# Induced current in braneworld model in high-dimensional AdS bulk in the
cosmic string spacetime

W. Oliveira dos Santos1 , E. R. Bezerra de Mello1

1Departamento de Física, Universidade Federal da Paraíba
58.059-970, Caixa Postal 5.008, João Pessoa, PB, Brazil E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we investigate the bosonic current induced by a brane and a
magnetic flux running along the idealized cosmic string in a
$(D+1)$-dimensional anti-de Sitter (AdS) background. We consider the brane is
parallel to the AdS boundary and the cosmic string is orthogonal to them.
Moreover, we assume that on the brane the charged bosonic field obeys the
Robin boundary condition. The brane divides the space into two regions with
different properties of the vacuum state. We show that the only nonzero
component of the current density is along the azimuthal direction in both
regions. In order to develop this analysis we calculate, for both regions, the
positive frequency Wightman functions. Both functions present a part
associated with the AdS in presence of a cosmic string only, and the other
part induced by the brane. In this paper we consider only the contributions
induced by the brane. We show that in both regions the azimuthal current
densities are odd functions of the magnetic flux along the string. Different
analytic and numerical analysis are performed and an application of our
results is provided for the Randall-Sundrum braneworld model with a single
brane.

PACS numbers: 98.80.Cq, 11.10.Gh, 11.27.+d

##  1 Introduction

One of the most fascinate topological object predicted by the Grand Unified
Theory as consequence of gauge symmetry breaking is the cosmic string [1, 2].
Although recent observational data on the cosmic microwave background have
discarded cosmic strings as the primary source for primordial density
perturbation, these objects are still candidate for the generation of a number
of interesting physical effects such as gamma ray bursts [3], gravitational
waves [4] and high energy cosmic rays [5]. The gravitational field produced by
a cosmic string may be approximated by a planar angle deficit in the two-
dimensional sub-space. The strength of gravitational interactions of cosmic
strings with matter is its tension, that is characterized by the dimensionless
parameter, $G\mu_{0}$, defined in natural units. In this expression $G$
represents the Newton’s gravitation constant and $\mu_{0}$ the linear mass
density of the string, which is proportional to the square of the energy scale
where the gauge symmetry is broken.

The anti-de Sitter (AdS) spacetime is a solution of the Einstein equation in
presence of negative cosmological constant. Being maximally symmetric
spacetime, it allowed us to solve many problems in quantum fields exactly
(see, for example [6]-[12]). Besides, the importance of this background has
increased when it was observed that it generically arises as a ground state in
extended supergravity and in string theories. Moreover, additional interest in
this geometry was generated by the appearance of two models where AdS
spacetime plays a special role. The first model, the AdS/CFT correspondence
(for a review see [13]), represents a realization of the holographic principle
and relates string theories or supergravity in the AdS bulk with a conformal
field theory living on its boundary. The second model is the braneworld
scenario with large extra dimensions. This model offers a solution to the
hierarchy energy scale problem associated with the gravitational and
electroweak interactions (for comprehensive discussions on braneworld gravity
and cosmology, see [14, 15]).

The analysis of the spacetime geometry due to a cosmic string in AdS bulk has
been considered in [16, 17]. There it was shown that at distances larger than
the string’s core radius, the gravitational effects due to the presence of the
string is well described by a planar deficit angle in the AdS metric,
similarly to the case in the Minkowskian bulk. This non-trivial topology by
its turn provides additional vacuum polarization effects. In this way the
combined effect of the curvature and non-trivial topology contribute to the
evaluation of the vacuum expectation value of several physical observables, as
the energy-momentum tensor.

The investigation of the vacuum expectation value (VEV) of the bosonic current
density, $\langle j^{\mu}\rangle$, and the energy-momentum tensor, $\langle
T^{\mu}_{\nu}\rangle$, induced by an idealized cosmic string carrying magnetic
flux running along its core in a $(D+1)-$dimensional AdS spacetime, have been
analyzed in [18] and [19], respectively. Moreover, in both papers it was
admitted a compactification of one dimension along the string, and the
presence of an extra magnetic flux running its center. Also the study of the
VEV of fermionic energy-momentum tensor and current density in a
$(1+4)-$dimensional AdS space with a compactified cosmic string have been
considered in [20] and [21], respectively. The analysis of the effects due to
a brane on the vacuum fermionic current, $\langle j^{\mu}\rangle$, and the
energy-momentum tensor, $\langle T^{\mu}_{\nu}\rangle$, with the brane
parallel to the AdS boundary, were studied in [22] and [23], respectively. The
analysis of the VEV of the energy-momentum tensor associated with a charged
bosonic field on the AdS background in the presence of a cosmic string
considering a brane parallel to the AdS boundary, was developed in [24]. Here
in this paper, we want to continue in the same line of investigation, but at
this time we will turn our attention to analyze the effects of the brane on
the VEV of the induced current in both regions defined by the brane.

The organization of the paper is the following: In section 2 we present the
geometry of the spacetime that we want to consider, and the complete set of
normalized positive energy solutions of the Klein-Gordon equation considering
the presence of a brane parallel to the AdS boundary. In section 3 we
construct the Wightman functions for both regions of the space separated by
the brane. These regions are: between the AdS boundary and the brane
($L$-region) and between the brane and AdS horizon ($R$\- region). The
corresponding Wightman functions are decomposed in a part due to the AdS
spacetime in presence of a cosmic sting in the absence of brane, plus the ones
induced by the brane. In section 4 we evaluate the VEV of the bosonic current
densities in both regions. Because the above mentioned decomposition of the
Wightman functions, the same happens for the current densities. Also in this
section, various asymptotic limits of the currents are considered and
numerical results are presented. In Section 6 we apply our analysis to the
Randal-Sundrum type model with a single brane. In Section 7 we summarize the
most relevant result obtained. Throughout the paper, we use natural units
$G=\hbar=c=1$.

##  2 Klein-Gordon equation

The main objective of this section is to provide the complete set of
normalized solutions of the Klein-Gordon equation associated with a massive
scalar charged quantum field propagating in a $(D+1)$-dimensional AdS
spacetime, with $D\geq 3$, in presence of a magnetic-flux-carrying cosmic
string and taking into account the presence of a brane parallel to the AdS
boundary. So, with this objective we present first, the line element, in
cylindrical coordinate, in $(1+3)-$dimensional AdS spacetime in the presence
of a cosmic stiring:

$ds^{2}=e^{-2y/a}[dt^{2}-dr^{2}-r^{2}d\phi^{2}]-dy^{2}\ .$ (1)

In the above line element the idealized cosmic string is along the $y-$axis,
$r\geqslant 0$ and $\phi\in[0,\ 2\pi/q]$ define the coordinates on the conical
geometry, $(t,\ y)\in(-\infty,\ \infty)$. The parameter $a$ is associated with
the curvature scale of the background spacetime; moreover, the parameter $q>1$
provides the planar angle deficit, $\delta\phi=2\pi(1-q^{-1})$, produced by
the cosmic string. In our analysis, we will use the Poincaré coordinate
defined by $w=ae^{y/a}$. In this case the line element above is expressed in
the form conformally related to the line element associated with a cosmic
string in Minkowski spacetime:

$ds^{2}=\left(\frac{a}{w}\right)^{2}[dt^{2}-dr^{2}-r^{2}d\phi^{2}-dw^{2}]\ .$
(2)

The new coordinate, $w$, is defined in the interval $[0,\ \infty)$. Two values
for this coordinates deserve to be mentioned: $w=0$ and $w=\infty$. They
correspond to the AdS boundary and horizon, respectively. The generalization
of (2) to $(D+1)$-dimensional, with $D>3$, is given by

$ds^{2}=\left(\frac{a}{w}\right)^{2}[dt^{2}-dr^{2}-r^{2}d\varphi^{2}-dw^{2}-\sum_{i=4}^{D}(dx^{i})^{2}]\
,$ (3)

with $x^{i}\in(-\infty,\ \infty)$.

The curvature scale $a$ in (3) is related to the cosmological constant,
$\Lambda$, and the Ricci scalar, $R$, by the formulas

$\Lambda=-\frac{D(D-1)}{2a^{2}}\ ,\ \ R=-\frac{D(D+1)}{a^{2}}\ .$ (4)

###  2.1 Klein-Gordon equation

The field equation which governs the quantum dynamics of a charged bosonic
field with mass $m$, in a curved background and in the presence of an
electromagnetic potential vector, $A_{\mu}$, reads

$({\mathcal{D}}^{2}+m^{2}+\xi R)\varphi(x)=0\ ,$ (5)

where the differential operator in the field equation reads

$\displaystyle{\mathcal{D}}^{2}=\frac{1}{\sqrt{|g|}}{\mathcal{D}}_{\mu}\left(\sqrt{|g|}g^{\mu\nu}{\mathcal{D}}_{\nu}\right),\
{\mathcal{D}}_{\mu}=\partial_{\mu}+ieA_{\mu}\ \ {\rm with}\ \
g=\det(g_{\mu\nu})\ .$ (6)

In the above equation, we also consider the presence of a non-minimal
coupling, $\xi$, between the field and the geometry represented by the Ricci
scalar, $R$. Two specific values for the curvature coupling are of special
interest: the value $\xi=0$ corresponds to minimal coupling, and and
$\xi=\frac{D-1}{4D}$, the conformal coupling, respectively. Moreover, we
consider only the component vector potential $A_{\phi}=-q\Phi_{\phi}/(2\pi)$
different from zero. This vector potential corresponds to a magnetic flux,
$\Phi_{\phi}$, running along the string’s axis.

In this work we admit the presence of a codimension one flat boundary,
hereafter named brane, located at $w=w_{0}$ and parallel to the AdS boundary,
and that the the field operator obeys the gauge invariant Robin boundary
condition,

$(1+\beta n^{\mu}{\mathcal{D}}_{\mu})\varphi(x)=0,\ w=w_{0}\ .$ (7)

The parameter $\beta$ in the equation above is a constant and it encodes the
properties of the brane. In particular, the special cases $\beta=0$ and
$\beta=\infty$ correspond to the Dirichlet and Neumann boundary conditions,
respectively. In addition, $n^{\mu}$ is the inward pointing (with respect to
the region under consideration) normal to the brane at $w=w_{0}$. In the
region $0\leq w\leq w_{0}$ (referred to as the $L$(left)-region this normal is
given as $n^{\mu}=-\delta_{3}^{\mu}a/w$ and in the region $w_{0}\leq
w\leq\infty$ $R$(right)-region as $n^{\mu}=\delta_{3}^{\mu}a/w$.

In the geometry defined by (3) and in the presence of the azimuthal vector
potentials, $A_{\phi}$, the Klein-Gordon (KG) equation (5) becomes

$\displaystyle\left[\frac{\partial^{2}}{\partial
t^{2}}-\frac{\partial^{2}}{\partial r^{2}}-\frac{1}{r}\frac{\partial}{\partial
r}-\frac{1}{r^{2}}\left(\frac{\partial}{\partial\phi}+ieA_{\phi}\right)^{2}-\frac{\partial^{2}}{\partial
w^{2}}-\frac{(1-D)}{w}\frac{\partial}{\partial w}\right.$
$\displaystyle\left.+\frac{M(D,m,\xi)}{w^{2}}-\sum_{i=4}^{D}\frac{\partial^{2}}{\partial(x^{i})^{2}}\right]\varphi(x)=0\
,$ (8)

with $M(D,m,\xi)=a^{2}m^{2}-\xi D(D+1)$.

On basis of previous analysis presented in [18], the positive energy solution
of (8) can be expressed by

$\varphi_{\sigma}(x)=C_{\sigma}w^{\frac{D}{2}}W_{\nu}(pw)J_{q|n+\alpha|}(\lambda
r)e^{-iEt+iqn\phi+i\vec{k}\cdot\vec{x}_{\parallel}}\ ,$ (9)

with the function $W_{\nu}(pw)$,

$W_{\nu}(pw)=C_{1}J_{\nu}(pw)+C_{2}Y_{\nu}(pw)\ ,$ (10)

being given by a linear combination of the Bessel and Neumann functions [25].
The order of these functions is,

$\nu=\sqrt{\frac{D^{2}}{4}+a^{2}m^{2}-\xi D(D+1)}\ .$ (11)

The enrgy and the parameter $\alpha$ in the Bessl function associated with the
radial coordinate are:

$\displaystyle E$ $\displaystyle=$
$\displaystyle\sqrt{\lambda^{2}+p^{2}+\vec{k}^{2}},$ $\displaystyle\alpha$
$\displaystyle=$
$\displaystyle\frac{eA_{\phi}}{q}=-\frac{\Phi_{\phi}}{\Phi_{0}}\ ,$ (12)

being $\Phi_{0}=\frac{2\pi}{e}$, the quantum flux. Moreover, in (9)
$\vec{x}_{\parallel}$ corresponds the coordinates defined in the $(D-3)$ extra
dimensions, and $\vec{k}$ the corresponding momentum. In (9), $\sigma$
represents the set of quantum numbers $(n,\lambda,p,\vec{k})$, with $n=0,\pm
1,\pm 2,\ldots$, $\lambda\geq 0$, $-\infty<k^{j}<\infty$ for $j=4,...,D$. As
to the quantum number $p$, it is determined, separately, for each region
divided by the brane.

The coefficient $C_{\sigma}$ in (9) is determined from the normalization
condition

$\displaystyle\int
d^{D}x\sqrt{|g|}g^{00}\varphi_{\sigma^{\prime}}^{*}(x)\varphi_{\sigma}(x)=\frac{1}{2E}\delta_{\sigma,\sigma^{\prime}}\
,$ (13)

with delta symbol on the right-hand side is understood as Dirac delta function
for the continuous quantum number, and Kronecker delta for the discrete one.

####  2.1.1 Normalized wave-functions in $R$-region

First let us consider the $R$-region. Due to the Robin boundary condition (7)
on the flat boundary, we obtain the relation
$C_{2}/C_{1}=-\bar{J}_{\nu}(pw_{0})/\bar{Y}_{\nu}(pw_{0})$ for the
coefficients in (10).

From now on, we use the compact notation

$\bar{F}(x)=A_{0}F(x)+B_{0}xF^{\prime}(x)\ ,$ (14)

with the coefficients

$A_{0}=1+\frac{D\beta}{2a},\quad B_{0}=\beta/a\ ,$ (15)

in the expressions of the wave-functions.

The normalized solutions of KG equation in $R$-region compatible with the
boundary condition (7), can be written presented as,

$\varphi_{(R)\sigma}(x)=C_{(R)\sigma}w^{\frac{D}{2}}g_{\nu}(pw_{0},pw)J_{q|n+\alpha|}(\lambda
r)e^{-iEt+iqn\phi+i\vec{k}\cdot\vec{x}_{\parallel}}\ ,$ (16)

where we have introduced the function

$g_{\nu}(u,v)=J_{\nu}(v)\bar{Y}_{\nu}(u)-\bar{J}_{\nu}(u)Y_{\nu}(v)\ .$ (17)

Due to the continuous values assumed by the quantum number $p$, the
normalization condition (13), provides

$|C_{(R)\sigma}|^{2}=\frac{(2\pi)^{2-D}qp\lambda}{2Ea^{D-1}[\bar{J}_{\nu}^{2}(pw_{0})+\bar{Y}_{\nu}^{2}(pw_{0})]}\
.$ (18)

####  2.1.2 Normalized wave-functions in $L$-region

In the $L$-region, the integration over $w$ in (13), is restricted in the
interval $0\leq w\leq w_{0}$. In the case of $C_{2}\neq 0$ in (10), the
integral over $w$ diverges at the lower limit $w=0$ for the case with $\nu\geq
1$. Consequently, in that region, we should take $C_{2}=0$. On the other hand,
in the interval $0\leq\nu<1$, the solution (10) with $C_{2}\neq 0$ is
normalizable and in order to uniquely define the mode functions an additional
boundary condition at the AdS boundary is required [26, 27]. Here, we assume
the Dirichlet boundary condition on $w=0$ which implies $C_{2}=0$. Thus, with
this choice, the mode function in the $L$-region are given by

$\varphi_{(L)\sigma}(x)=C_{(L)\sigma}w^{\frac{D}{2}}J_{\nu}(pw)J_{q|n+\alpha|}(\lambda
r)e^{-iEt+iqn\phi+i\vec{k}\cdot\vec{x}_{\parallel}}\ .$ (19)

According to the Robin boundary condition (7), the eigenvalues of the quantum
number $p$ obey the relation:

$\bar{J}_{\nu}(pw_{0})=0\ ,$ (20)

being $\bar{J}_{\nu}(x)$ given by (14), with

$A_{0}=1-\frac{D\beta}{2a},\quad B_{0}=-\beta/a\ .$ (21)

The eigenvalues of (20) are given by $p=p_{\nu,i}/w_{0}$, with $p_{\nu,i}$
being the positive zeros of the function $\bar{J}_{\nu}(x)$, enumerated by
$i=1,2,...$. We can observe that the roots $p_{\nu,i}$ do not depend on the
location of the brane. Considering now the normalization condition (13), with
$\delta_{p,p^{\prime}}=\delta_{i,i^{\prime}}$, and integrating over $w$ in the
interval $[0,w_{0}]$, after some algebraic manipulations involving the Bessel
functions, we obtain

$|C_{(L)\sigma}|^{2}=\frac{(2\pi)^{2-D}qp_{\nu,i}\lambda
T_{\nu}(p_{\nu,i})}{w_{0}a^{D-1}\sqrt{p_{\nu,i}^{2}+w_{0}^{2}(\lambda^{2}+\vec{k}^{2}})}\
.$ (22)

In the above expression we have for the function $T_{\nu}(z)$, the following
result:

$\displaystyle
T_{\nu}(z)=z[(z^{2}-\nu^{2})J_{\nu}^{2}(z)+z^{2}(J_{\nu}^{\prime}(z))^{2}]^{-1}\
.$ (23)

##  3 Wightman Function

In this section we want to obtain the positive frequency Wightman function
induced by the brane for both regions, $R$ and $L$, in a closed form. The
positive frequency Wightman function is defined by $W(x,x^{\prime})=\langle
0|\hat{\varphi}(x)\hat{\varphi}^{\dagger}(x^{\prime})|0\rangle$, where
$|0\rangle$ stands for the vacuum state. Here we assume that the field
operator, $\hat{\varphi}(x)$, is prepared in the Poincaré vacuum state. To
evaluate this function, we use the mode sum formula

$W(x,x^{\prime})=\sum_{\sigma}\varphi_{\sigma}(x)\varphi_{\sigma}^{\ast}(x^{\prime})\
.$ (24)

###  3.1 $R$-region

The normalized positive energy solution of the KG equation in $R$-region, is
given by combining previous results, (16), (17) and (18). It reads,

$\varphi_{(R)\sigma}(x)=\sqrt{\frac{(2\pi)^{2-D}qp\lambda}{2Ea^{D-1}[\bar{J}_{\nu}^{2}(pw_{0})+\bar{Y}_{\nu}^{2}(pw_{0})]}}w^{\frac{D}{2}}g_{\nu}(pw_{0},pw)J_{q|n+\alpha|}(\lambda
r)e^{-iEt+iqn\phi+i\vec{k}\cdot\vec{x}_{\parallel}}\ .$ (25)

Substituting the above expression into (24), we get:

$\displaystyle W_{R}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(ww^{\prime})^{D/2}}{2a^{D-1}(2\pi)^{D-2}}\sum_{\sigma}\frac{p\lambda}{E}\frac{g_{\nu}(pw_{0},pw)g_{\nu}(pw_{0},pw^{\prime})}{(\bar{J}_{\nu}^{2}(pw_{0})+\bar{Y}_{\nu}^{2}(pw_{0}))}J_{q|n+\alpha|}(\lambda
r)J_{q|n+\alpha|}(\lambda r^{\prime})$ (26) $\displaystyle\times$
$\displaystyle
e^{-iE(t-t^{\prime})+iqn(\phi-\phi^{\prime})+i\vec{k}\cdot(\vec{x}_{\parallel}-\vec{x}^{\prime}_{\parallel})}\
,$

where we are using the compact notation below for the summation over $\sigma$:

$\sum_{\sigma}=\sum_{n=-\infty}^{+\infty}\ \int dp\ \int_{0}^{\infty}\
d\lambda\ \int d{\vec{k}}\ .$ (27)

Now performing a Wick rotation on the time coordinate and using the identity

$\frac{e^{-E\Delta\tau}}{E}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}dse^{-s^{2}E^{2}-(\Delta\tau)^{2}/(4s^{2})}\
,$ (28)

being $E=\sqrt{\lambda^{2}+p^{2}+\vec{k}^{2}}$, we can integrate over
$\lambda$ and $\vec{k}$ with the help of [28]. The final result result is

$\displaystyle W_{R}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{qrr^{\prime}}{2(2\pi)^{D/2}a^{D-1}}\left(\frac{ww^{\prime}}{rr^{\prime}}\right)^{D/2}\int_{0}^{\infty}dvv^{\frac{D}{2}-2}e^{-\frac{r^{2}+r^{\prime
2}+\Delta\vec{x}_{\parallel}^{2}-\Delta
t^{2}}{2rr^{\prime}}v}\sum_{n}e^{inq\Delta\phi}I_{q|n+\alpha|}(v)$ (29)
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dppe^{-\frac{rr^{\prime}}{2v}p^{2}}\frac{g_{\nu}(pw_{0},pw)g_{\nu}(pw_{0},pw^{\prime})}{\bar{J}_{\nu}^{2}(pw_{0})+\bar{Y}_{\nu}^{2}(pw_{0})}\
,$

where we have introduced a new variable, $v=rr^{\prime}/(2s^{2})$.

The above Wightman function contains the contributions coming from the cosmic
string in AdS spacetime without boundary, more the contribution induced by the
boundary. Because, in this paper, we are interested to calculate the VEV of
the current induced by the presence of the brane, let us subtract form (29)
the corresponding Wightman induced by the cosmic string only in AdS. The
latter can be obtained from (29), by taking the limit $w_{0}\to 0$. So we can
write:

$W_{b(R)}(x,x^{\prime})=W_{R}(x,x^{\prime})-W_{cs}(x,x^{\prime})\ .$ (30)

The Wightman function, $W_{b}(x,x^{\prime})$, is obtained by using the the
following identity:

$\displaystyle\frac{g_{\nu}(pw_{0},pw)g_{\nu}(pw_{0},pw^{\prime})}{\bar{J}_{\nu}^{2}(pw_{0})+\bar{Y}_{\nu}^{2}(pw_{0})}-J_{\nu}(pw)J_{\nu}(pw^{\prime})=-\frac{1}{2}\sum_{l=1}^{2}\frac{\bar{J}_{\nu}(pw_{0})}{\bar{H}_{\nu}^{(l)}(pw_{0})}H_{\nu}^{(l)}(pw)H_{\nu}^{(l)}(pw^{\prime})\
,$ (31)

being $H_{\nu}^{(l)}(x)$, $l=1,2$, the Hankel functions [25]. So, we get,

$\displaystyle W_{b(R)}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{qrr^{\prime}}{4(2\pi)^{D/2}a^{D-1}}\left(\frac{ww^{\prime}}{rr^{\prime}}\right)^{D/2}\int_{0}^{\infty}dvv^{\frac{D}{2}-2}e^{-\frac{r^{2}+r^{\prime
2}+\Delta\vec{x}_{\parallel}^{2}-\Delta
t^{2}}{2rr^{\prime}}v}\sum_{n}e^{inq\Delta\phi}I_{q|n+\alpha|}(v)$ (32)
$\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dppe^{-\frac{rr^{\prime}}{2v}p^{2}}\sum_{l=1}^{2}\frac{\bar{J}_{\nu}(pw_{0})}{\bar{H}_{\nu}^{(l)}(pw_{0})}H_{\nu}^{(l)}(pw)H_{\nu}^{(l)}(pw^{\prime})\
.$

The parameter $\alpha$ can be written in the form

$\alpha=n_{0}+\alpha_{0},\ \textrm{with}\ |\alpha_{0}|<\frac{1}{2},$ (33)

with $n_{0}$ being an integer number; moreover, the sum over the quantum
number $n$ in (32), has been developed in [29]. The result is given below:

$\displaystyle\sum_{n=-\infty}^{\infty}e^{iqn\Delta\phi}I_{q|n+\alpha|}(v)=\frac{1}{q}\sum_{k}e^{v\cos(2\pi
k/q-\Delta\phi)}e^{i\alpha(2\pi
k-q\Delta\phi)}-\frac{e^{-iqn_{0}\Delta\phi}}{2\pi i}$ (34)
$\displaystyle\times$ $\displaystyle\sum_{j=\pm 1}je^{ji\pi
q|\alpha_{0}|}\int_{0}^{\infty}dy\frac{(\cosh{[qy(1-|\alpha_{0}|)]}-\cosh{(|\alpha_{0}|qy)e^{-iq(\Delta\phi+j\pi)}})e^{-v\cosh(y)}}{\cosh{(qy)}-\cos{(q(\Delta\phi+j\pi)}}\
,$

with the parameter $k$ varying in the interval:

$-\frac{q}{2}+\frac{\Delta\phi}{\Phi_{0}}\leq
k\leq\frac{q}{2}+\frac{\Delta\phi}{\Phi_{0}}\ .$ (35)

Substituting (34) into (32), it is possible to perform the integration over
$v$ using [28]. The result is

$\displaystyle W_{b(R)}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{(ww^{\prime})^{D/2}}{2(2\pi)^{D/2}a^{D-1}}\Biggl{\\{}\sum_{k}\frac{e^{i\alpha(2\pi
k-q\Delta\phi)}}{u_{k}^{\frac{D}{2}-1}}\int_{0}^{\infty}dpp^{D/2}\sum_{l=1}^{2}\frac{\bar{J}_{\nu}(pw_{0})}{\bar{H}_{\nu}^{(l)}(pw_{0})}$
(36) $\displaystyle\times$ $\displaystyle
H_{\nu}^{(l)}(pw)H_{\nu}^{(l)}(pw^{\prime})K_{\frac{D}{2}-1}(pu_{k})-\frac{qe^{-iqn_{0}\Delta\phi}}{2\pi
i}\sum_{j=\pm 1}je^{ji\pi q|\alpha_{0}|}$ $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dy\frac{\cosh{[qy(1-|\alpha_{0}|)]}-\cosh{(|\alpha_{0}|qy)e^{-iq(\Delta\phi+j\pi)}}}{u_{y}^{\frac{D}{2}-1}\big{[}\cosh{(qy)}-\cos{(q(\Delta\phi+j\pi))}\big{]}}\int_{0}^{\infty}dpp^{D/2}\sum_{l=1}^{2}\frac{\bar{J}_{\nu}(pw_{0})}{\bar{H}_{\nu}^{(l)}(pw_{0})}$
$\displaystyle\times$ $\displaystyle
H_{\nu}^{(l)}(pw)H_{\nu}^{(l)}(pw^{\prime})K_{\frac{D}{2}-1}(pu_{y})\Biggr{\\}}\
.$

In the above expression we have introduced the notation

$\displaystyle u_{k}^{2}$ $\displaystyle=$ $\displaystyle r^{2}+r^{\prime
2}-2rr^{\prime}\cos{(2\pi
k/q-\Delta\phi)}+\Delta\vec{x}_{\parallel}^{2}-\Delta t^{2}$ $\displaystyle
u_{y}^{2}$ $\displaystyle=$ $\displaystyle r^{2}+r^{\prime
2}+2rr^{\prime}\cosh{(y)}+\Delta\vec{x}_{\parallel}^{2}-\Delta t^{2}.$ (37)

Finally we rotate the contour integration over $p$ by the angle $\pi/2$
$(-\pi/2)$ for the term $l=1$ $(l=2)$. Using the relations involving Bessel
functions with imaginary argument [25], the result is

$\displaystyle W_{b(R)}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{(ww^{\prime})^{D/2}}{(2\pi)^{D/2}a^{D-1}}\int_{0}^{\infty}dpp^{D-1}\frac{\bar{I}_{\nu}(pw_{0})}{\bar{K}_{\nu}(pw_{0})}K_{\nu}(pw)K_{\nu}(pw^{\prime})$
(38) $\displaystyle\times$ $\displaystyle\Biggl{\\{}\sum_{k}e^{i\alpha(2\pi
k-q\Delta\phi)}f_{\frac{D}{2}-1}(pu_{k})-\frac{qe^{-iqn_{0}\Delta\phi}}{2\pi
i}\sum_{j=\pm 1}je^{ji\pi q|\alpha_{0}|}$ $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dy\frac{\cosh{[qy(1-|\alpha_{0}|)]}-\cosh{(|\alpha_{0}|qy)e^{-iq(\Delta\phi+j\pi)}}}{\cosh{(qy)}-\cos{(q(\Delta\phi+j\pi))}}f_{\frac{D}{2}-1}(pu_{y})\Biggl{\\}}\
,$

where we have introduced the notation

$f_{\mu}(x)=\frac{J_{\mu}(x)}{x^{\mu}}\ .$ (39)

###  3.2 $L$-Region

Now we turn our attention to calculate the positive frequency Wightman
function in the $L$-region. Substituting the respective wave function
solutions (19), with the coefficient (22), into (24), we get

$\displaystyle W_{L}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(ww^{\prime})^{D/2}}{a^{D-1}(2\pi)^{D-2}w_{0}^{2}}\sum_{\sigma}\frac{\lambda
p_{\nu,i}}{\sqrt{(p_{\nu,i}/w_{0})^{2}+\lambda^{2}+\vec{k}^{2}}}T_{\nu}(p_{\nu,i})J_{\nu}(p_{\nu,i}w)J_{\nu}(p_{\nu,i}w^{\prime})$
(40) $\displaystyle\times$ $\displaystyle J_{q|n+\alpha|}(\lambda
r)J_{q|n+\alpha|}(\lambda
r^{\prime})e^{inq\Delta\varphi+i\vec{k}\cdot\Delta\vec{x}-iE\Delta t}\ ,$

where now, we have

$\sum_{\sigma}=\int_{0}^{\infty}d\lambda\sum_{i=1}^{\infty}\sum_{n}\int
d\vec{k}\ .$ (41)

Using the identity (28), we can integrate over $\lambda$ and $\vec{k}$ using
the formulas of [28], obtaining the following expression:

$\displaystyle W_{L}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{q(ww^{\prime})^{D/2}}{a^{D-1}(2\pi)^{D/2}w_{0}^{2}(rr^{\prime})^{D/2-1}}\int_{0}^{\infty}dvv^{D/2-2}e^{-\frac{r^{2}+r^{\prime
2}+\Delta\vec{x}_{\parallel}^{2}-\Delta
t^{2}}{2rr^{\prime}}v}\sum_{n=-\infty}^{\infty}e^{inq\Delta\varphi}I_{q|n+\alpha_{0}|}(v)$
(42) $\displaystyle\times$
$\displaystyle\sum_{i=1}^{\infty}p_{\nu,i}T_{\nu}(p_{\nu,i})J_{\nu}(p_{\nu,i}w/w_{0})J_{\nu}(p_{\nu,i}w^{\prime}/w_{0})e^{-\frac{r^{2}p_{\nu,i}^{2}}{2w_{0}^{2}v}}\
,$

where we have introduced a new variable $v=r^{2}/(2s^{2})$.

At this point we substitute (34) into the above expression. After procedure
the integral $v$, we obtain,

$\displaystyle W_{L}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{2(ww^{\prime})^{D/2}}{(2\pi)^{D/2}a^{D-1}w_{0}^{\frac{D}{2}+1}}\Biggl{\\{}\sum_{k}\frac{e^{i\alpha(2\pi
k-q\Delta\phi)}}{u_{k}^{\frac{D}{2}-1}}\sum_{i=1}^{\infty}p_{\nu,i}^{D/2}T_{\nu}(p_{\nu,i})J_{\nu}(p_{\nu,i}w/w_{0})J_{\nu}(p_{\nu,i}w^{\prime}/w_{0})$
(43) $\displaystyle\times$ $\displaystyle
K_{\frac{D}{2}-1}(u_{k}p_{\nu,i}/w_{0})-\frac{qe^{-iqn_{0}\Delta\phi}}{2\pi
i}\sum_{j=\pm 1}je^{ji\pi q|\alpha_{0}|}$ $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dy\frac{\cosh{[qy(1-|\alpha_{0}|)]}-\cosh{(|\alpha_{0}|qy)e^{-iq(\Delta\phi+j\pi)}}}{u_{y}^{\frac{D}{2}-1}\big{[}\cosh{(qy)}-\cos{(q(\Delta\phi+j\pi))}\big{]}}$
$\displaystyle\times$
$\displaystyle\sum_{i=1}^{\infty}p_{\nu,i}^{D/2}T_{\nu}(p_{\nu,i})J_{\nu}(p_{\nu,i}w/w_{0})J_{\nu}(p_{\nu,i}w^{\prime}/w_{0})K_{\frac{D}{2}-1}(u_{y}p_{\nu,i}/w_{0})\Biggl{\\}}\
.$

Again, we use the definition (37) to the variable $u_{k}$ and $u_{y}$.

In order to obtain an expression to the Wightman function in the $L$-region
more convenient for the extraction of the brane induced part, we apply to the
series over $i$ a variant of the generalized Abel-Plana formula [30]

$\sum_{i=1}^{\infty}T_{\nu}(p_{\nu,i})f(p_{\nu,i})=\frac{1}{2}\int_{0}^{\infty}dzf(z)-\frac{1}{2\pi}\int_{0}^{\infty}dz\frac{\bar{K}_{\nu}(z)}{\bar{I}_{\nu}(z)}\left[e^{-i\nu
z}f(iz)+e^{i\nu z}f(-iz)\right]\ .$ (44)

For the problem that we are analyzing the function $f(z)$ is given below,

$f(z)=z^{D/2}J_{\nu}(zw/w_{0})J_{\nu}(zw^{\prime}/w_{0})K_{\frac{D}{2}-1}(2uz/w_{0})\
.$ (45)

The first term provided by (44) corresponds the Wightman function in the
absence of brane, while the second is induced by the boundary. As mentioned in
the previous subsection, here we are interested in the brane-induced Whigthman
function. Therefore, after some intermediate steps, we obtain

$\displaystyle W_{b(L)}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{(ww^{\prime})^{D/2}}{(2\pi)^{D/2}a^{D-1}}\int_{0}^{\infty}dpp^{D/2}\frac{\bar{K}_{\nu}(pw_{0})}{\bar{I}_{\nu}(pw_{0})}I_{\nu}(pw)I_{\nu}(pw^{\prime})$
(46) $\displaystyle\times$ $\displaystyle\Biggl{\\{}\sum_{k}e^{i\alpha(2\pi
k-q\Delta\phi)}f_{\frac{D}{2}-1}(pu_{k})-\frac{qe^{-iqn_{0}\Delta\phi}}{2\pi
i}\sum_{j=\pm 1}je^{ji\pi q|\alpha_{0}|}$ $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dy\frac{\cosh{[qy(1-|\alpha_{0}|)]}-\cosh{(|\alpha_{0}|qy)e^{-iq(\Delta\phi+j\pi)}}}{\cosh{(qy)}-\cos{(q(\Delta\phi+j\pi))}}f_{\frac{D}{2}-1}(pu_{y})\Biggl{\\}}\
,$

where we have made change of variable $z=pw_{0}$.

##  4 Bosonic Current Density

The VEV of the bosonic current density can be expressed in terms of the
positive frequency Wightman function by,

$\langle j_{\mu}\rangle=ie\lim_{x^{\prime}\rightarrow
x}\\{(\partial_{\mu}-\partial_{\mu^{\prime}})W(x,x^{\prime})+2ieA_{\mu}W(x,x^{\prime})\\}$
(47)

Because the analysis of the induced bosonic current in the $(1+D)-$dimensional
AdS space in the presence of a carrying-magnetic-flux cosmic string has been
given in [18], here we are mainly interested to calculate the bosonic current
induced by the presence of the brane in both regions, i.e., for $0\leq w\leq
w_{0}$ and $w_{0}\leq w<\infty$.

As we will see in the next subsections, the only non-zero current density
components are the azimuthal ones, and it is a periodic function of the
magnetic flux along the string, $\Phi_{\phi}$, with period equal to the
quantum flux.

###  4.1 Charge density

Let us begin with the calculation of the charge density. Since $A_{0}=0$, we
have

$\langle j_{0}(x)\rangle_{b(J)}=ie\lim\limits_{x^{\prime}\rightarrow
x}(\partial_{t}-\partial^{\prime}_{t})W_{b(J)}(x,x^{\prime})\ ,$ (48)

with $J=R,L$, that represents the $R$ and $L$ regions.

The analysis of the charge density depends on the behavior of the time
derivative of the function $f_{D/2-1}(pu_{\sigma})$, with $\sigma=j,\ y$, that
appear in (38) and (46). Using the fact that

$\displaystyle\partial_{z}f_{\mu}(z)=-zf_{\mu+1}(z)\ ,$ (49)

and knowing that the arguments of the function $f_{D/2-1}(pu_{\sigma})$ depend
on the time variable with $(t-t^{\prime})$, we can see that

$\displaystyle\partial_{t}f_{D/2-1}(pu_{\sigma})=p^{2}(t-t^{\prime})f_{D/2}(pu_{\sigma})\
.$ (50)

Now taking the coincidence limit on the function $f_{D/2}(pu_{\sigma})$, we
can verify that for the case where $\sigma=k=0$, this function goes to a
finite value for $u_{0}\to 0$. As to the case $\sigma=k\neq 0$, $u_{k}\to
2r\sin(k\pi/q)$ and for $\sigma=y$, $u_{y}\to 2r\cosh(y/2)$. For both last
cases the function $f_{D/2}(pu_{\sigma})$ assumes a finite value, even for
$r\to 0$. Finally taking $t^{\prime}\to t$, the above expression vanishes.
Consequently we conclude that there is no induced charge density.

Following similar procedure we can prove that there are no induced current
density along the radial coordinate and extra dimensions, i.e., $\langle
j^{r}\rangle=\langle j^{i}\rangle=0$. As to the induced current along $w$,
$\langle j^{w}\rangle$, we can promptly verify that it is zero.

##  5 Azimuthal Current

In this subsection we will proceed the calculations of the induced azimuthal
currents in the regions $R$ and $L$, induced by the brane.

###  5.1 $R$-region

Let us start the analysis in $R$-region. Although the Wightman in this region
was given in (38), to develop the calculation of the azimuthal current
density, it is more convenient to use the Wightman function provided in (32).
So, substituting this function into the formal expression for the VEV of the
current density operator,

$\displaystyle\langle j_{\phi}(x)\rangle_{b(R)}=ie\lim_{x^{\prime}\to
x}[(\partial_{\phi}-\partial_{\phi^{\prime}})W_{b(R)}(x^{\prime},x)+2ieA_{\phi}W_{b(R)}(x^{\prime},x)]\
,$ (51)

using $A_{\mu}=\delta_{\mu}^{\phi}A_{\phi}=q\alpha/e$, and taking the
coincidence limit also in the angular variable, we get

$\displaystyle\langle j_{\phi}(x)\rangle_{b(R)}$ $\displaystyle=$
$\displaystyle\frac{qew^{D}}{2a^{D-1}(2\pi)^{D/2}r^{D-2}}\int_{0}^{\infty}{dv}{v^{D/2-2}}e^{-v}\sum_{n}q(n+\alpha)I_{q|n+\alpha|}(v)$
(52) $\displaystyle\times$
$\displaystyle\int_{0}^{\infty}dppe^{-\frac{p^{2}r^{2}}{2v}}\sum_{l=1}^{2}\frac{\bar{J}_{\nu}^{2}(pw_{0})}{\bar{H}_{\nu}^{(l)}(pw_{0})}\left(H_{\nu}^{(l)}(pw)\right)^{2}\
.$

In [31], a compact expression for the summation over the quantum number $n$
above has been derived. We reproduce this result here,

$\displaystyle\sum_{n=-\infty}^{\infty}(n+\alpha)I_{q|n+\alpha|}(v)$
$\displaystyle=$
$\displaystyle\frac{2v}{q^{2}}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}e^{v\cos(2\pi k/q)}$ (53) $\displaystyle+$
$\displaystyle\frac{v}{q\pi}\int_{0}^{\infty}dy\sinh{(y)}\frac{e^{-v\cosh{(y)}}g(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(\pi
q)}}\ ,$

where $[q/2]$ represents the integer part of $q/2$, and the prime on the sign
of the summation means that in the case $q=2p$ the term $k=q/2$ should be
taken with the coefficient $1/2$. Moreover the function, $g(q,\alpha_{0},y)$,
is defined as

$g(q,\alpha_{0},y)=\sin{(q\pi\alpha_{0})}\sinh{((1-|\alpha_{0}|)qy)}-\sinh{(q\alpha_{0}y)}\sin{((1-|\alpha_{0}|)\pi
q)}.$ (54)

Substituting the above result into (52) and with the help of [28], after a few
straightforward steps, we get

$\displaystyle\langle j_{\phi}(x)\rangle_{b(R)}$ $\displaystyle=$
$\displaystyle\frac{2ew^{D}}{(4\pi)^{D/2}a^{D-1}r^{D/2-2}}\int_{0}^{\infty}dpp^{D/2+1}\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}}{s_{k}^{D/2}}K_{\frac{D}{2}}(2prs_{k})$ (55)
$\displaystyle+$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh^{D/2}(y/2)[\cosh(qy)-\cos(q\pi)]}K_{\frac{D}{2}}(2pr\cosh(y/2))\Bigg{]}$
$\displaystyle\times$
$\displaystyle\sum_{l=1}^{2}\frac{\bar{J}_{\nu}^{2}(pw_{0})}{\bar{H}_{\nu}^{(l)}(pw_{0})}\left(H_{\nu}^{(l)}(pw)\right)^{2}\
,\ s_{k}=\sin(\pi k/q)\ .$

As the next step we rotate the contour of the integration over $p$ by the
angle $\pi/2$ $(-\pi/2)$ for the term with $l=1$ $(l=2)$. Using the relations
between Bessel functions of imaginary argument, the contravariant component of
the azimuthal current reads

$\displaystyle\langle j^{\phi}\rangle_{b(R)}$ $\displaystyle=$
$\displaystyle-\frac{4e}{(2\pi)^{D/2}a^{D+1}}\int_{0}^{\infty}dzz^{D+1}\frac{\bar{I}_{\nu}(zw_{0}/w)}{\bar{K}_{\nu}(zw_{0}/w)}K_{\nu}^{2}(z)$
(56) $\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}f_{\frac{D}{2}}(2z(r/w)s_{k})$ $\displaystyle+$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}f_{\frac{D}{2}}(2z(r/w)\cosh(y/2))\Bigg{]}\
,$

where we have defined $z=pw$. Note that this VEV depends on the ratio $r/w$,
which is related to the proper distance from the string, and the ratio
$w/w_{0}$, which is related to the proper distance from the brane
$w/w_{0}=e^{(y-y_{0})/a}$.

Let us now investigate this VEV in some special and asymptotic cases. In the
conformal massless field case, we have $\nu=1/2$ according to (11), and using
the corresponding expressions of the Bessel functions for this particular
order, we obtain

$\displaystyle\langle j^{\phi}\rangle_{b(R)}$ $\displaystyle=$
$\displaystyle-\frac{4e}{(2\pi)^{D/2}a^{D+1}}\int_{0}^{\infty}dzz^{D}e^{(-2+\frac{w_{0}}{w})z}$
(57) $\displaystyle\times$
$\displaystyle\frac{2B_{0}(w_{0}/w)z\cosh(zw_{0}/w)+(2A_{0}-B_{0})\sinh(zw_{0}/w)}{2A_{0}-B_{0}(1+2zw_{0}/w)}$
$\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}f_{\frac{D}{2}}(2z(r/w)s_{k})$ $\displaystyle+$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}f_{\frac{D}{2}}(2z(r/w)\cosh(y/2))\Bigg{]}\
.$

For distances from the brane much larger compared with the AdS radius,
$w/w_{0}\gg 1$, we make use of the formulae for the modified Bessel functions
for small values of the argument [25], with the assumption that $A_{0}-\nu
B_{0}\neq 0$, to the leading order, we have

$\displaystyle\langle j^{\phi}\rangle_{b(R)}$ $\displaystyle\approx$
$\displaystyle-\frac{2^{3-2\nu-D/2}e}{\pi^{D/2}\Gamma(\nu)\Gamma(\nu+1)a^{D+1}}\left(\frac{A_{0}+\nu
B_{0}}{A_{0}-\nu
B_{0}}\right)\left(\frac{w_{0}}{w}\right)^{2\nu}\int_{0}^{\infty}dzz^{D+2\nu+1}K_{\nu}^{2}(z)$
(58) $\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}f_{\frac{D}{2}}(2z(r/w)s_{k})$ $\displaystyle+$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}f_{\frac{D}{2}}(2z(r/w)\cosh(y/2))\Bigg{]}\
.$

Another interesting limiting case is the Minkowskian limit. In this asymptotic
limit, we take $a\rightarrow\infty$ with fixed $y$, and the geometry under
consideration is reduced to the background of a cosmic string in
$(D+1)$-dimensional Minkowski spacetime. As we approach the Minkowskian limit,
the coordinate $w$ in the arguments of the Bessel functions become large and
one has $w\approx a+y$. By taking into account that also the order becomes
large in this limit, we can use the corresponding uniform asymptotic expansion
of the Bessel function in (56). After a few intermediate steps, to the leading
order, we obtain

$\displaystyle\langle j^{\phi}\rangle_{b}^{(\rm{M})}$ $\displaystyle=$
$\displaystyle-\frac{2e}{(2\pi)^{D/2}}\int_{m}^{\infty}du(u^{2}-m^{2})^{D/2}\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}$ (59) $\displaystyle\times$ $\displaystyle
f_{\frac{D}{2}}(2rs_{k}\sqrt{u^{2}-m^{2}})+\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}$
$\displaystyle\times$ $\displaystyle
f_{\frac{D}{2}}(2r\cosh(y/2)\sqrt{u^{2}-m^{2}})\Bigg{]}\frac{1+\beta
u}{1-\beta u}e^{-2u(y-y_{0})}\ .$

For the Neumann BC, $\beta\rightarrow\infty$, we can perform the last
integration by making the change of variable $v=\sqrt{u^{2}-m^{2}}$ and using
the identity (28), the expression given in (70) is reduced to

$\displaystyle\langle j^{\phi}\rangle_{b}^{(\rm{M})}$ $\displaystyle=$
$\displaystyle\frac{4em^{D+1}}{(2\pi)^{(D+1)/2}}\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}$ (60) $\displaystyle\times$
$\displaystyle\frac{K_{\frac{D+1}{2}}\left(2m[r^{2}s_{k}^{2}+(y-y_{0})^{2}]\right)}{\left(2m[r^{2}s_{k}^{2}+(y-y_{0})^{2}]\right)^{\frac{D+1}{2}}}+\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}$
$\displaystyle\times$
$\displaystyle\frac{K_{\frac{D+1}{2}}\left(2m[r^{2}\cosh^{2}(y/2)+(y-y_{0})^{2}]\right)}{\left(2m[r^{2}\cosh^{2}(y/2)+(y-y_{0})^{2}]\right)^{\frac{D+1}{2}}}\Bigg{]}\
,$

which coincides exactly with the result reported in [32] for $y_{0}=0$. On the
other hand, for Dirichlet BC, $\beta=0$, the corresponding result differs from
(60) by the sign.

In Fig. 1 is exhibit the behavior of the current density in $R$-region as
function of the magnetic flux, $\alpha_{0}$, (left panel) and the ratio
$w/w_{0}$ (right panel), considering Dirichlet and Neumann boundary conditions
with different values of the parameter associated with the deficit angle, $q$.
From the left panel we can see that the current density is an odd function of
$\alpha_{0}$, as expected. From the right panel we observe that the VEV is
finite on the brane location and goes to zero as it approaches the AdS
horizon, in accordance to our asymptotic analysis. Moreover, note that in both
plots the intensities increase with $q$ and are higher for Neumann BC.

Figure 1: The VEV of the current density in the $R$-region for $D=3$ is
plotted as function of the magnetic flux, $\alpha_{0}$, (left panel) and the
proper distance from the brane in units of $a$, $w/w_{0}$, considering
Dirichlet and Neumann boundary conditions and different values of $q$. Both
graphs are plotted for $D=3$ and $r/w=0.5$. Moreover, in the left panel we
have fixed $w/w_{0}=2$ and in the right one, $\alpha_{0}=0.4$.

###  5.2 $L$-region

To calculate the induced azimuthal current density in the $L$-region, we start
by substituting the Wightman function given in (42) into the expression
bellow,

$\displaystyle\langle j_{\phi}(x)\rangle_{b(L)}=ie\lim_{x^{\prime}\to
x}[(\partial_{\phi}-\partial_{\phi^{\prime}})W_{b(L)}(x^{\prime},x)+2ieA_{\phi}W_{b(L)}(x^{\prime},x)]\
,$ (61)

using $A_{\mu}=\delta_{\mu}^{\phi}A_{\phi}=q\alpha/e$. After some steps we
get,

$\displaystyle\langle j^{\phi}\rangle_{b(L)}$ $\displaystyle=$
$\displaystyle-\frac{2eq^{2}w^{D}}{a^{D-1}(2\pi)^{D/2}w_{0}^{2}r^{D-2}}\int_{0}^{\infty}dvv^{D/2-2}e^{-v}\sum_{n=-\infty}^{\infty}(n+\alpha)I_{q|n+\alpha_{0}|}(v)$
(62) $\displaystyle\times$
$\displaystyle\sum_{i=1}^{\infty}p_{\nu,i}T_{\nu}(p_{\nu,i})J_{\nu}^{2}(p_{\nu,i}w/w_{0})e^{-\frac{r^{2}p_{\nu,i}^{2}}{2w_{0}^{2}v}}\
.$

Substituting the summation over $n$ in the modified Bessel function by (53),
we have,

$\displaystyle\langle j_{\phi}\rangle_{b(L)}$ $\displaystyle=$
$\displaystyle-\frac{4ew^{D}r^{2-D}}{a^{D-1}(2\pi)^{D/2}w_{0}^{2}}\int_{0}^{\infty}dvv^{D/2-1}e^{-v}\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}e^{v\cos(2\pi k/q)}\right.$ (63) $\displaystyle+$
$\displaystyle\left.\frac{q}{2\pi}\int_{0}^{\infty}dy\sinh{(y)}\frac{e^{-v\cosh{(y)}}g(q,\alpha_{0},y)}{\cosh{(qy)}-\cos{(\pi
q)}}\right]\sum_{i=1}^{\infty}p_{\nu,i}T_{\nu}(p_{\nu,i})J_{\nu}^{2}(p_{\nu,i}w/w_{0})e^{-\frac{r^{2}p_{\nu,i}^{2}}{2w_{0}^{2}v}}.$

Now we are in position to integrate over the variable $v$. Doing this we get,

$\displaystyle\langle j_{\phi}\rangle_{b(L)}$ $\displaystyle=$
$\displaystyle-\frac{8ew^{D}}{a^{D-1}(4\pi)^{D/2}r^{D/2-2}w_{0}^{D/2+2}}\sum_{i=1}^{\infty}p^{D/2+1}_{\nu,i}T_{\nu}(p_{\nu,i})J_{\nu}^{2}(p_{\nu,i}w/w_{0})$
(64) $\displaystyle\times$
$\displaystyle\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{\sin{(2\pi
k/q)}\sin{(2\pi
k\alpha_{0})}}{s_{k}^{D/2}}K_{\frac{D}{2}}(2s_{k}rp_{\nu,i}/w_{0})\right.$
$\displaystyle+$
$\displaystyle\left.\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh^{D/2}(y/2)[\cosh{(qy)}-\cos{(\pi
q)}]}K_{\frac{D}{2}}(2\cosh(y/2)rp_{\nu,i}/w_{0})\right]\ .$

Now to procedure the summation over the quantum number $i$, we use the
generalized Abel-Plana summation formula (44), with

$\displaystyle f(z)=z^{D/2+1}J^{2}_{\nu}(z(w/w_{0}))K_{\frac{D}{2}}(2\rho
z(r/w_{0}))\ {\rm with}\ \rho=(\sin(\pi k/q),\ \cosh(y/2))\ .$ (65)

Because we want to calculate the azimuthal current induced by the brane, only
the second term on the right hand side of the summation formula is of our
interest. Using the relations between the Bessel functions with imaginary
arguments [25], and after some intermediate steps, we obtain

$\displaystyle\langle j_{\phi}\rangle_{b(L)}$ $\displaystyle=$
$\displaystyle\frac{4ew^{D}}{(4\pi)^{D/2}a^{D-1}}\frac{1}{w_{0}^{D/2+2}r^{D/2-2}}\int_{0}^{\infty}dzz^{D/2+1}\frac{\bar{K}_{\nu}(z)}{\bar{I}_{\nu}(z)}I_{\nu}^{2}(z(w/w_{0}))$
(66) $\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}}{s_{k}^{D/2}}J_{\frac{D}{2}}(2z(r/w_{0})s_{k})$
$\displaystyle+$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh^{D/2}(y/2)[\cosh(qy)-\cos(q\pi)]}J_{\frac{D}{2}}(2z(r/w_{0})\cosh(y/2))\Bigg{]}\
.$

Now changing the variable $z\to z(w_{0}/w)$, the contravariant component of
the azimuthal current reads

$\displaystyle\langle j^{\phi}\rangle_{b(L)}$ $\displaystyle=$
$\displaystyle-\frac{4e}{(2\pi)^{D/2}a^{D+1}}\int_{0}^{\infty}dzz^{D+1}\frac{\bar{K}_{\nu}(zw_{0}/w)}{\bar{I}_{\nu}(zw_{0}/w)}I_{\nu}^{2}(z)$
(67) $\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}f_{\frac{D}{2}}(2z(r/w)s_{k})$ $\displaystyle+$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}f_{\frac{D}{2}}(2z(r/w)\cosh(y/2))\Bigg{]}\
.$

Comparing the above expression with (56), we observe that the brane-induced
azimuthal current in the $L$-region is obtained from the corresponding one in
$R$-region by the replacements $I\rightarrow K$, $K\rightarrow I$ of the
modified Bessel functions.

Proceeding similarly as in the $R$-region, we now investigate some special and
limiting cases. In the conformal massless field case, with $\nu=1/2$, we use
the corresponding expressions of the Bessel functions, and after a few
algebraic manipulations, we get

$\displaystyle\langle j^{\phi}\rangle_{b(L)}$ $\displaystyle=$
$\displaystyle-\frac{4e}{(2\pi)^{D/2}a^{D+1}}\int_{0}^{\infty}dzz^{D}e^{-zw_{0}/w}$
(68) $\displaystyle\times$
$\displaystyle\frac{[2A_{0}-B_{0}(1+2zw_{0}/w)]\sinh^{2}(z)}{2B_{0}(w_{0}/w)z\cosh(zw_{0}/w)+(2A_{0}-B_{0})\sinh(zw_{0}/w)}I_{\nu}^{2}(z)$
$\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}f_{\frac{D}{2}}(2z(r/w)s_{k})$ $\displaystyle+$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}f_{\frac{D}{2}}(2z(r/w)\cosh(y/2))\Bigg{]}\
.$

For points near the AdS boundary ($w=0$), $w\ll w_{0}$, the argument of the
modified Bessel function $I_{\nu}(z)$ is small and using the corresponding
asymptotic function in the leading term [25], we get

$\displaystyle\langle j^{\phi}\rangle_{b(L)}$ $\displaystyle\approx$
$\displaystyle-\frac{2^{2-2\nu-D/2}e}{\pi^{D/2}\Gamma^{2}(\nu+1)a^{D+1}}\left(\frac{w}{w_{0}}\right)^{2\nu}\int_{0}^{\infty}dzz^{D+2\nu+1}\frac{\bar{K}_{\nu}(zw_{0}/w)}{\bar{I}_{\nu}(zw_{0}/w)}$
(69) $\displaystyle\times$
$\displaystyle\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}f_{\frac{D}{2}}(2z(r/w)s_{k})$ $\displaystyle+$
$\displaystyle\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}f_{\frac{D}{2}}(2z(r/w)\cosh(y/2))\Bigg{]}\
.$

In the Minkowskian limit, we follow the same procedure described for the
current density in the $R$-region, obtaining the following result:

$\displaystyle\langle j^{\phi}\rangle_{b(L)}^{(\rm{M})}$ $\displaystyle=$
$\displaystyle-\frac{2e}{(2\pi)^{D/2}}\int_{m}^{\infty}du(u^{2}-m^{2})^{D/2}\Bigg{[}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\sin{(2\pi
k/q)}\sin{(2\pi k\alpha_{0})}$ (70) $\displaystyle\times$ $\displaystyle
f_{\frac{D}{2}}(2rs_{k}\sqrt{u^{2}-m^{2}})+\frac{q}{2\pi}\int_{0}^{\infty}dy\frac{\sinh{(y)}g(q,\alpha_{0},y)}{\cosh(qy)-\cos(q\pi)}$
$\displaystyle\times$ $\displaystyle
f_{\frac{D}{2}}(2r\cosh(y/2)\sqrt{u^{2}-m^{2}})\Bigg{]}\frac{1+\beta
u}{1-\beta u}e^{-2u(y_{0}-y)}\ ,$

which is similar to the result we found for the $R$-region with $y-y_{0}$
replaced by $y_{0}-y$. This similarity is expected since in the Minkowskian
limit the VEV is symmetric to the brane. Moreover, for Neumann BC we obtain
the same result present in (60) with $y-y_{0}$ replaced by $y_{0}-y$. On the
other hand, for Dirichlet BC the corresponding expression differs by the sign.

In Fig. 2 is displayed the dependence of the current density in the $L$-region
as function of the magnetic flux, $\alpha_{0}$, (left panel) and the ratio
$w/w_{0}$ (right panel), considering Dirichlet and Neumann boundary conditions
with different values of the parameter associated with the deficit angle, $q$.
Similar to the $R$-region, from the left panel we observe that the current
density in the $L$-region is an odd function of $\alpha_{0}$. On the other
hand, from the right panel we can see that the VEV rapidly goes to zero near
the AdS boundary and is finite on the brane. Moreover, in both plots the
intensities increase with the parameter $q$ and, differently from the
$R$-region, are higher for Dirichlet BC.

Figure 2: The same as in Fig. 1 for the $L$-region.

##  6 Application to RSII model

The formulas for the VEV of the current density in the setup problem under
consideration and present in the previous section can be applied to the study
of the influence of the cosmic string in $Z_{2}$-symmetric braneworlds models.
In this paper we apply them to the Randall-Sundrum model with a single brane
(RSII) [33]. In this model, the four-dimensional world is understood as a
$Z_{2}$-symmetric brane with positive tension embedded in a higher dimensional
AdS bulk. Moreover the existence of bulk fields is a common feature in models
motivated by string theories. In the present setup problem, we consider the
cosmic string perpendicular to the brane. The latter is located at $y=0$ (or
$w_{0}=a$ in the $w$ coordinate) and divides the background geometry in two
copies of the $R$-region which are identified by the $Z_{2}$-symmetry
transformation $y\longleftrightarrow-y$. The corresponding line element is
obtained from (1) by the replacement $e^{-2y/a}\rightarrow e^{-2|y|/a}$. The
fields in the regions $-\infty<y<0$ and $0<y<\infty$ are related by the
$Z_{2}$-symmetry of the model. It worths to note that for an observer located
at $y=0$, the corresponding line element for the described geometry is reduced
to the one for a cosmic string in $(D+1)$-dimensional flat spacetime.

The bulk scalar field obeys the boundary conditions at the position of the
brane and are obtained by integration of the field equations about $y=0$ [34,
35, 36]. Following this procedure for an untwisted scalar field (even under
reflection with respect to the brane location), one can see that the received
boundary condition has the Robin form (7), with coefficient $\beta=a/4D\xi$.
In the particular case of a minimally coupled field, $\xi=0$, the boundary
condition is reduced to the Neumann one ($\beta\rightarrow\infty$). On the
other hand, for twisted scalar fields (odd under reflection with respect to
the brane), the boundary condition has the Dirichlet form.

In models with $Z_{2}$ symmetry, the range of integration over the extra
dimension $y$ varies from $-\infty$ to $\infty$ in the normalization
condition. This results in an additional factor of $1/2$ when compared to the
one obtained previously for the $R$-region, which has half of interval range,
$0\leq y<\infty$, with the brane location at $y_{0}=0$. Thus, we can conclude
that, similar to the VEVs of the field squared and the energy-momentum tensor
[24], the formulas for the VEV of the current density induced by a cosmic
string in the generalized RSII braneworld model are obtained from those
expressions present in subsection 5.1 by directly putting $w_{0}=a$ with an
additional overall factor of $1/2$.

##  7 Conclusions

In the present paper we have studied the combined effects of curvature,
conical topology and the presence of a brane on the VEV of the current density
for a massive charged scalar field propagating in the background of a
$(D+1)$-dimensional AdS spacetime. Along the cosmic string we assume the
existence of a magnetic flux. Moreover, on the brane we impose that the field
operator obeys the general Robin boundary conditions, and we consider that the
string is perpendicular to the brane which is parallel to the AdS boundary,
which divides the manifold into two regions, $L$-region ($0\leq w\leq w_{0}$)
and $R$-region ($w_{0}\leq w<\infty)$. In this setup the scalar modes are
obtained for both regions and the corresponding Wightman functions. They are
presented in closed form in Section 3 for the $R$-region (38) and $L$-region
(46).

In Section 4 we have calculated the VEVs of the current density for the
$R$-region and $L$-region. We have shown that the only non-vanishing component
is induced along the azimuthal direction. In Subsection 5.1 we have developed
the VEV of the azimuthal current density in the $R$-region and a closed
expression is present in (56). For a massless quantum scalar field, the
azimuthal current density is given by (57) for general Robin boundary
conditions. For distances from the brane much larger compared with the AdS
radius, $w/w_{0}\gg 1$, we found that the $\langle j^{\phi}\rangle_{b(R)}$
decays as $(w_{0}/w)^{2\nu}$. Also we have obtained its Minkowskian limit
given by the expression (59) for the general Robin boundary condition. For the
particular cases of Neumann and Dirichet boundary condition we found that the
VEV in this limiting case differs by the sign. In order to provide a better
understanding of this induced current, in figure 1 we exhibit the plots for
behaviors of the azimuthal current density, considering $D=3$ and fixed $r/w$,
in the $R$-region, $\langle j^{\phi}\rangle_{b(R)}$, as function of the
magnetic flux along the string, $\alpha_{0}$, (left panel) and the ratio
$w/w_{0}$ (right panel). The left panel shows that this VEV is an odd function
of the magnetic flux. The right panel shows that $\langle
j^{\phi}\rangle_{b(R)}$ is finite on the brane and goes to zero as it
approaches the AdS horizon, confirming the asymptotic analysis. In addition,
both plots show that the intensities increase with parameter associated with
deficit angle, $q$, and are higher for Neumann BC.

In Subsection 5.2 we have developed the azimuthal current density induced in
the $L$-region. The corresponding formula in closed form is given by (67) and
is obtained from the result for $R$-region by the replacement $I\rightarrow
K$, $K\rightarrow I$ of the Bessel functions. For a conformal massless scalar
field, $\langle j^{\phi}\rangle_{b(L)}$ is present in (68). For points close
to the AdS boundary, $w\ll w_{0}$, we found the expression (69) which shows
that the azimuthal current density in this region decays as
$(w/w_{0})^{2\nu}$. In the Minkowskian limit we have obtained the expression
present in (70), which is similar to the result we found in the $R$-region
with $y-y_{0}$ replaced by $y_{0}-y$. Moreover, in the particular cases of
Neumann and Dirichlet boundary conditions, the VEV in this limiting case
differs by the sign. For further investigation, in figure 2 we have plotted
the azimuthal current density given by (67) as function of $\alpha_{0}$ (left
panel) and $w/w_{0}$ (right panel), considering $D=3$ and fixed $r/w$. Similar
to the $R$-region, the left panel shows that $\langle j^{\phi}\rangle_{b(L)}$
is an odd function of the magnetic flux long the string, $\alpha_{0}$, as
expected. The right panel shows that azimuthal current density rapidly
vanishes near the AdS boundary, in accordance to the asymptotic analysis, and
is finite on the brane. Moreover, both plots show that the intensities of the
azimuthal current density increase with the parameter $q$ and, in contrast
with the $R$-region, are higher for Dirichlet BC.

In the last section we have applied the results found in the $R$-region to
investigate the cosmic string induced effects in the generalized RSII model.
By integration of the field equations about the brane location, $y=0$, we have
found that the boundary conditions in this $Z_{2}$-symmetric model has the
Robin type with the coefficient $\beta=a/4D\xi$ for untwisted scalar field and
it is reduced to the Neumann BC for a minimally coupled field, $\xi=0$. On the
other hand for a twisted scalar field, we have the Dirichlet BC. As a final
remark, we have concluded that the VEV of the azimuthal current density
induced by a cosmic string in the RSII model can be obtained from those
present in the Section 6 by putting $w_{0}=a$ with an additional factor 1/2.

## Acknowledgment

W.O.S is supported under grant 2022/2008, Paraíba State Research Foundation
(FAPESQ). E.R.B.M is partially supported by CNPq under Grant no
301.783/2019-3.

## References

  * [1] T. W. Kibble, J. Phys. A 9, 1387 (1976).
  * [2] A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, England, 1994).
  * [3] V. Berezinski, B. Hnatyk and A. Vilenkin, Phys. Rev. D 64, 043004 (2001).
  * [4] T. Damour and A. Vilenkin, Phys. Rev. Lett. 85, 3761 (2000).
  * [5] P. Bhattacharjee and G. Sigl, Phys. Rep. 327, 109 (2000).
  * [6] C. Fronsdal, Phys. Rev. D 10, 589 (1974).
  * [7] C. Fronsdal and R. B. Haugen, Phys. Rev. D 12, 3810 (1975).
  * [8] J. S. Avis, C. J. Isham and D. Storey, Phys. Rev. D 18, 3565 (198).
  * [9] B. Allen and T. Jacobson, Commun. Math. Phys. 103, 669 (1986).
  * [10] R. Camporesi, Phys. Rev. D 43, 3958 (1991).
  * [11] R. Camporesi and H. Higuchi, Phys. Rev. D 45, 3951 (1992).
  * [12] M. M. Caldareli, Nuc. Phys. B 549, 499 (1999) .
  * [13] O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys. Rep. 323, 183 (2000).
  * [14] P. Brax and C. Van de Bruck, Classical Quantum Gravity 20, R201(2003).
  * [15] R. Maartens, Living Rev. Relativity 13, 5 (2010).
  * [16] M. H. Dehghani, A. M. Ghezelbash and R.B. Mann, Nucl. Phys. B 625, 389 (2002).
  * [17] C. A. Ballon Bayona, C.N. Ferreira and V.J. Vasquez Otoya, Class. Quantum Grav. 28, 015011 (2011).
  * [18] W. Oliveira dos Santos, H.F. Mota and E.R. Bezerra de Mello, Phys. Rev. D 99, 045005 (2019).
  * [19] W. Oliveira dos Santos, E.R. Bezerra de Mello and H. F. Mota, Eur.Phys. J. Plus 135, 27 (2020).
  * [20] S. Bellucci, W. Oliveira dos Santos and E.R. Bezerra de Mello, Eur. Phys. J. C 80, 963 (2020).
  * [21] S. Bellucci, W. Oliveira dos Santos, E.R. Bezerra de Mello and A. A. Saharian, JCAP 01 (2022) 010.
  * [22] S. Bellucci, W. Oliveira dos Santos, E.R. Bezerra de Mello and A. A. Saharian, JHEP 02 (2021) 190.
  * [23] S. Bellucci, W. Oliveira dos Santos, E.R. Bezerra de Mello and A. A. Saharian, JHEP 05 (2022) 021.
  * [24] W. Oliveira dos Santos and E.R. Bezerra de Mello, Eur. Phys. J. C 83, 726 (2023).
  * [25] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  * [26] P. Breitenlohner, D.Z. Freedman, Ann. Phys. (NY) 1982, 144 (2), 249-281.
  * [27] S. J. Avis, C. J. Isham and D. Storey, Phys. Rev. D 18, 3565-3576 (1978).
  * [28] I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series and Products (Academic Press, New York, 1980).
  * [29] E. R. Bezerra de Mello, V. B. Bezerra, A. A. Saharian, and H. H. Harutyunyan, Phys. Rev. D 91, 064034 (2015).
  * [30] A. A. Saharian, The Generalized Abel-Plana Formula with Applications to Bessel Functions and Casimir Effect (Yerevan State University Publishing House, Yerevan, 2008); Preprint ICTP/2007/082; arXiv:0708.1187.
  * [31] E. A. F. Bragança, H. F. Santana Mota, E. R. Bezerra de Mello, Int. J. Mod. Phys. D 24, 1550055 (2015).
  * [32] E. A. F. Bragança and E. R. Bezerra de Mello, Eur. Phys. J. Plus 136, 50 (2021)
  * [33] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999).
  * [34] A. A. Saharian, Nucl. Phys. B 712, 196-228 (2005).
  * [35] T. Gherghetta, A. Pomarol, Nucl. Phys. B 586, 41 (2000).
  * [36] A. Flachi, D. J. Toms, Nucl. Phys. B 610, 144 (2001).