# New brane-like solutions in modified four-dimensional Einstein-Gauss-Bonnet
gravity

D. Bazeia  R. Menezes  A. Yu. Petrov  P. J. Porfírio 11footnotetext:
Corresponding author.

###### Abstract

We investigate solutions of a new $4D$ Einstein-Gauss-Bonnet gravity. We first
describe the bulk vacuum solution, then we add a massive probe scalar field,
and we follow considering a self-interacting scalar field which acts as a
source to support thick brane solutions in the four-dimensional Einstein-
Gauss-Bonnet scenario with a single extra dimension of infinite extent. We
illustrate our results with some distinct brane-like configurations
engendering controllable thickness.

##  1 Introduction

Fundamental discoveries of last years, such as late-time cosmic acceleration
[1, 2], detection of gravitational waves (see f.e. [3]), and obtaining the
image of the shadow of a black hole (see f.e. [4]), called strongest attention
to studies of gravity. The known difficulties faced by general relativity (GR)
(namely, its non-renormalizability, at the quantum level, and the
impossibility of explaining the accelerated expansion of the Universe and
formation of large-scale structures in the Universe, without postulating the
existence of an amount of unseen matter and energy) clearly elevate the
importance of search for its consistent extension. Several modified gravity
models have been considered in this context, those ones involving modification
of geometric sector only (f.e. $f(R)$ gravity, see [5] for a review) or those
ones including extra fields treated as ingredients of a complete gravity
description rather than as matter fields (f.e. scalar-tensor or vector-tensor
gravities). An excellent review on applying modified gravity models within the
cosmological context is presented in [6, 7, 8], and a wide list of classes of
extended gravity models is presented in [9].

Among various modifications of gravity, one of the very interesting examples
is given by the four-dimensional Einstein-Gauss-Bonnet $(EGB)$ gravity [10].
This theory possesses the following advantage – it allows to involve only
equations of motion of second order, thus avoiding the presence of ghosts, and
introducing, at the same time, a richer structure in the action. Actually,
this theory involves the additive Gauss-Bonnet (GB) term with a special
rescaling of its coupling constant, $\alpha\rightarrow{\alpha}/{({\cal
D}-4)}$, and, simultaneously, taking the limit $\mathcal{D}\rightarrow 4$,
with ${\cal D}$ being the space-time dimension; this procedure converts the
topological term to a physical one and effectively introduce some kind of
dimensional reduction similar to what happens in quantum field theory (cf.
[11]). In fact, the origin of the resultant non-trivial GB term in the field
equations lies, in part, motivated by quantum corrections to the energy-
momentum tensor [12, 13]. However, it should be mentioned some formal
inconsistencies in taking the limit $\mathcal{D}\rightarrow 4$ at the level of
field equations, since the regularization process is in contradiction to the
Lovelock theorem (see [14, 15, 16, 17, 18] for more details). Nevertheless,
this formal issue was circumvented by carrying out the dimensional reduction
procedure at the level of action (see [19, 20, 21]), then rendering it a non-
pathological theory. Actually, following this approach, it can be seen in a
more transparent way that $EGB$, upon Kaluza-Klein dimensional reduction down
to four dimensions, reduces to a scalar-tensor theory of gravity [22, 23, 24,
25, 26]; thereby possessing a scalar field degree of freedom in addition to
the gravitational ones. A variety of exact solutions has been obtained within
the framework of four-dimensional $EGB$ [10, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40].

One of the natural tasks here is to find other solutions, and this will be the
main concern of the present work. In particular, we will also focus on the
construction of thick brane solutions in this new environment. Our motivation
here is that the presence of thin and thick brane-like configurations [41, 42]
in the four-dimensional $AdS$ bulk with the inclusion of the GB contribution
may also bring novelties concerning old results on black holes [43, 44, 45],
an issue that may lead us to ask new questions about strong gravity on thick
branes. Numerous studies addressing brane-like solutions have been carried out
in the context of other modified theories of gravity. For example, thin and
thick branes solutions in higher-dimensional $f(R)$ gravity have been found in
[46, 47, 50, 49, 48] and in references therein; see also [51, 52, 53, 54] for
issues within the context of supergravity. Another intrinsic motivation is
rooted in the realm of quantum gravity, specifically within the framework of
string theory, where brane-like solutions naturally emerge, highlighting the
crucial role played by $p$-branes in this context (see [55, 56] for a detailed
review). In particular, it has been shown in [57] that a broader class of
$p$-branes interpolates between $AdS_{p+2}\times S^{\mathcal{D}-(p+2)}$, where
$S^{n}$ is a $n$-sphere embedded in a $\mathcal{D}$-dimensional spacetime, in
the near-horizon geometry limit and a $\mathcal{D}$-dimensional Minkowski
space in the asymptotic limit (at spatial infinity). Due to the geometrical
features of such branes in the near-horizon limit, a holographic description
was established, providing a powerful theoretical framework for the study of
strongly coupled gauge theories (see f.e. the original references [58, 59, 60]
). On the other hand, given the mathematical difficulties associated with
higher-dimensional gravity theories [61], lower-dimensional theories are
frequently employed to investigate fundamental issues in quantum gravity [61]
or to mimic certain four-dimensional setups in laboratory settings [62, 63].
In braneworld scenarios, akin to the Dvali-Gabadadze-Porrati (DGP) approach
[64], $(2+1)$-dimensional induced gravity on a $2$-brane embedded in a flat
four-dimensional bulk has been explored in the literature [65, 66, 67]. Their
holographic properties have also been explored within the context of AdS3/CFT2
duality [68, 69]; as well as in lower dimensions through the AdS2/CFT1
correspondence [70].

In view of the above considerations, we organize the work as follows: in
Section 2, we write down the action of our theory and the modified Einstein
equations. After, in Section 3 we describe the main results. There we first
deal with the vacuum solution and then find the results for the massive probe
scalar field in this gravitational background. We also investigate and solve
the modified Einstein equation for the scalar field acting as a source field
to generate thick brane solutions in the bulk $AdS_{4}$, in a way similar to
the case of thick brane in the $AdS_{5}$ geometry with a single extra spatial
dimension of infinite extent. We end the investigation in Section 4, where we
summarize our results and add some new possibilities of continuation of the
work.

##  2 Four-dimensional Einstein-Gauss-Bonnet gravity

Recently, a new four-dimensional $EGB$ gravity was proposed [10]. Its action
is initially defined in a $\mathcal{D}$-dimensional space, and then one takes
the limit $\mathcal{D}\to 4$. Thus, $EGB$ action looks like

$\displaystyle S=\int
d^{\mathcal{D}}x\sqrt{-g}\left(\frac{1}{2\kappa^{2}}R-\frac{\alpha}{{\cal
D}-4}{\cal G}+{\cal L}_{m}\right),$ (2.1)

where $\kappa^{2}=8\pi G$ is related to Newton’s constant,
$\mathcal{G}=\left(R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\right)$
is the GB scalar invariant, and the other geometrical entities are: the Ricci
scalar $\left(R=g^{\mu\nu}R_{\mu\nu}\right)$, the Ricci tensor
$\left(R_{\mu\nu}\right)$ and the Riemann tensor
$\left(R^{\mu}_{\,\,\,\nu\alpha\beta}\right)$. The matter sources are
minimally coupled with the metric through the Lagrangian, $\mathcal{L}_{m}$.
Conversely, to the usual $EGB$ gravity, this novel $4D$ version of the theory
is achieved by rescaling the GB coupling constant to
$\dfrac{\alpha}{\mathcal{D}-4}$, where $\alpha$ is a dimensionless constant.
It has been shown in [10] that, by taking the limit $\mathcal{D}\rightarrow
4$, the theory provides non-trivial contributions to the gravitational field
equations stemming from the GB action, as we shall see in what follows.

By varying the action (2.1) with respect to the metric, we obtain the
following equations of motion

$\displaystyle\frac{1}{\kappa^{2}}G^{\mu}_{\nu}+\frac{2\alpha}{{\cal
D}-4}\mathcal{H}^{\mu}_{\nu}=T^{\mu}_{\nu},$ (2.2)

where $G^{\mu}_{\nu}$ is the usual Einstein tensor,

$\mathcal{H}^{\mu}_{\nu}=\left(-2R^{\mu\alpha}_{\phantom{\mu\alpha}\rho\sigma}R^{\rho\sigma}_{\phantom{\rho\sigma}\nu\alpha}+4R^{\mu\alpha}_{\phantom{\mu\alpha}\nu\beta}R^{\beta}_{\alpha}+4R^{\mu}_{\alpha}R^{\alpha}_{\nu}-2RR^{\mu}_{\nu}+\frac{1}{2}{\cal
G}\delta^{\mu}_{\nu}\right)$ (2.3)

is the contribution stemming from the Gauss-Bonnet term and

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta
g^{\mu\nu}},$ (2.4)

is the stress-energy tensor of the matter sources.

Taking the trace of $\mathcal{H}^{\mu}_{\nu}$, one finds the general relation

$\mathcal{H}^{\mu}_{\mu}\equiv\mathcal{H}=\frac{\mathcal{D}-4}{2}\mathcal{G},$
(2.5)

which, due to the constant factor $(\mathcal{D}-4)$ accompanying $\mathcal{G}$
in the above equation, provides a finite contribution to the trace equation of
(2.2). However, the question as to whether this property holds or not for the
tensorial equation (2.2) is a key point. It has been addressed in several
works, see f.e. [13, 14] and the original one itself [10]. To shed more light
on this, we call attention to the fact that $\mathcal{H}_{\mu\nu}$ can be
split into two pieces [14], namely,

$\mathcal{H}_{\mu\nu}=2\left(\mathcal{L}_{\mu\nu}+\mathcal{Z}_{\mu\nu}\right),$
(2.6)

where

$\displaystyle\mathcal{L}_{\mu\nu}=C_{\mu\alpha\beta\gamma}C_{\nu}^{\phantom{a}\alpha\beta\gamma}-\frac{1}{4}g_{\mu\nu}C_{\alpha\beta\gamma\sigma}C^{\alpha\beta\gamma\sigma},$
(2.7)

with $C_{\mu\alpha\beta\gamma}$ being the Weyl tensor and

$\displaystyle\mathcal{Z}_{\mu\nu}=\frac{\left(\mathcal{D}-4\right)\left(\mathcal{D}-3\right)}{\left(\mathcal{D}-1\right)\left(\mathcal{D}-2\right)}\bigg{[}$
$\displaystyle-$
$\displaystyle\frac{2\left(\mathcal{D}-1\right)}{\left(\mathcal{D}-1\right)}C_{\mu\rho\nu\sigma}R^{\rho\sigma}-\frac{2\left(\mathcal{D}-1\right)}{\left(\mathcal{D}-1\right)}R_{\mu\lambda}R^{\lambda}_{\nu}+\frac{\mathcal{D}}{\left(\mathcal{D}-2\right)}R_{\mu\nu}R$
(2.8) $\displaystyle+$
$\displaystyle\frac{1}{\left(\mathcal{D}-2\right)}g_{\mu\nu}\left((\mathcal{D}-1)R_{\alpha\beta}R^{\alpha\beta}-\frac{\left(\mathcal{D}+2\right)}{4}R^{2}\right)\bigg{]}.$

It is straightforward to see that $\mathcal{Z}_{\mu\nu}$ is well-defined when
taking the limit $\mathcal{D}\to 4$, while $\mathcal{L}_{\mu\nu}$ is not, as
discussed in [13]. In fact, for $\mathcal{D}\leq 4$, $\mathcal{L}_{\mu\nu}$
vanishes completely, therefore it cannot be written in the form
$\mathcal{L}_{\mu\nu}=\left(\mathcal{D}-4\right)\mathcal{Y}_{\mu\nu}$, for a
generic metric. Yet, for particular classes of metrics, such as
$\mathcal{D}$-dimensional Friedmann-Robertson-Walker,
$\mathcal{D}$-dimensional maximally symmetric and spherically symmetric
spacetimes, the limit $\mathcal{D}\to 4$ is well-behaved. Some other
subtleties have also been discussed in the literature, see e.g. [14], e.g., a
well-defined perturbative scheme for the field equations, which points out
that Glavan’s procedure is not well-defined [10]. Fortunately, there are
covariant alternative regularizations at the action level to circumvent this
problem, namely: conformal regularization [23] and regularization via Kaluza-
Klein reduction [22, 24, 26]. After performing the regularization, the
remaining action is such that it belongs to a particular Horndeski class of
scalar-tensor theories [71, 72]. In the following, we obtain the field
equations using Glavan’s method [10]; however, we also present a completely
well-defined version of $4D$ $EGB$, which is based on a Kaluza-Klein
regularization at the action level in Appendix A.

As a first attempt, we can consider the toy model of the brane-like metric
where one of the axes (say $w$) plays the role of the extra dimension. Then,
the metric ansatz reads

$\displaystyle ds^{2}=e^{2A(w)}\eta_{ab}dx^{a}dx^{b}-dw^{2},$ (2.9)

where $\eta_{ab}$ is the $(\mathcal{D}-1)$-dimensional Minkowski metric.
Recalling that we pick the following convention: small Greek letters to label
bulk coordinates $(\mu,\nu=0,1...D)$, with $D=\mathcal{D}-1$, and small Latin
letters to label “parallel” coordinates to the brane $(a,b=0...D-1)$. For the
metric (2.9), the relevant non-trivial geometrical quantities are

$\displaystyle\Gamma^{a}_{bD}$ $\displaystyle=$ $\displaystyle
A^{\prime}\delta^{a}_{b};\quad\,\Gamma^{D}_{bc}=\eta_{bc}e^{2A}A^{\prime};$
$\displaystyle R^{a}_{\phantom{a}bcd}$ $\displaystyle=$
$\displaystyle(\delta^{a}_{c}\eta_{bd}-\delta^{a}_{d}\eta_{bc})(A^{\prime})^{2}e^{2A};\quad\,R^{D}_{\phantom{D}bDd}=\eta_{bd}e^{2A}(A^{\prime\prime}+(A^{\prime})^{2});$
$\displaystyle R_{bd}$ $\displaystyle=$ $\displaystyle\eta_{bd}e^{2A}[({\cal
D}-1)(A^{\prime})^{2}+A^{\prime\prime}];\quad\,R_{DD}=-({\cal
D}-1)[A^{\prime\prime}+(A^{\prime})^{2}];$ $\displaystyle R$ $\displaystyle=$
$\displaystyle({\cal D}-1)[2A^{\prime\prime}+{\cal D}(A^{\prime})^{2}].$
(2.10)

Here, the prime stands for derivative with respect to $w$. These expressions
will be used to construct our equations. Notice that the components of the
Einstein tensor can be decomposed into two pieces, namely,

$\displaystyle G^{a}_{b}=(2-{\cal D})\delta^{a}_{b}[(\frac{{\cal
D}-1}{2})(A^{\prime})^{2}+A^{\prime\prime}];\quad\,G^{D}_{D}=({\cal
D}-1)(\frac{2-{\cal D}}{2})(A^{\prime})^{2}.$ (2.11)

Further,

$\displaystyle R_{a\nu}R^{\nu c}$ $\displaystyle=$
$\displaystyle\delta_{a}^{c}[({\cal
D}-1)(A^{\prime})^{2}+A^{\prime\prime}]^{2},\quad\,R_{D\nu}R^{\nu D}=({\cal
D}-1)^{2}(A^{\prime\prime}+(A^{\prime})^{2})^{2};$ $\displaystyle
R^{a\alpha}_{\phantom{a\alpha}b\beta}R^{\beta}_{\alpha}$ $\displaystyle=$
$\displaystyle\delta^{a}_{b}[D^{2}(A^{\prime})^{4}+(3D-1)A^{\prime\prime}(A^{\prime})^{2}+D(A^{\prime\prime})^{2}],$
$\displaystyle R^{D\alpha}_{\phantom{D\alpha}D\beta}R^{\beta}_{\alpha}$
$\displaystyle=$ $\displaystyle
D^{2}(A^{\prime})^{4}+D(D+1)A^{\prime\prime}(A^{\prime})^{2}+D(A^{\prime\prime})^{2};$
$\displaystyle R^{a\mu\nu\lambda}R_{b\mu\nu\lambda}$ $\displaystyle=$
$\displaystyle\delta^{a}_{b}[2D(A^{\prime})^{4}+2(A^{\prime\prime})^{2}+4A^{\prime\prime}(A^{\prime})^{2}];$
$\displaystyle R^{D\mu\nu\lambda}R_{D\mu\nu\lambda}$ $\displaystyle=$
$\displaystyle
2D[(A^{\prime})^{4}+2A^{\prime\prime}(A^{\prime})^{2}+(A^{\prime\prime})^{2}];$
$\displaystyle\mathcal{G}$ $\displaystyle=$ $\displaystyle
D(D-1)(D-2)[(D+1)(A^{\prime})^{4}+4A^{\prime\prime}(A^{\prime})^{2}].$ (2.12)

Substituting Eqs.(2.11) and (2.12) into Eq.(2.2) and, after that, taking the
limit $\mathcal{D}\rightarrow 4$, we find

$\begin{split}\kappa^{2}T^{a}_{b}&=\left[-3(A^{\prime})^{2}-2A^{\prime\prime}+2\alpha\kappa^{2}\left(3(A^{\prime})^{4}+4A^{\prime\prime}(A^{\prime})^{2}\right)\right]\delta^{a}_{b};\\\
\kappa^{2}T^{3}_{3}&=-3(A^{\prime})^{2}+6\alpha\kappa^{2}(A^{\prime})^{4},\end{split}$
(2.13)

where now $a,b=0...2$ and $D=3$. The same field equations are obtained via
regularized Kaluza-Klein reduction, as we explicitly show in Appendix A.

##  3 Results

Let us now consider some specific situations, where we can investigate and
find results of current interest. We shall first deal with the bulk vacuum
solution, then include a massive probe scalar field, and then make the scalar
field to support self-interaction, taking it as a source to study its ability
to support brane-like configurations with a single extra dimension of infinite
extent.

###  3.1 The bulk vacuum solution

We now consider the absence of source fields. Different from Einstein’s
theory, $4D$ $EGB$ gravity supports a non-trivial solution in the presence of
a Randall-Sundrum-like thin brane in $(3+1)$-dimensional bulk [41, 42]. In
this scenario, $T^{3}_{3}=0$ and $T^{a}_{b}=\delta(w)\delta^{a}_{b}$, where
the brane is placed at $w=0$ (we have taken the brane tension to be equal to
one). In this way, by solving the field equations (2.13), one obtains the
following solution

$A(w)=-\dfrac{1}{\sqrt{2\alpha}\kappa}|w|,$ (3.1)

which leads to the bulk metric

$ds^{2}=e^{-\sqrt{\frac{2}{\alpha}}\frac{|w|}{\kappa}}\eta_{ab}dx^{a}dx^{b}-dw^{2},$
(3.2)

where $\alpha$ should be a positive constant.

Note that this solution describes an $AdS_{4}$ spacetime. The particular $|w|$
term reflects the $\mathrm{Z}_{2}$-symmetry around the brane. In fact, in
order to see that in more detail, let us perform the following coordinate
transformation:
$e^{-\sqrt{\frac{1}{2\alpha}}\frac{|w|}{\kappa}}=\dfrac{r_{{}_{AdS}}}{z}$,
where $0<z<+\infty$ and $r_{{}_{AdS}}^{2}=-\dfrac{3}{\Lambda}$, with
$\Lambda<0$ being the $AdS_{4}$ radius. The conformal boundary of $AdS$ is by
definition located at $z=0$, where the warped factor blows up. Then, the bulk
metric can be written in the upper half plane representation of $AdS_{4}$ [73]
by using the new coordinate $z$, namely,

$ds^{2}=\frac{r^{2}_{AdS}}{z^{2}}\left(\eta_{ab}dx^{a}dx^{b}-dz^{2}\right),$
(3.3)

in which the effective cosmological constant is defined by
$\Lambda=-\dfrac{3}{2\kappa^{2}\alpha}<0$. Therefore, the vacuum solution
(3.3) emerges as a result of the non-trivial contribution of the GB term for
the field equations in four dimensions, so that the physical effect appears as
an effective negative cosmological constant which, in turn, is sourced by the
coupling constant $\alpha$. For the sake of convenience, let us proceed with a
further coordinate transformation: $u=\dfrac{r_{AdS}^{2}}{z}$. By doing so,
the $AdS$ line element takes the form

$ds^{2}=\frac{u^{2}}{r_{AdS}^{2}}\eta_{ab}dx^{a}dx^{b}-\frac{r_{AdS}^{2}}{u^{2}}d\lambda^{2},$
(3.4)

where the conformal boundary is now placed at $u=+\infty$.

###  3.2 Massive probe scalar field in Gaussian null coordinates

We here investigate a massive probe scalar field in the background (3.3). The
scalar field action is given by

$S[\Phi]=-\frac{1}{2}\int
d^{4}x\sqrt{-g}\bigg{(}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-m^{2}\Phi^{2}\bigg{)}.$
(3.5)

As it is well known $AdS_{4}$ spaces are not compact manifolds, in other
words, they possess infinite volume. Hence, a priori, the above action can
diverge. To circumvent this problem, as usual, one temporally shifts the
boundary from $z=0$ to $z=\epsilon$, where $\epsilon$ is the cutoff parameter
to regularize the $AdS_{4}$ volume. By varying the action with respect to
$\Phi$, one gets

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int
d^{3}x\int_{\epsilon}^{\infty}dz\left\\{\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\delta\Phi\partial_{\nu}\Phi\right)-\left[\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)+m^{2}\Phi\right]\sqrt{-g}\delta\Phi\right\\}$
(3.6) $\displaystyle=$
$\displaystyle\\!-\\!\int\\!d^{3}x\\!\int_{\epsilon}^{\infty}\\!\\!\\!dz\\!\left[\frac{1}{\sqrt{-g}}\partial_{\mu}\\!\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)\\!+\\!m^{2}\Phi\right]\\!\sqrt{-g}\delta\Phi+$
$\displaystyle+$ $\displaystyle\int
d^{3}x\sqrt{-h}\,n^{\mu}\partial_{\mu}\Phi\delta\Phi\bigg{|}_{z=\epsilon}^{z=\infty},$

where we have defined: $h$ is the determinant of the induced metric,
$h_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}$, defined at the brane, and $n^{\mu}$ is
the unit normal vector to the brane. Note that the first integral in the
former equation is just the l.h.s. of the Klein-Gordon equation; consequently,
it vanishes on-shell, and the second term is a boundary one, which must vanish
under certain special boundary conditions in order to ensure a well-defined
variation principle. Therefore, upon fixing the boundary conditions, one just
needs to solve the Klein-Gordon equation in the bulk.

It should be noted that the metric (3.4) in Poincare coordinates is singular
at the horizon ($u=0$), so it is more instructive to define a coordinate
system where the metric is completely non-singular at the horizon. The
suitable set of coordinates for this task is the so-called Gaussian null
coordinates [74, 75], where such new coordinates in the neighborhood of the
horizon are, explicitly, $(v,r,y^{A})$, with $v$ is the incoming null
coordinate, $y^{A}$, with $A=1,2$, parameterizes the $2$-dimensional spatial
section of the whole space, $r$ is the radial coordinate and, in particular,
$0\leq y^{(1)}=\eta<+\infty$. The Gaussian null coordinates are related to the
Poincare ones by the following relation:

$u=r\cosh{\eta},\,\,\,\,t=\left(v+\frac{1}{r}\right)r_{AdS}^{2},\,\,\,\,x^{A}=\left(\frac{\mu^{A}\tanh{\eta}}{r}\right)r_{AdS}^{2},$
(3.7)

where we have defined $\mu^{A}$ as a quantity to parameterize the
$2$-dimensional section of the full space and must fulfill the requirement
$\mu^{A}\mu^{A}=1$.

The metric in Gaussian null coordinates looks like

$ds^{2}_{N}=r_{AdS}^{2}\left[\cosh{\eta}\left(r^{2}dv^{2}-2dvdr\right)-d\eta^{2}-\sinh^{2}{\eta}\,d\theta^{2}\right],$
(3.8)

where $\theta$ is the angular coordinate. The horizon is placed at $r=0$, then
the metric is regular at the horizon.

The $AdS_{4}$ metric, as expressed in terms of Gaussian null coordinates,
Eq.(3.8), can be viewed as a warped product of $AdS_{2}$ with a
$2$-dimensional hyperbolic space, $\mathcal{H}^{2}$. In these coordinates, the
boundary is located at $\eta=+\infty$. From the another perspective, using the
conformal symmetry of $AdS$ spaces, by making the coordinate transformation
$\tan{\beta}=\sinh{\eta}$, the boundary can be brought to a finite point, i.e,
$\beta=\pi/2$. Explicitly, we have

$ds^{2}_{N}=\frac{r_{AdS}^{2}}{\cos{\beta}^{2}}\left[r^{2}dv^{2}-2dvdr-d\beta^{2}-\sin^{2}{\beta}\,d\theta^{2}\right],$
(3.9)

where $0\leq\beta<\pi/2$. In this description, the $AdS_{4}$ space can be seen
as a warped geometry of $AdS_{2}\times S^{2}$, whose topology of the conformal
boundary (located at $\beta=\pi/2$) is simply $AdS_{2}\times S^{1}$, which is
locally equivalent (isometric) to $\mathbf{R}^{3}$. Fig. 1 displays the
$AdS_{4}$ topology. Note that the vertical axis, located at $\beta=0$,
describes the section of the full space corresponding to $AdS_{2}$, whilst the
bulk of the “cylinder” represents $AdS_{4}$.

Figure 1: This plot schematically displays $AdS_{4}$ topology. $S^{1}$
represents circles of constant radii. The dashed vertical axis placed at
$\beta=\pi/2$ stands for the boundary localization. At $\beta=0$, we have the
vertical axis which represents $AdS_{2}$.

Having obtained a non-singular description of the $AdS_{4}$ metric, we now
solve the Klein-Gordon equation on this background (3.9), which is explicitly
given by

$\left(\Box+m^{2}\right)\Phi=\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Phi\right)+m^{2}\Phi=0,$
(3.10)

with $\sqrt{-g}=\sin{\beta}\left(\dfrac{r_{AdS}^{2}}{\cos{\beta}}\right)^{4}$
and

$g_{\mu\nu}=\left(\begin{array}[]{cccc}\dfrac{r_{AdS}^{2}}{\cos^{2}{\beta}}r^{2}&-\dfrac{r_{AdS}^{2}}{\cos^{2}{\beta}}&0&0\\\
-\dfrac{r_{AdS}^{2}}{\cos^{2}{\beta}}&0&0&0\\\
0&0&-\dfrac{r_{AdS}^{2}}{\cos^{2}{\beta}}&0\\\
0&0&0&-r_{AdS}^{2}\tan^{2}{\beta}\end{array}\right),$ (3.11)

$g^{\mu\nu}=\left(\begin{array}[]{cccc}0&-\dfrac{\cos^{2}{\beta}}{r_{AdS}^{2}}&0&0\\\
-\dfrac{\cos^{2}{\beta}}{r_{AdS}^{2}}&-\dfrac{\cos^{2}{\beta}}{r_{AdS}^{2}}r^{2}&0&0\\\
0&0&-\dfrac{\cos^{2}{\beta}}{r_{AdS}^{2}}&0\\\
0&0&0&-\dfrac{\cot^{2}{\beta}}{r_{AdS}^{2}}\end{array}\right),$ (3.12)

where $g^{\mu\nu}$ is the inverse metric of $g_{\mu\nu}$. Plugging Eqs.(3.11)
and (3.12) into Eq.(3.10) and upon some algebraic manipulations, one finds

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-2\frac{\sin{\beta}}{\cos^{2}{\beta}}\partial_{v}\partial_{r}\Phi-\frac{\sin{\beta}}{\cos^{2}{\beta}}\partial_{r}\left(r^{2}\partial_{r}\Phi\right)-\partial_{\beta}\left(\frac{\sin{\beta}}{\cos^{2}\beta}\partial_{\beta}\Phi\right)-\frac{1}{\sin{\beta}\cos^{2}{\beta}}\partial^{2}_{\theta}\Phi+$
(3.13) $\displaystyle+$ $\displaystyle
m^{2}\frac{\sin{\beta}}{\cos^{4}{\beta}}\Phi.$

In order to solve it, we decompose the scalar field in modes as shown in the
ansatz below:

$\Phi=\sum\limits_{l,m^{\prime}}\Psi_{m^{\prime}}(v,r)Y_{l}(\theta)\phi_{l,m^{\prime}}(\beta),$
(3.14)

where $l,m^{\prime}$ are quantum numbers featuring the general solution and
the spherical harmonic function $Y_{l}$ satisfies

$\Box_{S^{1}}Y_{l}(\theta)=-l^{2}Y_{l}(\theta)$ (3.15)

where $\Box_{S^{1}}=\partial^{2}_{\theta}$. By substituting the expansion in
modes (3.14), Eq.(3.13) can be set into the form

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{\Psi(v,r)}\left[2\partial_{v}\partial_{r}\Psi(v,r)+\partial_{r}\left(r^{2}\partial_{r}\Psi(v,r)\right)\right]+\frac{\cos^{2}\beta}{\sin\beta}\frac{1}{\phi(\beta)}\partial_{\beta}\left(\frac{\sin\beta}{\cos^{2}\beta}\partial_{\beta}\phi({\beta})\right)-$
(3.16) $\displaystyle-$
$\displaystyle\frac{l^{2}}{\sin^{2}\beta}-\frac{m^{2}}{\cos^{2}\beta}.$

The labels $l$ and $m^{\prime}$ were omitted above for simplicity. It is
noteworthy that Eq.(3.16) might be conveniently rewritten as two differential
equations by introducing a separation constant ($\xi^{2})$, namely,

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\partial_{v}\partial_{r}\Psi(v,r)+\partial_{r}\left(r^{2}\partial_{r}\Psi(v,r)\right)-\xi^{2}\Psi(v,r);$
(3.17) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{\beta}\left(\frac{\sin\beta}{\cos^{2}\beta}\partial_{\beta}\phi(\beta)\right)-\left(\frac{l^{2}}{\sin\beta\cos^{2}\beta}+\frac{m^{2}\sin\beta}{\cos^{4}\beta}-\xi^{2}\frac{\sin\beta}{\cos^{2}\beta}\right)\phi(\beta).$
(3.18)

Eq.(3.17) is the partial differential equation (PDE) related to the
$AdS_{2}$-section of the whole space ($AdS_{4}$) whilst Eq.(3.18) is the
ordinary differential equation concerned with the radial function,
$\phi(\beta)$. We shall deal only with the latter since the former has already
been discussed in the literature [76, 77].

Our goal now is to provide an analytical solution for the radial equation
(3.18). To begin with, it is convenient to introduce the new coordinate
$y=\sin^{2}\beta$ in order to use the Sturm-Liouville theory [78]. By doing
so, the radial equation looks like

$\displaystyle L[\phi(y)]=\omega^{2}r(y)\phi(y),$ (3.19)

where $L$ is the Sturm-Liouville operator. More explicitly,

$\displaystyle L[\phi(y)]$ $\displaystyle=$
$\displaystyle-\partial_{y}\left(p(y)\partial_{y}\phi(y)\right)+q(y)\phi(y);$
(3.20) $\displaystyle p(y)$ $\displaystyle=$
$\displaystyle\frac{y}{(1-y)^{1/2}};$ (3.21) $\displaystyle q(y)$
$\displaystyle=$
$\displaystyle\frac{m^{2}}{4(1-y)^{5/2}}+\frac{l^{2}}{4}\frac{1}{y(1-y)^{3/2}};$
(3.22) $\displaystyle r(y)$ $\displaystyle=$
$\displaystyle\frac{1}{(1-y)^{3/2}};$ (3.23) $\displaystyle\omega^{2}$
$\displaystyle=$ $\displaystyle\frac{\xi^{2}}{4}.$ (3.24)

Note that $y=0$ ($\beta=0$) and $y=1$ ($\beta=\frac{\pi}{2}$) are singular
points. Furthermore, the function $q(y)$ should be interpreted as an effective
potential; so that now the “mass” term contains curvature contributions (the
second term in the r.h.s of Eq.(3.22)) arising from the dimensional reduction
procedure, apart from the usual squared mass $m^{2}$. The following general
boundary conditions should be fulfilled

$\displaystyle\alpha_{1}\phi(0)+\alpha_{2}\phi^{\prime}(0)=0,\,\,\,\,\gamma_{1}\phi(1)+\gamma_{2}\phi^{\prime}(1)=0,$
(3.25)

with $\alpha$’s and $\gamma$’s arbitrary constants, and the prime stands for
derivative with respect to $y$. Since we are interested in well-defined
solutions at the singular points it makes necessary imposing regularity
conditions. This is pursued by investigating the behavior of the solutions
around each of two singular points ($y=0$, origin, and $y=1$, boundary).

Proceeding with the redefinition

$\phi(y)=(1-y)^{\lambda}y^{\sigma}\varphi(y),$ (3.26)

to facilitate the algebraic manipulations, Eq.(3.19) reduces to the
hypergeometric differential equation [79]

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
y(1-y)\varphi^{\prime\prime}+\left(\left(-2\sigma-2\lambda-\frac{1}{2}\right)y+2\sigma+1\right)\varphi^{\prime}+$
(3.27) $\displaystyle+$
$\displaystyle\left(-\left(\sigma+\lambda\right)^{2}+\frac{1}{2}(\lambda+\sigma)+\frac{\xi^{2}}{4}\right)\varphi,$

where the coefficients must be identified as

$\displaystyle 4\sigma^{2}$ $\displaystyle=$ $\displaystyle l^{2};$ (3.28)
$\displaystyle 4\lambda^{2}-6\lambda$ $\displaystyle=$ $\displaystyle m^{2},$
(3.29)

whose solutions are

$\displaystyle\sigma_{\pm}$ $\displaystyle=$ $\displaystyle\pm\frac{l}{2};$
(3.30) $\displaystyle\lambda_{\pm}$ $\displaystyle=$
$\displaystyle\frac{3}{4}\pm\frac{1}{4}\sqrt{9+4m^{2}}.$ (3.31)

Recalling that the mass should satisfy the Breitenlohner-Freedman (BF) bound
[80]

$m^{2}\geq-\frac{9}{4},$ (3.32)

to guarantee real solutions. Moreover, there are two linear independent
hypergeometric solutions for Eq.(3.27) depending on the parameter $\sigma$ and
$\lambda$. It is straightforward to check the leading behavior of the solution
near the single points. Around the origin ($y=0$), the function $\phi(y)$
falls off as $y^{\sigma_{\pm}}$ while, near the boundary ($y=1$), $\phi(y)$
falls off as $(1-y)^{\lambda_{\pm}}$. To shed more light on this, we fix
$\lambda=\lambda_{+}$ without loss of generality since the asymptotic behavior
at the origin depends only on $\sigma_{\pm}$. In this case, using the general
formulas of [81] and restoring the coordinate $\beta$, we are able to find the
full analytical solutions of Eq. (3.19) near the origin $\beta=0$, namely,

$\displaystyle\phi_{1}(\beta)$ $\displaystyle=$
$\displaystyle\sin^{l}(\beta)\cos^{2\lambda_{+}}(\beta)\,{}_{2}F_{1}\left(a_{+},b_{+};l+1;\sin^{2}(\beta)\right);$
(3.33) $\displaystyle\phi_{2}(\beta)$ $\displaystyle=$
$\displaystyle\phi_{1}(\beta)\ln{\left(\sin^{2}(\beta)\right)}-\bigg{[}\sum_{k=1}^{l}\frac{l!(k-1)!}{(l-k)!(1-a_{+})_{k}(1-b_{+})_{k}}\left(-\sin^{2k}(\beta)\right)+$
(3.34) $\displaystyle+$
$\displaystyle\sum_{k=0}^{\infty}\frac{(a_{+})_{k}(b_{+})_{k}}{(l+1)_{k}k!}g_{k}\sin^{2k}(\beta)\bigg{]},$

where

$\begin{split}a_{+}&=\lambda_{+}-\frac{1}{4}\left(1-\sqrt{1+4\xi^{2}}\right)+\frac{l}{2};\\\
b_{+}&=\lambda_{+}-\frac{1}{4}\left(1+\sqrt{1+4\xi^{2}}\right)+\frac{l}{2},\end{split}$
(3.35)

and the other quantities are defined by

$\begin{split}\psi(x)&=\frac{\Gamma^{\prime}(x)}{\Gamma(x)};\\\
g_{k}&=\psi(a_{+}+k)+\psi(b_{+}+k)-\psi(1+k)-\psi(l+1+k);\\\
(a_{+})_{k}&=\frac{\Gamma(a_{+}+k)}{\Gamma(k)},\end{split}$ (3.36)

similar to [81].

As a result of the AdS/CFT property [73, 82], the boundary action must vanish
by imposing regularity conditions at the origin $y=0$. In other words,

$S_{boundary}=\int
d^{3}x\sqrt{-h}\,g^{\beta\beta}\Phi\partial_{\beta}\Phi\big{|}_{\beta=0}=0,$
(3.37)

which just holds for the solution $\phi_{1}(\beta)$ while for
$\phi_{2}(\beta)$ does not. Hence, $\phi_{2}(\beta)$ is an unacceptable
solution.

We now turn our attention to the solutions near the boundary
$y=1\,(\beta=\pi/2)$. In this case, it is convenient to rewrite the radial
equation in terms of the coordinate $y^{{}^{\prime}}=\cos^{2}\beta$ or
$y^{{}^{\prime}}=1-y$. Note however that, by using the linear properties of
hypergeometric functions [79], one recovers a similar equation to Eq.(3.19)
with the difference that $y$ should be replaced by $(1-y)$. In this scenario,
the general solution looks like

$\phi(\beta)=A\phi_{3}(\beta)+B\phi_{4}(\beta),$ (3.38)

where $A$ and $B$ are arbitrary constants and $\phi_{3}(\beta)$ and
$\phi_{4}(\beta)$ are linear independent solutions near the boundary. As it
was discussed before, the asymptotic behavior of $\phi(\beta)$ falls off as
$\cos^{2\lambda_{\pm}}(\beta)$ near the boundary, thereby we can pick
$\sigma=\sigma_{+}$ without loss of generality. Moreover, it is worth
mentioning that the explicit solutions will depend on the difference between
$\lambda_{+}$ and $\lambda_{-}$, and here we define it by
$\nu=(\lambda_{+}-\lambda_{-})$; thus, by using Eq. (3.31), it is easy to see
that

$\nu=\frac{1}{2}\sqrt{9+4m^{2}}.$ (3.39)

As a result of the BF bound (3.32), we have $\nu\geq 0$, which ensures stable
solutions. In this way, the general solution can be split into three different
cases depending on $\nu$:

  1. 1.

The $\nu$ is a non-integer number. In this case,

$\displaystyle\phi(\beta)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(l+1)\Gamma(-\nu)}{\Gamma(a_{-})\Gamma(b_{-})}\sin^{l}(\beta)\cos^{2\lambda_{+}}(\beta)\,{}_{2}F_{1}\left(a_{+},b_{+};(1+\nu);\cos^{2}(\beta)\right)+$
(3.40) $\displaystyle+$
$\displaystyle\frac{\Gamma(l+1)\Gamma(\nu)}{\Gamma(a_{+})\Gamma(b_{+})}\sin^{l}(\beta)\cos^{2\lambda_{-}}(\beta)\,{}_{2}F_{1}\left(a_{-},b_{-};(1-\nu);\cos^{2}(\beta)\right),$

where

$\displaystyle a_{-}$ $\displaystyle=$
$\displaystyle\lambda_{-}-\frac{1}{4}\left(1-\sqrt{1+4\xi^{2}}\right)+\frac{l}{2};$
(3.41) $\displaystyle b_{-}$ $\displaystyle=$
$\displaystyle\lambda_{-}-\frac{1}{4}\left(1+\sqrt{1+4\xi^{2}}\right)+\frac{l}{2};$
(3.42) $\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{\Gamma(l+1)\Gamma(-\nu)}{\Gamma(a_{-})\Gamma(b_{-})};$
(3.43) $\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{\Gamma(l+1)\Gamma(\nu)}{\Gamma(a_{+})\Gamma(b_{+})}.$
(3.44)

  2. 2.

The $\nu$ is an integer number. In this situation, the general solution is
given by

$\displaystyle\phi(\beta)$ $\displaystyle=$
$\displaystyle\sin^{l}(\beta)\cos^{2\lambda_{-}}(\beta)\frac{\Gamma(l+1)\Gamma(\nu)}{\Gamma(a_{+})\Gamma(b_{+})}\sum_{k=0}^{\nu-1}\left(\frac{(a_{+}-\nu)_{k}(b_{+}-\nu)_{k}}{k!(1-\nu)_{k}}\cos^{2k}(\beta)\right)-$
(3.45) $\displaystyle-$
$\displaystyle\sin^{l}(\beta)\cos^{2\lambda_{+}}(\beta)\frac{(-1)^{\nu}\Gamma(l+1)\Gamma(\nu)}{\Gamma(a_{+}-\nu)\Gamma(b_{+}-\nu)}\sum_{k=0}^{\infty}\frac{(a_{+})_{k}(b_{+})_{k}}{k!(k+\nu)!}\cos^{2k}(\beta)\times$
$\displaystyle\times$ $\displaystyle\big{[}\ln{(\cos^{2}(\beta))}-$
$\displaystyle-$
$\displaystyle\psi(k+1)-\psi(k+\nu+1)+\psi(a_{+}+k)+\psi(b_{+}+k)\big{]},$
(3.46)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{(-1)^{\nu}\Gamma(l+1)\Gamma(\nu)}{\Gamma(a_{+}-\nu)\Gamma(b_{+}-\nu)};$
(3.47) $\displaystyle B$ $\displaystyle=$
$\displaystyle\frac{\Gamma(l+1)\Gamma(\nu)}{\Gamma(a_{+})\Gamma(b_{+})}.$
(3.48)

  3. 3.

The case $\nu=0$. The solution reduces to

$\displaystyle\phi(\beta)$ $\displaystyle=$
$\displaystyle\sin^{l}(\beta)\cos^{3/2}(\beta)\bigg{\\{}\frac{\Gamma(l+1)}{\Gamma(a_{+})\Gamma(b_{+})}\sum_{k=0}^{\infty}\frac{(a_{+})(b_{+})}{(k!)^{2}}\bigg{[}2\psi(k+1)-\psi(a_{+}+k)-$
(3.49) $\displaystyle-$
$\displaystyle\psi(b_{+}+k)-\ln{(\cos^{2}(\beta))}\bigg{]}\cos^{2k}(\beta)\bigg{\\}}$

with

$\displaystyle A=B=\frac{\Gamma(l+1)}{\Gamma(a_{+})\Gamma(b_{+})}.$ (3.50)

Now, it is important to investigate the behavior of the general solution near
the boundary. For the first case ($\nu$ is a non-integer number), it is
straightforward to check from the properties of the hypergeometric functions
that $\phi(\beta)$ falls off as $\cos^{2\lambda_{\pm}}(\beta)$ near the
boundary. The fluctuations modes corresponding to $\lambda_{-}$ fail to be
square integrable, as a consequence, they are classified as non-normalizable.
On the other hand, the modes associated with $\lambda_{+}$ are normalizable,
as a result, they are classified as normalizable modes. Therefore, near the
boundary, the asymptotic behavior of the general solution can be cast into the
following compact form

$\Phi(Z,\beta)\sim\Phi_{1}(Z)\cos^{2\lambda_{-}}(\beta)+\,\Phi_{2}(Z)\cos^{2\lambda_{+}}(\beta),$
(3.51)

where $Z=(v,r,\theta)$ is a shorthand notation to describe the three-
dimensional boundary coordinates and $\Phi_{1}(Z)$ and $\Phi_{2}(Z)$ are
arbitrary fields living in the boundary.

For our purpose in this work, we must only select the normalizable modes since
it is well known that the non-normalizable modes lead to a non-trivial
contribution to the boundary action [83]. For $\nu>1$, this is achieved by
imposing the vanishing of the reciprocal gamma functions of the non-
normalizable modes, such requirement leads to the following quantization
condition on the separation constant

$\xi=\pm\sqrt{\left(2\lambda_{+}+l+2n\right)\left(2\lambda_{+}+l+2n-1\right)},\,\,\,\mbox{with}\,\,\,n=0,1,2...$
(3.52)

In such case, the coefficient $B$ vanishes for both cases: when $\nu$ is an
integer number and a non-integer number. Other than for $0<\nu<1$ in which
both modes are normalizable. In this case, the mode proportional to
$\cos^{2\lambda_{-}}(\beta)$ is dominant near the boundary, such mode is
selected by the following quantization condition

$\xi=\pm\sqrt{\left(2\lambda_{-}+l+2n\right)\left(2\lambda_{-}+l+2n-1\right)},\,\,\,\mbox{with}\,\,\,n=0,1,2...$
(3.53)

When $\nu=0$, both modes coincide, $\lambda_{-}=\lambda_{+}=3/4$, and are
normalizable.

###  3.3 Presence of a self-interacting scalar field

In this subsection, unlike the previous one, we allow for the backreaction of
the scalar field on the background geometry. In other words, the energy-
momentum tensor of the scalar field is not neglected in the gravitational
field equations. Furthermore, we also include scalar field self-interactions
through the scalar potential $V(\Phi)$. Explicitly, the action of the scalar
field is now given by

$S[\Phi]=-\frac{1}{2}\int
d^{\mathcal{D}}x\sqrt{-g}\bigg{(}g^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi-2V(\Phi)\bigg{)},$
(3.54)

leading to the equation of motion for $\Phi$ field in the form
$\Box\Phi+V_{\Phi}=0$, with $V_{\Phi}=dV/d\Phi$.

In this new environment, we want to search for the presence of brane-like
configurations having a thickness due to the self-interacting scalar, as it
happens in the standard $AdS_{5}$ case with a single extra dimension of
infinite extent [84, 85, 86, 87]. Towards this goal, let us consider the
scalar field $\Phi$ depending only on the extra dimension $w$. The non-
vanishing components of the energy-momentum tensor are then

$T^{a}_{b}=\delta^{a}_{b}\left(\frac{1}{2}\Phi^{\prime
2}+V(\Phi)\right);\,\,\,T^{D}_{D}=-\frac{1}{2}\Phi^{\prime 2}+V(\Phi).$ (3.55)

Substituting these results in the field equations (2.13) and proceeding with
the regularization, $\mathcal{D}\rightarrow 4$, we are able to find

$\displaystyle\kappa^{2}\left(\frac{1}{2}\Phi^{\prime 2}+V(\Phi)\right)$
$\displaystyle=$
$\displaystyle-3(A^{\prime})^{2}-2A^{\prime\prime}+2\alpha\kappa^{2}\left(3(A^{\prime})^{4}+4A^{\prime\prime}(A^{\prime})^{2}\right),$
(3.56) $\displaystyle\kappa^{2}\left(-\frac{1}{2}\Phi^{\prime
2}+V(\Phi)\right)$ $\displaystyle=$
$\displaystyle-3(A^{\prime})^{2}+6\alpha\kappa^{2}(A^{\prime})^{4},$ (3.57)

or yet

$\displaystyle V(\Phi)$ $\displaystyle=$
$\displaystyle-\frac{1}{\kappa^{2}}A^{\prime\prime}-\frac{3}{\kappa^{2}}A^{\prime
2}+6\alpha A^{\prime 4}+4\alpha A^{\prime 2}A^{\prime\prime},$ (3.58)
$\displaystyle\Phi^{\prime 2}$ $\displaystyle=$
$\displaystyle-\frac{2}{\kappa^{2}}A^{\prime\prime}+8\alpha A^{\prime
2}A^{\prime\prime}.$ (3.59)

Using the metric (2.9), the equation of motion for $\Phi$ becomes

$\Phi^{\prime\prime}+3A^{\prime}\phi^{\prime}+V_{\Phi}=0.$ (3.60)

In order to get a first-order framework, we follow [88] and introduce an
arbitrary function of the scalar field $W(\Phi)$ such that

$A^{\prime}=-\frac{1}{3}\;{W(\Phi)}.$ (3.61)

We use this in Eq. (3.59) to get

$\Phi^{\prime}=\frac{2}{3\kappa^{2}}W_{\Phi}-\frac{8\alpha}{27}W(\Phi)^{2}W_{\Phi},$
(3.62)

where $W_{\Phi}=dW/d\Phi$. We see that the term dependent on $\alpha$ brings
the possibility of new types of solutions. We substitute the equations Eqs.
(3.61) and (3.62) in Eq. (3.58) to obtain the potential

$V(\Phi)=\frac{2}{9\kappa^{4}}W_{\Phi}^{2}-\frac{1}{3\kappa^{2}}W(\Phi)^{2}+\frac{2\alpha}{81\kappa^{2}}W(\Phi)^{2}\left(3W(\Phi)^{2}\kappa^{2}-8W_{\Phi}^{2}\right)+\frac{32\alpha^{2}}{729}W(\Phi)^{4}W_{\Phi}^{2}.$
(3.63)

There are homogeneous solutions of the first order equation given by Eq.
(3.62). We can in particular consider the two possibilities, obeying

$\displaystyle W_{\Phi}$ $\displaystyle=$ $\displaystyle 0,$ (3.64a)
$\displaystyle W^{2}(\Phi)$ $\displaystyle=$
$\displaystyle\frac{9}{4\alpha\kappa^{2}}.$ (3.64b)

In the first case, solutions $\bar{\Phi}$ that obey the relation (3.64a) lead
to the potential with the following behavior

$\displaystyle V(\bar{\Phi})$ $\displaystyle=$
$\displaystyle-\frac{1}{3\kappa^{2}}W(\bar{\Phi})^{2}+\frac{2\alpha}{27}W(\bar{\Phi})^{4},$
(3.65a) $\displaystyle V_{\Phi}(\bar{\Phi})$ $\displaystyle=$ $\displaystyle
0,$ (3.65b) $\displaystyle V_{\Phi\Phi}(\bar{\Phi})$ $\displaystyle=$
$\displaystyle\frac{2}{729\kappa^{4}}W_{\Phi\Phi}(4\alpha\kappa^{2}W^{2}-9)(2(4\alpha\kappa^{2}W^{2}-9)W_{\Phi\Phi}+27W\kappa^{2}).$
(3.65c)

The critical values have characteristics that may depend on the GB
contribution. Moreover, in the second case the system only makes sense in the
presence of the Gauss term $(\alpha\neq 0)$. Here one observes that the
potential has the following features

$\displaystyle V(\bar{\Phi})$ $\displaystyle=$
$\displaystyle-\frac{3}{8\kappa^{4}\alpha},$ (3.66a) $\displaystyle
V_{\Phi}(\bar{\Phi})$ $\displaystyle=$ $\displaystyle 0,$ (3.66b)
$\displaystyle V_{\Phi\Phi}(\bar{\Phi})$ $\displaystyle=$
$\displaystyle\frac{4W_{\Phi}^{2}}{81\kappa^{2}}(27+16\alpha W_{\Phi}^{2}),$
(3.66c)

in this case always identifying minima that lead to negative values of the
potential.

The above general results motivate us to further explore specific models,
investigating how adding new gravitational term changes the standard scenario.
Let us now study two specific situations, controlled via the function $W$,
with

$\displaystyle W_{I}(\Phi)$ $\displaystyle=$ $\displaystyle a\,\Phi,$ (3.67a)
$\displaystyle W_{II}(\Phi)$ $\displaystyle=$ $\displaystyle a\sin(b\Phi).$
(3.67b)

These functions correspond to type-I and type-II models, where $a$ and $b$ are
real parameters.

####  3.3.1 Type-I model

The first example that we consider is described by the simplest choice for
$W(\Phi)$, the linear function. Using Eq. (3.67a), the potential (3.63) is
written in the form

$V(\Phi)=\frac{2a^{2}}{9\kappa^{4}}-\frac{a^{2}(16a^{2}\alpha+27)}{81\kappa^{2}}\Phi^{2}+\frac{2a^{4}\alpha(16a^{2}\alpha+27)}{729}\Phi^{4}.$
(3.68)

The first order equation given by Eq. (3.61) becomes

$\Phi^{\prime}=\frac{8a^{3}\alpha}{27}\left(\frac{9}{4a^{2}\kappa^{2}\alpha}-\Phi^{2}\right).$
(3.69)

Using the Eq. (3.64b), we find the asymptotic limits of the solution at
$\Phi_{\pm}=\pm 3/2\kappa a\sqrt{\alpha}$, which are the minima of the
potential, whose values of $V_{\Phi\Phi}$ are given by Eq. (3.65c). We can see
from the potential in Eq. (3.68), for instance, that it engenders spontaneous
symmetry breaking, so it support kink-like solutions. In order to explicitly
describe this case, we go on and solve Eq. (3.69), leading to the kink
solution

$\Phi(w)=\frac{3}{2\kappa
a\sqrt{\alpha}}\tanh\left(\frac{4\sqrt{\alpha}a\,w}{9\kappa}\right),$ (3.70)

where we have taken the positive sign and chosen the integration constant such
that $\phi(0)=0$. Knowing the field profile, we can rewrite equation (3.61) as

$A^{\prime}=-\frac{1}{2\kappa\sqrt{\alpha}}\tanh\left(\frac{4\sqrt{\alpha}a\,w}{9\kappa}\right),$
(3.71)

which gives the warp function

$A(w)=\frac{9}{8a^{2}\alpha}\ln\left({\rm
sech}\left(\frac{4\sqrt{\alpha}a\,w}{9\kappa}\right)\right),$ (3.72)

with $A(0)=0$. As we can see, this model leads to analytical solutions for
both the scalar field and the warp factor. Interestingly, we notice that for
$|w|>>1$ one gets

$\lim_{w\to\pm\infty}A(w)=-\dfrac{1}{\sqrt{2\alpha}\kappa}|w|.$ (3.73)

As expected, we have obtained the same result shown in Eq. (3.1). Furthermore,
using the metric (2.9), which is here controlled by $A(w)$ in (3.72), we see
that the warp factor $e^{2A}$ has the adequate profile, with its thickness
increasing as we increase the parameter $\alpha$ that responds for the
modification introduced by the GB contribution. This is illustrated in Fig. 2.

Figure 2: The warp factor depicted with $A(w)$ in Eq. (3.72), for $a=1$,
$\kappa=1$ and $\alpha=1/4$ (red), $1$ (blue), and $4$ (black).

####  3.3.2 Type-II model

Now, look at the second model presented in equation (3.67b), which is an
extension of the sine-Gordon model. In this case, using (3.63), (3.69) and
(3.61), we obtain

$\displaystyle V(\Phi)=$ $\displaystyle-\frac{(9-2\alpha
a^{2}\kappa^{2})a^{2}}{27\kappa^{2}}+\frac{(9-4\alpha
a^{2}\kappa^{2})(27\kappa^{2}+18b^{2}-8a^{2}b^{2}\alpha\kappa^{2})a^{2}}{729\kappa^{4}}\cos^{2}(b\Phi)$
(3.74a) $\displaystyle+\frac{2\alpha a^{4}(27\kappa^{2}+72b^{2}-32a^{2}\alpha
b^{2}\kappa^{2})}{729\kappa^{2}}\cos^{4}(b\Phi)+\frac{32a^{6}\alpha^{2}b^{2}}{729}\cos^{6}(b\Phi),$
$\displaystyle\Phi^{\prime}=$
$\displaystyle\frac{2ab}{3\kappa^{2}}\cos(b\Phi)\left(1-\frac{4a^{2}\alpha\kappa^{2}}{9}\sin^{2}(b\Phi)\right),$
(3.74b) $\displaystyle A^{\prime}=$ $\displaystyle-\frac{a}{3}\sin(b\Phi).$
(3.74c)

When $\alpha=0$, we get
$V(\Phi)=-a^{2}/3\kappa^{2}+(3\kappa^{2}+2b^{2})a^{2}/9\kappa^{4}\cos^{2}(b\Phi)$
and its first order equation $\Phi^{\prime}={2ab}/{3\kappa^{2}}\cos(b\Phi)$,
as in the original sine-Gordon model. An interesting case is when
$\alpha=\alpha_{1}$, where $\alpha_{1}=9/4a^{2}\kappa^{2}$. Under this
condition, Eqs. (3.74a) and (3.74b) give

$\displaystyle\Phi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2ab}{3\kappa^{2}}\cos^{3}(b\Phi),$ (3.75a) $\displaystyle
V(\Phi)$ $\displaystyle=$
$\displaystyle-\frac{a^{2}}{6\kappa^{2}}+\frac{a^{2}}{6\kappa^{2}}\cos^{4}(b\Phi)+\frac{2a^{2}b^{2}}{9\kappa^{4}}\cos^{6}(b\Phi).$
(3.75b)

To determine the asymptotic values of the solutions of Eq. (3.74b), we can use
the two ways to make it vanish. By applying (3.64a) and (3.64b), respectively,
we get

$\displaystyle\Phi_{n}$ $\displaystyle=$ $\displaystyle\frac{(2n+1)\pi}{2b},$
(3.76a) $\displaystyle\Phi_{m_{\pm}}$ $\displaystyle=$
$\displaystyle\frac{m\pi}{b}\pm\frac{1}{b}{\arcsin\left(\frac{3}{2\kappa
a\sqrt{\alpha}}\right)},$ (3.76b)

where $n$ and $m$ are integer numbers. For the $\Phi_{n}$ case, we get

$\displaystyle V_{\Phi\Phi}(\Phi_{n})$ $\displaystyle=$
$\displaystyle\frac{2a^{2}b^{2}(4\alpha
a^{2}\kappa^{2}-9)(8a^{2}b^{2}\alpha\kappa^{2}-18b^{2}-27\kappa^{2})}{729\kappa^{4}}$
(3.77) $\displaystyle=$
$\displaystyle\frac{9(\alpha-\alpha_{1})(\alpha-\alpha_{2})}{4\alpha_{1}\kappa^{2}(\alpha_{1}-\alpha_{2})^{2}},$
(3.78)

with $\alpha_{2}=\alpha_{1}(1+(3\kappa^{2}/2b^{2}))$. From the above
expression, we can determine that $\Phi_{n}$ represents the local maxima of
the potential for $\alpha_{1}<\alpha<\alpha_{2}$. In other regions of
parameter values, $\Phi_{n}$ corresponds to the minima of the potential.
Furthermore, the existence of $\Phi_{m_{\pm}}$ is limited to values of
$\alpha$ greater than or equal to $\alpha_{1}$, and they always represent
minima of the potential. In the special case of $\alpha=\alpha_{1}$, these
minima collapse to the values given by Eq. (3.76a).

Considering the conditions obtained above, we display in Fig. 3, the critical
points in terms of $\alpha$, the green and brown curves representing
$\Phi_{m_{\pm}}$ and $\Phi_{n}$, respectively. Minima and maxima are
represented by solid and dotted lines, respectively. In the Fig. 4, we display
the potential $V(\phi)$ for some values for $\alpha$. The vertical brown lines
denote regions where $\Phi_{n}$ are minima (solid lines) or maxima (dotted
lines).

Figure 3: The structure of the critical points $\Phi_{n}$ (brown) and
$\Phi_{m_{\pm}}$(green) associated with the potential $V(\Phi)$ in Eq. (3.74a)
as a function of $\alpha$. The solid (dotted) lines represent values of
minimum (maximum). Figure 4: The potential $V(\phi)$ given by Eq. (3.75b) with
$\kappa=1$, $b=1$, $a=1$ and some values of $\alpha$ parameter. The vertical
brown line segments represent the regions where the critical points $\Phi_{n}$
are minimum (solid line) and maximum (dotted line).

We can write the scalar field as

$\Phi=\frac{m\pi}{b}\pm\frac{\arcsin(\chi)}{b},$ (3.79)

with $m$ integer. With this, we rewrite the Eq.(3.74b) as

$\frac{d\chi}{dw}=\pm\frac{8a^{3}b^{2}}{27}(1-\chi^{2})\left(\frac{9}{4a^{2}\alpha\kappa^{2}}-\chi^{2}\right).$
(3.80)

This equation can be solved numerically, but it is not possible to find
analytical solutions for the equation for arbitrary choices of parameters.
However, using results from Ref. [89], we can obtain analytical solutions for
two specific situations. For $\alpha=\alpha_{1}/4=9/16a^{2}\kappa^{2}$,

$\chi(w)=2\cos\left(\frac{\pi+\arccos\left(\tanh\left(\frac{ab^{2}w}{\kappa^{2}}\right)\right)}{3}\right).$
(3.81)

that connects the minima $m\pi/2$ to $(m+1)\pi/2$, with $m$ integer. The warp
function equation given by Eq. (3.74c) is $A^{\prime}=-a\chi/3$, therefore

$A(w)=-\frac{2a}{3}\bigintsss^{w}\cos\left(\frac{\pi+\arccos\left(\tanh\left(\frac{ab^{2}\tilde{w}}{\kappa^{2}}\right)\right)}{3}\right)d\tilde{w}.$
(3.82)

In Fig. 5, we depict the warp factor $e^{2A(w)}$ for $a=1$, and $\kappa=1$,
and $b=1/2,1,2$. It shows that the thickness of the configuration decreases as
$b$ increases.

Figure 5: The warp factor depicted with $A(w)$ in Eq. (3.82) for
$\alpha=\alpha_{1}/4.$ We take $a=1$, $\kappa=1$ and $b=1/2$ (red), $1$
(blue), and $2$ (black).

The second possibility is for $\alpha=4\alpha_{1}=9/(a^{2}\kappa^{2})$. In
this case, we have two kinds of solutions

$\displaystyle\chi_{I}(w)$ $\displaystyle=$
$\displaystyle\cos\left(\frac{\pi+\arccos\left(\tanh\left(\frac{2ab^{2}w}{\kappa^{2}}\right)\right)}{3}\right),$
(3.83) $\displaystyle\chi_{II}(w)$ $\displaystyle=$
$\displaystyle\cos\left(\frac{\arccos\left(\tanh\left(\frac{2ab^{2}w}{\kappa^{2}}\right)\right)}{3}\right).$
(3.84)

For $b^{2}>\kappa^{2}/2$, these solutions connect two minima. The
corresponding warp functions are

$\displaystyle A_{I}(w)$ $\displaystyle=$
$\displaystyle-\frac{a}{3}\bigintsss^{w}\cos\left(\frac{\pi+\arccos\left(\tanh\left(\frac{2ab^{2}\tilde{w}}{\kappa^{2}}\right)\right)}{3}\right)d\tilde{w},$
(3.85) $\displaystyle A_{II}(w)$ $\displaystyle=$
$\displaystyle-\frac{a}{3}\bigintsss^{w}\cos\left(\frac{\arccos\left(\tanh\left(\frac{2ab^{2}\tilde{w}}{\kappa^{2}}\right)\right)}{3}\right)d\tilde{w}.$
(3.86)

In Fig. (6), we display the warp factors $e^{2A_{I}(w)}$ and $e^{2A_{II}(w)}$
for $a=1$, and $\kappa=1$, and $b=1/2,1,2$. Notice that only solutions with
$A_{I}$ arise connecting $AdS$ geometries, with the thickness decreasing as
$b$ increases.

Figure 6: The warp factor using $A_{I}$ (left) and $A_{II}$ (right) given by
Eqs. (3.85) and (3.86) for $\alpha=4\alpha_{1}.$ We take $a=1$, $\kappa=1$ and
$b=1/2$ (red), $1$ (blue), and $2$ (black).

##  4 Summary and conclusion

In this paper, we have tackled $2$-brane solutions in four-dimensional $EGB$
gravity. First, by solving the $EGB$ field equations for a flat domain-wall
ansatz in the absence of matter fields, we found that its worldvolume is
described by an $AdS_{4}$ metric, with the cosmological constant entirely
sourced by a positive GB coupling constant, $\alpha>0$. As expected, this
solution has no analog within GR, once the limit $\alpha\rightarrow 0$ is not
well defined.

In order to capture some properties of the aforementioned solution, we added a
massive probe scalar field and then solved the corresponding Klein-Gordon
equation in the bulk metric. However, since the $AdS_{4}$ metric possesses a
singularity at the horizon in the usual Poincarè coordinates, we have obtained
its description in Gaussian null coordinates which, in turn, has made it clear
that $AdS_{4}$ can be viewed as a warped product between $AdS_{2}$ and
$\mathcal{H}^{2}$, and also its regularity at the horizon. By solving the wave
equation in this non-singular coordinate system, we were able to find the
normalizable and non-normalizable modes that play an important role in the
choice of the boundary conditions to eliminate the contributions stemming from
the boundary. In fact, we have found that only the normalizable modes lead to
the vanishing of the boundary term. Such a requirement is attained by imposing
a quantization condition on the separation constant that appears in Eqs.
(3.52) and (3.53).

Furthermore, we have investigated the system in the presence of a source
scalar field having self-interaction, searching for the possibility to
construct braneworld solutions within the new $AdS_{4}$ geometry under the
modified GB gravity. In this case, after including an auxiliary function
$W(\Phi)$, we have been able to introduce a first-order procedure, with both
the warp function $A$ and the scalar field being controlled by the first-order
equations (3.61) and (3.62), which solve the equations of motion when the
potential has the form displayed in (3.63). These general results show that
the GB modification nicely contributes to the change of the warp factor,
making the configuration represent a thick brane. We have illustrated the
results with several distinct models, in particular, with the type-I model
which supports analytical solutions for both the scalar field and the warp
factor, so it can serve as a good model to be considered for extensions that
add other modifications. The presence of $W(\Phi)$ and the accompanying first-
order equations suggest that we investigate issues related to fake
supergravity [90] and the first-order Hamilton–Jacobi equations [91, 92], to
see if novelties appear in the present scenario.

The models studied in the present work can be enlarged to include other
fields, in particular, another scalar field to see if the brane can support
internal modification, in a way similar to the cases considered in Refs. [93,
94], where mechanisms to control the internal structure of the brane were
developed. We can also study the entrapment of fermions and other fields in
the brane; see, e.g., Refs. [95, 96, 97]. It is also possible to introduce
other geometric modifications, similar to the ones considered in Refs. [98,
99] in the five-dimensional spacetime. Another interesting possibility is to
change the metric in Eq. (2.9) to the more general case, having the bent brane
profile

$\displaystyle ds^{2}=e^{2A(w)}g_{ab}dx^{a}dx^{b}-dw^{2},$ (4.1)

where $g_{ab}$ may now engender nontrivial three-dimensional geometry. This
may make a direct connection to the study described in Refs. [100], in which
one considers Born-Infeld extension of general relativity formulated in
metric-affine spaces in $2+1$ dimensions to find analytical solutions and
also, to the old results described in Ref. [43, 44], in which the authors
discovered that in the case of the bulk $AdS_{4}$, a class of brane-localized
black hole metrics is possible. These and other specific issues are currently
under consideration, and we hope to report on them in the near future.

##  Appendix A Regularized Kaluza-Klein reduction for $4D$ $EGB$

This Appendix is devoted to deriving the field equations of regularized $4D$
$EGB$ via Kaluza-Klein dimensional reduction. To begin with, let us consider
the ansatz (2.9), which describes a brane-like metric. Hence upon Kaluza-Klein
dimensional reduction, (2.1) reduces to the effective action below

$\displaystyle S_{{}_{\mbox{eff}}}$ $\displaystyle=$
$\displaystyle\int\,d^{\mathcal{D}-1}x\int\,dw\,e^{(\mathcal{D}-1)A}\bigg{[}\frac{(\mathcal{D}-1)(\mathcal{D}(A^{\prime})^{2}+2A^{\prime\prime})}{2\kappa^{2}}-\alpha(\mathcal{D}-1)(\mathcal{D}-2)(\mathcal{D}-3)\times$
(A.1) $\displaystyle\times$
$\displaystyle\bigg{(}\mathcal{D}(A^{\prime})^{4}+4A^{\prime\prime}(A^{\prime})^{2}\bigg{)}\bigg{]}.$

For the sake of convenience, let us redefine $\mathcal{D}=4+\epsilon$, then
the former equation becomes

$\displaystyle S_{{}_{\mbox{eff}}}$ $\displaystyle=$
$\displaystyle\int\,d^{3+\epsilon}x\int\,dw\,e^{(3+\epsilon)A}\bigg{[}\frac{(3+\epsilon)\left((4+\epsilon)(A^{\prime})^{2}+2A^{\prime\prime}\right)}{2\kappa^{2}}-\alpha(3+\epsilon)(2+\epsilon)(1+\epsilon)\times$
(A.2) $\displaystyle\times$
$\displaystyle\bigg{(}(4+\epsilon)(A^{\prime})^{4}+4A^{\prime\prime}(A^{\prime})^{2}\bigg{)}\bigg{]}.$

In order to have a non-trivial contribution to the GB term by taking the limit
$\mathcal{D}\rightarrow 4$ or $\epsilon\rightarrow 0$, we must add a
counterterm to the former action. Its explicit form is given by

$S_{{}_{\mbox{c.t.}}}=\alpha\int\,d^{3}x\int\,dw\,e^{3A}\mathcal{G}=24\alpha\int\,d^{3}x\int\,dw\,e^{3A}\left((A^{\prime})^{4}+A^{\prime\prime}(A^{\prime})^{2}\right).$
(A.3)

Therefore, by adding (A.3) to the action (A.1), the regularized action, after
proceeding with the redefinitions
$\alpha\rightarrow\dfrac{\alpha}{\mathcal{D}-4}=\dfrac{\alpha}{\epsilon}$ and
$\mathcal{D}\rightarrow 4$ or $\epsilon\rightarrow 0$, looks like

$\displaystyle
S_{{}_{\mbox{reg}}}=\int\,d^{3}x\int\,dw\,\frac{3}{2\kappa^{2}}e^{3A}\left[4(A^{\prime})^{2}+2A^{\prime\prime}\right]+2\alpha\int\,d^{3}x\int\,dw\,e^{3A}(A^{\prime})^{4},$
(A.4)

where we got rid of the boundary terms to find the aforementioned regularized
action in four dimensions. Note that the three-dimensional section of the bulk
can be omitted since the effective dynamics of the warped function $A=A(w)$ is
restricted to the “extra” dimension coordinate only; thereby, the only
effective dynamical degree of freedom is $A$.

Varying (A.4) with respect to the scalar field $A$ and disregarding the
boundary terms we are able to find

$\displaystyle\delta S_{{}_{\mbox{reg}}}=6\int d^{3}x\int
dw\,e^{3A}\left[\frac{1}{2\kappa^{2}}\left(3(A^{\prime})^{2}+2A^{\prime\prime}\right)-\alpha\left(3(A^{\prime})^{4}+4A^{\prime\prime}(A^{\prime})^{2}\right)\right]\delta
A.\;\;$ (A.5)

Now, varying the matter action with respect to $A$, considering the scalar
field of the subsection (3.3) as the matter source, we obtain

$\delta S_{m}=3\int d^{3}x\int dw\,e^{3A}\left[\frac{1}{2}\Phi^{\prime
2}+V(\Phi)\right]\delta A.$ (A.6)

Defining the total action as $S=S_{{}_{\mbox{reg}}}+S_{m}$ and using Eqs.
(A.5) and (A.6), the variational principle tells us that

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\frac{3}{\kappa^{2}}\int d^{3}x\int
dw\,e^{3A}\bigg{[}\left(3(A^{\prime})^{2}+2A^{\prime\prime}\right)-2\alpha\kappa^{2}\left(3(A^{\prime})^{4}+4A^{\prime\prime}(A^{\prime})^{2}\right)+$
$\displaystyle+$ $\displaystyle\kappa^{2}\left(\frac{1}{2}\Phi^{\prime
2}+V(\Phi)\right)\bigg{]}\delta A,$ (A.8)

providing the following field equation

$\kappa^{2}\left(\frac{1}{2}\Phi^{\prime
2}+V(\Phi)\right)=-3(A^{\prime})^{2}-2A^{\prime\prime}+2\alpha\kappa^{2}\left(3(A^{\prime})^{4}+4A^{\prime\prime}(A^{\prime})^{2}\right),$
(A.9)

which coincides with Eq. (3.56) found via Glavan’s procedure [10]. Now, to
find the other field equation, we must trace Eq. (2.2)222As discussed in
Section 2, the trace of Eq. (2.2) is well-defined in the limit
$\mathcal{D}\rightarrow 4$ or $\epsilon\rightarrow 0$ and for the Kaluza-Klein
regularization method such an equation holds, see [22].  and take the limit
$\mathcal{D}\rightarrow 4$ or $\epsilon\rightarrow 0$, then we arrive at

$\kappa^{2}\left(\Phi^{\prime
2}+4V(\Phi)\right)=-6\left[2(A^{\prime})^{2}+A^{\prime\prime}\right]+24\alpha\kappa^{2}\left((A^{\prime})^{4}+(A^{\prime})^{2}A^{\prime\prime}\right),$
(A.10)

combining with Eq. (A.9), one has

$-3(A^{\prime})^{2}+6\alpha\kappa^{2}(A^{\prime})^{4}=\kappa^{2}\left(-\frac{1}{2}\Phi^{\prime
2}+V(\Phi)\right),$ (A.11)

which is the same equation (3.57) found via Glavan’s procedure [10].
Therefore, for the ansatz (2.9), the $4D$ $EGB$ performing the Kaluza-Klein
regularization shares the same field equations as the original formulation
[10].

## Acknowledgments

This work is partially supported by Conselho Nacional de Desenvolvimento
Científico e Tecnológico, Grants 303469/2019-6 (DB), 310994/2021-7 (RM),
301562/2019-9 (AYP), and 307628/2022-1 (PJP) and by Paraiba State Research
Foundation, Grants 0015/2019 (DB and AYP), 0003/2019 (RM) and 150891/2023-7
(AYP and PJP).

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