}). This seems to offer an interesting window to
investigate the effects of the bulk geometry on the
four-dimensional physics of the brane.
\section{Conclusions}
\setcounter{equation}{0}
\def\theequation{\thesection.\arabic{equation}}
In this paper we have analysed the full set of coupled equations
governing the evolution of scalar fluctuations in a dilatonic
brane-world background, supporting a flat 3-brane rigidly located
at the fixed point of $Z_2$ symmetry. We have diagonalized the
system of dynamical equations, and found four decoupled but
self-interacting variables representing, in a five-dimensional
bulk, the four independent degrees of freedom of scalar
excitations of the gravi-dilaton background.
We have then restricted our discussion to the class of background
solutions characterized by a brane of positive tension, by a
decreasing warp-factor as we move away from the brane, and by the
absence of naked singularities \cite{27}. Such a class of
backgrounds can be characterized by a real parameter $\Da$ ranging
from $-8/3$ to $-2$ or, alternatively, by a real parameter $\nu_0$
ranging from $2$ to $+\infty$. The limiting case $\Da =-8/3$,
$\nu_0=2$, corresponds to the ``pure-gravity" AdS$_5$ background
\cite{9a}.
We have presented the exact solutions of
the canonical perturbation equation for all the scalar degrees of
freedom, and we have discussed, in this class of backgrounds, the
effects of their massless and massive spectrum for the scalar
interactions on the brane, taking into account the appropriate
parity under $Z_2$ symmetry, the normalization condition for the
bound states of the spectrum, and the first-order differential
constraints arising from the dynamics of scalar perturbations.
We have found that, for all backgrounds, there is one propagating
massless mode localized on the brane, associated with a
long-range dilatonic interaction in four dimensions. Only very
exceptionally (i.e. for a RS background and vanishing dilatonic
charges) this ``fifth force" disappears. This interaction is
always affected by higher-dimensional corrections, due to the
scalar massive modes that are not confined on the brane and can
freely propagate in the bulk space-time. The amplitudes of such
massive modes are constrained by the scalar perturbation
equations and, in general, only two amplitudes can be
independently assigned. The scalar fluctuation spectra are in
general different from the corresponding spectra of the spin-2
degrees of freedom. In the weak and static limit, however, the
leading-order short-range corrections to the scalar force have the
same radial dependence as in the case of pure tensor interactions.
In particular, we have found a non-trivial massive spectrum of
scalar metric fluctuations even in the pure Randall--Sundrum
scenario with AdS$_5$ metric \cite{9a}. In more general
backgrounds we have found that there are also scalar contributions
to the long-range interactions of two massive bodies, even in the
absence of specific ``dilatonic" charges, with a resulting
effective scalar-tensor interaction on the brane. In this sense,
the bulk geometry seems to affect not only the radial dependence,
but also the spin content of the effective gravitational forces.
We believe that this effect is potentially interesting for further
applications of (and constraints on)
the brane-world scenario.
\section*{Acknowledgements}
It is a plesure to thank P. Bin\'etruy, M. Giovannini, D. Langlois
and C. Ungarelli for helpful conversations. G.V. wishes to
acknowledge the support of a ``Chaire Internationale Blaise
Pascal", administered by the ``Fondation de L'Ecole Normale
Sup\'erieure'', during most of this work.
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