# Vacuum densities and the Casimir forces for branes
orthogonal to the AdS boundary

S. Bellucci1, A. A. Saharian2, V. Kh. Kotanjyan2,3

1 INFN, Laboratori Nazionali di Frascati,
Via Enrico Fermi 54, 00044 Frascati, Italy
2Department of Physics, Yerevan State University,
1 Alex Manoogian Street, 0025 Yerevan, Armenia
3Institute of Applied Problems of Physics NAS RA,
25 Hrachya Nersissyan Street, 0014 Yerevan, Armenia E-mail:
<EMAIL_ADDRESS>E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For a massive scalar field with general curvature coupling we evaluate the
Wightman function in the geometry of two parallel branes perpendicular to the
AdS boundary. On the separate branes, the field operator is constrained by
Robin boundary conditions, in general, with different coefficients. In the
region between the branes their contribution to the Wightman function is
explicitly separated. By using this decomposition, the brane-induced effects
on the vacuum expectation values (VEVs) for the field squared and energy-
momentum tensor are investigated. The behavior of those expectation values is
studied in various asymptotic regions of the parameters. The vacuum energy-
momentum tensor in addition to the diagonal components has a nonzero off-
diagonal stress. Depending on the boundary conditions and also on the distance
from the branes, the vacuum energy density can be either positive or negative.
The Casimir forces acting on the branes have two components. The first one
corresponds to the standard normal force and the second one is parallel to the
branes and presents the vacuum shear force. Unlike to the problem of parallel
plates in the Minkowski bulk, the normal Casimir forces acting on separate
branes differ if the boundary conditions on the branes are different. They can
be either repulsive or attractive. In a similar way, depending on the
coefficients in the boundary conditions, the shear force is directed toward or
from the AdS boundary. The separate components may also change their signs as
functions of the interbrane separation. At large proper separations between
the branes, compared to the AdS curvature radius, both the components of the
Casimir forces exhibit a power-law decay. For a massive scalar field this
behavior is in contrast to that for the Minkowski bulk, where the decrease is
exponential.

Keywords: Casimir effect, AdS spacetime, branes

##  1 Introduction

Motivated by fundamental questions and applications in condensed matter
physics, cosmology and in high-energy physics, the Casimir effect (for reviews
see [1]) remains to be an active field of research in quantum field theory.
The effect is an interesting manifestation of quantum fluctuations of fields
influenced by the presence of boundaries or by nontrivial spatial topology.
Depending on the model under consideration the physical nature of the
boundaries can be different. Examples are the interfaces of macroscopic bodies
in quantum electrodynamics, boundaries separating different phases of the
system, horizons in gravitational physics, branes in string theories and in
cosmological models of braneworld type, etc. The boundary and periodicity
conditions imposed on the operator of a quantum field modify the spectrum of
fluctuations and result in the shift of the expectation values of physical
quantities such as energy-momentum tensor or current densities for charged
fields.

In addition to the boundary or periodicity conditions imposed on the field,
the properties of quantum fluctuations are sensitive to the presence of
background classical fields. Those fields reduce the symmetry in respective
problems and exact results for physical characteristics in the Casimir effect
are obtained for highly symmetric bulk and boundary geometries only. In the
present paper we consider the influence of the background gravitational field
on the properties of the scalar vacuum in the geometry of two parallel branes
in background of anti-de Sitter (AdS) spacetime. That geometry is the
maximally symmetric solution of the Einstein field equations with a negative
cosmological constant as the only source of the gravitational field. As it
will be shown below, this high symmetry allows to obtain closed analytic
expressions for the expectation values characterizing the local properties of
the vacuum state. In addition to the high symmetry, our choice of AdS
spacetime as the background geometry is motivated by its important role in two
exciting developments of theoretical physics during the last decade, namely,
AdS/conformal field theory (CFT) correspondence and braneworld scenarios with
large extra dimensions. The AdS/CFT correspondence (for reviews see, e.g.,
[2]) establishes duality between two different theories: supergravity or
string theory on asymptotically AdS bulk from one side and conformal field
theory on AdS boundary from another one. Those theories live in different
numbers of spacetime dimensions and the correspondence is an example of
holographic duality. It provides an important possibility to investigate
strong coupling non-perturbative effects in one theory by mapping them to weak
coupling region of dual theory and has been applied in different physical
settings including the variety of condensed matter systems. The braneworld
paradigm [3] naturally arises in the context of supergravity and string
theories and presents an alternative to Kaluza-Klein compactification of extra
dimensions. The models formulated on AdS bulk provide a geometrical solution
for the hierarchy problem between the electroweak and gravitational energy
scales and also new perspectives and different interpretations for various
problems in particle physics and cosmology.

In the Randall-Sundrum-type realizations of the braneworld models [4] the
branes are parallel to the AdS boundary. Motivated by the radion stabilization
and the generation of cosmological constant on branes, the Casimir effect in
that setup has been widely investigated in the literature for scalar [5, 6],
fermionic [7] and vector [8] fields. In the main part of the papers, as a
physical characteristic of the vacuum, global quantities, such as the Casimir
energy or the effective potential, are investigated by using various
regularization schemes. Local observables carry more detailed information
about the properties of the vacuum state. In particular, being a source of
gravity in semiclassical Einstein equations, the vacuum expectation value
(VEV) of the energy-momentum tensor is of special importance. It is
investigated in [9, 10, 11] for scalar, fermionic and electromagnetic fields.
The combined effects of a brane and topological defect of a cosmic string type
on the local characteristics of the fermionic vacuum in AdS spacetime have
been recently considered in [12]. For charged fields, another important local
characteristic of the vacuum state, bilinear in the field, is the VEV of the
current density. It has been studied in [13] for scalar and fermionic fields
in the geometry of branes parallel to the boundary of locally AdS spacetime
with a part of spatial dimensions compactified to a torus.

Motivated by an increase of interest to conformal field theories in the
presence of boundaries (see, for example, references given in [14]), in recent
studies the AdS/CFT correspondence is extended to the problems where
boundaries are present in the conformal field theory side. In the
corresponding setup the boundary CFT is dual to a theory in AdS bulk with
additional boundaries intersecting the AdS boundary at the locations of
boundaries in CFT (AdS/BCFT correspondence) [15]. Problems with surfaces in
the AdS bulk crossing the AdS boundary have been considered in recent studies
of entanglement entropy in the context of AdS/CFT correspondence [16] (for
reviews see [17, 18]). A geometric classical procedure is suggested for
evaluation of the entanglement entropy of quantum systems living on the AdS
boundary. In accordance of that procedure, the entanglement entropy for a
bounded region in CFT is expressed in terms of the area of the minimal surface
in the AdS bulk anchored at the boundary of that region.

In the papers cited above, the physical characteristics in the Casimir effect
with branes serving as constraining boundaries, have been considered in the
context of Randall-Sundrum-type models with branes parallel to the AdS
boundary. Motivated by recent developments for physical models on AdS bulk
with boundaries crossing the spacetime boundary and continuing the
investigation started in [19], in the present paper we consider a problem with
two branes orthogonal to the AdS boundary for a massive scalar field with
general curvature coupling parameter. Though this problem is less symmetric
than the setups with branes parallel to the AdS boundary, as it will be seen
below, it is still exactly solvable.

The organization of the paper is as follows. In the next section we fix the
problem setup and present the complete set of mode functions for a scalar
field in the region between the branes. By using those functions, the positive
frequency Wightman function is evaluated in Section 3. The brane-induced
contribution is explicitly separated. Taking the coincidence limit of the
arguments in that contribution, the mean field squared is investigated in
Section 4. The behavior of the VEV in various asymptotic regions for the
values of the parameters is discussed. Similar investigations for the VEV of
the energy-momentum tensor are presented in Section 5. The Casimir forces
acting on the branes are discussed in Section 6. It is shown that for Robin
boundary conditions, in addition to the normal component, those forces have a
nonzero component parallel to the branes (shear force). The nature of the
forces is studied depending on the boundary conditions. The main results are
summarized in Section 7. In Appendix A, by using a variant of the generalized
Abel-Plana formula we provide an integral representation for the series in the
mode-sum over the eigenvalues of the quantum number describing the degree of
freedom along the direction normal to the branes.

##  2 Problem setup and the field modes

We consider a scalar field $\varphi(x)$ on the background of a
$(D+1)$-dimensional AdS spacetime with the curvature radius $\alpha$. In
Poincaré coordinates the corresponding line element is given by

$ds^{2}=g_{ik}dx^{i}dx^{k}=e^{-2y/\alpha}\left[dt^{2}-\left(dx^{1}\right)^{2}-d\mathbf{x}^{2}\right]-dy^{2},$
(2.1)

where the coordinates $\mathbf{x}=(x^{2},\ldots,x^{D-1})$ are separated for
the future convenience. In addition to the coordinate $y$,
$-\infty<y<+\infty$, we will also use the coordinate $z$, defined as $z=\alpha
e^{y/a}$, $0<z<\infty$, in terms of which the line element is written in a
manifestly conformally flat form

$ds^{2}=\left(\frac{\alpha}{z}\right)^{2}\left[dt^{2}-\left(dx^{1}\right)^{2}-d\mathbf{x}^{2}-dz^{2}\right].$
(2.2)

The AdS boundary and horizon are presented by the hypersurfaces $z=0$ and
$z=\infty$, respectively. The Ricci scalar and the cosmological constant are
expressed in terms of the AdS curvature radius by the relations
$R=-D(D+1)/\alpha^{2}$ and $\Lambda=-D(D-1)/(2\alpha^{2})$.

The operator of the scalar field with the curvature coupling constant $\xi$
obeys the equation

$\left(g^{ik}\nabla_{i}\nabla_{k}+m^{2}+\xi R\right)\varphi(x)=0.$ (2.3)

The most popular special cases correspond to minimally and conformally coupled
fields with $\xi=0$ and $\xi=\xi_{D}=(D-1)/(4D)$, respectively. We are
interested in the effects of two branes located at $x^{1}=a_{1}$ and
$x^{1}=a_{2}$ on the local properties of vacuum state for the field
$\varphi(x)$ (see Figure 1). It is assumed that on the brane at $x^{1}=a_{j}$,
$j=1,2$, the field obeys Robin boundary condition

$(A_{j}+B_{j}n_{j}^{i}\nabla_{i})\varphi(x)=0,$ (2.4)

where $n_{j}^{i}$ is the normal to the brane. The discussion in what follows
will be mainly focused on the VEVs in the region between the branes,
$a_{1}\leq x^{1}\leq a_{2}$, with
$n_{j}^{i}=(-1)^{j-1}\delta_{1}^{i}z/\alpha$. We will consider the special
case with $B_{j}/A_{j}=\alpha\beta_{j}/z$, where $\beta_{j}$, $j=1,2$, are
constants. With this choice, the boundary condition (2.4) in the region
between the branes is written as

$(1+(-1)^{j-1}\beta_{j}\partial_{1})\varphi(x)=0.$ (2.5)

Note that for a given $z$, the physical coordinate that measures the proper
distance from the branes is given by $x_{(p)}^{1}=\alpha x^{1}/z$ and the
condition (2.4) is presented as
$(1+\beta_{j}n_{j}^{1}\partial_{x_{(p)}^{1}})\varphi(x)=0$. This means that
the coefficient in the Robin boundary condition written in terms of the
coordinate $x_{(p)}^{1}$ is constant. The results for Dirichlet and Neumann
boundary conditions are obtained in the special cases $\beta_{j}=0$ and
$\beta_{j}=\infty$, respectively.

Our use of the term ”brane” for the boundaries is, in some sense, conditional.
Fundamental branes in string theory or phenomenological branes in braneworld
scenarios are among the possible physical realizations of the boundary
conditions (2.4). For example, in Randall-Sundrum type models they follow from
the $Z_{2}$-symmetry with respect to the branes and the corresponding Robin
coefficients are expressed in terms of constants in the brane mass terms of
the part of the action located on the branes (see [6, 10]). The Robin
conditions also arise on boundaries separating spatial regions with different
geometries (this type of setup on the AdS bulk has been considered in [11] to
model the finite thickness of branes). In this case the Robin coefficients are
expressed in terms of geometric characteristics of the contacting regions. The
Robin type boundary conditions were used to model the finite penetration of
the field into the boundary with the penetration length determined by the
coefficient in the boundary condition.

Figure 1: The geometry of the problem with branes intersecting the AdS
boundary.

The properties of the vacuum state in the problem under consideration are
encoded in two-point functions. Those functions are presented in the form of
sums over complete set of the field modes obeying the boundary conditions.
Those modes for a scalar field in AdS spacetime when the branes are absent are
well known in the literature. We will denote by $\lambda$ the quantum number
corresponding to the coordinate $x^{1}$. For the problem at hand the
corresponding solutions are found by combining the factors $e^{i\lambda
x^{1}}$ and $e^{-i\lambda x^{1}}$ with the relative coefficient that will be
determined from the boundary conditions. Denoting by $\sigma$ the set of
quantum numbers specifying the modes, in the region between the branes the
mode functions are written in the form

$\varphi_{\sigma}(x)=C_{\sigma}z^{D/2}J_{\nu}(\gamma
z)\cos[\lambda|x^{1}-a_{j}|+\alpha_{j}(\lambda)]e^{i\mathbf{k}\mathbf{x}-iEt},$
(2.6)

where $J_{\nu}(u)$ is the Bessel function, $\mathbf{k}=(k^{2},\dots,k^{D-1})$,
$-\infty<k^{l}<+\infty$, $0\leq\lambda<+\infty$, and the energy is expressed
as

$E=\sqrt{\lambda^{2}+\mathbf{k}^{2}+\gamma^{2}}.$ (2.7)

In (2.6) and in what follows we use the notation

$\nu=\sqrt{\frac{D^{2}}{4}-D(D+1)\xi+m^{2}\alpha^{2}},$ (2.8)

assuming that $\nu\geq 0$. This condition is dictated by the stability of the
vacuum state [20]. With the mode functions (2.6) the set of quantum numbers is
specified as $\sigma=(\lambda,\mathbf{k},\gamma)$.

From the boundary condition at $x^{1}=a_{j}$ it follows that

$e^{2i\alpha_{j}(\lambda)}=\frac{i\lambda\beta_{j}-1}{i\lambda\beta_{j}+1}.$
(2.9)

We have $\alpha_{j}(\lambda)=\pi/2$ and $\alpha_{j}(\lambda)=0$ for Dirichlet
and Neumann conditions, respectively. The boundary condition on the second
brane gives the equation that determines the eigenvalues for the quantum
number $\lambda$:

$\left(b_{1}+b_{2}\right)u\cos u+\left(u^{2}b_{1}b_{2}-1\right)\sin u=0,$
(2.10)

where $u=\lambda a$ and $b_{j}=\beta_{j}/a$. This eigenvalue equation is the
same as the corresponding equation for two parallel plates in the Minkowski
bulk, considered in [21]. As it has been discussed in [21], depending on the
values of $b_{j}$, the equation (2.10) may have single or two purely imaginary
roots with respect to $u$. In the presence of those roots and for the part of
the modes with $\mathbf{k}^{2}+\gamma^{2}<|u|^{2}/a^{2}$ the energy becomes
imaginary which signals about the instability of the vacuum state. Note that
here the situation is different from that in the corresponding problem on the
Minkowski bulk. In the latter problem the mass enters in the expression for
the energy, and for imaginary modes with $|u|/a<m$ the energy is positive for
all the modes and the vacuum is stable. To have a stable vacuum state, we will
assume the values of the parameters $b_{1}$ and $b_{2}$ for which all the
roots of the equation (2.10) are real. Those values belong to the region in
the plane $(b_{1},b_{2})$ given by $\\{b_{1}+b_{2}\geq 1,b_{1}b_{2}\leq
0\\}\cup\\{b_{1,2}\leq 0\\}$ (see [21]). We will denote by $u=u_{n}$,
$n=1,2,\ldots$, the positive roots of the equation (2.10). For the eigenvalues
of the quantum number $\lambda$ one has $\lambda=\lambda_{n}=u_{n}/a$. For
Dirichlet and Neumann boundary conditions the eigenvalue equation is reduced
to $\sin u=0$ with the modes $\lambda_{n}=\pi n/a$, where $n=1,2,\ldots$, and
$n=0,1,2,\ldots$, for the first and second cases, respectively. Note the
presence of an additional zero mode for Neumann condition. For Dirichlet
condition on one brane and the Neumann one on another from (2.10) we get $\cos
u=0$ and $\lambda_{n}=\pi(n-1/2)/a$, $n=1,2,\ldots$.

The constant $C_{\sigma}$ in (2.6) is determined from the normalization
condition

$\int
d^{D}x\sqrt{|g|}g^{00}\varphi_{\sigma}(x)\varphi_{\sigma^{\prime}}^{\ast}(x)=\frac{\delta_{nn^{\prime}}}{2E}\delta(\mathbf{k}-\mathbf{k}^{\prime})\delta(\gamma-\gamma^{\prime}).$
(2.11)

For the mode functions (2.6) this gives

$\left|C_{\sigma}\right|^{2}=\frac{\left(2\pi\right)^{2-D}\gamma}{\alpha^{D-1}aEN_{n}},$
(2.12)

with the notation

$N_{n}=1+\frac{\sin
u_{n}}{u_{n}}\cos\left[u_{n}+2\alpha_{j}(\lambda_{n})\right].$ (2.13)

Note that
$\cos\left[u_{n}+2\alpha_{1}(\lambda_{n})\right]=\cos\left[u_{n}+2\alpha_{2}(\lambda_{n})\right]$.
Having fixed the complete set of modes we pass to the evaluation of the
Wightman function.

##  3 Wightman function

As a two-point function we consider the positive frequency Wightman function
defined as the VEV $W(x,x^{\prime})=\langle
0|\varphi(x)\varphi(x^{\prime})|0\rangle$. Expanding the operators
$\varphi(x)$ and $\varphi(x^{\prime})$ in terms of the complete set
$\\{\varphi_{\sigma}(x),\varphi_{\sigma}^{\ast}(x)\\}$ and using the
definition of the vacuum state, it is written in the form of the following sum
over the modes:

$W(x,x^{\prime})=\int_{0}^{\infty}d\gamma\int
d\mathbf{k}\sum_{n=1}^{\infty}\varphi_{\sigma}(x)\varphi_{\sigma}^{\ast}(x^{\prime}).$
(3.1)

The problem is homogeneous in the subspace $(t,\mathbf{x})$ and we expect that
the dependence on the arguments in that subspace will enter in the form of the
differences $\Delta t=t-t^{\prime}$ and
$\Delta\mathbf{x=x}-\mathbf{x}^{\prime}$. Substituting the functions (2.6) and
the expression (2.12) for the normalization coefficient, the Wightman function
is expressed as

$W(x,x^{\prime})=\frac{2(zz^{\prime})^{D/2}}{(2\pi\alpha)^{D-1}a}\int
d\mathbf{k\,}e^{i\mathbf{k}\Delta\mathbf{x}}\int_{0}^{\infty}d\gamma\,\gamma
J_{\nu}(\gamma z)J_{\nu}(\gamma z^{\prime})S(b,\Delta t,x^{1},x^{\prime 1}),$
(3.2)

where $b=\sqrt{\gamma^{2}+\mathbf{k}^{2}}$, and

$S(b,\Delta t,x^{1},x^{\prime
1})=\pi\sum_{n=1}^{\infty}\frac{e^{-i\sqrt{\lambda_{n}^{2}+b^{2}}\Delta
t}}{\sqrt{\lambda_{n}^{2}+b^{2}}N_{n}}\cos{[\lambda_{n}|x}^{1}{-a_{j}|+\alpha_{j}(\lambda_{n})]}\cos{[\lambda_{n}{|x}}^{\prime}{{}^{1}{-a_{j}|}+\alpha_{j}(\lambda_{n})]}.$
(3.3)

An equivalent representation for the series over $n$ is obtained by using the
definition (2.9) for the function ${\alpha_{j}(\lambda)}$:

$S(b,\Delta t,x^{1},x^{\prime
1})=\frac{\pi}{4}\sum_{n=1}^{\infty}\frac{e^{-i\sqrt{\lambda_{n}^{2}+b^{2}}\Delta
t}}{\sqrt{\lambda_{n}^{2}+b^{2}}N_{n}}\left[2\cos\left({\lambda_{n}{\Delta
x}^{1}}\right){+}\sum_{l=\pm
1}\left({e}^{i{\lambda_{n}|{{x}^{1}{+x}}^{\prime}{{}^{1}-2{a_{j}}}|}}\frac{i\lambda\beta_{j}-1}{i\lambda\beta_{j}+1}\right)^{l}\right].$
(3.4)

For boundary conditions different from Dirichlet or Neumann ones on both the
branes, the eigenvalues ${\lambda_{n}}$ are given implicitly, as roots of
(2.10), and the representation (3.2) with (3.3) or (3.4) is not convenient for
the investigation of the local VEVs in the coincidence limit.

In order to get around this inconvenience and also to separate explicitly the
divergence in the coincidence limit, the integral representation (A.7) for the
function (3.4) is obtained in Appendix A by using the generalized Abel-Plana
formula from [22]. Substituting (A.7) in (3.2), the Wightman function is
decomposed as

$\displaystyle W(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
W_{j}(x,x^{\prime})+\frac{(zz^{\prime})^{\frac{D}{2}}}{(2\pi\alpha)^{D-1}}\int
d\mathbf{k\,}e^{i\mathbf{k}\Delta\mathbf{x}}\int_{0}^{\infty}d\gamma\,\gamma
J_{\nu}(\gamma z)J_{\nu}(\gamma z^{\prime})$ (3.5)
$\displaystyle\times\int_{b}^{\infty}d\lambda\frac{\cosh(\sqrt{\lambda^{2}-b^{2}}\Delta
t)}{\sqrt{\lambda^{2}-b^{2}}}\frac{2\cosh\left({\lambda{x}_{-}^{1}}\right){+}\sum_{l=\pm
1}\left[{e}^{{|{{x}_{+}^{1}-2{a_{j}}}|\lambda}}c_{j}(\lambda)\right]^{l}}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1},$

where and in what follows $x_{\pm}^{1}=x^{1}\pm x^{\prime 1}$ and

$c_{j}(\lambda)=\frac{\beta_{j}\lambda-1}{\beta_{j}\lambda+1}.$ (3.6)

Note that $c_{j}(\lambda)=\tilde{c}_{j}(\lambda a)$, with the functions
$\tilde{c}_{j}(u)$ defined in Appendix A after formula (A.1). In (3.5) we have
defined the two-point function

$\displaystyle W_{j}(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
W_{0}(x,x^{\prime})+\frac{(zz^{\prime})^{\frac{D}{2}}}{(2\pi\alpha)^{D-1}}\int
d\mathbf{k\,}e^{i\mathbf{k}\Delta\mathbf{x}}\int_{0}^{\infty}d\gamma\,\gamma
J_{\nu}(\gamma z)J_{\nu}(\gamma z^{\prime})$ (3.7)
$\displaystyle\times\int_{b}^{\infty}d\lambda\frac{\cosh(\sqrt{\lambda^{2}-b^{2}}\Delta
t)}{\sqrt{\lambda^{2}-b^{2}}}\frac{e^{-{|{{x}_{+}^{1}-2{a_{j}}}|\lambda}}}{c_{j}(\lambda)}.$

Here, the part $W_{0}(x,x^{\prime})$ comes from the term $S_{0}(b,\Delta
t,{{x}_{-}^{1}})$ in (A.4) and is given by

$W_{0}(x,x^{\prime})=\frac{(zz^{\prime})^{\frac{D}{2}}}{2(2\pi\alpha)^{D-1}}\int
d\mathbf{K\,}e^{i\mathbf{K}\Delta\mathbf{X}}\int_{0}^{\infty}d\gamma\,\gamma
J_{\nu}(\gamma z)J_{\nu}(\gamma
z^{\prime})\frac{e^{-i\sqrt{\gamma^{2}+\mathbf{K}^{2}}\Delta
t}}{\sqrt{\gamma^{2}+\mathbf{K}^{2}}},$ (3.8)

with $\mathbf{X}=(x^{1},\mathbf{x})$,
$\mathbf{K}=(k^{1},k^{2},\ldots,k^{D-1})$. The integration over the angular
coordinates of the vector $\mathbf{k}$ in (3.5) and (3.7) (and in a similar
way for (3.8)) can be done by using the formula

$\int
d\mathbf{k\,}e^{i\mathbf{k}\Delta\mathbf{x}}g(k)=\frac{\left(2\pi\right)^{\frac{D}{2}-1}}{|\Delta\mathbf{x}|^{\frac{D}{2}-2}}\int_{0}^{\infty}dk\,k^{\frac{D}{2}-1}J_{\frac{D}{2}-2}(k|\Delta\mathbf{x}|)g(k),$
(3.9)

for a given function $g(k)$, where $k=|\mathbf{k}|$.

The separate terms in (3.5) and (3.7) have clear physical interpretation. The
function $W_{0}(x,x^{\prime})$ is the Wightman function in AdS spacetime in
the absence of the branes. Its expression in terms of the hypergeometric
function is well known from the literature (see below). As it has been
mentioned in Appendix A, the last term in (3.5) vanishes in the limit
$(-1)^{j^{\prime}}a_{j^{\prime}}\rightarrow+\infty$, where $j^{\prime}=1$ for
$j=2$ and $j^{\prime}=2$ for $j=1$. Hence, the function $W_{j}(x,x^{\prime})$
corresponds to the Wightman function in the problem with a single brane at
$x^{1}=a_{j}$. It has been obtained in [19]. The last term in (3.5) is
interpreted as a contribution induced by the second brane at
$x^{1}=a_{j^{\prime}}$ when we add it to the geometry with a brane at
$x^{1}=a_{j}$. The representation of the Wightman function with combined
contributions from the branes is obtained from (A.8):

$\displaystyle W(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
W_{0}(x,x^{\prime})+\frac{(zz^{\prime})^{\frac{D}{2}}}{(2\pi\alpha)^{D-1}}\int
d\mathbf{k\,}e^{i\mathbf{k}\Delta\mathbf{x}}\int_{0}^{\infty}d\gamma\,\gamma
J_{\nu}(\gamma z)J_{\nu}(\gamma z^{\prime})$ (3.10)
$\displaystyle\times\int_{b}^{\infty}d\lambda\frac{\cosh(\sqrt{\lambda^{2}-b^{2}}\Delta
t)}{\sqrt{\lambda^{2}-b^{2}}}\frac{2\cosh\left({\lambda{x}_{-}^{1}}\right){+}\sum_{j=1,2}{e}^{{|{{x}_{+}^{1}-2{a_{j}}}|\lambda}}c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}.$

For the angular part of the integral over $\mathbf{k}$ we can use the relation
(3.9).

In the representations (3.5) and (3.10) the explicit knowledge of the
eigenvalues $\lambda_{n}$ is not required and they are well adapted for the
investigation of local VEVs. Those representations give the Wightman function
in the region between the branes. In the regions $x^{1}<a_{1}$ and
$x^{1}>a_{2}$ the Wightman functions coincide with that for the problem with a
single brane and they are given by (3.7) with $j=1$ and $j=2$ in the first and
second regions respectively.

For the special cases of Dirichlet and Neumann boundary conditions we have
$c_{j}(\lambda)=-\delta_{\mathrm{J}}$, where
$\mathrm{J}=\mathrm{D},\mathrm{N}$ correspond to Dirichlet and Neuamann
boundary conditions with $\delta_{\mathrm{D}}=1$, $\delta_{\mathrm{N}}=-1$.
The part with the exponential function is reduced to $1/(e^{2a\lambda}-1)$.
Presenting this function as the series $\sum_{n=1}^{\infty}e^{-2na\lambda}$,
the integral over $\lambda$ is expressed in terms of the modified Bessel
function $K_{0}(u)$ (see [23]). Next, we use the result (3.9) for the integral
over the angular coordinates of $k$. The integral over $k$ is expressed
through the associated Legendre function $Q_{\beta}^{\mu}(x)$ and the final
expression reads

$W(x,x^{\prime})=W_{0}(x,x^{\prime})+\frac{\alpha^{1-D}}{2^{\frac{D}{2}+\nu+1}\pi^{\frac{D}{2}}}\sum_{n=1}^{\infty}\left[\sum_{l=\pm
1}f_{\nu}(u_{l,n}^{(-)}){-}\delta_{\mathrm{J}}\sum_{j=1,2}f_{\nu}(u_{j,n}^{(+)})\right],$
(3.11)

with the notations

$\displaystyle u_{l,n}^{(-)}$ $\displaystyle=$ $\displaystyle
1+\frac{\left(2nla-{{x}_{-}^{1}}\right)^{2}+|\Delta\mathbf{x}|^{2}+\Delta
z^{2}-\Delta t^{2}}{2zz^{\prime}},$ $\displaystyle u_{j,n}^{(+)}$
$\displaystyle=$ $\displaystyle
1+\frac{\left(2na-{|{{x}_{+}^{1}-2{a_{j}}}|}\right)^{2}+|\Delta\mathbf{x}|^{2}+\Delta
z^{2}-\Delta t^{2}}{2zz^{\prime}},$ (3.12)

and $\Delta z=z-z^{\prime}$. In (3.11) we have defined the function

$\displaystyle f_{\nu}(u)$ $\displaystyle=$
$\displaystyle\frac{2^{\nu+\frac{1}{2}}}{\sqrt{\pi}}e^{-\frac{D-1}{2}\pi
i}\frac{Q_{\nu-\frac{1}{2}}^{\frac{D-1}{2}}(u)}{\left(u^{2}-1\right)^{\frac{D-1}{4}}}$
(3.13) $\displaystyle=$
$\displaystyle\frac{\Gamma\left(\nu+\frac{D}{2}\right)}{\Gamma\left(\nu+1\right)u^{\nu+\frac{D}{2}}}\,F\left(\frac{2+2\nu+D}{4},\frac{2\nu+D}{4};\nu+1;\frac{1}{u^{2}}\right),$

with $F(a,b;c;x)\equiv\,_{2}F_{1}(a,b;c;x)$ being the hypergeometric function.

The Wightman function for a scalar field in the brane-free AdS spacetime is
expressed in terms of the function $f_{\nu}(u)$ as

$W_{0}(x,x^{\prime})=\frac{\alpha^{1-D}f_{\nu}(u_{0,0}^{(-)})}{2^{\frac{D}{2}+\nu+1}\pi^{\frac{D}{2}}},$
(3.14)

and the formula (3.11) presents the Wightman function in the region between
the branes in the form of the image sum. In the spacetime region
$\left({{x}_{-}^{1}}\right)^{2}+|\Delta\mathbf{x}|^{2}+\Delta z^{2}>\Delta
t^{2}$ one has the relation
$u_{0,0}^{(-)}=\cosh\left(\sigma(x,x^{\prime})/\alpha\right)$ with
$\sigma(x,x^{\prime})$ being the geodesic distance between the spacetime
points $x$ and $x^{\prime}$. In the cases of Dirichlet and Neumann boundary
conditions, the Wightman function for the geometry of a single brane at
$x^{1}=a_{j}$ is obtained from (3.11) in the limit
$(-1)^{j^{\prime}}a_{j^{\prime}}\rightarrow+\infty$. In the series over $n$
the contribution of the term $n=1$, $j=j^{\prime}$ survives only and we get
the result obtained in [19]:

$W_{j}(x,x^{\prime})=W_{0}(x,x^{\prime})-\frac{\delta_{\mathrm{J}}\alpha^{1-D}}{2^{\frac{D}{2}+\nu+1}\pi^{\frac{D}{2}}}f_{\nu}\left(1+\frac{\left|{{{x}_{+}^{1}}}-2a_{j}\right|^{2}+|\Delta\mathbf{x}|^{2}+\Delta
z^{2}-\Delta t^{2}}{2zz^{\prime}}\right).$ (3.15)

We can also consider the problem with Dirichlet boundary condition on the
brane $x^{1}=a_{1}$ and Neumann condition on the second brane. In this case
$c_{j}(\lambda)=(-1)^{j}$ and the Wightman function is obtained in a way
similar to the cases of Dirichlet and Neumann conditions on both the branes.
It can be seen that the corresponding expression is obtained from (3.11) by
the replacements (the replacement of $\delta_{\mathrm{J}}$ should be made
after the summation sign $\sum_{j=1,2}$)

$\sum_{n=1}^{\infty}\rightarrow\sum_{n=1}^{\infty}(-1)^{n},\;\delta_{\mathrm{J}}\rightarrow(-1)^{j-1}.$
(3.16)

In the regions $x^{1}<a_{1}$ and $x^{1}>a_{2}$ the Wightman functions for the
Dirichlet-Neumann combination of boundary conditions are given by (3.15) with
$\delta_{\mathrm{J}}=1$ and $\delta_{\mathrm{J}}=-1$, respectively.

##  4 VEV of the field squared

In this section we investigate the VEV of the field squared $\langle
0|\varphi^{2}|0\rangle\equiv\langle\varphi^{2}\rangle$. It is obtained taking
the coincidence limit of the arguments in the Wightman function. Of course,
that limit is divergent and a renormalization procedure is required to extract
finite physical values. Here we are interested in the effects induced by the
branes. For points outside the branes the local geometry is the same as that
for AdS spacetime without branes. The divergences in the coincidence limit are
determined by the local geometrical characteristics and we conclude that for
$x^{1}\neq a_{j}$, $j=1,2$, they are the same as in AdS spacetime. Having
extracted the part in the Wightman function corresponding to the latter
geometry (the function $W_{0}(x,x^{\prime})$), the renormalization is reduced
to that for brane-free AdS spacetime. That procedure for the VEVs of the field
squared and of the energy-momentum tensor is well investigated in the
literature.

Taking the coincidence limit $x^{\prime}\rightarrow x$ in (3.10), the VEV of
the field squared is presented as

$\displaystyle\langle\varphi^{2}\rangle$ $\displaystyle=$
$\displaystyle\langle\varphi^{2}\rangle_{0}+\frac{2^{2-D}\alpha^{1-D}z^{D}}{\pi^{\frac{D}{2}}\Gamma\left(\frac{D}{2}-1\right)}\int_{0}^{\infty}dk\,\,k^{D-3}\int_{0}^{\infty}d\gamma\,\gamma
J_{\nu}^{2}(\gamma z)$ (4.1)
$\displaystyle\times\int_{b}^{\infty}d\lambda\,\frac{2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|\lambda}}c_{j}(\lambda)}{\left[c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1\right]\sqrt{\lambda^{2}-b^{2}}},$

where $\langle\varphi^{2}\rangle_{0}$ is the renormalized VEV in AdS spacetime
when the branes are absent. Because of the maximal symmetry of AdS geometry
the part $\langle\varphi^{2}\rangle_{0}$ does not depend on the spacetime
point and it is well investigated in the literature. For further
transformation of the brane-induced contribution in (4.1), instead of
$\lambda$ we introduce a new integration variable $\chi=$
$\sqrt{\lambda^{2}-b^{2}}$ and then pass to polar coordinates in the plane
$(k,\chi)$. After integrating over the angular part one finds

$\langle\varphi^{2}\rangle=\langle\varphi^{2}\rangle_{0}+\frac{\left(2\sqrt{\pi}\alpha\right)^{1-D}z^{D}}{\Gamma\left(\frac{D-1}{2}\right)}\int_{0}^{\infty}dr\,r^{D-2}\,\int_{0}^{\infty}d\gamma\,\frac{\gamma}{\lambda}\frac{2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|\lambda}}c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}J_{\nu}^{2}(\gamma
z),$ (4.2)

where $\lambda=\sqrt{\gamma^{2}+r^{2}}$. Introducing polar coordinates in the
plane $(r,\gamma)$, for the angular integral we use the result [24]

$\int_{0}^{1}dxx(1-x^{2})^{\mu-3/2}J_{\nu}^{2}(ux)=\frac{\Gamma(\mu-1/2)}{2^{2\nu+1}}u^{2\nu}F_{\nu}^{\mu}(u),$
(4.3)

with the function

$F_{\nu}^{\mu}(u)=\frac{\,{}_{1}F_{2}(\nu+\frac{1}{2};\mu+\nu+\frac{1}{2},1+2\nu;-u^{2})}{\Gamma(\mu+\nu+\frac{1}{2})\Gamma(1+\nu)}.$
(4.4)

Here, ${}_{1}F_{2}(a;b,c;z)$ is the hypergeometric function. The final
expression reads

$\langle\varphi^{2}\rangle=\langle\varphi^{2}\rangle_{0}+\frac{\left(\sqrt{\pi}\alpha\right)^{1-D}}{2^{D+2\nu}}\int_{0}^{\infty}dx\,x^{D+2\nu-1}F_{\nu}^{D/2}(x)\frac{2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|x/z}}c_{j}(x/z)}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}.$
(4.5)

In a similar way, by making use of the formula (3.5), we can obtain the
representation

$\langle\varphi^{2}\rangle=\langle\varphi^{2}\rangle_{j}+\frac{\left(\sqrt{\pi}\alpha\right)^{1-D}}{2^{D+2\nu}}\int_{0}^{\infty}dx\,x^{D+2\nu-1}F_{\nu}^{D/2}(x)\frac{2{+}\sum_{l=\pm
1}\left[{e}^{2{|{{x}^{1}-{a_{j}}}|}x/z}c_{j}(x/z)\right]^{l}}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1},$
(4.6)

where the VEV in the geometry of a single brane at $x^{1}=a_{j}$ is expressed
as (see [19])

$\langle\varphi^{2}\rangle_{j}=\langle\varphi^{2}\rangle_{0}+\frac{\left(\sqrt{\pi}\alpha\right)^{1-D}}{2^{D+2\nu}}\int_{0}^{\infty}dx\,x^{D+2\nu-1}F_{\nu}^{D/2}(x)\frac{e^{-2{|{{x}^{1}-{a_{j}}}|x/z}}}{c_{j}({x/z})}.$
(4.7)

Note that the product $\alpha^{D-1}\langle\varphi^{2}\rangle$ depends on the
quantities having dimension of length ($x^{1}$, $a_{j}$, $\beta_{j}$) and on
the coordinate $z$ through the ratios $x^{1}/z$, $a_{j}/z$, $\beta_{j}/z$.
Those ratios are the proper values of the quantities measured by an observer
with fixed $z$ in units of the curvature radius $\alpha$. This feature is a
consequence of the AdS maximal symmetry.

For a conformally coupled massless field one has $\nu=1/2$ and

$F_{1/2}^{\mu}(u)=\frac{2}{\sqrt{\pi}u^{2}}\left[\frac{1}{\Gamma\left(\mu\right)}-\frac{J_{\mu-1}(2u)}{u^{\mu-1}}\right].$
(4.8)

For the VEV of the field squared this gives

$\langle\varphi^{2}\rangle=\langle\varphi^{2}\rangle_{0}+\left(z/\alpha\right)^{D-1}\langle\varphi^{2}\rangle_{\mathrm{(M)}},$
(4.9)

where

$\langle\varphi^{2}\rangle_{\mathrm{(M)}}=\frac{1}{2^{D}\pi^{\frac{D}{2}}}\int_{0}^{\infty}d\lambda\,\lambda^{D-2}\left[\frac{1}{\Gamma(D/2)}-\frac{J_{D/2-1}(2z\lambda)}{(z\lambda)^{D/2-1}}\right]\frac{2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|\lambda}}c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}.$
(4.10)

The background geometry under consideration is conformally flat and the last
term in (4.9) exhibits the standard conformal relation between the boundary-
induced parts of the VEVs in two conformally related problems (see, for
example, [25]). The geometry of two branes in AdS spacetime is conformally
connected to the problem in the Minkowski spacetime with the line element

$ds_{\mathrm{M}}^{2}=dt^{2}-\left(dx^{1}\right)^{2}-d\mathbf{x}^{2}-dz^{2},$
(4.11)

involving two parallel Robin plates at $x^{1}=a_{1}$ and $x^{1}=a_{2}$
intersected by the plate $z=0$ with Dirichlet boundary condition. The latter
plate is the conformal image of the AdS boundary. Note that the part in (4.10)
coming from the first term in the square brackets gives the mean field squared
in the region between two parallel plates in the Minkowski spacetime (the
boundary at $z=0$ is absent) and the part with the second term is induced by
the Dirichlet plate at $z=0$.

Note that the Dirichlet boundary condition at $z=0$ in the conformally related
problem on the Minkowski bulk is related to the condition we have imposed for
the scalar modes (2.6) on the AdS boundary. For the values of the parameter
$\nu$ in the range $0\leq\nu<1$ the general normalizable solution of the field
equation has the form (2.6) with the Bessel function replaced by the linear
combination $J_{\nu}(\gamma z)+b_{\sigma}Y_{\nu}(\gamma z)$, where
$Y_{\nu}(x)$ is the Neumann function. In this case an additional boundary
condition is required on the AdS boundary for unique fixation of the set of
modes. Our choice in (2.6) corresponds to Dirichlet condition. In the
literature the Neumann and more general Robin boundary conditions have been
considered as well (for recent discussions see [26]). In the conformally
related problem on the Minkowski spacetime, the boundary condition on the
$z=0$ image is determined by the respective condition on the AdS boundary.
Note that the different boundary conditions will correspond to different
conformal field theories in the context of the AdS/CFT correspondence.

Let us consider the Minkowskian limit of the problem at hand. It corresponds
to the limit $\alpha\rightarrow\infty$ for fixed value of the coordinate $y$
in (2.1). Introducing in (4.5) a new integration variable $\lambda=x/z$ and by
taking into account that in the limit under consideration $z\approx\alpha$ and
$\nu\approx m\alpha$, we see that both the argument and the order $\nu$ of the
function $F_{\nu}^{D/2}(\lambda z)$ are large. The uniform asymptotic
expansion is obtained in [19] by using the corresponding expansion for the
Bessel function in (4.3). It has been shown that for large $\nu$ and
$\lambda<m$ the function $F_{\nu}^{D/2}(\nu\lambda/m)$ is exponentially small.
The VEV of the field squared is dominated by the contribution of the integral
coming from the region $\lambda>m$. In that region the leading term in the
expansion over $1/\nu$ is given by [19]

$F_{\nu}^{\mu}\left(\frac{\nu}{m}\lambda\right)\approx\frac{\left(\lambda^{2}-m^{2}\right)^{\mu-1}(2m/\nu)^{2\nu+1}}{2\sqrt{\pi}\Gamma(D/2)\lambda^{2\mu+2\nu-1}}.$
(4.12)

With this estimate we get
$\lim_{\alpha\rightarrow\infty}\langle\varphi^{2}\rangle=\langle\varphi^{2}\rangle_{\mathrm{(M)}}^{(0)}$,
where

$\langle\varphi^{2}\rangle_{\mathrm{(M)}}^{(0)}=\frac{\left(4\pi\right)^{-\frac{D}{2}}}{\Gamma(D/2)}\int_{m}^{\infty}d\lambda\,\left(\lambda^{2}-m^{2}\right)^{D/2-1}\frac{2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|\lambda}}c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1},$
(4.13)

is the mean field squared in the region between two Robin plates in background
of Minkowski spacetime with the line element (4.11). This result for a massive
field was presented in [22]. For a massless field it is reduced to the result
derived in [21].

In order to find the mean field squared on AdS bulk in the special cases of
Dirichlet and Neumann boundary conditions we can use the representation (3.11)
for the Wightman function. The corresponding expression reads

$\langle\varphi^{2}\rangle=\langle\varphi^{2}\rangle_{0}+\frac{\alpha^{1-D}}{2^{\frac{D}{2}+\nu+1}\pi^{\frac{D}{2}}}\sum_{n=1}^{\infty}\left[2f_{\nu}(u_{n}){-}\delta_{\mathrm{J}}\sum_{j=1,2}f_{\nu}(u_{j,n})\right],$
(4.14)

with the notations

$\displaystyle u_{n}$ $\displaystyle=$ $\displaystyle
1+2\left(na/z\right)^{2},$ $\displaystyle u_{j,n}$ $\displaystyle=$
$\displaystyle 1+\frac{2}{z^{2}}\left(na-{|{{x}^{1}-{a_{j}}}|}\right)^{2}.$
(4.15)

An alternative representation is obtained from (4.5) expanding the function
$1/(e^{2ax/z}-1)$. The integral is evaluated by using the formula from [27]:

$\int_{0}^{\infty}dx\,x^{2\mu+2\nu-1}e^{-2cx}F_{\nu}^{\mu}(x)=\frac{h_{\nu}^{\mu}(c)}{2\sqrt{\pi}},$
(4.16)

where the function in the right-hand side is defined as

$h_{\nu}^{\mu}(u)=\frac{\Gamma\left(\mu+\nu\right)}{\Gamma(\nu+1)}\frac{1}{u^{2\left(\mu+\nu\right)}}F\left(\nu+\frac{1}{2},\mu+\nu;1+2\nu;-\frac{1}{u^{2}}\right).$
(4.17)

The VEV is presented as

$\langle\varphi^{2}\rangle=\langle\varphi^{2}\rangle_{0}+\frac{\alpha^{1-D}}{2^{D+2\nu+1}\pi^{\frac{D}{2}}}\sum_{n=1}^{\infty}\left[2h_{\nu}^{\frac{D}{2}}\left(\frac{na}{z}\right){-}\delta_{\mathrm{J}}\sum_{j=1,2}h_{\nu}^{\frac{D}{2}}\left(\frac{na-{|{{x}^{1}-{a_{j}}}|}}{z}\right)\right].$
(4.18)

By employing the linear and quadratic transformation formulas for the
hypergeometric function (see, for example, [28]) we can see that

$h_{\nu}^{\frac{D}{2}}(x)=2^{\frac{D}{2}+\nu}f_{\nu}(1+2x^{2}).$ (4.19)

This relation shows the equivalence of the representations (4.14) and (4.18).
Note that for a conformally coupled massless field

$h_{1/2}^{\mu}(x)=\frac{4\Gamma\left(\mu+1/2\right)}{\sqrt{\pi}\left(2\mu-1\right)}\left[x^{1-2\mu}-\left(x^{2}+1\right)^{\frac{1}{2}-\mu}\right].$
(4.20)

For Dirichlet or Neumann boundary conditions, the VEV in the problem with a
single brane is obtained from (4.18) taking the limit
$a_{1}\rightarrow-\infty$ or $a_{2}\rightarrow+\infty$:

$\langle\varphi^{2}\rangle_{j}=\langle\varphi^{2}\rangle_{0}-\frac{\delta_{\mathrm{J}}\alpha^{1-D}}{2^{D+2\nu+1}\pi^{\frac{D}{2}}}h_{\nu}^{\frac{D}{2}}\left(\frac{\left|{{{x}^{1}}}-a_{j}\right|}{z}\right).$
(4.21)

In problems with two scalar fields with Dirichlet and Neumann conditions on a
single brane, the brane-induced mean field squared vanishes as a result of
cancelations of contributions from Dirichlet and Neumann scalars. In
particular, for $D=3$, the electromagnetic field with perfectly conducting
boundary condition on the brane is reduced to two scalar modes with Dirichlet
and Neumann conditions and their contributions in the vacuum energy density
cancel each other. An equivalent representation for the single brane mean
field squared $\langle\varphi^{2}\rangle_{j}$, given in [19], is derived from
(3.15) in the coincidence limit. In (4.14) and (4.18), the parts corresponding
to the contribution of the brane at $x^{1}=a_{j^{\prime}\text{,}}$ when the
second brane is absent, are presented by the term $n=1$, $j=2$ for
$j^{\prime}=1$ and by the term $n=1$, $j=1$ for $j^{\prime}=2.$

In the case of Dirichlet boundary condition on the brane $x^{1}=a_{1}$ and
Neumann boundary condition on $x^{1}=a_{2}$ the expression for the mean field
squared in the region between the branes is obtained from (4.18) making the
replacements (3.16).

Now let us consider the behavior of the VEV $\langle\varphi^{2}\rangle$ in
asymptotic regions of the parameters. The VEV diverges on the branes. The
divergences come from the single brane contributions: in the representation
(4.6) the divergence at $x^{1}=a_{j}$ is contained in the part
$\langle\varphi^{2}\rangle_{j}$ (in the last term of (4.7)). Near the brane,
for $\left|{{{x}^{1}}}-a_{j}\right|\ll z$, the total VEV
$\langle\varphi^{2}\rangle$ is dominated by the last term in (4.7). Assuming
additionally $\left|{{{x}^{1}}}-a_{j}\right|\ll|\beta_{j}|$ (non-Dirichlet
boundary conditions), the leading term in the expansion over the distance from
the brane reads [19]

$\langle\varphi^{2}\rangle\approx\frac{\Gamma\left(\frac{D-1}{2}\right)}{\left(4\pi\right)^{\frac{D+1}{2}}}\left(\frac{z}{\alpha\left|{{{x}^{1}}}-a_{j}\right|}\right)^{D-1}.$
(4.22)

For the Dirichlet boundary condition the corresponding asymptotic differs from
(4.22) by the sign of the right-hand side. The last term in the representation
(4.6) is finite on the brane ${{{x}^{1}}}=a_{j}$.

For points near the AdS boundary, $z\ll|x^{1}-a_{j}|$, $j=1,2$, the main
contribution to the integral in (4.5) comes from the region with small values
of $x$. By using the asymptotic expression $F_{\nu}^{\mu}(x)\approx
F_{\nu}^{\mu}(0)\left(1+\mathcal{O}(x^{2})\right)$ with

$F_{\nu}^{\mu}(0)=\frac{1}{\Gamma\left(\nu+1\right)\Gamma\left(\mu+\nu+\frac{1}{2}\right)},$
(4.23)

in the leading order, for the brane-induced contribution we get

$\langle\varphi^{2}\rangle\approx\langle\varphi^{2}\rangle_{0}+\frac{F_{\nu}^{\frac{D}{2}}(0)z^{D+2\nu}}{2^{D+2\nu}\left(\sqrt{\pi}\alpha\right)^{D-1}}\int_{0}^{\infty}d\lambda\,\lambda^{D+2\nu-1}\frac{2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|\lambda}}c_{j}({\lambda})}{c_{1}({\lambda})c_{2}({\lambda})e^{2a{\lambda}}-1}.$
(4.24)

Hence, for points near the AdS boundary and not too close to the branes, the
brane-induced part in the mean field squared tends to zero like $z^{D+2\nu}$.
For points near the horizon, $z\gg a$, the integral in (4.5) is dominated by
the contribution coming from the region with large values of $x$. For those
$x$ one has [19]

$F_{\nu}^{\mu}(x)\approx\frac{2^{2\nu}}{\sqrt{\pi}\Gamma\left(\mu\right)x^{2\nu+1}},\;x\gg
1,$ (4.25)

and the VEV of the field squared is approximated by

$\langle\varphi^{2}\rangle\approx\langle\varphi^{2}\rangle_{0}+\left(z/\alpha\right)^{D-1}\langle\varphi^{2}\rangle_{\mathrm{(M)}}^{(0)}|_{m=0},$
(4.26)

where $\langle\varphi^{2}\rangle_{\mathrm{(M)}}^{(0)}|_{m=0}$ (see (4.13)) is
the corresponding VEV for a massless scalar field between two parallel plates
in the Minkowski bulk with separation $a$ [21]. Note that the latter is
obtained from (4.10) in the limit $z\rightarrow\infty$. As seen, near the
horizon the effects of the curvature on the brane-induced VEV are weak. Note
that for a given $a$ and large $z$ the proper separation between the branes is
much smaller than the curvature radius, $a_{p}=\alpha a/z\ll\alpha$, and the
main contribution to the brane-induced VEV comes from the vacuum fluctuations
with the wavelengths much smaller than the curvature radius. The influence of
the gravitational field on those fluctuations is weak.

In Figure 2 the brane-induced VEV of the field squared,
$\langle\varphi^{2}\rangle_{\mathrm{b}}=\langle\varphi^{2}\rangle-\langle\varphi^{2}\rangle_{0}$,
is plotted in the region between the branes as a function of the proper
distance from the brane at $x^{1}=0$ (in units of the AdS curvature radius
$\alpha$). For the location of the second brane we have taken $a_{2}/z=5$. The
graphs are plotted for $D=4$ conformally (left panel) and minimally (right
panel) coupled massive scalar fields with $m\alpha=0.5$. The same boundary
conditions are imposed on the branes ($\beta_{1}=\beta_{2}$) and the numbers
near the curves indicate the respective values of the ratio $\beta_{1}/z$. The
graphs for Dirichlet and Neumann boundary conditions are presented as well
(Dir and Neu, respectively). The brane-induced mean field squared is negative
for the Dirichlet case and positive for the Neumann one. For Robin conditions
with sufficiently small values of $|\beta_{j}|/z$, the VEV is positive near
the branes and negative in the region near the center with respect to the
branes. With increasing $|\beta_{j}|/z$, started from some critical value, the
VEV $\langle\varphi^{2}\rangle_{\mathrm{b}}$ becomes positive everywhere in
the region between the branes. For the example presented in Figure 2, for the
critical values one has $\beta_{j}/z\approx-1.08$ and
$\beta_{j}/z\approx-0.70$ for conformally and minimally coupled scalars,
respectively. The critical value for $|\beta_{j}|/z$ decreases with decreasing
$a/z$.

|
---|---
Figure 2: The brane-induced mean field squared in the region between the
branes as a function of the ratio $x^{1}/z$ for $D=4$ fields with conformal
and minimal couplings (left and right panels, respectively). The graphs are
plotted for $m\alpha=0.5$ and the numbers near the curves correspond to the
values of the ratio $\beta_{1}/z=\beta_{2}/z$. The graphs for Robin boundary
conditions are located in the region between the curves corresponding to
Dirichlet and Neumann conditions.

##  5 Energy-momentum tensor

Another important characteristic of the vacuum state is the VEV of the energy-
momentum tensor. With the known Wightman function and the VEV of the field
squared, it is evaluated by using the formula

$\langle T_{ik}\rangle=\frac{1}{2}\lim_{x^{\prime}\rightarrow
x}\left({\partial_{i}\partial_{k}^{\prime}+\partial_{k}\partial_{i}^{\prime}}\right)W(x,x^{\prime})+\hat{B}_{ik}\langle\varphi^{2}\rangle,$
(5.1)

where the operator acting on the VEV of the field squared is defined by

$\hat{B}_{ik}=\left(\xi-\frac{1}{4}\right)g_{ik}g^{lm}\nabla_{l}\nabla_{m}-\xi\left(\nabla_{i}\nabla_{k}+R_{ik}\right),$
(5.2)

with $R_{ik}=-Dg_{ik}/\alpha^{2}$ being the Ricci tensor for AdS spacetime. In
the geometry at hand one gets

$\hat{B}_{00}=\left(\frac{1}{4}-\xi\right)\left(\partial_{1}^{2}+\partial_{z}^{2}-\frac{D-1}{z}\partial_{z}\right)-\frac{\xi}{z}\partial_{z}+\frac{D}{z^{2}}\xi,$
(5.3)

and the spatial diagonal components are expressed as
$\hat{B}_{ll}=-\hat{B}_{00}-\hat{C}_{ll}$, $l=1,2,\ldots,D$, where

$\displaystyle\hat{C}_{11}$ $\displaystyle=$
$\displaystyle\xi\partial_{1}^{2},\;\hat{C}_{ll}=0,\;l=2,\ldots,D-1,$
$\displaystyle\hat{C}_{DD}$ $\displaystyle=$
$\displaystyle\xi\left(\partial_{z}^{2}+\frac{2}{z}\partial_{z}\right).$ (5.4)

In addition to the diagonal components, the action of the operator (5.2) on
$\langle\varphi^{2}\rangle$ gives a nonzero off-diagonal component
$\hat{B}_{1D}\langle\varphi^{2}\rangle$ with the operator

$\hat{B}_{1D}=-\xi\left(\partial_{z}+\frac{1}{z}\right)\partial_{1}.$ (5.5)

By using the representation (3.10), the coincidence limit of the bitensor
${\partial_{i}\partial_{k}^{\prime}}W(x,x^{\prime})$ is evaluated in a way we
have described above for the mean field squared. For the diagonal components
$\langle T_{ll}\rangle$ with $l\neq D$ the angular integrals at the last step
are expressed in terms of the functions $F_{\nu}^{D/2}(u)$ and
$F_{\nu}^{D/2+1}(u)$. For the component $\langle T_{DD}\rangle$ the integral
is reduced to

$\int_{0}^{1}dx\,x\left(1-x^{2}\right)^{\frac{D-3}{2}}\left[\partial_{u}\left(u^{\frac{D}{2}}J_{\nu}(ux)\right)\right]^{2}.$
(5.6)

By making use of the equation for the Bessel function this integral is
expressed in terms of the functions $F_{\nu}^{D/2}(u)$, $F_{\nu}^{D/2+1}(u)$
and of the first and second derivatives of $F_{\nu}^{D/2}(u)$.

After long but straightforward calculations, the VEVs of the diagonal
components of the energy-momentum tensor are written in the form (no summation
over $i$)

$\displaystyle\langle T_{i}^{i}\rangle$ $\displaystyle=$ $\displaystyle\langle
T_{i}^{i}\rangle_{0}-\frac{\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\,x\left\\{\frac{E_{i}x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x)}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}\right.$
(5.7)
$\displaystyle\left.+\frac{2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|x/z}}c_{j}(x/z)}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}\left[A_{i}x^{D+2\nu}F_{\nu}^{\frac{D}{2}+1}(x)+\hat{B}_{i}x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x)\right]\right\\}.$

Here we have defined the operators

$\displaystyle\hat{B}_{1}$ $\displaystyle=$
$\displaystyle\left(\xi-\frac{1}{4}\right)\partial_{x}^{2}+\left[\frac{D-1}{4}-\left(D-2\right)\xi\right]\frac{\partial_{x}}{x}-\frac{D\xi}{x^{2}},$
$\displaystyle\hat{B}_{i}$ $\displaystyle=$
$\displaystyle\hat{B}_{1}+4\xi-1,\;i=0,2,\ldots,D-1,$
$\displaystyle\hat{B}_{D}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\partial_{x}^{2}-D\left(\xi+\xi_{D}\right)\frac{\partial_{x}}{x}+\frac{D^{2}\xi-m^{2}\alpha^{2}}{x^{2}}+4\xi,$
(5.8)

and the coefficients

$\displaystyle E_{i}$ $\displaystyle=$ $\displaystyle
2\left(1-4\xi\right),\;i=0,2,\ldots,D,\;E_{1}=-2,$ $\displaystyle A_{i}$
$\displaystyle=$
$\displaystyle\frac{1}{2},\;i=0,2,\ldots,D-1,\;A_{D}=\frac{1-D}{2},$ (5.9)

and $A_{1}=0$. The nonzero off-diagonal component is expressed as

$\langle
T_{D}^{1}\rangle=-\frac{2\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\frac{\sum_{j=1,2}\left(-1\right)^{j}{e}^{2{|{{x}^{1}-{a_{j}}}|}x/z}c_{j}(x/z)}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}\left[\left(\xi-\frac{1}{4}\right)x\partial_{x}+\xi\right]x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x).$
(5.10)

In (5.7), the part $\langle T_{i}^{k}\rangle_{0}$ corresponds to the vacuum
energy-momentum tensor in the brane-free AdS spacetime. Similar to the case of
the VEV $\langle\varphi^{2}\rangle_{0}$, that part is well-known from the
literature. From the maximal symmetry of the AdS geometry one has $\langle
T_{i}^{k}\rangle_{0}=\mathrm{const}\cdot\delta_{i}^{k}$. The components
$\langle T_{0}^{0}\rangle$ and $\langle T_{i}^{i}\rangle$, $i=2,\ldots,D-1$,
determining the energy density and stresses along the directions parallel to
the branes (except the component $i=D$), are equal. Of course, that is a
consequence of the problem symmetry. As another consequence of the symmetry,
the VEV of the energy-momentum tensor depends on the variables $x^{1}$,
$a_{j}$, $\beta_{j}$, $z$ in terms of the combinations $x^{1}/z$, $a_{j}/z$,
$\beta_{j}/z$. The first and second derivatives of the product
$x^{D+2\nu}F_{\nu}^{D/2}(x)$, appearing in (5.7) and (5.10), are expressed in
terms of the functions $F_{\nu}^{D/2}(x)$, $F_{\nu}^{D/2-1}(x)$, and
$F_{\nu}^{D/2-2}(x)$. The corresponding relations can be found in [19].

Let us denote by $\langle T_{i}^{k}\rangle_{\mathrm{b}}=\langle
T_{i}^{k}\rangle-\langle T_{i}^{k}\rangle_{0}$ the brane-induced contribution
to the vacuum energy-momentum tensor. We can check the following relation for
the corresponding trace:

$\langle
T_{i}^{i}\rangle_{\mathrm{b}}=D\left(\xi-\xi_{D}\right)\nabla_{l}\nabla^{l}\langle\varphi^{2}\rangle_{\mathrm{b}}+m^{2}\langle\varphi^{2}\rangle_{\mathrm{b}},$
(5.11)

where the brane-induced part in the VEV of the field squared is given by the
last term in (4.5). The trace is zero for a conformally coupled massless
field. Another relation expected from general arguments is the covariant
conservation equation $\nabla_{k}\langle T_{i}^{k}\rangle_{\mathrm{b}}=0$. The
latter is a necessary condition for $\langle T_{i}^{k}\rangle_{\mathrm{b}}$ to
be a source in the Einstein field equations. From the equations with $i=1$ and
$i=D$ the following two relations are obtained between the separate components
(see also [19] for the corresponding relations in the geometry of a single
brane):

$\displaystyle\partial_{1}\langle T_{1}^{1}\rangle_{\mathrm{b}}$
$\displaystyle=$ $\displaystyle-z^{D+1}\partial_{z}\left(\frac{\langle
T_{1}^{D}\rangle_{\mathrm{b}}}{z^{D+1}}\right),$
$\displaystyle\partial_{1}\langle T_{D}^{1}\rangle_{\mathrm{b}}$
$\displaystyle=$ $\displaystyle-z^{D}\partial_{z}\left(\frac{\langle
T_{D}^{D}\rangle_{\mathrm{b}}}{z^{D}}\right)-\frac{1}{z}\sum_{k=0}^{D-1}\langle
T_{k}^{k}\rangle_{\mathrm{b}}.$ (5.12)

The first equation shows that the dependence of the normal stress on the
coordinate $x^{1}$ is related to the presence of the nonzero off-diagonal
component.

The VEV of the energy-momentum tensor in the geometry of a single brane at
$x^{1}=a_{j}$ is obtained from (5.7) and (5.10) in the limit
$(-1)^{j^{\prime}}a_{j^{\prime}}\rightarrow\infty$ with $j^{\prime}\neq j$.
For the diagonal components this gives (no summation over $i$)

$\langle T_{i}^{i}\rangle_{j}=\langle
T_{i}^{i}\rangle_{0}-\frac{\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\,x\frac{{e}^{-2{|{{x}^{1}-{a_{j}}}|x/z}}}{c_{j}(x/z)}\left[A_{i}x^{D+2\nu}F_{\nu}^{\frac{D}{2}+1}(x)+\hat{B}_{i}x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x)\right].$
(5.13)

The corresponding expression for the off-diagonal component reads

$\langle
T_{D}^{1}\rangle_{j}=\frac{2\left(-1\right)^{j}\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\frac{{e}^{-2{|{{x}^{1}-{a_{j}}}|}x/z}}{c_{j}(x/z)}\left[\left(\xi-\frac{1}{4}\right)x\partial_{x}+\xi\right]x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x).$
(5.14)

The formulae (5.13) and (5.14) were obtained in [19] from the Wightman
function (3.7) by using (5.1). Note that (5.14) presents the VEV in the region
$x^{1}>a_{1}$ for $j=1$ and in the region $x^{1}<a_{2}$ for $j=2$. Making the
replacement

$\left(-1\right)^{j}\rightarrow\mathrm{sgn}(a_{j}-x^{1}),$ (5.15)

in (5.14) we obtain the expression for a single brane at $x^{1}=a_{j}$ that is
valid for both regions $x^{1}<a_{j}$ and $x^{1}>a_{j}$.

Extracting the single brane contributions from the VEVs we can obtain the
following equivalent representations for the components of the vacuum energy-
momentum tensor (no summation over $i$):

$\displaystyle\langle T_{i}^{i}\rangle$ $\displaystyle=$ $\displaystyle\langle
T_{i}^{i}\rangle_{j}-\frac{\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\,x\left\\{\frac{E_{i}x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x)}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}\right.$
(5.16) $\displaystyle\left.+\frac{2{+}\sum_{l=\pm
1}\left[{e}^{2{|{{x}^{1}-{a_{j}}}|x/z}}c_{j}(x/z)\right]^{l}}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}\left[A_{i}x^{D+2\nu}F_{\nu}^{\frac{D}{2}+1}(x)+\hat{B}_{i}x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x)\right]\right\\},$

and

$\displaystyle\langle T_{D}^{1}\rangle$ $\displaystyle=$ $\displaystyle\langle
T_{D}^{1}\rangle_{j}-\frac{\left(-1\right)^{j}\alpha^{-1-D}}{2^{D+2\nu-1}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\frac{\sum_{l=\pm
1}l\left[{e}^{2{|{{x}^{1}-{a_{j}}}|}x/z}c_{j}(x/z)\right]^{l}}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}$
(5.17)
$\displaystyle\times\left[\left(\xi-\frac{1}{4}\right)x\partial_{x}+\xi\right]x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x).$

The last terms in these representations are the contributions induced by the
brane at $x^{1}=a_{j^{\prime}}$ when we add it to the problem with a single
brane at $x^{1}=a_{j}$. Those terms are finite on the brane $x^{1}=a_{j}$ and
the divergences on that brane come from the single brane contribution $\langle
T_{i}^{k}\rangle_{j}$. For points near the brane the total VEV is dominated by
the single brane contribution. Under the conditions
$\left|{{{x}^{1}}}-a_{j}\right|\ll z,|\beta_{j}|$, the corresponding leading
terms in the expansion over the distance from the brane are given in [19]:

$\langle T_{0}^{0}\rangle_{\mathrm{b}}\approx\frac{\left(1-D\right)\langle
T_{1}^{1}\rangle_{\mathrm{b}}}{\left(|x^{1}-a_{j}|/z\right)^{2}}\approx\frac{z\langle
T_{D}^{1}\rangle_{\mathrm{b}}}{x^{1}-a_{j}}\approx\frac{2D\left(\xi_{D}-\xi\right)\Gamma\left(\frac{D+1}{2}\right)}{\pi^{\frac{D+1}{2}}\left(2\alpha|x^{1}-a_{j}|/z\right)^{D+1}}.$
(5.18)

For Dirichlet boundary condition, $\beta_{j}=0$, the leading terms are given
by the same expressions with opposite signs. As seen, the divergence on the
branes is weaker for the normal stress and off-diagonal component. For
conformal coupling the leading terms vanish and the next terms in the
expansion should be kept.

In the case of a conformally coupled massless field, by using the expression
(4.8) for the function $F_{\nu}^{\mu}(x)$, the vacuum energy-momentum tensor
is presented in the form

$\langle T_{k}^{i}\rangle=\langle
T_{k}^{i}\rangle_{0}+\left(z/\alpha\right)^{D+1}\langle
T_{k}^{i}\rangle_{\mathrm{(M)}},$ (5.19)

where $\langle T_{i}^{k}\rangle_{\mathrm{(M)}}$ is the corresponding VEV in
the region $a_{1}<x^{1}<a_{2}$, $z>0$ for the geometry of plates at
$x^{1}=a_{1},a_{2}$ and $z=0$ in the Minkowski spacetime with the line element
(4.11). On the plates $x^{1}=a_{1},a_{2}$ the field obeys Robin boundary
condition (2.5) and on the plate $z=0$ the Dirichlet boundary condition is
imposed. For the diagonal components the Minkowskian VEV is given by (no
summation over $i$)

$\displaystyle\langle T_{i}^{i}\rangle_{\mathrm{M}}$ $\displaystyle=$
$\displaystyle\langle
T_{i}^{i}\rangle_{\mathrm{M}}^{(0)}+\frac{\pi^{-\frac{D}{2}}}{2^{D+1}D}\mathbf{\,}\int_{0}^{\infty}d\lambda\,\frac{\lambda^{D}}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}\Bigg{\\{}a^{(i)}g_{\frac{D}{2}-1}(\lambda
z)$ (5.20)
$\displaystyle+\left[2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|\lambda}}c_{j}(\lambda)\right]\left[b^{(i)}g_{\frac{D}{2}-1}(\lambda
z)+c^{(i)}g_{\frac{D}{2}}(\lambda z)\right]\Bigg{\\}},$

with the coefficients

$\displaystyle(a^{(0)},a^{(1)},a^{(D)})$ $\displaystyle=$
$\displaystyle(4,-4D,4),$ $\displaystyle(b^{(0)},b^{(1)},b^{(D)})$
$\displaystyle=$ $\displaystyle(0,2,-2),$
$\displaystyle(c^{(0)},c^{(1)},c^{(D)})$ $\displaystyle=$
$\displaystyle(1,1-D,0).$ (5.21)

In (5.20) we have introduced the function

$g_{\mu}(x)=x^{-\mu}J_{\mu}(2x),$ (5.22)

and (no summation over $i$)

$\langle
T_{i}^{k}\rangle_{\mathrm{M}}^{(0)}=-\delta_{i}^{k}\frac{(-D)^{\delta_{i}^{1}}(4\pi)^{-\frac{D}{2}}}{\Gamma\left(\frac{D}{2}+1\right)}\int_{0}^{\infty}d\lambda\,\frac{\lambda^{D}}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}$
(5.23)

is the corresponding VEV in the problem where the plate $z=0$ is absent.
Hence, the last term in (5.20) is induced by the Dirichlet plate $z=0$ added
to the geometry of two parallel plates. For the off-diagonal component in the
Minkowski bulk we obtain

$\langle
T_{D}^{1}\rangle_{\mathrm{(M)}}=\frac{2z^{1-\frac{D}{2}}}{2^{D+2\nu}\pi^{\frac{D}{2}}D}\mathbf{\,}\int_{0}^{\infty}d\lambda\,\lambda^{\frac{D}{2}+1}\frac{\sum_{j=1,2}\left(-1\right)^{j}{e}^{2{|{{x}^{1}-{a_{j}}}|}\lambda}c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}J_{\frac{D}{2}}(2\lambda
z).$ (5.24)

For the VEV of the energy-momentum tensor, the consideration of the
Minkowskian limit, corresponding to large values of the curvature radius
$\alpha$, is similar to that for the mean field squared. By taking into
account that both $\nu$ and $z$ are large, we use the asymptotic (4.12) for
the functions $F_{\nu}^{D/2}(x)$ and $F_{\nu}^{D/2+1}(x)$ in (5.7) and (5.10).
For the diagonal components, to the leading order over $1/\alpha$ one gets
$\langle T_{i}^{i}\rangle\approx\langle
T_{i}^{i}\rangle_{\mathrm{(M)}}^{(0)}$, where (no summation over $i$)

$\displaystyle\langle T_{i}^{i}\rangle_{\mathrm{(M)}}^{(0)}$ $\displaystyle=$
$\displaystyle-\frac{\left(4\pi\right)^{-\frac{D}{2}}}{D\Gamma(D/2)}\mathbf{\,}\int_{m}^{\infty}d\lambda\,\frac{\left(\lambda^{2}-m^{2}\right)^{D/2}}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}\,$
(5.25)
$\displaystyle\times\left[2{+}\frac{4D\left(\xi-\xi_{D}\right)w^{2}-m^{2}}{\lambda^{2}-m^{2}}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|}\lambda}c_{j}(\lambda)\right],$

for the components $i\neq 1$ and

$\langle
T_{1}^{1}\rangle_{\mathrm{(M)}}^{(0)}=\frac{2\left(4\pi\right)^{-\frac{D}{2}}}{\Gamma(D/2)}\mathbf{\,}\int_{m}^{\infty}d\lambda\,\frac{\lambda^{2}\left(\lambda^{2}-m^{2}\right)^{D/2-1}}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1},$
(5.26)

for the normal stress. These results coincide with the expressions given in
[22] for the VEV of the energy-momentum tensor between two plates in the
Minkowski bulk. In the massless limit they are reduced to the expressions in
[21]. Note that the distribution of the normal stress is uniform. For the off-
diagonal component the leading order term in the expansion over $1/\alpha$ is
expressed as

$\displaystyle\langle T_{D}^{1}\rangle$ $\displaystyle\approx$
$\displaystyle-\frac{2\left(4\pi\right)^{-\frac{D}{2}}}{\Gamma(D/2)\alpha}\mathbf{\,}\int_{m}^{\infty}d\lambda\,\frac{\sum_{j=1,2}\left(-1\right)^{j}{e}^{2{|{{x}^{1}-{a_{j}}}|}\lambda}c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}$
(5.27)
$\displaystyle\times\lambda\left(\lambda^{2}-m^{2}\right)^{D/2-2}\left[D\left(\xi-\xi_{D}\right)\lambda^{2}-\left(2\xi-\frac{1}{4}\right)m^{2}\right].$

Of course, this component vanishes in the Minkowskian limit.

Now let us consider the special cases of Dirichlet and Neumann boundary
conditions. Similar to the discussion for the field squared, we expand the
function $1/(e^{2ax/z}-1)$ in (5.7). The resulting integral over $x$ is
presented in terms of the integral (4.16) and its first and second order
derivatives with respect to $c$. In this way we can show that the VEVs of the
diagonal components of the energy-momentum tensor are presented as (no
summation over $i$)

$\langle T_{i}^{i}\rangle=\langle
T_{i}^{i}\rangle_{0}-\frac{\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D}{2}}}\mathbf{\,}\sum_{n=1}^{\infty}\left\\{\left[\frac{E_{i}}{8}\partial_{c}^{2}h_{\nu}^{\frac{D}{2}}(c)+q_{\nu}^{(i)}(c)\right]_{c=\frac{na}{z}}{-}\frac{\delta_{\mathrm{J}}}{2}\sum_{j=1,2}\left.q_{\nu}^{(i)}(c)\right|_{c=\frac{na-{|{{x}^{1}-{a_{j}}}|}}{z}}\right\\}.$
(5.28)

Here we have defined the function

$q_{\nu}^{(i)}(c)=\left[\left(w_{2}^{(i)}c^{2}+w^{(i)}\right)\partial_{c}^{2}+w_{1}^{(i)}c\partial_{c}+w_{0}^{(i)}\right]h_{\nu}^{\frac{D}{2}}(c)+A_{i}h_{\nu}^{\frac{D}{2}+1}(c),$
(5.29)

with the coefficients

$(w_{2}^{(i)},w_{1}^{(i)},w_{0}^{(i)},w^{(i)})=\left(\xi-\frac{1}{4},D\xi-\frac{D+1}{4},-D\xi,\left(\xi-\frac{1}{4}\right)\delta_{1}^{i}\right),$
(5.30)

for $i\neq D$ and

$(w_{2}^{(D)},w_{1}^{(D)},w_{0}^{(D)},w^{(D)})=\left(\frac{1}{4},D\xi+\frac{D+1}{4},D^{2}\xi-m^{2}\alpha^{2},\xi\right).$
(5.31)

For the off-diagonal component we get

$\langle
T_{D}^{1}\rangle=\frac{\delta_{\mathrm{J}}\alpha^{-1-D}}{2^{D+2\nu+3}\pi^{\frac{D}{2}}}\sum_{n=1}^{\infty}\sum_{j=1,2}\left(-1\right)^{j}\left[\left(4\xi-1\right)c\partial_{c}-1\right]\partial_{c}\left.h_{\nu}^{\frac{D}{2}}(c)\right|_{c=\frac{na-{|{{x}^{1}-{a_{j}}}|}}{z}}.$
(5.32)

This component has opposite signs for Dirichlet and Neumann boundary
conditions. Note that for the system of two scalars with Dirichlet and
Neaumann boundary conditions and with the same mass the total energy-momentum
tensor is diagonal and does not depend on the coordinate $x^{1}$.

The VEVs for a single brane with Dirichlet or Neumann boundary conditions are
obtained from (5.28) in the limit when the location of the second brane tends
to infinity. For the diagonal components this gives (no summation over $i$)

$\langle T_{i}^{i}\rangle=\langle
T_{i}^{i}\rangle_{0}+\frac{\delta_{\mathrm{J}}\alpha^{-1-D}}{2^{D+2\nu+1}\pi^{\frac{D}{2}}}\mathbf{\,}q_{\nu}^{(i)}\left(\frac{{|{{x}^{1}-{a_{j}}}|}}{z}\right).$
(5.33)

In a similar way the expression for the off-diagonal component reads

$\langle
T_{D}^{1}\rangle=\mathrm{sgn}(x^{1}-a_{j})\frac{\delta_{\mathrm{J}}\alpha^{-1-D}}{2^{D+2\nu+3}\pi^{\frac{D}{2}}}\left[\left(4\xi-1\right)c\partial_{c}-1\right]\partial_{c}\left.h_{\nu}^{\frac{D}{2}}(c)\right|_{c={|{{x}^{1}-{a}}}_{j}|/z}.$
(5.34)

Alternative representations for the VEVs (5.33) and (5.34) in terms of the
function (3.13) are provided in [19].

The VEVs for the components of the energy-momentum tensor in the special case
of Dirichlet boundary condition on the brane $x^{1}=a_{1}$ and Neumann
condition on $x^{1}=a_{1}$ are obtained from (5.28) and (5.32) by the
replacements (3.16).

Let us consider the behavior of the vacuum energy-momentum tensor near the AdS
boundary and horizon. Near the AdS boundary, assuming that $z\ll|x^{1}-a_{j}|$
for $j=1,2$, the contributions of small $x$ dominate in (5.7) and (5.10). To
the leading order, making the replacement $F_{\nu}^{\frac{D}{2}}(x)\approx
F_{\nu}^{\frac{D}{2}}(0)$, with $F_{\nu}^{\frac{D}{2}}(0)$ given by (4.23), we
get (no summation over $i$)

$\displaystyle\langle T_{i}^{i}\rangle$ $\displaystyle\approx$
$\displaystyle\langle
T_{i}^{i}\rangle_{0}-\frac{B_{\nu}F_{\nu}^{\frac{D}{2}}(0)z^{D+2\nu}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}\alpha^{D+1}}\left[2\nu-\left(D+2\nu\right)\delta_{i}^{D}\right]\mathbf{\,}\int_{0}^{\infty}d\lambda\,\lambda^{D+2\nu-1}\frac{2{+}\sum_{j=1,2}{e}^{2{|{{x}^{1}-{a_{j}}}|}\lambda}c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1},$
$\displaystyle\langle T_{D}^{1}\rangle$ $\displaystyle\approx$
$\displaystyle-\frac{2B_{\nu}F_{\nu}^{\frac{D}{2}}(0)z^{D+2\nu+1}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}\alpha^{D+1}}\mathbf{\,}\int_{0}^{\infty}d\lambda\lambda^{D+2\nu}\frac{\sum_{j=1,2}\left(-1\right)^{j}{e}^{2{|{{x}^{1}-{a_{j}}}|}\lambda}c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1},$
(5.35)

where we have defined

$B_{\nu}=\left(D+2\nu+1\right)\xi-\frac{D+2\nu}{4}.$ (5.36)

Under the conditions assumed, all the components tend to zero in the limit
$z\rightarrow 0$. Note that the coefficient $B_{\nu}$ is negative for
minimally and conformally coupled fields.

For points tending to the horizon the coordinate $z$ is large. Assuming that
$z\gg a$ we use the asymptotic (4.25) for the function
$F_{\nu}^{\frac{D}{2}}(x)$. For the diagonal components this gives (no
summation over $i$)

$\langle T_{i}^{i}\rangle\approx\langle
T_{i}^{i}\rangle_{0}+(z/\alpha)^{D+1}\langle
T_{i}^{i}\rangle_{\mathrm{(M)}}^{(0)}|_{m=0},$ (5.37)

with $\langle T_{i}^{i}\rangle_{\mathrm{(M)}}^{(0)}$ being the corresponding
VEV for two parallel plates in Minkowski spacetime given by (5.25) for a
massive field. The leading term in the expansion of the off-diagonal component
is obtained from (5.27) taking $m=0$ and multiplying by $(z/\alpha)^{D}$. For
a non-conformally coupled field it behaves like $(z/\alpha)^{D}$. For the
conformal coupling the next term in the expansion should be kept.

Figure 3 presents the brane-induced energy density for conformally (left
panel) and minimally (right panel) coupled scalar fields in the region between
the branes versus the proper distance from the brane (in units of $\alpha$).
The graphs are plotted for $a_{1}=0$, $a_{2}/z=5$, $m\alpha=0.5$ and for the
same Robin boundary conditions on the branes ($\beta_{1}=\beta_{2}$). The
numbers near the graphs correspond to the values of the ratio $\beta_{1}/z$.
We have also plotted the graphs for Dirichlet and Neumann boundary conditions.
In accordance with the asymptotic (5.18), for a minimally coupled field and
for non-Dirichlet boundary conditions the vacuum energy density is positive
near the branes. For Dirichlet boundary condition the energy density is
negative. The behavior of the energy density near the center with respect to
the branes depends on the Robin coefficients. For $\beta_{j}/z<0$ and
sufficiently close to zero the brane-induced energy density is negative near
the center. With increasing value of $|\beta_{j}|/z$, started from certain
critical value $\beta_{j}^{(c)}$, that depends on $a/z$, it becomes positive
everywhere in the region between the branes. For the values of the parameters
corresponding to Figure 3, the critical values are given by
$\beta_{j}^{(c)}/z\approx-1.12$ and $\beta_{j}^{(c)}/z\approx-0.69$ in the
cases of conformal and minimal couplings, respectively. The critical values
$|\beta_{j}^{(c)}|/z$ are increasing functions of $a/z$.

|
---|---
Figure 3: The vacuum energy densities for $D=4$ conformally (left panel) and
minimally (right panel) coupled scalar fields induced by the branes in the
region $0<x^{1}/z<5$. The graphs are plotted in the cases of Dirichlet,
Neumann and Robin boundary conditions (with the values of
$\beta_{1}/z=\beta_{1}/z$ given near the curves) for the locations of the
branes $a_{1}=0$, $a_{2}/z=5$ and for $m\alpha=0.5$.

In this section we have considered the local densities induced by the branes.
They are well defined for points away from the branes and do not contain
renormalization ambiguities. The global quantities, such as the total vacuum
energy in the region between the branes (per unit surface of the branes), are
also of physical interest. However, because of the surface divergences, it
cannot be obtained by direct integration of the vacuum energy density: an
additional renormalization is required. This problem is well-known from the
theory of the Casimir effect for curved boundaries in flat spacetime. It is
worth mentioning that for general Robin boundary conditions the vacuum energy
obtained by the integration of the bulk energy density, in general, does not
coincide with the total vacuum energy evaluated as the sum of the ground state
energies for elementary oscillators. As it has been discussed in [29] for
general case of the bulk and boundary geometries, the reason for that is the
presence of surface energy density located on constraining boundaries. For a
scalar field with general curvature coupling parameter the expression for the
surface energy-momentum tensor is obtained in [29] by using the standard
variational procedure. Similar to the case of the integrated bulk energy, the
corresponding VEV requires an additional renormalization. As an example we can
use the approach based on the generalized zeta function approach. We plan to
address these points in a separate publication.

##  6 The Casimir forces

The $i$th component of the force acting on the surface element $dS$ of the
brane at $x^{1}=a_{j}$ is given by $-\langle
T_{l}^{i}\rangle_{x^{1}=a_{j}+0}n_{(+)j}^{l}dS$ in the region $x^{1}\geq
a_{j}+0$ and by $-\langle T_{l}^{i}\rangle_{x^{1}=a_{j}-0}n_{(-)j}^{l}dS$ in
the region $x^{1}\leq a_{j}-0$, where $n_{(\pm)j}^{l}=\pm\delta_{1}^{l}$. For
the resulting force we get

$dF_{(j)}^{i}=\langle T_{1}^{i}\rangle|_{x^{1}=a_{j}+0}^{x^{1}=a_{j}-0}dS.$
(6.1)

Due to the nonzero off-diagonal stress $\langle T_{1}^{D}\rangle$, in addition
to the normal component $dF_{(j)}^{1}$, this force has nonzero component
parallel to the brane (shear force), $dF_{(j)}^{D}$. First we will consider
the normal force.

###  6.1 Normal force

For the normal force acting on the brane at $x^{1}=a_{j}$ one has
$dF_{(j)}^{1}=\langle T_{1}^{1}\rangle|_{z=a_{j}+0}^{z=a_{j}-0}dS$. For
$\langle T_{1}^{1}\rangle$ we have the decomposition (5.16) in the region
between the branes and $\langle T_{1}^{1}\rangle=\langle T_{1}^{1}\rangle_{j}$
in the remaining regions. The parts $\langle T_{1}^{1}\rangle_{j}$ are the
same on the left and right-hand sides of the brane and they do not contribute
to the net force. The nonzero contribution comes from the part $\langle
T_{1}^{1}\rangle-\langle T_{1}^{1}\rangle_{j}$ (given by the last term in
(5.16)) in the region between the branes. Hence, for the vacuum effective
pressure on the brane $x^{1}=a_{j}$, given as $P_{j}=-\left(\langle
T_{1}^{1}\rangle-\langle T_{1}^{1}\rangle_{j}\right)_{x^{1}=a_{j}}$, one gets

$P_{j}=\frac{\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\,x\frac{-2+\left[2{+}c_{j}(x/z)+1/c_{j}(x/z)\right]\hat{B}_{1}}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x).$
(6.2)

The corresponding Casimir forces act on the sides $x^{1}=a_{1}+0$ and
$x^{1}=a_{2}-0$. They are attractive for $P_{j}<0$ and repulsive for
$P_{j}>0$. In the special cases of Dirichlet or Neumann boundary conditions
the Casimir forces are obtained directly from (5.28) with $i=1$:

$P_{j}=-\frac{\alpha^{-1-D}}{2^{D+2\nu+2}\pi^{\frac{D}{2}}}\mathbf{\,}\sum_{n=1}^{\infty}\left[\partial_{c}^{2}h_{\nu}^{\frac{D}{2}}(c)-4\left(1{-}\delta_{\mathrm{J}}\right)q_{\nu}^{(1)}\left(c\right)\right]_{c=na/z}.$
(6.3)

For Dirichlet boundary condition on the brane $x^{1}=a_{1}$ and Neumann
condition for $x^{1}=a_{2}$ the corresponding formula is obtained from (6.3)
by the replacement (3.16).

The expression for the Casimir normal force in the Minkowskian limit directly
follows from (5.26). The corresponding effective pressure is expressed as
$P_{j}^{\mathrm{(M)}}=-\langle T_{1}^{1}\rangle_{\mathrm{(M)}}^{(0)}$. Note
that for the Minkowskian bulk the forces acting on separate plates coincide
regardless of the values of the Robin coefficients. As seen from (6.2), in
general this is not the case for the AdS geometry.

For small separations between the branes, $a\ll z$, the dominant contribution
to the integral in (6.2) comes from the region with large $x$ and we use the
asymptotic (4.25) for the function $F_{\nu}^{\mu}(x)$. The leading term in the
expansion of the force comes from the part with -2 in the numerator of the
integrand in (6.2) and we get

$P_{j}\approx-\frac{2\left(z/\alpha\right)^{D+1}}{(4\pi)^{\frac{D}{2}}\Gamma\left(\frac{D}{2}\right)}\int_{0}^{\infty}d\lambda\,\frac{\lambda^{D}}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}.$
(6.4)

If additionally one has $a\ll|\beta_{l}|$, $l=1,2$, we note that the integral
in (6.4) is dominated by the contribution from the region $\lambda\lesssim
1/a$ and in that region $c_{1}(\lambda)c_{2}(\lambda)\approx 1$. The estimate
(6.4) is further simplified as

$P_{j}\approx-\frac{D\zeta(D+1)}{\left(2\sqrt{\pi}\alpha
a/z\right)^{D+1}}\Gamma\left(\frac{D+1}{2}\right),$ (6.5)

where $\zeta(x)$ is the Riemann zeta function. For Dirichlet boundary
conditions on both the branes $c_{1}(\lambda)c_{2}(\lambda)=1$ and we get the
same leading term. For Dirichlet boundary condition on one brane and non-
Dirichlet condition on the other, with the modulus of the Robin coefficient
much larger than $a$, we have $c_{1}(\lambda)c_{2}(\lambda)\approx-1$. In this
case (6.4) is reduced to

$P_{j}\approx\frac{D\zeta(D+1)}{\left(2\sqrt{\pi}\alpha
a/z\right)^{D+1}}\left(1-\frac{1}{2^{D}}\right)\Gamma\left(\frac{D+1}{2}\right).$
(6.6)

The approximations (6.5) and (6.6) are obtained from the corresponding
asymptotics for Robin plates in the Minkowski spacetime replacing the
separation between the plates by the proper separation $\alpha a/z$ in the AdS
bulk. The asymptotics show that for small separations between the branes
($a\ll z$ and $a\ll|\beta_{l}|$ for non-Dirichlet boundary conditions) the
Casimir normal forces are repulsive for Dirichlet boundary condition on one
brane and non-Dirichlet condition on the other (formula (6.6)). In the
remaining cases the forces are attractive. In the asymptotic region under
consideration with the proper separation much smaller than the curvature
radius, the effects of gravity on the Casimir forces are small and the results
are similar to those for the Minkowski bulk.

We expect that the influence of the gravity will be essential for proper
separations larger than the AdS curvature radius. In the limit $a/z\gg 1$ the
integral in (6.2) is dominated by the contribution from the region with small
$x$. Expanding the function $F_{\nu}^{D/2}(x)$ in (6.2) one finds

$P_{j}\approx\frac{2\pi^{\frac{1-D}{2}}\alpha^{-1-D}\left(z/2\right)^{D+2\nu}}{\Gamma\left(\frac{D+1}{2}+\nu\right)\Gamma\left(1+\nu\right)}\mathbf{\,}\int_{0}^{\infty}d\lambda\,\lambda^{D+2\nu-1}\frac{\nu
B_{\nu}\left[2{+}c_{j}(\lambda)+1/c_{j}(\lambda)\right]-\lambda^{2}z^{2}}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1},$
(6.7)

where $B_{\nu}$ is defined by (5.36). Under additional conditions
$a\gg|\beta_{l}|$, $l=1,2$ (non-Neumann boundary conditions on both the
branes), we further expand the integrand over the small ratio $|\beta_{l}|/a$
with the result

$P_{j}\approx-\frac{2\left(D+2\nu+1\right)\left(4\nu
B_{\nu}\beta_{j}^{2}/z^{2}+1\right)}{\pi^{\frac{D}{2}}\Gamma\left(1+\nu\right)\alpha^{D+1}\left(2a/z\right)^{D+2\nu+2}}\zeta\left(D+2\nu+2\right)\Gamma\left(\frac{D}{2}+\nu+1\right),$
(6.8)

For $a\gg|\beta_{j}|$ and for Neumann boundary condition on the second brane
($c_{j^{\prime}}(\lambda)=1$, as before, $j^{\prime}=1$ for $j=2$ and
$j^{\prime}=2$ for $j=1$) from (6.7) we get

$\displaystyle P_{j}$ $\displaystyle\approx$
$\displaystyle\frac{2\left(D+2\nu+1\right)\left(4\nu
B_{\nu}\beta_{j}^{2}/z^{2}+1\right)}{\pi^{\frac{D}{2}}\Gamma\left(1+\nu\right)\alpha^{D+1}\left(2a/z\right)^{D+2\nu+2}}\mathbf{\,}$
(6.9)
$\displaystyle\times\left(1-\frac{1}{2^{D+2\nu+1}}\right)\zeta\left(D+2\nu+2\right)\Gamma\left(\frac{D}{2}+\nu+1\right).$

The forces corresponding to (6.8) and (6.9) have opposite signs. As it has
been already mentioned before, the coefficient $B_{\nu}$ is negative for
minimally and conformally coupled fields. Then, from (6.8) we see that,
depending on the boundary conditions, the Casimir forces can be either
attractive or repulsive at large distances. The sign of the forces is
determined by the factor $4\nu B_{\nu}\beta_{j}^{2}/z^{2}+1$. This factor is
positive near the horizon and is negative near the AdS boundary if
$B_{\nu}<0$. This shows that, for given values of the parameters, the vacuum
pressure changes the sign as a function of $z$.

For Neumann boundary condition on the brane at $x^{1}=a_{j}$ and at large
separations, to the leading order, we can ignore the term $\lambda^{2}z^{2}$
in (6.7). For non-Neumann boundary condition on the second brane, assuming
$a\gg|\beta_{j^{\prime}}|$, this gives

$P_{j}\approx-\frac{4\nu
B_{\nu}\left(1-2^{1-D-2\nu}\right)\zeta(D+2\nu)}{\pi^{\frac{D}{2}}\Gamma\left(1+\nu\right)\alpha^{D+1}\left(2a/z\right)^{D+2\nu}}\Gamma\left(\frac{D}{2}+\nu\right).$
(6.10)

By taking into account that $B_{\nu}<0$ for minimal and conformal couplings,
this result shows that for Neumann boundary condition on the brane
$x^{1}=a_{j}$ and for non-Neumann condition on the second brane the force is
repulsive at large separations. For Neumann boundary condition on both the
branes the leading term is expressed as

$P_{j}\approx\frac{4\nu
B_{\nu}\zeta\left(D+2\nu\right)\Gamma\left(D/2+\nu\right)}{\pi^{\frac{D}{2}}\Gamma\left(1+\nu\right)\alpha^{D+1}\left(2a/z\right)^{D+2\nu}},$
(6.11)

and the force is attractive for $B_{\nu}<0$. The decay of the normal force at
large proper separations between the branes is power-law for both massless and
massive cases. In the Minkowski bulk and for massive fields the corresponding
suppression is exponential. The leading term is found from (5.26),
$P_{j}^{\mathrm{(M)}}\propto a^{-D/2}e^{-2ma}$.

As seen from the analysis given above, for the brane with Neumann boundary
condition the Casimir force on that brane decays at large separations like
$(z/a)^{D+2\nu}$ regardless the boundary condition on the second brane (except
the special case with $B_{\nu}=0$). For non-Neumann boundary conditions on the
brane at $x^{1}=a_{j}$ and for $a\gg|\beta_{j}|$ the corresponding force
behaves as $(z/a)^{D+2\nu+2}$ and the suppression is stronger. As an example
let us consider the case of Dirichlet boundary condition for $x^{1}=a_{1}$ and
Neumann condition for $x^{1}=a_{2}$. At large separations the Casimir pressure
on the brane $x^{1}=a_{1}$ is given by (6.9) with $j=1$ and $\beta_{j}=0$. It
corresponds to a repulsive force. The leading term for the Casimir force on
the brane $x^{1}=a_{2}$ is obtained from (6.10) with $j=2$. The force is
repulsive for $B_{\nu}<0$ and attractive for $B_{\nu}>0$. This shows that, in
principle, we can have a situation when the force has an attractive nature for
one brane and repulsive nature for another.

In Figure 4 we have displayed the normal Casimir force versus the proper
separation between the branes, in units of the AdS curvature radius, for $D=4$
minimally coupled scalar field. The same boundary conditions are imposed on
the branes. The numbers near the curves are the values for $\beta_{1}/z=$
$\beta_{2}/z$. The dashed and dotted curves correspond to Dirichlet and
Neumann boundary conditions, respectively. The graphs are plotted for
$m\alpha=0.5$. The presented graphs demonstrate the feature already seen from
asymptotic analysis: the forces attractive at small separations may become
repulsive for larger distances.

Figure 4: The Casimir normal force for $D=4$ minimally coupled field with
$m\alpha=0.5$ as a function of the interbrane separation. The graphs are
plotted for Dirichlet and Neumann boundary conditions (dashed and dotted
curves), and for Robin boundary conditions with the coefficients
$\beta_{1}/z=\beta_{2}/z$ given near the corresponding graphs.

Figure 5 presents the dependence of the Casimir normal force acting on the
brane at $x^{1}=a_{1}$, given by (6.2) with $j=1$, on the coefficient in the
Robin boundary condition on that brane. The left and right panels correspond
to $D=4$ conformally and minimally coupled fields with $m\alpha=0.5$. For the
proper separation between the branes we have taken $a/z=1$. The graphs are
plotted for different boundary conditions on the second brane: Dirichlet and
Neumann conditions (Dir and Neu, respectively), Robin boundary conditions with
$\beta_{2}/z$ given near the curves. The dashed lines correspond to Dirichlet
and Neumann conditions on both the branes (DD and NN), Dirichlet (Neumann)
condition at $x^{1}=a_{1}$ and Neumann (Dirichlet) condition at $x^{1}=a_{2}$,
indicated as DN (ND). The graphs show that depending on the coefficient in the
Robin boundary conditions the force can be either attractive or repulsive.

|
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Figure 5: The Casimir normal force per unit surface of the brane $x^{1}=a_{1}$
as a function of the Robin coefficient in the boundary condition on that brane
for $D=4$ conformally (left panel) and minimally (right panel) coupled fields.
The graphs are plotted for $m\alpha=0.5$, $a/z=1$, and for different boundary
conditions on the second brane (see the text).

###  6.2 Shear force

As it has been emphasized above, in the problem at hand in addition to the
normal Casimir force one has a nonzero shear force along the $z$-direction,
$dF_{(j)}^{D}=f_{(j)}dS$, where $f_{(j)}$ is the shear force per unit surface
of the plate at $z=z_{j}$. The latter is given by $f_{(j)}=\langle
T_{1}^{D}\rangle|_{x^{1}=a_{j}+0}^{x^{1}=a_{j}-0}$. In accordance with the
decomposition (5.17), the shear force contains two contributions. The first
part comes from the term $\langle T_{1}^{D}\rangle_{j}$ and corresponds to the
force acting on the brane at $x^{1}=a_{j}$ when the second brane is absent. We
will call this part the self-acting shear force and will denote by
$f_{j}^{\mathrm{(s)}}$. Those forces acting on the sides $x^{1}=a_{j}-0$ and
$x^{1}=a_{j}+0$ coincide and we get

$f_{j}^{\mathrm{(s)}}=\langle
T_{1}^{D}\rangle_{j}|_{x^{1}=a_{j}+0}^{x^{1}=a_{j}-0}=\frac{4\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\,\frac{{1}}{c_{j}(x/z)}\left[\left(\xi-\frac{1}{4}\right)x\partial_{x}+\xi\right]x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x).$
(6.12)

By using the asymptotic (4.25), we see that for nonconformally coupled fields
and for large $x$ the integrand in (6.12) behaves like $x^{D-1}$ and the
integral is divergent in the upper limit. For the conformal coupling the next
to the leading term should be kept and the integral is still divergent. Of
course, the divergence comes from the surface divergences in the single brane
contributions to the VEVs. The renormalization of the divergence in the self-
action shear force can be considered in the same line as that for the total
and surface Casimir energies and will be discussed elsewhere. Here we will be
focused on the contribution to the shear force that is induced by the second
brane. This part acts on the sides $x^{1}=a_{1}+0$ and $x^{1}=a_{2}-0$ and is
determined from the last term in (5.17). Denoting it by
$f_{j}^{\mathrm{(int)}}$, we get

$f_{j}^{\mathrm{(int)}}=-\frac{2\alpha^{-1-D}}{2^{D+2\nu}\pi^{\frac{D-1}{2}}}\mathbf{\,}\int_{0}^{\infty}dx\frac{c_{j}(x/z)-1/c_{j}(x/z)}{c_{1}(x/z)c_{2}(x/z)e^{2ax/z}-1}\left[\left(\xi-\frac{1}{4}\right)x\partial_{x}+\xi\right]x^{D+2\nu}F_{\nu}^{\frac{D}{2}}(x).$
(6.13)

This part acting on the brane at $x^{1}=a_{j}$ vanishes for Dirichlet and
Neumann boundary conditions on that brane regardless of boundary conditions on
the second brane. The shear force is directed toward the horizon for
$f_{j}^{\mathrm{(int)}}>0$ and toward the AdS boundary for
$f_{j}^{\mathrm{(int)}}<0$.

The asymptotic behavior of the shear force is found in a way similar to that
for the normal force. At small proper separations compared with the curvature
radius, $a/z\ll 1$, one gets

$f_{j}^{\mathrm{(int)}}\approx-\frac{2D\left(\xi-\xi_{D}\right)z^{D}}{2^{D}\pi^{\frac{D}{2}}\Gamma\left(\frac{D}{2}\right)\alpha^{D+1}}\int_{0}^{\infty}d\lambda\,\lambda^{D-1}\frac{c_{j}(\lambda)-1/c_{j}(\lambda)}{c_{1}(\lambda)c_{2}(\lambda)e^{2a\lambda}-1}.$
(6.14)

For a conformally coupled field the leading term vanishes and the next term in
the expansion should be kept. If additionally $\left|\beta_{l}\right|\gg a$,
$l=1,2$ (the condition with $l=j^{\prime}$ is required only for non-Dirichlet
boundary conditions on the brane at $x^{1}=a_{j^{\prime}}$), the further
expansion gives

$f_{j}^{\mathrm{(int)}}\approx\frac{4D\left(\xi-\xi_{D}\right)\zeta(D-1)}{\pi^{\frac{D+1}{2}}\alpha^{D+1}\left(2a/z\right)^{D}b_{j}}\Gamma\left(\frac{D-1}{2}\right)\left(2^{2-D}-1\right)^{\delta_{0b_{j^{\prime}}}},$
(6.15)

where the last factor is present only for Dirichlet boundary condition at
$x^{1}=a_{j^{\prime}}$. Note that under the specified conditions one has
$|b_{j}|\gg 1$. As seen, at small separations, the shear component of the
force has opposite signs for Dirichlet and non-Dirichlet boundary conditions
on the second brane. For a minimally coupled field with $b_{j}<0$ and for
small separations the shear force acting on the brane at $x^{1}=a_{j}$ is
directed toward the AdS horizon for non-Dirichlet boundary conditions on the
second brane and toward the AdS boundary for Dirichlet condition.

At large proper separations, $a/z\gg 1$, the interaction force is approximated
by

$f_{j}^{\mathrm{(int)}}\approx-\frac{2\pi^{\frac{1-D}{2}}B_{\nu}\alpha^{-1-D}\left(z/a\right)^{D+2\nu+1}}{2^{D+2\nu}\Gamma\left(\nu+1\right)\Gamma\left(\frac{D+1}{2}+\nu\right)}\mathbf{\,}\int_{0}^{\infty}dx\,\frac{c_{j}(x/a)-1/c_{j}(x/a)}{c_{1}(x/a)c_{2}(x/a)e^{2x}-1}x^{D+2\nu}.$
(6.16)

This estimate is further simplified under the condition
$\left|\beta_{l}\right|\ll a$, $l=1,2$ (the condition for $l=j^{\prime}$ is
required only for non-Neumann boundary conditions at $x^{1}=a_{j^{\prime}})$:

$f_{j}^{\mathrm{(int)}}\approx-\frac{4b_{j}B_{\nu}\left(D+2\nu+1\right)\zeta\left(D+2\nu+2\right)}{\pi^{\frac{D}{2}}\Gamma\left(\nu+1\right)\alpha^{D+1}\left(2a/z\right)^{D+2\nu+1}}\Gamma\left(\frac{D}{2}+\nu+1\right)\left(\frac{1}{2^{D+2\nu+1}}-1\right)^{\delta_{\infty
b_{j^{\prime}}}},$ (6.17)

where $B_{\nu}$ is defined by (5.36) and $|b_{j}|\ll 1$. The force (6.17) has
opposite signs for Neumann and non-Neumann boundary conditions on the brane
$x^{1}=a_{j^{\prime}}$. For conformally and minimally coupled fields one has
$B_{\nu}<0$. In those cases, for $b_{j}<0$ and at large separations between
the branes the shear force acting on the brane $x^{1}=a_{j}$ is directed
toward the AdS horizon for Neumann boundary condition on the second brane and
toward the AdS boundary for non-Neumann conditions.

The interaction part of the shear force acting on the brane $x^{1}=a_{1}$
versus the distance between the branes is depicted in Figure 6 for $D=4$
conformally and minimally coupled field (left and right panels, respectively).
The graphs are plotted for $m\alpha=0.5$ and for different values of the ratio
$\beta_{1}/z=\beta_{2}/z$ (the numbers near the curves). In both cases the
shear force is directed toward the horizon at small separations between the
branes and toward the AdS boundary at large separations. For a minimally
coupled field this is in agreement with the asymptotic analysis presented
above.

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Figure 6: The interaction contribution to the shear force per unit surface of
the brane $x^{1}=a_{1}$ versus the interbrane separation for different values
of $\beta_{1}/z=\beta_{2}/z$ (the numbers given near the curves). The graphs
are plotted for $D=4$ conformally (left panel) and minimally (right panel)
coupled fields with $m\alpha=0.5$.

The interaction part of the shear force per unit surface of the brane
$x^{1}=a_{1}$ is plotted in Figure 7 as a function of the ratio $\beta_{1}/z$
for different boundary conditions on the second brane (Dirichlet, Neumann and
Robin conditions with the values for $\beta_{2}/z$ presented near the curves).
The left and right panels correspond to conformally and minimally coupled
fields in (4+1)-dimensional AdS spacetime. The graphs are plotted for
$m\alpha=0.5$ and $a/z=1$.

|
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Figure 7: The interaction shear force acting on the brane $x^{1}=a_{1}$ versus
the corresponding Robin coefficient for $D=4$ conformally (left panel) and
minimally (right panel) coupled fields. The numbers near the curves are the
values of the ratio $\beta_{2}/z$ and the graphs are plotted for
$m\alpha=0.5$, $a/z=1$.

##  7 Conclusion

In this paper we have investigated the influence of two parallel branes,
orthogonal to the AdS boundary, on the local properties of the scalar vacuum
in background of $(D+1)$-dimensional AdS spacetime. Robin boundary conditions
are imposed, in general, with different coefficients on separate branes. We
consider a free field theory and the properties of the vacuum are completely
determined by the two-point functions. As a two-point function, the positive
frequency Wightman function is chosen. The local VEVs are obtained in the
coincidence limit of the arguments of that function and its derivatives. For
the evaluation of the Wightman function the direct summation over the complete
set of scalar modes is used. In the region between the branes the mode
functions are given by (2.6) with the function $\alpha_{j}(\lambda)$ defined
as (2.9). The eigenvalues of the quantum number $\lambda$ are discretized by
the boundary conditions and they are expressed in terms of the roots of
equation (2.10). The geometry of the subspace $y=\mathrm{const}$, parallel to
the AdS boundary, is Minkowskian and the eigenvalue equation coincides with
that in the Casimir problem for two Robin plates in flat spacetime. For
general Robin boundary conditions the eigenvalues of $\lambda$ are given
implicitly and for the summation of the corresponding series in the mode sum
of the Wightman function we have employed the Abel-Plana-type formula (A.1).
This has two advantages: (i) an integral representation is provided for which
the explicit knowledge of the eigenvalues is not required and (ii) the parts
corresponding to the brane-free and single brane geometries are explicitly
extracted. In particular, on the basis of (ii), the renormalization of local
VEVs for points outside the branes is reduced to the one in the brane-free
problem.

As a local characteristic of the vacuum state we have considered the mean
field squared. Based on the Wightman function decomposition, the VEV is
presented in two equivalent forms, (4.5) and (4.6). In the second one the
contribution corresponding to the problem with a single brane is separated.
For special cases of Dirichlet and Neumann boundary conditions the VEVs are
further simplified to (4.14). An alternative representation for those cases is
given by (4.18). For a conformally coupled massless scalar field the problem
under consideration is conformally related to the problem with parallel Robin
plates in the Minkowski spacetime orthogonally intersected by a Dirichlet
plate, the latter being the conformal image of the AdS boundary. The Dirichlet
boundary condition on the conformal image is related to the condition for the
field modes (2.6) imposed on the AdS boundary. In the Minkowskian limit we
recover the result for a massive scalar field in the geometry of two parallel
plates, previously considered in [21] and [22] for massless and massive
fields, respectively. For points near the branes and not too close to the AdS
boundary the dominant contribution to the VEV comes from quantum fluctuations
with wavelengths smaller than the curvature radius and the influence of the
gravity is weak. The leading term in the expansion over the distance from the
brane coincides with that for a plate in the Minkowski bulk with the distance
from the plate replaced by the proper distance in the AdS bulk. The brane-
induced contribution vanishes on the AdS boundary. For points not too close to
the branes the corresponding asymptotic is given by (4.24). In the opposite
near-horizon limit, for fixed value of the coordinate distance $a$, the proper
separation between the branes is small compared to the curvature radius and
the brane-induced VEV is well approximated by the Minkowskian expression (see
(4.26)). Depending on the boundary conditions, the mean field squared, as a
function of the distance from the brane, may change the sign.

The vacuum energy density and stresses in the region between the branes have
been discussed in Section 5. The diagonal components of the vacuum energy-
momentum tensor are given by the formula (5.7). The only nonzero off-diagonal
component corresponds to the stress $\langle T_{D}^{1}\rangle$, expressed as
(5.10). The generation of this component is a pure brane-induced effect and
gives rise to a shear force acting on the branes. As expected, the brane-
induced contribution obeys the trace relation (5.11) and the covariant
conservation equation. Single brane contributions in the components of the
vacuum energy-momentum tensor are explicitly separated in the representations
(5.16) and (5.17). In the Minkowskian limit we recover the results of Refs.
[21] and [22] for massless and massive scalar fields. In the special case of a
conformally coupled massless field the brane-induced part has a conformal
connection with the corresponding vacuum energy-momentum tensor for two
parallel plates with Robin boundary conditions intersected by the third plate
with Dirichlet boundary condition. The respective VEVs are given by (5.20) and
(5.24). For special cases of Dirichlet and Neumann boundary conditions
equivalent representations are given by formulae (5.28) and (5.32). Near the
branes and near the horizon, for fixed value of the separation $a$, the
effects of the gravity on the brane-induced VEVs of the components $\langle
T_{i}^{i}\rangle$ with $i\neq 1$ are weak and the leading terms in the
corresponding expansions coincide with those for the Minkowski bulk. The
brane-induced contributions in the diagonal components vanish on the AdS
boundary like $z^{D+2\nu}$. The decay for the off-diagonal component is
stronger, as $z^{D+2\nu+1}$. The numerical investigation for the distribution
of the vacuum energy density is presented for the case when the boundary
conditions imposed on separate branes are the same. The brane-induced vacuum
energy density in the region between the branes is negative for Dirichlet
boundary conditions and positive for Neumann conditions. For Robin conditions
there is a critical value of the coefficient $\beta_{j}=\beta_{j}^{(c)}<0$
that separates two qualitatively different distributions. For
$\beta_{j}<\beta_{j}^{(c)}$ the behavior of the energy density is of Neuman-
type: the energy density is positive everywhere in the region between the
branes. In the range $\beta_{j}^{(c)}<\beta_{j}<0$ the energy density is
positive near the branes and negative near the center with respect to the
brane locations. This type of beahvior is depicted in Figure 3.

The Casimir forces acting on the branes have two components. The first one
corresponds to the normal force which is considered in the literature for
different bulk and boundary geometries. Interpreted in terms of the vacuum
pressure on the brane at $x^{1}=a_{j}$, it is given by the expression (6.2) or
by an alternative representation (6.3) for Dirichlet and Neumann boundary
conditions. Unlike the problem in the Minkowskian bulk, the forces for
Dirichlet and Neumann boundary conditions are different. Another difference is
that the forces acting on separate branes differ if the coefficients in the
Robin boundary conditions on them are different. Depending on the boundary
conditions and on the separation between the branes the normal forces can be
either attractive or repulsive. At small separations the effects of background
curvature are weak and the force is well approximated by the corresponding
result for the Minkowski bulk. They are repulsive for Dirichlet boundary
condition on one brane and non-Dirichlet condition on the other and attractive
in the remaining cases. The influence of gravity is essential for proper
separations larger than the AdS curvature radius. The decay of forces at large
separations is power-law for both cases of massless and massive fields. For
massive fields this results is in contrast to the exponential decay in the
Minkowski bulk. The Casimir normal force acting on the brane decays at large
separations like $(z/a)^{D+2\nu}$ for Neumann boundary condition on that brane
and behaves as $(z/a)^{D+2\nu+2}$ for non-Neumann boundary conditions with
$|\beta_{j}|\ll a$. The large-distance asymptotic beahvior of the vacuum
effective pressure on the brane at $x^{1}=a_{j}$ is given by (6.8) and (6.9)
for non-Neumann boundary conditions on that brane and by (6.10) and (6.11) for
Neumann condition. The sign of the force at large separations depend on the
parameter $B_{\nu}$, defined by (5.36). For given values of the parameters,
the vacuum pressure may also change its sign as a function of $z$. This means
that the forces acting on different parts of the brane may differ by sign.

A qualitatively different feature of the problem in the AdS bulk is the
presence of the vacuum shear force on the branes. The corresponding part
induced by the second brane is expressed as (6.13). It vanishes for the brane
with Dirichlet or Neumann boundary conditions regardless of the condition on
the second brane. Depending on the coefficients in the boundary conditions, on
the separation between the branes and also on the distance from the AdS
boundary, the shear component of the force can be either positive or negative.
At small separations, the leading term in the expansion of the shear force is
given by (6.15). In particular, for a minimally coupled field with negative
value of the Robin coefficient $\beta_{j}$ the shear force on the brane
$x^{1}=a_{j}$ is directed toward the AdS horizon for non-Dirichlet boundary
conditions on the second brane and toward the AdS boundary for Dirichlet
condition. At large separations the asymptotic for the interaction part of the
shear force is given by (6.17). For minimally and conformally coupled fields
and for $\beta_{j}<0$, at large separations the force $f_{j}^{\mathrm{(int)}}$
is directed toward the AdS horizon for Neumann boundary condition on the
second brane and toward the AdS boundary for non-Neumann conditions.

## Acknowledgments

A.A.S. was supported by the grants No. 20RF-059 and No. 21AG-1C047 of the
Science Committee of the Ministry of Education, Science, Culture and Sport RA.
V.Kh.K. was supported by the grant No. 21AG-1C069 of the Science Committee of
the Ministry of Education, Science, Culture and Sport RA. A.A.S. gratefully
acknowledges the hospitality of the INFN, Laboratori Nazionali di Frascati
(Frascati, Italy), where a part of this work was done.

##  Appendix A Integral representation for the series over eigenvalues

In this section we provide an integral representation for the series
$S(b,\Delta t,x^{1},x^{\prime 1})$, given by (3.4). The transformation will be
based on the summation formula [21]

$\displaystyle\sum_{n=1}^{\infty}\frac{f(u_{n})}{N_{n}}$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\int_{0}^{\infty}du\,f(u)+\frac{i}{\pi}\int_{0}^{\infty}du\frac{f(e^{\pi
i/2}u)-f(e^{-\pi i/2}u)}{\tilde{c}_{1}(u)\tilde{c}_{2}(u)e^{2u}-1}$ (A.1)
$\displaystyle-\frac{f(0)/2}{1-b_{2}-b_{1}}-\frac{\theta(b_{j})}{2b_{j}}\left[h_{1}(e^{\pi
i/2}/b_{j})+h_{1}(e^{-\pi i/2}/b_{j})\right],$

where $\theta(x)$ is the Heaviside step function,
$\tilde{c}_{j}(u)=(b_{j}u-1)/(b_{j}u+1)$ and $h(u)=(b_{j}^{2}u^{2}+1)f(u)$. In
(A.1) it is assumed that the function $f(u)$ is analytic in the right half-
plane $\mathrm{Re\,}u>0$. For the series in (3.2) the function $f(u)$ is given
by the expression

$f(u)=\frac{e^{-i\sqrt{u^{2}/a^{2}+b^{2}}\Delta
t}}{\sqrt{u^{2}/a^{2}+b^{2}}}\left[2\cos\left(\frac{{u}}{a}{{\Delta
x}^{1}}\right){+}\sum_{l=\pm
1}\left({e}^{i{|{{x}^{1}{+x}}^{\prime}{{}^{1}-2{a_{j}}}|u/a}}\frac{iub_{j}-1}{iub_{j}+1}\right)^{l}\right],$
(A.2)

with $f(0)=0$. By taking into account that for $x>0$ one has

$\sqrt{\left(e^{\pm\pi
i/2}x\right)^{2}+b^{2}}=\left\\{\begin{array}[]{cc}\sqrt{b^{2}-x^{2}},&x<b\\\
e^{\pm\pi i/2}\sqrt{x^{2}-b^{2}},&x>b\end{array}\right.,$ (A.3)

and introducing a new integration variable $\lambda=u/a$, the function (3.4)
is presented as

$\displaystyle S(b,\Delta t,x^{1},x^{\prime 1})$ $\displaystyle=$
$\displaystyle\frac{a}{2}S_{0}(b,\Delta
t,{{x}_{-}^{1}})+\frac{a}{4}S_{j}(b,\Delta
t,x_{+}^{1})+a\frac{\pi\theta(\beta_{j})}{2\beta_{j}}e^{-{|{{x}_{+}^{1}-2{a_{j}}}|/\beta}_{j}}\sum_{l=\pm
1}\frac{e^{-i\sqrt{(li/\beta_{j})^{2}+b^{2}}\Delta
t}}{\sqrt{(li/\beta_{j})^{2}+b^{2}}}$ (A.4)
$\displaystyle+\frac{a}{2}\int_{b}^{\infty}d\lambda\frac{2\cosh\left({\lambda{x}_{-}^{1}}\right){+}\sum_{l=\pm
1}\left[{e}^{{|{{x}_{+}^{1}-2{a_{j}}}|\lambda}}c_{j}(\lambda
a)\right]^{l}}{\left[c_{1}(\lambda a)c_{2}(\lambda
a)e^{2a\lambda}-1\right]\sqrt{\lambda^{2}-b^{2}}}\cosh\left(\sqrt{\lambda^{2}-b^{2}}\Delta
t\right),$

where $x_{\pm}^{1}=x^{1}\pm x^{\prime 1}$ and

$\displaystyle S_{0}(b,\Delta t,{{x}_{-}^{1}})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\lambda\,\frac{e^{-i\sqrt{\lambda^{2}+b^{2}}\Delta
t}}{\sqrt{\lambda^{2}+b^{2}}}\cos\left({\lambda{x}_{-}^{1}}\right),$
$\displaystyle S_{j}(b,\Delta t,x_{+}^{1})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\lambda\,\frac{e^{-i\sqrt{\lambda^{2}+b^{2}}\Delta
t}}{\sqrt{\lambda^{2}+b^{2}}}\sum_{l=\pm
1}\left({e}^{i{|{{x}_{+}^{1}-2{a_{j}}}|\lambda}}\frac{i\lambda\beta_{j}-1}{i\lambda\beta_{j}+1}\right)^{l}.$
(A.5)

For the further transformation of the function $S_{j}(b,\Delta t,x_{+}^{1})$
we rotate the integration contour by the angle $\pi/2$ for the $l=1$ term and
by the angle $-\pi/2$ for the term with $l=-1$. This choice for the
integration contours is dictated by the behavior of the integrands in the
upper and lower half-planes of the complex variable $\lambda$. The poles
$\lambda=\pm i/\beta_{j}$ for $\beta_{j}>0$ are excluded by semicircles in the
right half-plane $\mathrm{Re\,}\lambda\geq 0$ with small radius. Again, by
using (A.3), this gives

$\displaystyle S_{j}(b,\Delta t,x_{+}^{1})$ $\displaystyle=$ $\displaystyle
2\int_{b}^{\infty}d\lambda\frac{\cosh\left(\sqrt{\lambda^{2}-b^{2}}\Delta
t\right)}{\sqrt{\lambda^{2}-b^{2}}}\frac{e^{-{\lambda|{{x}_{+}^{1}-2{a_{j}}}|}}}{c_{j}(\lambda
a)}$ (A.6) $\displaystyle-\frac{2\pi}{\beta_{j}}\theta(\beta_{j})\sum_{l=\pm
1}\frac{e^{-i\sqrt{\left(i/\beta_{j}\right)^{2}+b^{2}}\Delta
t}}{\sqrt{\left(i/\beta_{j}\right)^{2}+b^{2}}}e^{-{|{{x}_{+}^{1}-2{a_{j}}}|/}\beta_{j}}.$

Substituting this in (A.4) we see that the terms with the Heaviside step
function are cancelled out and the function (3.4) is expressed as

$\displaystyle S(b,\Delta t,x^{1},x^{\prime 1})=\frac{a}{2}S_{0}(b,\Delta
t,{{\Delta
x}^{1}})+\frac{a}{2}\int_{b}^{\infty}d\lambda\frac{\cosh\left(\sqrt{\lambda^{2}-b^{2}}\Delta
t\right)}{\sqrt{\lambda^{2}-b^{2}}}\frac{e^{-{\lambda|{{x}_{+}^{1}-2{a_{j}}}|}}}{c_{j}(\lambda
a)}$
$\displaystyle\qquad+\frac{a}{2}\int_{b}^{\infty}d\lambda\frac{2\cosh\left({\lambda{x}_{-}^{1}}\right){+}\sum_{l=\pm
1}\left[{e}^{{|{{x}_{+}^{1}-2{a_{j}}}|\lambda}}c_{j}(\lambda
a)\right]^{l}}{\left[c_{1}(\lambda a)c_{2}(\lambda
a)e^{2a\lambda}-1\right]\sqrt{\lambda^{2}-b^{2}}}\cosh\left(\sqrt{\lambda^{2}-b^{2}}\Delta
t\right).$ (A.7)

Another representation, symmetric with respect to the branes, is obtained from
(A.7) combining the integrals:

$\displaystyle S(b,\Delta t,x^{1},x^{\prime 1})$ $\displaystyle=$
$\displaystyle\frac{a}{2}S_{0}(b,\Delta t,{{\Delta
x}^{1}})+\frac{a}{2}\int_{b}^{\infty}d\lambda\frac{\cosh\left(\sqrt{\lambda^{2}-b^{2}}\Delta
t\right)}{\sqrt{\lambda^{2}-b^{2}}}$ (A.8)
$\displaystyle\times\frac{2\cosh\left({\lambda{x}_{-}^{1}}\right){+}\sum_{j=1,2}{e}^{{|{{x}_{+}^{1}-2{a_{j}}}|\lambda}}c_{j}(\lambda
a)}{c_{1}(\lambda a)c_{2}(\lambda a)e^{2a\lambda}-1}.$

Note that in the limit $(-1)^{j^{\prime}}a_{j^{\prime}}\rightarrow+\infty$,
with $j^{\prime}=1$ for $j=2$ and $j^{\prime}=2$ for $j=1$, the last term in
(A.7) goes to zero and the first two terms in the right-hand side determine
the contribution to the Wightman function in the geometry of a single bran at
$x^{1}=a_{j}$.

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