# Vacuum fermionic currents in braneworld models
on AdS bulk with a cosmic string

S. Bellucci1, W. Oliveira dos Santos1,2, E. R. Bezerra de Mello2, A. A.
Saharian3

1INFN, Laboratori Nazionali di Frascati,
Via Enrico Fermi 40, 00044 Frascati, Italy
2Departamento de Física, Universidade Federal da Paraíba,
58059-900, Caixa Postal 5008, João Pessoa, PB, Brazil
3 Department of Physics, Yerevan State University,
1 Alex Manoogian Street, 0025 Yerevan, Armenia

###### Abstract

We investigate the effects of a brane and magnetic-flux-carrying cosmic string
on the vacuum expectation value (VEV) of the current density for a charged
fermionic field in the background geometry of (4+1)-dimensional anti–de Sitter
(AdS) spacetime. The brane is parallel to the AdS boundary and the cosmic
string is orthogonal to the brane. Two types of boundary conditions are
considered on the brane that include the MIT bag boundary condition and the
boundary conditions in $Z_{2}$-symmetric braneworld models. The brane divides
the space into two regions with different properties of the vacuum state. The
only nonzero component of the current density is along the azimuthal direction
and in both the regions the corresponding VEV is decomposed into the brane-
free and brane-induced contributions. The latter vanishes on the string and
near the string the total current is dominated by the brane-free part. At
large distances from the string and in the region between the brane and AdS
horizon the decay of the brane-induced current density, as a function of the
proper distance, is power-law for both massless and massive fields. For a
massive field this behavior is essentially different from that in the
Minkowski bulk. In the region between the brane and AdS boundary the large-
distance decay of the current density is exponential. Depending on the
boundary condition on the brane, the brane-induced contribution is dominant or
subdominant in the total current density at large distances from the string.
By using the results for fields realizing two inequivalent irreducible
representations of the Clifford algebra, the vacuum current density is
investigated in $C$\- and $P$-symmetric fermionic models. Applications are
given for a cosmic string in the Randall-Sundrum-type braneworld model with a
single brane.

Keywords: Cosmic string, AdS space, Brane, Vacuum polarization

##  1 Introduction

The anti-de Sitter (AdS) spacetime is one of the most interesting solutions of
General Relativity. The relevance of this maximally symmetric geometry from
the point of view of quantum field theory resides in the fact that a large
number of field theoretical problems are exactly soluble on its background.
This allows to reveal information on the influence of gravitational field on
quantum matter in less symmetric geometries. In addition, the length scale
related to the AdS constant curvature, can serve as a regularization parameter
for infrared divergences in interacting quantum field theories without
reducing the number of symmetries [1]. The AdS spacetime has been the subject
of considerable attention due to the realization that it emerges as a stable
ground state solution in Kaluza-Klein models, extended supergravity and string
theories and approximates the near-horizon geometries of extremal black holes,
domain walls and $p$-brane configurations in string theories. In addition, the
AdS background geometry plays a crucial role in the AdS/CFT correspondence and
braneworlds models with large extra dimensions. These two exciting
developments of modern theoretical physics naturally appear in the
string/M-theory context and offer an interesting framework to approach several
problems in particle physics, cosmology and condensed matter physics.
Motivated by the points given above, quantum field theory in AdS background
has been extensively studied by several authors (see, for instance, [2]-[23]
and references therein).

In the present paper we investigate the effects of a cosmic string and brane
on the vacuum expectation value (VEV) of the fermionic current density induced
by a magnetic flux tube in AdS spacetime. Cosmic strings are among the most
interesting types of topological defects that might have formed as a
consequence of spontaneous symmetry breaking at phase transitions in the early
Universe [24, 25, 26]. As possible seeds for large scale structure formation
in the Universe, they have been extensively studied in the eighties and early
nineties of the last century. Although the observational data on the
temperature anisotropies of the cosmic microwave background radiation (CMB)
supported the inflationary scenario for the density perturbations, excluding
the cosmic strings as the main origin of structures, they are still sources
for a number of interesting physical effects. The latter include the
gravitational lensing, creation of small non-Gaussianities in the CMB, the
generation of gravitational waves, gamma ray bursts and high-energy cosmic
rays.

Initially, cosmic strings have been regarded as completely different from
fundamental superstrings. In string theory the latter are considered as the
basic building blocks of matter. A particularly interesting topic in recent
developments of the subject is the investigation of possibility for appearance
of low tension cosmic strings in string theory (see [27] for early
discussion). Models were proposed where, under certain conditions, the
fundamental strings can grow to macroscopic size with features similar to
those for cosmic strings. Among the examples of this kind of models is the
brane-inflation scenario [28] (for reviews see [29, 30, 31, 32]). In this
scenario the inflation appears naturally in the braneworld context and it is a
specific realization of the inflationary paradigm in the early Universe within
the braneworld picture in the string theory. Among other interesting
predictions in this theoretical framework, the production of cosmic strings
towards the end of inflation has attracted considerable attention [33]. It has
been suggested that the detection of cosmic strings could be a possible way to
verify the inflationary paradigm, providing an interesting testing ground into
string theory by probing the braneworld scenario before the inflationary phase
[29, 30, 31, 32]. Many other publications have investigated the effects of
braneworld gravity on the physics of cosmic strings [34, 35, 36, 37, 38, 39].
Specifically in [39], the authors have studied the consequences of the
braneworld modified gravity on the cosmic string phenomenology. By considering
the presence of a cosmic string they were able to put a strong constraint on
the braneworld tension, making it compatible with the observational data for
temperature anisotropies of CMB. They also have shown that the deficit angle
induced in the string’s outside region is significantly attenuated due to the
braneworld gravitational corrections, providing a possible explanation for the
null experimental results based on the cosmic string effects caused by its
conical geometry, such as lensing effects. This close relation between
braneworlds and cosmic strings gives us enough motivation to consider both
objects within a unified picture for studying vacuum quantum effects in the
present work.

The geometry associated with a cosmic string in AdS spacetime has been
analyzed in [40, 41]. There the authors have shown that, similar to the case
of Minkowskian spacetime, at distances from the string larger than its core
radius, the gravitational effects of the string can be approximated by a
planar angle deficit in the two-dimensional subspace orthogonal to the string
in the AdS line-element. The combination of non-trivial topology and the
spacetime curvature induce vacuum polarization effects for quantum fields. In
this sense the investigation of the influence of the presence of a cosmic
string in AdS bulk on the vacuum polarization of quantum fields, have been
investigated for scalar and fermionic fields in [42] and [43], respectively.
Taking into account the presence of a magnetic flux along the string core, in
[44] the vacuum current for a charged scalar field in AdS bulk was
investigated. The VEV of the corresponding energy-momentum tensor was analyzed
in [45]. Finally, in [46] the induced fermionic current in the AdS bulk in the
presence of a cosmic string was discussed.111In fact, in references [44], [45]
and [46], higher dimensional AdS spacetimes were considered, admitting that an
extra dimension coordinate was compactified.

Here we consider the VEV of the fermionic current density around magnetic-
flux-carrying cosmic string on AdS bulk in the presence of a brane parallel to
the AdS boundary. The cosmic string is orthogonal to the brane and on the
brane the fermionic field is constrained by two types of boundary conditions.
The first one corresponds to the MIT bag boundary condition and the second one
is obtained from the bag boundary condition changing the sign of the term with
the normal to the boundary. In braneworld models these boundary conditions are
dictated by the $Z_{2}$-symmetry under the reflection with respect to the
brane. Note that the vacuum current densities on locally AdS bulk with a part
of spatial dimensions compactified to a torus and in the presence of branes
have been investigated in [47, 48, 49] and [50, 51, 52] for charged scalar and
fermionic fields, respectively.

The paper is organized as follows. In section 2 we describe the setup of the
problem and present the complete set of positive and negative energy solutions
to the Dirac equation in the presence of a brane parallel to the AdS boundary.
In section 3, the VEV of the current density in the region between the brane
and the AdS horizon (R-region) is investigated. Various asymptotic limits are
considered and numerical results are presented. Similar investigations for the
region between the brane and AdS boundary (L-region) are presented in section
4. In section 5 we consider the second type of boundary condition and analyse
the effects of the brane on the current densities in the R- and L-regions.
Discussion of our results in the context of $C$\- and $P$-symmetric models is
presented in section 6. In section 7, an application to the Randall-Sundrum
model with a single brane is provided. Section 8 summarizes the most relevant
results obtained. Throughout the paper, we use natural units $G=\hbar=c=1$.

##  2 Problem setup and the fermionic modes

We consider a fermionic quantum field $\psi(x)$ in background of
$(4+1)$-dimensional spacetime with the metric tensor $g_{\mu\nu}$ defined by
the line element

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\left(\frac{a}{w}\right)^{2}\left(dt^{2}-dr^{2}-r^{2}d\phi^{2}-dw^{2}-dz^{2}\right)\
,$ (2.1)

where $-\infty<t,z<+\infty$, $r\geq 0$, $0\leq\phi\leq\phi_{0}$, $0\leq
w<\infty$. For $\phi_{0}=2\pi$, Eq. (2.1) presents the AdS spacetime described
in Poincaré coordinates (with polar coordinates $(r,\phi)$ in two-dimensional
subspace). The parameter $a$ defines the cooresponding curvature radius and is
related to negative cosmological constant $\Lambda$ and the Ricci scalar $R$
by formulas $\Lambda=-6/a^{2}$ and $R=-20/a^{2}$. The limiting values of the
$w$-coordinate, $w=0$ and $w=\infty$, determine the AdS boundary and the AdS
horizon, respectively. In what follows we will be concerned with the case
$\phi_{0}<2\pi$ that corresponds to two-dimensional conical subspace
$(r,\phi)$ with planar angle deficit $2\pi-\phi_{0}$. Note that the latter
does not change the local geometry for $r\neq 0$. However, the topology is
changed and this gives rise interesting effects in quantum field theory. The
line element (2.1) without the term $dz^{2}$ describes an idealized cosmic
string in background of AdS spacetime.

In the presence of a classical gauge field $A_{\mu}$, the Dirac equation for
the field $\psi(x)$ reads

$\left[i\gamma^{\mu}(\partial_{\mu}+\Gamma_{\mu}+ieA_{\mu})-sm\right]\psi=0,$
(2.2)

where $\Gamma_{\mu}$ is the spin connection. In $(4+1)$-dimensional spacetime
the irreducible representation of the Clifford algebra is realized by $4\times
4$ Dirac matrices $\gamma^{\mu}$. In odd spacetime dimensions there are two
inequivalent irreducible representations and the parameter $s=\pm 1$
corresponds to those representations (see the discussion in section 6 below).
We assume the presence of a codimension one brane parallel to the AdS boundary
and located at $w=w_{0}$. On the brane the field operator is constrained by
the MIT bag boundary condition

$(1+i\gamma^{\mu}n_{\mu})\psi(x)=0\ ,\ \ w=w_{0}\ ,$ (2.3)

where $n_{\mu}$ is the corresponding normal vector. In terms of the coordinate
$w$ the line element (2.1) is written in conformally flat form for $r\neq 0$.
It is conformally related to the line element associated with a cosmic string
in $(4+1)$-dimensional Minkowski spacetime. The physical distance from the
brane is measured in terms of the coordinate $y$, defined as $y=a\ln(w/a)$,
$-\infty<y<+\infty$. For the $y$-coordinate of the brane one has
$y_{0}=a\ln(w_{0}/a)$. The distance from the brane is expressed as
$|y-y_{0}|$. The brane separates the background space into two regions: $0\leq
w\leq w_{0}$ and $w\geq w_{0}$. We shall refer them as L(left)- and R(right)-
regions, respectively. In the coordinate system $x^{\mu}=(t,r,\phi,w,z)$, for
the normal in the boundary condition (2.3) one has
$n_{\mu}=\delta_{\mu}^{3}a/w$ in the L-region and
$n_{\mu}=-\delta_{\mu}^{3}a/w$ in the R-region.

In the discussion below, the gamma matrices will be taken in the following
representation:

$\gamma^{0}=\frac{w}{a}\left({\begin{array}[]{cc}0&-i\\\
i&0\end{array}}\right)\
,\quad\gamma^{l}=-i\frac{w}{a}\left({\begin{array}[]{cc}\sigma^{l}&0\\\
0&-\sigma^{l}\end{array}}\right)\
,\quad\gamma^{4}=\frac{w}{a}\left({\begin{array}[]{cc}0&i\\\
i&0\end{array}}\right)$ (2.4)

where $l=1,2,3$ correspond to the coordinates $r,\phi,w$. In the respective
expressions $2\times 2$ matrices $\sigma^{l}$ are defined as

$\sigma^{1}=\left({\begin{array}[]{cc}0&e^{-iq\phi}\\\
e^{iq\phi}&0\end{array}}\right)\ ,\
\sigma^{2}=\frac{i}{r}\left({\begin{array}[]{cc}0&-e^{-iq\phi}\\\
e^{iq\phi}&0\end{array}}\right)\ ,$ (2.5)

with $q=2\pi/\phi_{0}$, and $\sigma^{3}=\mathrm{diag}(1,-1)$. These are the
Pauli matrices in the coordinate system $(r,\phi,w)$. With this choice of the
gamma matrices, the product $\gamma^{\mu}\Gamma_{\mu}$ in the field equation
(2.2) takes the form

$\gamma^{\mu}\Gamma_{\mu}=\frac{1-q}{2r}\gamma^{1}-\frac{2}{w}\gamma^{3}.$
(2.6)

In the absence of the cosmic string one has $q=1$.

Our interest in the present paper is the VEV of the fermionic current density
$j^{\mu}=e\bar{\psi}\gamma^{\mu}\psi$, with
$\bar{\psi}=\psi^{\dagger}\gamma^{(0)}$ being the Dirac adjoint defined in
terms of the flat spacetime gamma matrix $\gamma^{(0)}=(a/w)\gamma^{0}$. This
VEV $\left\langle 0\right|j^{\mu}\left|0\right\rangle\equiv\left\langle
j^{\mu}\right\rangle$ can be presented as a mode sum over complete set of
positive and negative energy fermionic wave-functions
$\\{\psi_{\sigma}^{(+)},\psi_{\sigma}^{(-)}\\}$, where the set of quantum
numbers specifies the modes. Expanding the field operators in the expression
for $j^{\mu}$ over the functions $\psi_{\sigma}^{(\pm)}$ and by using the
definition of the vacuum state one can see that

$\langle
j^{\mu}\rangle=\frac{e}{2}\sum_{\sigma}[\bar{\psi}_{\sigma}^{(-)}\gamma^{\mu}\psi_{\sigma}^{(-)}-\bar{\psi}_{\sigma}^{(+)}\gamma^{\mu}\psi_{\sigma}^{(+)}]\
.$ (2.7)

For the gauge field in (2.2) we will assume a simple configuration
$A_{\mu}=\delta_{\mu}^{2}A_{2}$ with constant $A_{2}$. This corresponds to a
magnetic flux along the string’s core. Hence, for evaluation of the vacuum
current density one needs complete set of solutions to (2.2) for the special
case of the vector potential and obeying the boundary condition (2.3).

The general procedure for solving the Dirac equation is similar to that
already discussed in [43] for (3+1)-dimensional AdS spacetime and in [46] for
the problem on (4+1)-dimensional AdS bulk with compactified spatial dimension.
Here we will describe the main steps only. From the problem symmetry the
dependence of the mode functions on the spacetime coordinates $t$ and $z$ can
be taken in the standard plane wave form $e^{ik_{z}z\mp iEt}$, where $E>0$ is
the energy and the upper and lower signs correspond to the positive and
negative energy modes. Decomposing the four-component spinor $\psi$ into two-
component spinors $\varphi_{\uparrow}$ and $\varphi_{\downarrow}$,
$\psi=(\varphi_{\uparrow},\varphi_{\downarrow})^{T}$, from the Dirac equation
(2.2) we obtain two coupled first order differential equations for the upper
and lower components. From that system of equations the second order
differential equations are derived for the separate functions. Separating the
variables $r$ and $w$ in both these equations, the corresponding functions are
expressed in terms of the cylinder functions. Then, additional conditions,
previously discussed in [53], are imposed that relate different components of
the spinor. As a result all those manipulations, the positive and negative
energy fermionic mode functions are presented as

$\psi_{\sigma}^{(\pm)}(x)=C_{\sigma}^{(\pm)}w^{5/2}\left({\begin{array}[]{c}J_{\beta_{j}}(\lambda
r)W_{\nu_{1}}(pw)e^{-iq\phi/2}\\\
-\epsilon_{j}\kappa_{\eta}b_{\eta}^{(+)}J_{\beta_{j}+\epsilon_{j}}(\lambda
r)W_{\nu_{2}}(pw)e^{iq\phi/2}\\\ i\kappa_{\eta}J_{\beta_{j}}(\lambda
r)W_{\nu_{2}}(pw)e^{-iq\phi/2}\\\
i\epsilon_{j}b_{\eta}^{(+)}J_{\beta_{j}+\epsilon_{j}}(\lambda
r)W_{\nu_{1}}(pw)e^{iq\phi/2}\end{array}}\right)e^{iqj\phi+ik_{z}z\mp iEt}\ ,$
(2.8)

with $0\leq\lambda,p<\infty$, $j=\pm 1/2,\pm 3/2,\ldots$, and

$E=\sqrt{\lambda^{2}+p^{2}+k_{z}^{2}}.$ (2.9)

The orders of the functions are given by the expressions

$\beta_{j}=q|j+\alpha|-\epsilon_{j}/2\ ,\;\nu_{l}=ma+(-1)^{l}s/2,$ (2.10)

where $\alpha=eA_{2}/q$, $\epsilon_{j}=1$ for $j>-\alpha$ and
$\epsilon_{j}=-1$ for $j<-\alpha$. In (2.8),

$W_{\nu}(pw)=c_{1}J_{\nu}(pw)+c_{2}Y_{\nu}(pw),$ (2.11)

where $J_{\nu}(x)$ and $Y_{\nu}(x)$ are the Bessel and Neumann functions and
we have introduced the notations

$\displaystyle\kappa_{\eta}$ $\displaystyle=$
$\displaystyle\frac{-k_{z}+\eta\sqrt{k_{z}^{2}+p^{2}}}{p},\ \eta=\pm 1\ ,$
$\displaystyle b_{\eta}^{(\pm)}$ $\displaystyle=$
$\displaystyle\frac{E\mp\eta\sqrt{k_{z}^{2}+p^{2}}}{\lambda}.$ (2.12)

Note that one of the constants $c_{1}$ and $c_{2}$ in (2.11) can be absorbed
in the coefficient $C_{\sigma}^{(\pm)}$. With the definitions given above, the
set of quantum numbers $\sigma$ in (2.7) is specified as
$\sigma=(\lambda,p,j,k_{z},\eta)$. It can be checked that the functions (2.8)
are the eigenfunctions of the $w$-projection of the total angular momentum
$J_{3}$ with the eigenvalues $qj$:

$\hat{J}_{3}\psi_{\sigma}^{(\pm)}(x)=\left(-i\partial_{\phi}+\frac{q}{2}\Sigma^{3}\right)\psi_{\sigma}^{(\pm)}(x)=qj\psi_{\sigma}^{(\pm)}(x),$
(2.13)

where $\Sigma^{3}=\mathrm{diag}(\sigma^{3},\sigma^{3})$. The parameter
$\alpha$ in the definition of $\beta_{j}$ is expressed in terms of the
magnetic flux $\Phi$ along the string axis by the relation
$\alpha=-\Phi_{\phi}/\Phi_{0}$, where $\Phi_{0}=2\pi/e$ is the quantum of the
flux.

The coefficient $C_{\sigma}^{(\pm)}$ in (2.8) is determined from the
normalization condition

$\int_{0}^{\infty}dr\int_{0}^{\phi_{0}}d\phi\int_{-\infty}^{+\infty}dz\int
dw\,r(a/w)^{4}(\psi_{\sigma}^{(\pm)})^{\dagger}\psi_{\sigma^{\prime}}^{(\pm)}=\delta_{\sigma\sigma^{\prime}}\
,\ $ (2.14)

where $\delta_{\sigma\sigma^{\prime}}$ is understood as the Dirac delta
function for continuous components of $\sigma$ and the Kronecker delta for
discrete ones. The integration over $w$ in (2.14) goes over $[0,w_{0}]$ in the
L-region and over $[w_{0},\infty)$ in the R-region. The remaining coefficient
in the linear combination is determined from the boundary condition (2.3).

Note that along the azimuthal direction the mode functions (2.8) obey the
periodicity condition
$\psi_{\sigma}^{(\pm)}(t,r,\phi+\phi_{0},w,z)=\psi_{\sigma}^{(\pm)}(t,r,\phi,w,z)$.
We could consider a more general quasiperiodicity condition
$\psi(t,r,\phi+\phi_{0},w,z)=e^{2\pi i\chi}\psi(t,r,\phi,w,z)$ with a constant
phase $2\pi\chi$. By the gauge transformation
$A_{\mu}^{\prime}=A_{\mu}+\delta_{\mu}^{2}q\chi/e$,
$\psi^{\prime}(x)=\psi(x)e^{-iq\chi\phi}$, the set of the fields
$(A_{\mu},\psi(x))$ is transformed to the set
$(A_{\mu}^{\prime},\psi^{\prime}(x))$, where the new field $\psi^{\prime}(x)$
is periodic along the azimuthal direction. From here we conclude that the VEV
of the current density for the field obeying the quasiperiodicity condition
with the phase $2\pi\chi$ is obtained from the corresponding results given
below for periodic fields by the replacement $\alpha\rightarrow\alpha+\chi$.

The properties of the vacuum state are different in the L- and R-regions and
separate consideration for the fermionic current is required. We start our
investigation with the R-region.

##  3 Current density in the R-region

In this section we consider the current density in the region $w>w_{0}$
(R-region).

###  3.1 General formula

The corresponding modes are given by (2.8), where the function $W_{\nu}(pw)$
is defined as (2.11). From the boundary condition (2.3) it follows that the
coefficients in the corresponding linear combination can be taken as
$c_{1}=Y_{\nu_{2}}(pw_{0})$ and $c_{2}=-J_{\nu_{2}}(pw_{0})$. Hence, in the
R-region the mode functions have the form (2.8), where

$W_{\nu_{l}}(pw)=G_{\nu_{2},\nu_{l}}(pw_{0},pw),\;l=1,2.$ (3.1)

Here and in what follows we use the notation

$G_{\mu,\nu}(x,y)=Y_{\mu}(x)J_{\nu}(y)-J_{\mu}(x)Y_{\nu}(y).$ (3.2)

The eigenvalues of the quantum number $p$ are continuous and in the
normalization condition (2.14) the corresponding part in the right-hand side
is understood as the delta function $\delta(p-p^{\prime})$. In the evaluation
of the normalization integral we note that for $p=p^{\prime}$ the dominant
contribution to the integral over $w$ comes from large values of $w$ and we
can replace the cylinder functions with the arguments $pw$ by the
corresponding asymptotic expressions. In this way one can see that
$C_{\sigma}^{(\pm)}=C_{\mathrm{(R)}\sigma}^{(\pm)}$ with

$\left|C_{\mathrm{(R)}\sigma}^{(\pm)}\right|^{-2}=\frac{16\pi^{2}a^{4}\eta}{q\lambda^{2}p^{2}}E\sqrt{E^{2}-\lambda^{2}}\kappa_{\eta}b_{\eta}^{(\pm)}\left[J_{\nu_{2}}^{2}(pw_{0})+Y_{\nu_{2}}^{2}(pw_{0})\right],$
(3.3)

and with definitions from (2.12).

Given the mode functions, the VEV of the current density in the R-region is
evaluated by using the formula (2.7), where

$\sum_{\sigma}=\int_{0}^{\infty}d\lambda\int_{-\infty}^{+\infty}dk_{z}\int_{0}^{\infty}dp\sum_{\eta=\pm
1}\sum_{j=\pm 1/2,...}.$ (3.4)

First of all, we can see that for the mode functions (2.8)
$\bar{\psi}_{\sigma}^{(\pm)}\gamma^{1}\psi_{\sigma}^{(\pm)}=\bar{\psi}_{\sigma}^{(\pm)}\gamma^{3}\psi_{\sigma}^{(\pm)}=0$
and consequently $\langle j^{1}\rangle=\langle j^{3}\rangle=0$. For the
components $\mu=0,4$ it can be seen that the product
$\bar{\psi}_{\sigma}^{(\pm)}\gamma^{\mu}\psi_{\sigma}^{(\pm)}$ is an odd
function of $k_{z}$. By taking into account that in the formula for the VEV of
the current density one has integration $\int_{-\infty}^{+\infty}dk_{z}$, we
conclude that $\langle j^{0}\rangle=\langle j^{4}\rangle=0$. Hence, the only
nonzero component corresponds to the current density along the azimuthal
direction, $\langle j^{2}\rangle$. The terms in the mode sum with $\eta=+1$
and $\eta=-1$ give the same contribution and, after summation over $\eta$, the
corresponding physical component $\langle j_{\phi}\rangle=r\langle
j^{2}\rangle$ is expressed as

$\displaystyle\langle j_{\phi}\rangle$ $\displaystyle=$
$\displaystyle-\frac{eqw^{6}}{4\pi^{2}a^{5}}\int_{0}^{\infty}d\lambda\int_{0}^{\infty}dp\int_{-\infty}^{+\infty}dk_{z}\sum_{j=\pm
1/2,...}\epsilon_{j}\frac{\lambda^{2}p}{E}$ (3.5) $\displaystyle\times
J_{\beta_{j}}(\lambda r)J_{\beta_{j}+\epsilon_{j}}(\lambda
r)\frac{G_{\nu_{2},\nu_{1}}^{2}(pw_{0},pw)+G_{\nu_{2},\nu_{2}}^{2}(pw_{0},pw)}{J_{\nu_{2}}^{2}(pw_{0})+Y_{\nu_{2}}^{2}(pw_{0})},$

where the energy is given by (2.9). If we write the parameter $\alpha$ in the
form

$\alpha=n_{0}+\alpha_{0}\ ,\ |\alpha_{0}|<1/2\ ,$ (3.6)

where $n_{0}$ is an integer number, then, by shifting $j+n_{0}\rightarrow j$,
we can see that the VEV depends on $\alpha_{0}$ only.

For the transformation of the expression (3.5) we use the integral
representation

$\frac{1}{E}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}dv\
e^{-(\lambda^{2}+p^{2}+k_{z}^{2})v^{2}}\ .$ (3.7)

For the evaluation of the integral over $\lambda$ we use the formula

$\int_{0}^{\infty}d\lambda\,\lambda^{2}e^{-\lambda^{2}v^{2}}J_{\beta_{j}}(\lambda
r)J_{\beta_{j}+\epsilon_{j}}(\lambda
r)=\epsilon_{j}\frac{re^{-u}}{4v^{4}}\left[I_{\beta_{j}}(u)-I_{\beta_{j}+\epsilon_{j}}(u)\right]\
,$ (3.8)

with $u=r^{2}/(2v^{2})$. After evaluation of the $k_{z}$-integral and
introducing instead of $v$ a new integration variable $u=r^{2}/(2v^{2})$, the
current density is expressed as

$\displaystyle\langle j_{\phi}\rangle$ $\displaystyle=$
$\displaystyle-\frac{eqw^{6}}{4\pi^{2}a^{5}r^{3}}\int_{0}^{\infty}du\,ue^{-u}\mathcal{J}(q,\alpha_{0},u)\int_{0}^{\infty}dp\,p$
(3.9) $\displaystyle\times
e^{-p^{2}r^{2}/(2u)}\frac{G_{\nu_{2},\nu_{1}}^{2}(pw_{0},pw)+G_{\nu_{2},\nu_{2}}^{2}(pw_{0},pw)}{J_{\nu_{2}}^{2}(pw_{0})+Y_{\nu_{2}}^{2}(pw_{0})},$

where we have introduced the notation

$\mathcal{J}(q,\alpha_{0},u)=\sum_{j}\left[I_{\beta_{j}}(u)-I_{\beta_{j}+\epsilon_{j}}(u)\right].$
(3.10)

An alternative expression is obtained from (3.9) by using the integral
representation for the function $\mathcal{J}(q,\alpha_{0},u)$ [54, 55, 56].
The latter is obtained from the representation for the series
$\sum_{j}I_{\beta_{j}}(u)$ from [57]:

$\displaystyle\mathcal{J}(q,\alpha_{0},u)$ $\displaystyle=$
$\displaystyle\frac{4}{\pi}\int_{0}^{\infty}dx\frac{e^{-u\cosh(2x)}g(q,\alpha_{0},2x)\cosh
x}{\cosh(2qx)-\cos(q\pi)}$ (3.11)
$\displaystyle+\frac{4}{q}\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}(-1)^{k}s_{k}\sin\left(2\pi
k\alpha_{0}\right)e^{u\cos(2\pi k/q)},$

where $s_{k}=\sin\left(\pi k/q\right)$ and

$g(q,\alpha_{0},x)=\sum_{\chi=-,+}\chi\cos\left[q\pi\left(1/2+\chi\alpha_{0}\right)\right]\cosh\left[\left(1/2-\chi\alpha_{0}\right)qx\right].$
(3.12)

The prime on the summation sign in (3.11) means that for even values of $q$
the term with $k=q/2$ should be taken with an additional coefficient 1/2. With
(3.11), after evaluating the $u$-integral, from (3.9) we find

$\displaystyle\langle j_{\phi}\rangle$ $\displaystyle=$
$\displaystyle-\frac{ew^{6}}{2\pi^{2}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\frac{G_{\nu_{2},\nu_{1}}^{2}(pw_{0},pw)+G_{\nu_{2},\nu_{2}}^{2}(pw_{0},pw)}{J_{\nu_{2}}^{2}(pw_{0})+Y_{\nu_{2}}^{2}(pw_{0})}$
(3.13)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)K_{2}(2prs_{k})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}K_{2}(2pr\cosh x)\right],$

where $K_{\nu}(x)$ is the Macdonald function.

We are interested in the effect induced by the brane. In order to separate the
corresponding contribution one should subtract from (3.13) the current density
in the absence of the brane. The latter is evaluated by the formula (2.7) with
the mode functions $\psi_{\sigma}^{(\pm)}(x)$ replaced by the functions
$\psi_{(0)\sigma}^{(\pm)}(x)$ in the brane-free geometry. Those functions are
obtained from (2.8) with the function $W_{\nu}(pw)=J_{\nu}(pw)$ and the
normalization constant

$\left|C_{(0)\sigma}^{(\pm)}\right|^{-2}=\frac{16\pi^{2}a^{4}\eta}{q\lambda^{2}p^{2}}E\sqrt{E^{2}-\lambda^{2}}\kappa_{\eta}b_{\eta}^{(\pm)}.$
(3.14)

For the VEV of the azimuthal current density in the brane-free geometry this
gives

$\displaystyle\langle j_{\phi}\rangle_{0}$ $\displaystyle=$
$\displaystyle-\frac{eqw^{6}}{4\pi^{2}a^{5}}\int_{0}^{\infty}d\lambda\int_{0}^{\infty}dp\int_{-\infty}^{+\infty}dk_{z}\sum_{j=\pm
1/2,...}\epsilon_{j}\frac{\lambda^{2}p}{E}$ (3.15) $\displaystyle\times
J_{\beta_{j}}(\lambda r)J_{\beta_{j}+\epsilon_{j}}(\lambda
r)\left[J_{\nu_{1}}^{2}(pw)+J_{\nu_{2}}^{2}(pw)\right].$

By transformations similar to those we have described before, this VEV is
presented as

$\displaystyle\langle j_{\phi}\rangle_{0}$ $\displaystyle=$
$\displaystyle-\frac{ew^{6}}{2\pi^{2}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\left[J_{\nu_{1}}^{2}(pw)+J_{\nu_{2}}^{2}(pw)\right]$
(3.16)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)K_{2}(2prs_{k})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}K_{2}(2pr\cosh x)\right].$

Note that (3.16) is the same for $s=+1$ and $s=-1$ and the current densities
in the brane-free geometry coincide for two inequivalent representations of
the Clifford algebra.

We present the current density in the geometry with the brane in the
decomposed form

$\langle j_{\phi}\rangle=\langle j_{\phi}\rangle_{0}+\langle
j_{\phi}\rangle_{\mathrm{b}},$ (3.17)

where the part $\langle j_{\phi}\rangle_{\mathrm{b}}$ is induced by the brane.
The corresponding integral representation is obtained by subtracting (3.16)
from (3.13). We can further transform the expression for the brane-induced
contribution by making use of relation

$\frac{G_{\nu,\mu}^{2}(x,y)}{J_{\nu}^{2}(x)+Y_{\nu}^{2}(x)}-J_{\mu}^{2}(y)=-\frac{1}{2}\sum_{n=1,2}\frac{J_{\nu}(x)}{H_{\nu}^{(n)}(x)}H_{\mu}^{(n)2}(y),$
(3.18)

where $H_{\nu}^{(n)}(x)$, $n=1,2$, are the Hankel functions. This leads to the
expression

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle=$
$\displaystyle\frac{ew^{6}}{4\pi^{2}a^{5}r}\sum_{l=1,2}\int_{0}^{\infty}dp\,p^{3}\sum_{n=1,2}\frac{J_{\nu_{2}}(pw_{0})}{H_{\nu_{2}}^{(n)}(pw_{0})}H_{\nu_{l}}^{(n)2}(pw)$
(3.19)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)K_{2}(2prs_{k})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}K_{2}(2pr\cosh x)\right].$

As the next step we rotate the contour of the integration over $p$ by the
angle $\pi/2$ ($-\pi/2$) for the term with $n=1$ ($n=2$). Introducing the
modified Bessel functions one finds

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle=$
$\displaystyle\frac{ew^{6}}{2\pi^{2}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\frac{I_{\nu_{2}}(pw_{0})}{K_{\nu_{2}}(pw_{0})}\left[K_{\nu_{2}}^{2}(pw)-K_{\nu_{1}}^{2}(pw)\right]$
(3.20)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)J_{2}(2prs_{k})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}J_{2}(2pr\cosh x)\right].$

From (3.20) it follows that the current density is an odd function of
$\alpha_{0}$. For a massless field one has $\nu_{1}=-s/2$, $\nu_{2}=s/2$ and
the brane-induced contribution in the R-region vanishes. The charge flux
through the hypersurface $\phi=\mathrm{const}$ is expressed as
$n_{\phi}\langle j_{\phi}\rangle_{\mathrm{b}}$, where $n_{\phi}=a/w$ is the
normal to the hypersurface. Note that the quantity $n_{\phi}\langle
j_{\phi}\rangle_{\mathrm{b}}$ depends on $r,w,w_{0}$ through the ratios $r/w$
and $w_{0}/w$. The first one is the proper distance from the string

$r_{\mathrm{p}}=ar/w,$ (3.21)

measured in units of the AdS curvature radius, and the second one determines
the proper distance from the brane

$y-y_{0}=a\ln(w/w_{0}).$ (3.22)

For half-integer values of the ratio of the magnetic flux to the flux quantum
one finds

$\lim_{\alpha_{0}\rightarrow\pm 1/2}\langle
j_{\phi}\rangle_{\mathrm{b}}=\mp\frac{eqw^{6}}{2\pi^{3}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\frac{I_{\nu_{2}}(pw_{0})}{K_{\nu_{2}}(pw_{0})}\left[K_{\nu_{2}}^{2}(pw)-K_{\nu_{1}}^{2}(pw)\right]\int_{1}^{\infty}du\,\frac{J_{2}(2pru)}{u\sqrt{u^{2}-1}}.$
(3.23)

This shows that the brane-induced contribution is discontinuous at half
integer values of the magnetic flux in units of the flux quantum. The same is
the case for the brane-free contribution. It is of interest to note that the
limiting values $\lim_{\alpha_{0}\rightarrow\pm 1/2}\langle
j_{\phi}\rangle_{\mathrm{b}}$ are linear functions of the parameter $q$.

For $q=1$ the planar angle deficit is absent and from (3.19) we get

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle=$
$\displaystyle-\frac{e\sin\left(\pi\alpha_{0}\right)w^{6}}{4\pi^{3}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\frac{I_{\nu_{2}}(pw_{0})}{K_{\nu_{2}}(pw_{0})}\left[K_{\nu_{2}}^{2}(pw)-K_{\nu_{1}}^{2}(pw)\right]$
(3.24)
$\displaystyle\times\int_{0}^{\infty}dx\frac{\cosh(2\alpha_{0}x)}{\cosh(2x)+1}J_{2}(2pr\cosh
x).$

This expression describes the brane-induced contribution in the current
density generated by an idealized zero-thickness magnetic flux tube. We expect
that it will approximate the corresponding current density for a finite
thickness flux tube at distances from the tube larger than the core radius.

Note that the integral in the expression (3.16) for the current density
$\langle j_{\phi}\rangle_{0}$ is evaluated by using the formula [58] (in [58]
there is a missprint, instead of $(u^{2}-1)^{-(1\mp 2\nu)/4}$ should be
$(u^{2}-1)^{-(1\pm 2\nu)/4}$)

$\int_{0}^{\infty}dp\,p^{3}J_{\nu_{l}}^{2}(pw)K_{2}(pc)=\frac{c^{2}e^{-5\pi
i/2}Q_{\nu_{l}-1/2}^{5/2}(u)}{\sqrt{2\pi}w^{6}\left(u^{2}-1\right)^{5/4}},$
(3.25)

where $u=1+c^{2}/(2w^{2})$ and $Q_{\nu}^{\mu}(u)$ is the associated Legendre
function of the second kind. This gives (see also [46])

$\displaystyle\langle j_{\phi}\rangle_{0}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}er}{\pi^{5/2}a^{5}}\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}(-1)^{k}s_{k}\sin\left(2\pi
k\alpha_{0}\right)\mathcal{Z}_{ma}(1+2\left(rs_{k}/w\right)^{2})\right.$
(3.26)
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)\cosh
x}{\cosh(2qx)-\cos(q\pi)}\mathcal{Z}_{ma}(1+2\left(r\cosh
x/w\right)^{2})\right],$

with the notation

$\mathcal{Z}_{ma}(u)=e^{-5i\pi/2}\frac{Q_{ma}^{5/2}\left(u\right)+Q_{ma-1}^{5/2}\left(u\right)}{\left(u^{2}-1\right)^{5/4}}.$
(3.27)

Near the string the leading term in the expansion over the distance from the
string is given by

$\langle
j_{\varphi}\rangle_{0}\approx-\frac{3e(w/a)^{5}}{16\pi^{2}r^{4}}h_{4}(q,\alpha_{0}),$
(3.28)

where we have introduced the notation

$h_{\mu}(q,\alpha_{0})=\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}(-1)^{k}s_{k}^{-\mu}\sin\left(2\pi
k\alpha_{0}\right)+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)\cosh^{-\mu}x}{\cosh(2qx)-\cos(q\pi)}.$
(3.29)

At large distances from the string, $r\gg w$, the leading term in the
corresponding expansion is expressed as

$\langle
j_{\phi}\rangle_{0}\approx-e\frac{(2ma+3)(2ma+1)w}{\pi^{2}a^{5}\left(2r/w\right)^{2ma+4}}h_{2ma+4}(q,\alpha_{0}).$
(3.30)

For a massless field the expression of the current density is simplified to

$\langle
j_{\phi}\rangle_{0}=-\frac{3e(w/a)^{5}}{16\pi^{2}r^{4}}h_{4}(q,\alpha_{0}).$
(3.31)

Note that for a massive field the leading term in the expansion over the
distance from the string, given by (3.28), coincides with (3.31).

The VEV of the current density for cosmic string in the Minkowski bulk,
$\langle j_{\phi}\rangle_{0}^{\mathrm{(M)}}$, is obtained from $\langle
j_{\phi}\rangle_{0}$ in the limit $a\rightarrow\infty$ with fixed value of the
coordinate $y$. In this limit one has $w\approx a+y$ and for the Bessel
functions in (3.16) we can use the Debye’s asymptotic expansions [59]. This
leads to the expression

$\displaystyle\langle j_{\phi}\rangle_{0}^{\mathrm{(M)}}$ $\displaystyle=$
$\displaystyle-\frac{e}{\pi^{3}r}\int_{m}^{\infty}dp\,\frac{p^{3}}{\sqrt{p^{2}-m^{2}}}\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)K_{2}(2prs_{k})\right.$ (3.32)
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)}{\cosh(2qx)-\cos(q\pi)}\frac{K_{2}(2pr\cosh
x)}{\cosh x}\right].$

The integrals are evaluated by using the formula from [58] and we find

$\displaystyle\langle j_{\phi}\rangle_{0}^{\mathrm{(M)}}$ $\displaystyle=$
$\displaystyle-\frac{em^{5/2}}{2\pi^{5/2}r^{3/2}}\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}^{3/2}}\sin\left(2\pi
k\alpha_{0}\right)K_{5/2}\left(2mrs_{k}\right)\right.$ (3.33)
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)}{\cosh(2qx)-\cos(q\pi)}\frac{K_{5/2}(2mr\cosh
x)}{\cosh^{3/2}x}\right].$

For a massless field one gets $\langle
j_{\phi}\rangle_{0}^{\mathrm{(M)}}=-3eh_{4}(q,\alpha_{0})/(16\pi^{2}r^{4})$.
This result is conformally related to the expression (3.31) of the current
density for cosmic string in the AdS bulk. For a massive field, at large
distances from the string, $mr\gg 1$, the current density $\langle
j_{\phi}\rangle_{0}^{\mathrm{(M)}}$ is suppresses by the factor
$\exp[-2mr\sin(\pi/q)]$ for $q\geq 2$ and by the factor $e^{-2mr}$ for $1\leq
q<2$. Recall that for the AdS bulk and at large distances from the string, the
brane-free contribution in the current density decays according to inverse
power-law (see (3.30)).

The full curves in figure 1 present the quantity
$r_{\mathrm{p}}^{4}n_{\phi}\langle j_{\phi}\rangle_{0}/e$ for cosmic string in
the AdS bulk as a function of the proper distance from the string (3.21),
measured in units of the AdS curvature radius. The graphs are plotted for
$ma=1$, $\alpha_{0}=0.3$, and the numbers near the curves correspond to the
values of the parameter $q$. The same quantity for cosmic string in the
Minkowski is plotted in figure 1 by dashed curves. The corresponding line
element is given by the expression in the brackets on the right-hand side of
(2.1) and $r_{\mathrm{p}}=r$, $n_{\phi}=1$. For both background geometries
$r_{\mathrm{p}}^{4}n_{\phi}\langle
j_{\phi}\rangle_{0}/e|_{r=0}=-3h_{4}(q,\alpha_{0})/(16\pi^{2})$ and this
limiting value is equal to $0.00301$, $0.00578$, $0.0179$ for $q=1,1.5,2.5$,
respectively. As noted above, near the string the leading terms in the
corresponding asymptotic expansions for the AdS and Minkowski bulks coincide
and in that region the effects of the background curvature are weak. As seen
from the graphs and in accordance with the asymptotic analysis, those effects
are essential at large distances from the cosmic string. Note that for a
massless field one has $r_{\mathrm{p}}^{4}n_{\phi}\langle
j_{\phi}\rangle_{0}/e=-3h_{4}(q,\alpha_{0})/(16\pi^{2})$ (see (3.31)) for all
values of $r_{\mathrm{p}}$ in both AdS and Minkowski geometries. For
$q=1,1.5,2.5$ the corresponding graphs will be presented by horizontal lines
which touch the curves in figure 1 at $r_{\mathrm{p}}=0$.

Figure 1: The boundary-free parts in the current density for the AdS (full
curves) and Minkowski (dashed curves) background geometries as functions of
the proper distance from the string. The graphs are plotted for $ma=1$,
$\alpha_{0}=0.3$ and the numbers near the curves are the corresponding values
of the parameter $q$.

###  3.2 Asymptotics of the current density and numerical results

We turn to the investigation of the current density for the geometry with a
brane in asymptotic regions of the variables. Near the string, assuming that
$r\ll w-w_{0}$, we get

$\langle
j_{\phi}\rangle_{\mathrm{b}}\approx\frac{erh_{-1}(q,\alpha_{0})}{4\pi^{2}a^{5}}\int_{0}^{\infty}dx\,x^{5}\frac{I_{\nu_{2}}(xw_{0}/w)}{K_{\nu_{2}}(xw_{0}/w)}\left[K_{\nu_{2}}^{2}(x)-K_{\nu_{1}}^{2}(x)\right],$
(3.34)

and the boundary-induced VEV behaves as $\langle
j_{\phi}\rangle_{\mathrm{b}}\propto r$. Note that on the string the brane free
part $\langle j_{\phi}\rangle_{0}$ diverges and, hence, it dominates for
points near the string. For the investigation of the current density at large
distances from the string it is convenient to use the representation (3.13).
Introducing a new integration variable $u=pr$ and assuming that $r\gg w$, we
see that the dominant contribution comes from the part of the integrand with
the function $G_{\nu_{2},\nu_{1}}(uw_{0}/r,uw/r)$. By taking into account that

$G_{\nu_{2},\nu_{1}}(uw_{0}/r,uw/r)\approx-\frac{2sr}{\pi}\frac{(w/w_{0})^{s\nu_{1}}}{uw_{0}},$
(3.35)

to the leading order one finds

$\langle
j_{\phi}\rangle\approx-4e|\nu_{2}|\left(|\nu_{2}|+1\right)h_{2|\nu_{2}|+3}(q,\alpha_{0})\frac{w(w_{0}/w)^{2|\nu_{2}|-2s\nu_{2}}}{\pi^{2}a^{5}(2r/w)^{2|\nu_{2}|+3}}.$
(3.36)

As seen, the decay of the current density at large distances from the string
is power-law for both massless and massive fields. This is in contrast to the
case of the Minkowski bulk, where the decay for massive fields is exponential.
For a massless field the brane-induced contribution vanishes and (3.36) is
reduced to (3.31). For $s=1$ the leading term (3.36) coincides with (3.30)
and, hence, the current density at large distances is dominated by the brane-
free contribution. For the field with $s=-1$, at large distances from the
string the current density behaves as $\langle
j_{\phi}\rangle\propto(w/r)^{|2ma-1|+3}$ and the brane-induced contribution
dominates.

Now let us consider the behavior of the current density at large distances
from the brane, $w\gg w_{0}$. Introducing in (3.20) a new integration variable
$u=pw$, we use the asymptotic expressions for the modified Bessel functions
with the argument $uw_{0}/w$ for small values of the latter ratio [59]. It can
be seen that, to the leading order, the brane induced current density is
suppressed by the factor $(w_{0}/2w)^{|\nu_{2}|+\nu_{2}}$.This factor also
describes the behavior of the current density for fixed $w$ when the location
of the brane tends to the AdS boundary, $w_{0}\rightarrow 0$. If in addition
$w\gg w_{0}$ one has $w\gg r$, we use the asymptotic of the Bessel function
for small argument. After integrating over $u$ and for $\nu_{2}>0$ this gives

$\langle
j_{\phi}\rangle_{\mathrm{b}}\approx\frac{er(w_{0}/2w)^{2\nu_{2}}}{8\pi^{3/2}a^{5}\nu_{2}}\frac{\Gamma\left(3+\nu_{2}\right)h_{-1}(q,\alpha_{0})}{\Gamma^{2}(\nu_{2})\Gamma\left(7/2+\nu_{2}\right)}\left[2\Gamma\left(3+2\nu_{2}\right)-\Gamma\left(3+2\nu_{2}-s\right)\Gamma\left(3+s\right)\right].$
(3.37)

Considered as a function of the physical distance from the brane, $y-y_{0}$,
the brane-induced contribution is suppressed by the factor
$e^{-2\nu_{2}(y-y_{0})/a}$. For $\nu_{2}<0$ the leading term in the asymptotic
expansion is given by

$\langle
j_{\phi}\rangle_{\mathrm{b}}\approx\frac{2eh_{-1}(q,\alpha_{0})r}{15\pi^{2}a^{5}}\sum_{l=1,2}\left(4-\nu_{l}^{2}\right)\left(\nu_{l}^{2}-1\right)\nu_{l}.$
(3.38)

In this case, the brane-induced part tends to finite value on the AdS horizon.

For the investigation of the current density near the brane it is convenient
to use the representation (3.13). In particular, the current density on the
brane is directly obtained putting in the integrand $w=w_{0}$. By taking into
account that $G_{\nu_{2},\nu_{1}}(x,x)=-2s/(\pi x)$, we get

$\displaystyle\langle j_{\phi}\rangle|_{w=w_{0}}$ $\displaystyle=$
$\displaystyle-\frac{2ew_{0}^{2}}{\pi^{4}a^{5}r}\int_{0}^{\infty}dp\frac{\,p}{J_{\nu_{2}}^{2}(p)+Y_{\nu_{2}}^{2}(p)}$
(3.39)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)K_{2}\left(\frac{2pr}{w_{0}}s_{k}\right)\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}K_{2}\left(\frac{2pr}{w_{0}}\cosh x\right)\right].$

Unlike the fermion condensate and the VEV of the energy-momentum tensor, the
VEV of the current density is finite on the brane. This feature can be
understood as follows. In the absence of the magnetic flux the current density
vanishes in the geometry with the brane. In the presence of the localized
flux, in the region outside the localization the local geometry and the field
strength tensor for the external electromagnetic field are the same as in the
absence of the flux (in the problem at hand the field strength tensor is
zero). From here it follows that adding the localized flux does not change the
divergence structure in the VEVs outside the localization range. In
particular, the current density being finite in the absence of the flux
remains finite in the problem with a magnetic flux.

Finally, we consider the limit $a\rightarrow\infty$ with fixed value of the
coordinate $y$. As seen from (2.1), in that limit the geometry under
consideration is reduced to the geometry of a cosmic string in background of
(4+1)-dimensional Minkowski spacetime. For the coordinate $w$ in the arguments
of the modified Bessel function one has $w\approx a+y$. By taking into account
that $\nu_{l}\gg 1$, we see that in the limit under consideration both the
order and the argument of the modified Bessel functions in (3.20) are large
and we can use the corresponding uniform asymptotic expansions. From those
expansions, to the leading order, one gets

$\frac{I_{\nu_{2}}(pw_{0})}{K_{\nu_{2}}(pw_{0})}\sim\frac{1}{\pi}e^{2\nu_{2}\rho(pw_{0}/\nu_{2})},$
(3.40)

with $\rho(x)=\sqrt{1+x^{2}}+\ln[x/(1+\sqrt{1+x^{2}})]$, and

$K_{\nu_{1}}^{2}(pw)-K_{\nu_{2}}^{2}(pw)\sim\frac{\pi
e^{-2\nu_{2}\rho(pw/\nu_{2})}}{\nu_{2}(pw/\nu_{2})^{2}}\left(\frac{1}{\sqrt{1+(pw/\nu_{2})^{2}}}-s\right).$
(3.41)

In all the terms, except the arguments of the function $\rho(x)$, we can
replace $pw/\nu_{2},pw_{0}/\nu_{2}\rightarrow p/m$. Expanding the functions in
the exponents we get $\langle j_{\phi}\rangle_{\mathrm{b}}\rightarrow\langle
j_{\phi}\rangle_{\mathrm{b}}^{\mathrm{(M)}}$, where the current density in the
Minkowskian bulk is given by

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}^{\mathrm{(M)}}$
$\displaystyle=$
$\displaystyle\frac{em}{2\pi^{2}r}\int_{m}^{\infty}du\,e^{-2u\left(y-y_{0}\right)}\left(su-m\right)$
(3.42)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)J_{2}(2rs_{k}\sqrt{u^{2}-m^{2}})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}J_{2}(2r\cosh x\sqrt{u^{2}-m^{2}})\right],$

where $y=y_{0}$ specifies the location of the planar boundary in the Minkowski
bulk. For a massless field the corresponding boundary-induced contribution
vanishes. Note that the fermionic condensate, the charge and current densities
in (2+1)-dimensional conical spacetime with a single and two circular
boundaries have been investigated in [57, 60, 61, 62]. The geometry of a
planar ring has been considered in [63].

In figures below we plot the charge flux through the hypersurface
$\phi=\mathrm{const}$, given by $n_{\phi}\langle
j_{\phi}\rangle_{\mathrm{b}}$, measured in units of $e/a^{4}$. The numbers
near the curves will correspond to the values of the parameter $q$. Figure 2
presents the corresponding brane-induced contribution in the R-region versus
the parameter $\alpha_{0}$ that determines the fractional part of the ratio of
the magnetic flux to flux quantum. The left and right panels correspond to
fermionic fields with $s=1$ and $s=-1$, respectively. The graphs are plotted
for $ma=1$, $w/w_{0}=1.5$, $r/w_{0}=0.5$. As seen, the absolute value of the
current density increases with increasing planar angle deficit. As it has been
already mentioned, for $\alpha_{0}=\pm 1/2$ it is a linear function of the
parameter $q$. We also see that the absolute value of the brane-induced
current density for the case $s=-1$ is significantly larger compared to that
for the field with $s=1$.

|
---|---
Figure 2: The brane-induced contribution in the VEV of the azimuthal current
density as a function of the fractional part of the ratio of the magnetic flux
to flux quantum. The left and right panels correspond to the cases $s=1$ and
$s=-1$. The numbers near the curves are the values of $q$. Graphs are plotted
for $ma=1$, $w/w_{0}=1.5$, $r/w_{0}=0.5$.

In figure 3 we display the dependence of the brane-induced current density on
the ratio $w/w_{0}$ for fields with $s=1$ (left panel) and $s=-1$ (right
panel). For the parameters we have taken $ma=1$, $\alpha_{0}=0.3$,
$r/w_{0}=0.5$. Note that the proper distance from the brane $y-y_{0}$ is
expressed as $y-y_{0}=a\ln(w/w_{0})$. As it has been shown by the asymptotic
analysis, for $w/w_{0}\gg 1$ the brane-induced contribution decays as
$(w/w_{0})^{2ma+s}$. The decay is stronger for the field with $s=1$.

|
---|---
Figure 3: The current density induced by the brane as a function of the ratio
$w/w_{0}$ for fixed values $ma=1$, $\alpha_{0}=0.3$, $r/w_{0}=0.5$. The left
and right panels correspond to fields with $s=1$ and $s=-1$.

The dependence of the brane-induced current density on the distance from the
string is plotted in figure 4 for fixed values of the combinations $ma=1$,
$\alpha_{0}=0.3$, $w/w_{0}=1.5$. As before the left and right panels
correspond to $s=1$ and $s=-1$. The contribution of the brane linearly
vanishes on the string (see (3.34)). At large distances from the string it
behaves like $(r/w)^{2ma+3+s}$. The fall-off is stronger for the case $s=1$.

|
---|---
Figure 4: The brane-induced contribution in the VEV of the current density in
the R-region as a function of the distance from the string for fixed $ma=1$,
$\alpha_{0}=0.3$, $w/w_{0}=1.5$. The left and right panels correspond to $s=1$
and $s=-1$.

It is of interest to consider the dependence of the current density on the
mass of the field. Figure 5 displays that dependence for the brane-induced
contribution for $\alpha_{0}=0.3$, $w/w_{0}=1.5$, $r/w_{0}=0.5$ and for fields
with $s=1$ (left panel) and $s=-1$ (right panel). As it has been mentioned
above, the brane-induced VEV in the R-region vanishes for a massless field. We
also expect that for large masses the VEV should tend to zero and, hence, for
some intermediate value of the mass the absolute value of the current density
has a maximum. That is seen from the graphs.

|
---|---
Figure 5: The dependence of the brane-induced current on the field mass for
fixed values $\alpha_{0}=0.3$, $w/w_{0}=1.5$, $r/w_{0}=0.5$ and for fields
with $s=1$ and $s=-1$ (the left and right panels, respectively).

In the numerical examples above we have considered the brane-induced part in
the current density. To show its relative contribution, in figure 6 the total
current density in the R-region is plotted versus the distance from the string
for fixed values $ma=1$, $\alpha_{0}=0.3$, $w/w_{0}=1.5$. For these values the
brane-free contribution is positive and monotonically decays with increasing
$r$. For the field with $s=1$ (dashed curves), though the brane-induced part
is negative, the brane-free contribution dominates at all distances and the
total current density is positive. For the field with $s=-1$ (full curves)
both contributions are positive. For $s=-1$, the total current density near
the string is dominated by the brane-free part and at large distances the
brane-induced part dominates.

Figure 6: The total current density as a function of the radial coordinate for
fields with $s=1$ (dashed curves) and $s=-1$ (full curves). The graphs are
plotted for $ma=1$, $\alpha_{0}=0.3$, $w/w_{0}=1.5$.

##  4 Vacuum current in the L-region

In this section we consider the VEV of the current density in the region
between the AdS boundary and the brane, corresponding to $0\leq w\leq w_{0}$.

###  4.1 Integral representation

The complete set of fermionic modes has the form (2.8) with the function
$W_{\nu}(pw)$ given by (2.11). Now, in the normalization condition (2.14) the
integration goes over the region $w\in[0,w_{0}]$. For $ma\geq 1/2$ and
$c_{2}\neq 0$ the normalization integral diverges at the lower limit and the
corresponding mode functions are not normalizable. Hence, in this case the
normalizability condition implies that $c_{2}=0$ and $W_{\nu}(pw)=J_{\nu}(pw)$
(the constant $c_{1}$ is included in the normalization coefficient). For
$0\leq ma<1/2$ the modes with $c_{2}\neq 0$ are normalizable and an additional
boundary condition on the AdS boundary is required to uniquely define the
wave-functions. Here we will assume a special boundary condition corresponding
to $c_{2}=0$. Hence, in our consideration the fermionic mode functions are
given by (2.8) with $W_{\nu}(pw)=J_{\nu}(pw)$ for $ma\geq 0$.

From the boundary condition (2.3) it follows that the eigenvalues for the
quantum number $p$ are roots of the equation

$J_{\nu_{1}}(pw_{0})=0\ ,$ (4.1)

for both the cases $s=\pm 1$. We will denote the corresponding positive roots
with respect to $pw_{0}$ by $p_{i}=p_{i}(\nu_{1})=pw_{0}$, with $i=1,2,...$,
assuming that they are enumerated in the ascending order, $p_{i+1}>p_{i}$.
Note that the roots $p_{i}$ do not depend on the location of the brane.
Substituting the mode functions in (2.14) and using the corresponding standard
integral involving the products of the Bessel functions, for the normalization
constants in the L-region,
$C_{\sigma}^{(\pm)}=C_{\mathrm{(L)}\sigma}^{(\pm)}$, we get

$|C_{\mathrm{(L)}\sigma}^{(\pm)}|^{2}=\frac{\eta
q\lambda^{2}p_{i}J_{\nu_{2}}^{-2}(p_{i})}{8\pi^{2}a^{4}E\kappa_{\eta}b_{\eta}^{(\pm)}w_{0}^{2}\sqrt{p_{i}^{2}+w_{0}^{2}k_{z}^{2}}}\
,$ (4.2)

where the energy is given by
$E=\sqrt{\lambda^{2}+p_{i}^{2}/w_{0}^{2}+k_{z}^{2}}$. As seen, the
normalization coefficients are independent of the representation adopted,
$s=1$ or $s=-1$.

Having specified the complete set of modes we can start the investigation of
the current density by using the formula (2.7). Similar to the R-region, it
can be seen that the only nonzero component corresponds to the one along the
azimuthal direction. After the summation over $\eta$, we get

$\displaystyle\langle j_{\phi}\rangle$ $\displaystyle=$
$\displaystyle-\frac{eqw^{6}}{2\pi^{2}a^{5}w_{0}^{2}r}\int_{0}^{\infty}d\lambda\int_{-\infty}^{+\infty}dk_{z}\sum_{j}\sum_{i=1}^{\infty}\epsilon_{j}\frac{\lambda^{2}}{E}$
(4.3) $\displaystyle\times\frac{J_{\beta_{j}}(\lambda
r)J_{\beta_{j}+\epsilon_{j}}(\lambda
r)}{J_{\nu_{2}}^{2}\left(p_{i}\right)}\left[J_{\nu_{1}}^{2}(p_{i}w/w_{0})+J_{\nu_{2}}^{2}(p_{i}w/w_{0})\right]\
.$

By using the integral representation (3.7) and making transformations similar
to those we have described in the previous section for the R-region, the
following expression is obtained

$\displaystyle\langle j_{\phi}\rangle$ $\displaystyle=$
$\displaystyle-\frac{ew^{6}}{\pi^{2}a^{5}rw_{0}^{4}}\sum_{i=1}^{\infty}p_{i}^{2}\frac{J_{\nu_{1}}^{2}(pw)+J_{\nu_{2}}^{2}(pw)}{J_{\nu_{2}}^{2}(p_{i})}$
(4.4)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}(-1)^{k}\frac{\sin\left(2\pi
k\alpha_{0}\right)}{s_{k}}K_{2}\left(2rps_{k}\right)\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}K_{2}\left(2rp\cosh x\right)\right]\ ,$

where $p=p_{i}/w_{0}$.

In order to obtain a representation more convenient for the asymptotic and
numerical analysis, and for explicit extraction of the brane-induced
contribution, we apply to the series over $i$ a variant of the generalized
Abel-Plana formula [64],

$\displaystyle\sum_{i=1}^{\infty}\frac{f(p_{i})}{p_{i}J_{\nu_{2}}^{2}(p_{i})}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}duf(u)-\frac{1}{2\pi}\int_{0}^{\infty}du\frac{K_{\nu_{1}}(u)}{I_{\nu_{1}}(u)}$
(4.5) $\displaystyle\times[e^{-\nu_{1}\pi i}f(e^{\pi i/2}u)+e^{\nu_{1}\pi
i}f(e^{-\pi i/2}u)]\ ,$

valid for a function $f(u)$ analytic in the right half-plane of the complex
variable $u$ (function $f(u)$ may have branch points on the imaginary axis,
for the conditions imposed on this function see [64]). For the problem under
consideration the function $f(u)$ is specified as

$f(u)=u^{3}[J_{\nu_{1}}^{2}(uw/w_{0})+J_{\nu_{2}}^{2}(uw/w_{0})]K_{2}(2ru\gamma/w_{0}),$
(4.6)

with $\gamma=s_{k},\cosh x$. For this function one has

$e^{-\nu_{1}\pi i}f(e^{\pi i/2}u)+e^{\nu_{1}\pi i}f(e^{-\pi i/2}u)=\pi
u^{3}[I_{\nu_{1}}^{2}(uw/w_{0})-I_{\nu_{2}}^{2}(uw/w_{0})]J_{2}(2ru\gamma/w_{0}).$
(4.7)

The contribution to the current density coming from the first term in the
right-hand side of (4.5) coincides with the current density in the geometry
without the brane given by (3.16). As a result, the current density in the
L-region is decomposed as (3.17), where the brane-induced contribution, coming
from the last term in (4.5), is presented as

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle=$
$\displaystyle\frac{ew^{6}}{2\pi^{2}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\frac{K_{\nu_{1}}(pw_{0})}{I_{\nu_{1}}(pw_{0})}\left[I_{\nu_{1}}^{2}(pw)-I_{\nu_{2}}^{2}(pw)\right]$
(4.8)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)J_{2}(2prs_{k})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}J_{2}(2pr\cosh x)\right].$

It is an odd periodic function of the magnetic flux with the period of the
flux quantum.

Comparing with (3.20), we see that the brane-induced contribution in the
L-region is obtained from the corresponding quantity for the R-region by the
replacements $I\rightarrow K$, $K\rightarrow I$ of the modified Bessel
functions and the replacements $\nu_{1,2}\rightarrow\nu_{2,1}$ of the orders.
The expression for the brane-induced contribution in the L-region at half-
integer values of the ratio of the magnetic flux to flux quantum is obtained
from (3.23) by the same replacements. The limiting values
$\lim_{\alpha_{0}\rightarrow\pm 1/2}\langle j_{\phi}\rangle_{\mathrm{b}}$ are
linear functions of the parameter $q$ describing the planar angle deficit.
Another special case with $q=1$ corresponds to magnetic flux tube in AdS
spacetime. The corresponding expression reads

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle=$
$\displaystyle\frac{e\sin\left(\pi\alpha_{0}\right)w^{6}}{4\pi^{3}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\frac{K_{\nu_{1}}(pw_{0})}{I_{\nu_{1}}(pw_{0})}\left[I_{\nu_{2}}^{2}(pw)-I_{\nu_{1}}^{2}(pw)\right]$
(4.9)
$\displaystyle\times\int_{0}^{\infty}dx\,\frac{\cosh(2\alpha_{0}x)}{\cosh(2x)+1}J_{2}(2pr\cosh
x).$

For a massless field one has $K_{\nu_{1}}(x)/I_{\nu_{1}}(x)=\pi/(e^{2x}+s)$
and $I_{\nu_{2}}^{2}(x)-I_{\nu_{1}}^{2}(x)=-2s/(\pi x)$. Expanding
$1/(e^{2x}+s)$ and evaluating the integral over $p$ by using the result from
[58], from (4.8) we find

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle=$
$\displaystyle-\frac{3er}{8\pi^{2}(aw_{0}/w)^{5}}\sum_{l=1}^{\infty}(-s)^{l}\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}s_{k}\sin\left(2\pi
k\alpha_{0}\right)}{\left[l^{2}+(rs_{k}/w_{0})^{2}\right]^{5/2}}\right.$
(4.10)
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)}{\cosh(2qx)-\cos(q\pi)}\frac{\cosh
x}{\left[l^{2}+(r\cosh x/w_{0})^{2}\right]^{5/2}}\right].$

Massless fermionic field is conformally invariant and the expression (4.10) is
conformally related to the corresponding result in the Minkowski bulk:
$\langle j_{\phi}\rangle_{\mathrm{b}}=(w/a)^{5}\langle
j_{\phi}\rangle_{\mathrm{b}}^{\mathrm{(M)}}$. Here, $\langle
j_{\phi}\rangle_{\mathrm{b}}^{\mathrm{(M)}}$ is the boundary-induced current
density in the region between two boundaries located at $w=0$ and $w=w_{0}$ in
(4+1)-dimensional Minkowski spacetime with a cosmic string (the line element
is given by the expression in the brackets of (2.1)). The boundary $w=0$ in
the Minkowskian problem is the conformal image of the AdS boundary. The vacuum
polarization effects induced by planar boundaries orthogonal to cosmic string
in the Minkowski bulk have been investigated in [65, 66, 67, 68].

###  4.2 Asymptotic and numerical analysis

We start the asymptotic analysis considering the points near the string,
assuming that $r\ll w_{0}-w$. By taking into account that in this region the
argument of the Bessel function in (4.8) is small, for the leading order term
we find

$\langle
j_{\phi}\rangle_{\mathrm{b}}\approx\frac{erh_{-1}(q,\alpha_{0})}{4\pi^{2}a^{5}}\int_{0}^{\infty}dx\,x^{5}\frac{K_{\nu_{1}}(xw_{0}/w)}{I_{\nu_{1}}(xw_{0}/w)}\left[I_{\nu_{1}}^{2}(x)-I_{\nu_{2}}^{2}(x)\right],$
(4.11)

and the brane induced VEV linearly vanishes on the string. At large distances
from the string we use the representation (4.4). The dominant contribution
comes from the lowest eigenmode for $p_{i}$ and for $q\geq 2$ the VEV of the
current density is suppressed by the exponential factor
$\exp[-2rp_{1}\sin(\pi/q)/w_{0}]$. For $q<2$ the suppression is stronger, by
the factor $e^{-2rp_{1}/w_{0}}$. Recall that in the R-region the decay of the
vacuum current at large distances from the string is power-law for both
massless and massive fields.

For points close to the AdS boundary, $w/w_{0}\ll 1$, the dominant
contribution in the integral over $p$ given in (4.8) comes from the region of
integration where the argument of the functions $I_{\nu_{l}}(pw)$ is small. By
using the corresponding asymptotic expression [59], to the leading order one
has

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle\approx$
$\displaystyle\frac{se(w/w_{0})^{2ma+5}w_{0}^{2}}{2^{2ma}\pi^{2}a^{5}\Gamma^{2}(ma+1/2)r}\int_{0}^{\infty}dx\,x^{2ma+2}\frac{K_{\nu_{1}}(x)}{I_{\nu_{1}}(x)}$
(4.12)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)J_{2}(2xrs_{k}/w_{0})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}J_{2}(2xr\cosh x/w_{0})\right].$

As we observe, the brane-induced current on the AdS boundary vanishes as
$w^{2ma+5}$. Note that this suppression factor does not depend on the
parameter $s$. The expression in the right-hand side of (4.12) also describes
the behavior of the current in the limit when the location of the brane is
close to the AdS horizon for fixed $w$. If in addition $w_{0}\gg r$, that
expression is further simplified as

$\langle
j_{\phi}\rangle_{\mathrm{b}}\approx\frac{seh_{-1}(q,\alpha_{0})(w/w_{0})^{2ma+5}r}{2^{2ma+1}\pi^{2}a^{5}\Gamma^{2}(ma+1/2)}\int_{0}^{\infty}dx\,x^{2ma+4}\frac{K_{\nu_{1}}(x)}{I_{\nu_{1}}(x)}.$
(4.13)

For the evaluation of the current density on the brane it is more convenient
to use the representation (4.4) directly putting $w=w_{0}$:

$\displaystyle\langle j_{\phi}\rangle|_{w=w_{0}}$ $\displaystyle=$
$\displaystyle-\frac{ew_{0}^{2}}{\pi^{2}a^{5}r}\sum_{i=1}^{\infty}p_{i}^{2}\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)K_{2}\left(2rp_{i}s_{k}/w_{0}\right)\right.$ (4.14)
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}K_{2}\left(2rp_{i}\cosh x/w_{0}\right)\right]\ .$

The consideration of the Minkowskian limit is similar to that of the R-region.
The corresponding result is given by (3.42) with $y-y_{0}$ replaced by
$y_{0}-y$. Of course, in the Minkowskian limit the VEV is symmetric with
respect to the brane. For the AdS bulk, though the local geometrical
characteristics are the same in the R- and L-regions, the extrinsic curvature
tensor, $K_{\mu\nu}$, for the brane under consideration is different for those
regions ($K_{\mu\nu}=-g_{\mu\nu}/a$ in the R-region and
$K_{\mu\nu}=g_{\mu\nu}/a$ in the L-region). As a consequence, the physical
properties of the vacuum state, in particular, the current densities, differ
in those regions. The limiting values of the current densities on the brane
are different for the R- and L-regions. The discontinuity is related to the
idealized model for the brane with zero thickness that constraints all the
modes of the vacuum fluctuations. This idealization leads, for example, to
surface divergences in the fermion condensate and in the VEV of the energy-
momentum tensor. In more realistic models, the fluctuations with high
frequencies are relatively insensitive to the presence of the brane. We expect
that the formulas for the current density given above will correctly
approximate the corresponding results in more realistic models for distances
from the brane larger than the cutoff wavelength. The latter is determined by
the specific model for the brane.

Now we turn to the results of numerical investigations for the current density
in the L-region. As before, the numbers near the curves present the values of
the parameter $q$. The left and right panels in figures correspond to fields
with $s=1$ and $s=-1$, respectively. In figure 7 the brane-induced current
density is plotted as a function of $\alpha_{0}$ for $ma=1$, $w/w_{0}=0.8$,
$r/w_{0}=0.5$. For these values of the parameters the absolute value of the
current density increases with increasing $q$. However, this is not a general
feature. For points too close to the string we have an opposite behavior (see
figure 9 below).

|
---|---
Figure 7: The brane-induced current density in the L-region versus the
parameter $\alpha_{0}$. Graphs are plotted for $ma=1$, $w/w_{0}=0.8$,
$r/w_{0}=0.5$.

Figure 8 displays the dependence of the current density on the ratio $w/w_{0}$
for fixed $ma=1$, $\alpha_{0}=0.3$, $r/w_{0}=0.5$. As it has been already
shown by the asymptotic analysis, the current density tends to zero on the AdS
boundary like $w^{2ma+5}$ for both cases $s=1$ and $s=-1$.

|
---|---
Figure 8: The brane contribution in the current density as a function of
$w/w_{0}$ for $ma=1$, $\alpha_{0}=0.3$, $r/w_{0}=0.5$.

The brane-induced current density versus the distance from the string is
depicted in figure 9 for $ma=1$, $\alpha_{0}=0.3$, $w/w_{0}=0.8$. Recall that
in the L-region the suppression of the current density at large distances from
the string is stronger compared to the behavior in the R-region. One has an
exponential decay insted of power-law fall-off for the R-region.

|
---|---
Figure 9: The dependence of the brane-induced current density on the distance
from the string for $ma=1$, $\alpha_{0}=0.3$, $w/w_{0}=0.8$.

Figure 10 presents the brane-induced current density as a function of the mass
for the values $\alpha_{0}=0.3$, $w/w_{0}=0.8$, $r/w_{0}=0.5$. Unlike the
R-region, the current density for massless fields does not vanish. In the case
$s=1$ (left panel), the corresponding values of the combination
$10^{2}a^{4}\langle j_{\phi}\rangle_{\mathrm{b}}/e$, plotted in figure 10, are
equal $-0.131,-0.201,-0.295$ for $q=1,1.5,2.5$, respectively. For the field
$s=-1$ (right panel) the respective values are given by $0.153,0.23,0.323$.
Again, we see that the current density is not a monotonic function of the
mass. At some intermediate value of the mass its absolute value takes the
maximum. For the example given in figure 10 the absolute value of the current
density at the maximum is essentially larger compared to the current density
for a massless field.

|
---|---
Figure 10: The brane-induced current density in the L-region versus the field
mass for $\alpha_{0}=0.3$, $w/w_{0}=0.8$, $r/w_{0}=0.5$.

Finally, in figure 11 we plot the total current density versus the distance
from the string for $ma=1$, $\alpha_{0}=0.3$, $w/w_{0}=0.8$. Full and dashed
curves correspond to the fields with $s=-1$ and $s=1$, respectively. Similar
to the R-region, at large distances from the string the current density for
the first case is dominated by the brane-induced contribution.

Figure 11: The same as in figure 5 for the total current density in the
L-region for $w/w_{0}=0.8$.

##  5 Vacuum current for the second type of boundary condition

In this section we consider the brane-induced contribution to the VEV of the
azimuthal current density for the boundary condition

$(1-i\gamma^{\mu}n_{\mu})\psi(x)=0\ ,\ \ w=w_{0}\ ,$ (5.1)

The latter differs from (2.3) by the sign of the term containing the Dirac
matrices. As it will be seen below both the boundary conditions (2.3) and
(5.1) appear in Randall-Sundrum type models as a consequence of the
$Z_{2}$-symmetry with respect to the brane. The evaluation procedure for the
condition (5.1) is similar to that we have described in the previous sections
and the final results will be presented only.

###  5.1 R-region

We start with the R-region. The corresponding mode functions are given by
(2.8) where now

$W_{\nu_{l}}(pw)=G_{\nu_{1},\nu_{l}}(pw_{0},pw),\;l=1,2.$ (5.2)

The expression for the normalization coefficient is obtained from (3.3) by the
replacement $\nu_{2}\rightarrow\nu_{1}$ in the orders of the Bessel and
Neumann functions. The mode-sum for the VEV of the azimuthal current density
is expressed as

$\displaystyle\langle j_{\phi}\rangle$ $\displaystyle=$
$\displaystyle-\frac{eqw^{6}}{4\pi^{2}a^{5}}\int_{0}^{\infty}d\lambda\int_{0}^{\infty}dp\int_{-\infty}^{+\infty}dk_{z}\sum_{j=\pm
1/2,...}\epsilon_{j}\frac{\lambda^{2}p}{E}$ (5.3) $\displaystyle\times
J_{\beta_{j}}(\lambda r)J_{\beta_{j}+\epsilon_{j}}(\lambda
r)\frac{G_{\nu_{1},\nu_{1}}^{2}(pw_{0},pw)+G_{\nu_{1},\nu_{2}}^{2}(pw_{0},pw)}{J_{\nu_{1}}^{2}(pw_{0})+Y_{\nu_{1}}^{2}(pw_{0})},$

with $w\geq w_{0}$. It is decomposed as (3.17) with the brane-induced
contribution

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle=$
$\displaystyle\frac{ew^{6}}{2\pi^{2}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\frac{I_{\nu_{1}}(pw_{0})}{K_{\nu_{1}}(pw_{0})}\left[K_{\nu_{1}}^{2}(pw)-K_{\nu_{2}}^{2}(pw)\right]$
(5.4)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)J_{2}(2prs_{k})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\,\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}J_{2}(2pr\cosh x)\right].$

Now comparing these formulas with (3.5) and (3.20), we see that the VEVs of
the current density for $s=\pm 1$ and for the boundary condition (5.1)
coincide with the VEVs for the boundary condition (2.3) and for $s=\mp 1$.

###  5.2 L-region

In the L-region the mode functions for the boundary condition (5.1) are given
by (2.8) with $W_{\nu}(pw)=J_{\nu}(pw)$. The eigenvalues $p_{i}$ of the
quantum number $p$ are determined from the boundary condition and are roots of
the equation $J_{\nu_{2}}(pw_{0})=0$. The normalization coefficient is given
by (4.2) with the replacement $\nu_{2}\rightarrow\nu_{1}$ in the order of the
Bessel function. For the VEV of the azimuthal current density one gets

$\displaystyle\langle j_{\phi}\rangle$ $\displaystyle=$
$\displaystyle-\frac{eqw^{6}}{2\pi^{2}a^{5}w_{0}^{2}r}\int_{0}^{\infty}d\lambda\int_{-\infty}^{+\infty}dk_{z}\sum_{j}\sum_{i=1}^{\infty}\epsilon_{j}\frac{\lambda^{2}}{E}$
(5.5) $\displaystyle\times\frac{J_{\beta_{j}}(\lambda
r)J_{\beta_{j}+\epsilon_{j}}(\lambda
r)}{J_{\nu_{1}}^{2}\left(p_{i}\right)}\left[J_{\nu_{1}}^{2}(p_{i}w/w_{0})+J_{\nu_{2}}^{2}(p_{i}w/w_{0})\right]\
.$

In a way similar to that we have used in section 4, by using the summation
formula that is obtained from (4.5) by the replacements
$\nu_{2}\rightarrow\nu_{1}$, $\nu_{1}\rightarrow\nu_{2}$, for the brane-
induced contribution in the L-region we find

$\displaystyle\langle j_{\phi}\rangle_{\mathrm{b}}$ $\displaystyle=$
$\displaystyle\frac{ew^{6}}{2\pi^{2}a^{5}r}\int_{0}^{\infty}dp\,p^{3}\frac{K_{\nu_{2}}(pw_{0})}{I_{\nu_{2}}(pw_{0})}\left[I_{\nu_{2}}^{2}(pw)-I_{\nu_{1}}^{2}(pw)\right]$
(5.6)
$\displaystyle\times\left[\sideset{}{{}^{\prime}}{\sum}_{k=1}^{[q/2]}\frac{(-1)^{k}}{s_{k}}\sin\left(2\pi
k\alpha_{0}\right)J_{2}(2prs_{k})\right.$
$\displaystyle\left.+\frac{q}{\pi}\int_{0}^{\infty}dx\,\frac{g(q,\alpha_{0},2x)/\cosh
x}{\cosh(2qx)-\cos(q\pi)}J_{2}(2pr\cosh x)\right].$

For a given $s$, the VEV of the azimuthal current density for the boundary
condition (5.1) coincides with that for the condition (2.3) and for $s$
replaced by $-s$.

##  6 Current density in $C$\- and $P$-symmetric models

In odd-dimensional spacetimes the Clifford algebra has two inequivalent
irreducible representations. For the spatial dimension $D=4$ we consider here,
the corresponding sets of the flat spacetime gamma matrices can be taken in
the form
$\gamma_{(s)}^{(b)}=\\{\gamma^{(0)},\gamma^{(1)},\gamma^{(2)},\gamma^{(3)},s\gamma^{(4)}\\}$,
where $s=\pm 1$ specify the representations and the matrices $\gamma^{(b)}$
are obtained from the corresponding matrices in (2.4) omitting the factors
$w/a$. Note that one has the relation
$\gamma^{(4)}=-\gamma^{(0)}\gamma^{(1)}\gamma^{(2)}\gamma^{(3)}$. We introduce
fermionic fields $\psi_{(s)}$ corresponding to a given representation and
having the Lagrangian
$L_{(s)}=\bar{\psi}_{(s)}[i\gamma_{(s)}^{\mu}(\partial_{\mu}+\Gamma_{\mu}^{(s)})-m]\psi_{(s)}$,
where $\gamma_{(s)}^{\mu}=(w/a)\delta_{b}^{\mu}\gamma_{(s)}^{(b)}$ are the
curved spacetime gamma matrices and $\Gamma_{\mu}^{(s)}$ is the respective
spin connection. Note that, as it follows from (2.6), the product
$\gamma_{(s)}^{\mu}\Gamma_{\mu}^{(s)}$ is the same for $s=+1$ and $s=-1$. For
massive fields, the separate Lagrangians with a given $s$ are not invariant
under the charge conjugation ($C$) and parity transformation ($P$). The models
invariant under these transformations in the absence of external gauge field
can be constructed considering the set of two fields with $s=+1$ and $s=-1$
and with the Lagrangian $L=\sum_{s=\pm 1}L_{(s)}$. In these models the VEV of
the total current density, $\langle J^{\mu}\rangle$, is the sum of the VEVs
$\langle j_{(s)}^{\mu}\rangle$ corresponding to the separate fields: $\langle
J^{\mu}\rangle=\sum_{s=\pm 1}\langle j_{(s)}^{\mu}\rangle$, where
$j_{(s)}^{\mu}=e\bar{\psi}_{(s)}\gamma_{(s)}^{\mu}\psi_{(s)}$. Note that the
matrices $\gamma_{(+1)}^{\mu}=\gamma^{\mu}$ coincide with those we have used
in the previous sections.

Let us pass to a new set of fields $\psi_{(s)}^{\prime}$ by the
transformations $\psi_{(+1)}^{\prime}=\psi_{(+1)}$,
$\psi_{(-1)}^{\prime}=-\gamma^{(4)}\psi_{(-1)}$. By taking into account that
$\gamma^{(4)}\gamma_{(-1)}^{\mu}\gamma^{(4)}=\gamma^{\mu}$ and
$\gamma_{(s)}^{\mu}\Gamma_{\mu}^{(s)}=\gamma^{\mu}\Gamma_{\mu}$, the total
Lagrangian is presented in the form $L=\sum_{s=\pm
1}\bar{\psi}_{(s)}^{\prime}[i\gamma^{\mu}(\partial_{\mu}+\Gamma_{\mu})-sm]\psi_{(s)}^{\prime}$.
For the fields $\psi_{(s)}^{\prime}$ the Dirac equation is reduced to (2.2).
This shows that the parameter $s=\pm 1$ in the previous discussion corresponds
to different representations of the Clifford algebra. For the current
densities of the separate fields we get
$j_{(s)}^{\mu}=e\bar{\psi}_{(s)}^{\prime}\gamma^{\mu}\psi_{(s)}^{\prime}$.

Now let us turn to the boundary conditions on the brane. First we consider the
case when both the fields $\psi_{(s)}$ obey the boundary condition (2.3):

$(1+i\gamma_{(s)}^{\mu}n_{\mu})\psi_{(s)}=0,\;w=w_{0}.$ (6.1)

In terms of the new fields $\psi_{(s)}^{\prime}$ these conditions take the
form

$(1+si\gamma^{\mu}n_{\mu})\psi_{(s)}^{\prime}=0,\;w=w_{0}.$ (6.2)

We see that the field $\psi_{(+1)}^{\prime}$ obeys the field equation and the
boundary condition discussed in sections 3 and 4 for the case $s=1$. Hence,
the brane-induced contributions in the corresponding current densities in the
R- and L-regions are given by (3.20) and (4.8) with $s=1$ and
$\nu_{l}=ma+(-1)^{l}/2$. The field $\psi_{(-1)}^{\prime}$ obeys the field
equation (2.2) with $s=-1$ and the boundary condition (5.1). As it has been
discussed in the previous section, the corresponding VEVs of the current
density in the R- and L-regions coincide with those considered in sections 3
and 4 for the case $s=1$ and for the condition (2.3). Hence, we conclude that
in the case of the boundary conditions (6.1) the current densities
$j_{(s)}^{\mu}$ coincide for the fields corresponding to the representations
$s=1$ and $s=-1$. The total current density is obtained from the expressions
given in sections 3 and 4 with $s=1$ and with an additional factor 2.

From (6.2) we see that the boundary conditions for the fields
$\psi_{(+1)}^{\prime}$ and $\psi_{(-1)}^{\prime}$ differ by the sign of the
terms with the Dirac matrices. We can consider the case where those terms have
the same sign and the transformed fields $\psi_{(s)}^{\prime}$ obey the same
boundary condition $(1+i\gamma^{\mu}n_{\mu})\psi_{(s)}^{\prime}=0$. That
corresponds to the conditions $(1+si\gamma_{(s)}^{\mu}n_{\mu})\psi_{(s)}=0$
for the initial fields on the brane $w=w_{0}$. With these boundary conditions,
the current density for the field with $s=1$ remains the same as that for
(6.1) and is given by the expressions in sections 3 and 4 with $s=1$. However,
now the VEV $\langle j_{(-1)}^{\mu}\rangle$ is different. It is given by the
expressions from sections 3 and 4 with $s=-1$. In this case the fields
realizing two inequivalent irreducible representations of the Clifford algebra
give different contributions to the brane-induced part in the VEV of the total
current density.

##  7 Applications to Randall-Sundrum model

The results given above can be directly applied for the investigation of the
cosmic string induced effects in background of the Randall-Sundrum model with
a single brane (RSII model) [69, 70] (for quantum effects in higher-
dimensional generalizations of RSII model see [71] and references therein). In
the presence of cosmic string perpendicular to the brane, the corresponding
background geometry contains two copies of the R-region that are identified by
the $Z_{2}$-symmetry with respect to the brane located at $y=0$ (or at
$w=w_{0}=a$ in terms of the coordinate $w$). The corresponding line element is
expressed as

$ds^{2}=e^{-2|y|/a}\left(dt^{2}-dr^{2}-r^{2}d\phi^{2}-dz^{2}\right)-dy^{2}\ ,$
(7.1)

where, as before, $-\infty<y<+\infty$ and $0\leq\phi\leq\phi_{0}$. Note that
for an observer located on the brane $y=0$ the line element (7.1) is reduced
to the standard line element for a cosmic string in (3+1)-dimensional
Minkowski spacetime. The boundary condition for the field $\psi(x)$ on the
brane is obtained on the basis of the $Z_{2}$-symmetry (see discussion in [51,
72]).

Let $M$ be the matrix that relates the fields in the regions $-\infty<y<0$ and
$0<y<+\infty$:

$\psi(t,r,\phi,-y,z)=M\psi(t,r,\phi,y,z).$ (7.2)

We require the invariance of the fermionic field action under the $Z_{2}$
identification of the points in those region. This requirement leads to the
following conditions on the matrix $M$:

$\\{\gamma^{(0)},M\\}=0,\;\\{\gamma^{(0)}\gamma^{(3)},M\\}=0,\;[\gamma^{(0)}\gamma^{(b)},M]=0,$
(7.3)

where $b=1,2,4$. By direct substitution it can be checked that these relations
are obeyed by the matrix $M=\varepsilon\gamma^{(3)}$ with a constant
$\varepsilon$. For the latter from the relation $M^{2}=1$ we find
$\varepsilon=\pm i$ and the transformation matrix has the form

$M=\pm\left({\begin{array}[]{cc}\sigma^{3}&0\\\
0&-\sigma^{3}\end{array}}\right).$ (7.4)

Hence, we have two types of fermionic fields with the identification rule
(7.2) and with the upper and lower signs in (7.4).

On the brane at $y=0$ one gets the following boundary condition

$\left(1-M\right)\psi(x)=0,\;w=a.$ (7.5)

Now we can see that the boundary condition (7.5) with the upper sign in (7.4)
coincides with the condition (2.3) and the boundary condition (7.5) with the
lower sign coincides with the condition (5.1). Hence, for the fermionic field
corresponding to the upper sign in (7.4), the expressions for the VEV of the
current density induced by a cosmic string in the RSII model is obtained from
the formulas in section 3 taking $w_{0}=a$ and adding an additional factor
1/2. For the field with the lower sign in (7.4) the corresponding result is
obtained from the expression for the current density in section 5.1 for the
R-region (again, with a factor 1/2). The appearance of the factor 1/2 is
related to that the integration in the normalization condition for the mode
functions goes over the interval $-\infty<y<+\infty$ instead of the interval
$0\leq y<+\infty$ for the R-region in the discussion above (with $y_{0}=0$).

The current density induced on the brane by vacuum fluctuations of a bulk
fermionic field is a source of magnetic fields surrounding the cosmic string.
These fields are among the distinctive features of magnetic-flux-carrying
strings. Note that the vacuum current density will also be generated by
quantum fluctuations of fields living on the brane (for example, Standard
Model fields in braneworld scenario). However, the corresponding properties
are different from those we have discussed above (for fermionic currents in
the geometry of a cosmic string on background of (3+1)-dimensional Minkowski
spacetime see [55, 73, 74]). As a consequence, the structure of magnetic
fields around the cosmic string is sensitive to the existence of extra
dimensions.

##  8 Conclusion

We have investigated the combined effects of the gravitational filed and a
brane on the vacuum current density for a charged fermionic field around a
magnetic-flux-carrying cosmic string. In order to have an exactly solvable
problem we have considered highly symmetric bulk and boundary geometries,
namely, AdS spacetime and a planar brane perpendicular to the cosmic string.
Two types of boundary conditions have been discussed on the brane. The first
one corresponds to the MIT bag boundary condition and the second one is given
by (5.1). The current density for the field realizing the representation
$s=\pm 1$ and the boundary condition (5.1) coincides with the current density
for the representation $s=\mp 1$ and for the boundary condition (2.3) and we
have described the main steps of the evaluation procedure for the bag boundary
condition only. The VEV of the current density is expressed in the form of
mode-sum over the complete set of fermionic modes. The latter are given by
(2.8). The brane divides the space into two regions (R- and L-regions) with
different properties of the fermionic vacuum. In the R-region the function
$W_{\nu}(pw)$ in the mode functions is given by (3.1) and the spectrum of the
quantum number $p$ is continuous. In the L-region one has
$W_{\nu}(pw)=J_{\nu}(pw)$ and the eigenvalues for $p$ are determined by the
zeros of the function $J_{\nu_{1}}(pw_{0})$.

The only nonzero component of the vacuum current density corresponds to the
current along the $\phi$-direction (azimuthal current). It is decomposed into
brane-free and brane-induced parts. The brane-induced contributions in the VEV
of the azimuthal current density are given by the expressions (3.20) and (4.8)
for the R- and L-regions, respectively. For a massless field the brane-induced
contribution vanishes in the R-region. The current density is an odd periodic
function of the magnetic flux along the string axis, with the period equal the
flux quantum. It is discontinuous at half-integer values of the magnetic flux
in units of flux quantum. The corresponding limiting values are linear
functions of the parameter $q$ and are expressed as (3.23). In the special
case $q=1$ the formulas obtained provide the expressions for the current
densities induced by idealized magnetic flux tubes with zero thickness. In the
limit of large values for the curvature radius of the background spacetime, in
the leading order we have derived the expression (3.42) for the fermionic
current density induced by a planar boundary, perpendicular to cosmic string
in $(4+1)$-dimensional Minkowski spacetime.

To clarify the behavior of the azimuthal current density we have considered
various asymptotic regions of the parameters in the problem. Near the cosmic
string, for points not too close to the brane, the brane-induced contribution
is a linear function of the radial coordinate $r$ and vanishes on the string.
Near the string the brane-free part behaves as $\langle
j_{\phi}\rangle_{0}\propto 1/r^{4}$ and it dominates in the total VEV. At
large proper distances from the string, compared with the curvature radius of
the AdS spacetime, one has $r/w\gg 1$ and the brane-free VEV decays like
$\langle j_{\phi}\rangle_{0}\propto\left(w/r\right)^{2ma+4}$. The behavior of
the brane-induced contribution in the R-region at large distances from the
string is different for the fields realizing two different irreducible
representations of the Clifford algebra. For the representation with $s=1$ the
total current density at large distances is dominated by the brane-free part.
For the representation with $s=-1$ the brane-induced contribution behaves as
$\langle j_{\phi}\rangle_{\mathrm{b}}\propto(w/r)^{|2ma-1|+3}$ and it is
dominant in the total VEV. In all these cases, the decay of the current
density on AdS bulk, as a function of the proper distance from the string, is
power-law for both massless and massive fields. For massive fields this large-
distance asymptotic in the R-region is in clear contrast to that for the
Minkowski bulk where the decay is exponential. At large distances from the
string the current density in L-region is suppressed by the exponential factor
$e^{-2rp_{1}\sin(\pi/q)/w_{0}}$ for $q\geq 2$ and by the factor
$e^{-2rp_{1}/w_{0}}$ for $q<2$, with $p_{1}/w_{0}$ being the lowest eigenvalue
for the quantum number $p$.

An important difference of the vacuum current density from the VEV of the
energy-momentum tensor is its finiteness on the brane. The corresponding
limiting values for the R- and L-region are directly obtained from the
representations (3.13), (4.4) and they are given by (3.39) and (4.14). At
large distances from the brane, $y-y_{0}\gg a$, $w\gg r$, the brane-induced
current density in the R-region is suppressed by the factor
$e^{-\left(2ma+s\right)(y-y_{0})/a}$ for $ma>-s/2$. The suppression is
stronger for the representation $s=1$. For $s=-1$ and $ma<1/2$ the brane-
induced current density tends to finite value on the AdS horizon
($w\rightarrow\infty$), determined from (3.38). For a fixed location of the
brane and in the limit $w\rightarrow 0$, corresponding to points on the AdS
boundary, the brane-induced VEV behaves like $w^{2ma+5}$. The same is the case
for the brane-free contribution.

In $(4+1)$-dimensional spacetime the Lagrangian for a massive fermionic field
realizing the irreducible representation of the Clifford algebra is not
invariant under charge conjugation and parity transformation. $C$\- and
$P$-invariant fermionic models are constructed by combining the fields
$\psi_{(s)}$, $s=\pm 1$, corresponding to two inequivalent sets of Dirac
matrices $\gamma_{(s)}^{\mu}$. By the transformation of the fields, the model
with two fields $\psi_{(s)}$ is reduced to the model with the fields
$\psi_{(s)}^{\prime}$ such that the new fields obey the equation (2.2). When
the fields $\psi_{(s)}$ obey the same boundary conditions, the boundary
conditions for the fields $\psi_{(s)}^{\prime}$ differ by the sign of the term
containing the normal to the boundary. In this case the VEVs of the current
densities for separate fields $\psi_{(s)}$ coincide. In the second case the
boundary conditions are the same for the fields $\psi_{(s)}^{\prime}$ and
differ by the sign of the term containing the normal for the fields
$\psi_{(s)}$. In this case the vacuum currents differ and the corresponding
expressions are obtained from the formulas given in sections 3 and 4 with
$s=+1$ and $s=-1$.

In the Randall-Sundrum model with a single brane the boundary conditions on a
bulk fermionic field is dictated by the $Z_{2}$-symmetry. Depending on the
parity of the field, two boundary conditions are obtained at the location of
the brane. They correspond to the conditions (2.3) and (5.1) in our
discussion. Consequently, the VEV of the current density for a bulk fermionic
field, induced by magnetic-flux-carrying cosmic string RSII model, is obtained
from the results in sections 3 and 4 with an additional factor 1/2, related to
the presence of two copies of the R-region.

For the convenience of the reader, in table 1 we summarize the main formulas
for the VEV of the azimuthal current density.

Table 1: Summary of the expressions for the vacuum current density. Brane-free part of the current density | (3.16),(3.26)
---|---
Boundary-free contribution in the Minkowski bulk | (3.33)
Boundary-induced contribution in the Minkowski bulk | (3.42)
Total current density in the R-region | (3.13)
Brane-induced contribution to the current density in the R-region | (3.20)
Total current density in the L-region | (4.4)
Brane-induced contribution to the current density in the L-region | (4.8)
Brane-induced contribution to the current density in the L-region | (4.10)
for a massless field |

## Acknowledgments

This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. E.R.B.M is
partially supported by CNPq under Grant no 301.783/2019-3.

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