# Gravity Localization on Intersecting Thick Braneworlds

Henrique Matheus Gauy<EMAIL_ADDRESS>Alex E. Bernardini
<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de São Carlos,
São Carlos, 13565-905 SP, Brazil

(August 28, 2024)

###### Abstract

The localization of gravity for $(5+1)$-dimensional intersecting thick
braneworld models is thoroughly investigated. Departing from preliminary
results for five independent proposals, from $I$ to $V$, gravity is shown to
be localized in the brane. In particular, for models from $I$ to $III$, only
the zero modes are analytically determined. On the other hand, for models $IV$
and $V$, massive modes are all obtained: a finite number of massive states is
identified for model $IV$, while an infinite but discrete number of massive
states bounded from below is exhibited by model $V$. Considering that the
discreteness of the graviton modes implies that gravity does not propagate in
the co-dimensions, the naked singularities of models $IV$ and $V$, at edges of
space, are made harmless. To conclude, the phenomenological implications for
sphere models, constructed out of models $III$ and $IV$, are also discussed.
According to such an overall classification, if the internal space is a
sphere, model $IV$ is shown to exhibit normalizable modes. More relevantly,
for the sphere model assembled out of model $III$, the subtle property of
reproducing a consistent Newtonian limit is identified.

###### pacs:

04.20.-q, 95.30.Sf, 98.20.-d

##  I Introduction

The idea that our universe is embedded in some higher dimensional space has
been interpreted as a viable paradigm since it was seminally introduced by
Randall and Sundrum (RS) RS-1 ; RS-2 through a self-gravitating brane model
which would work as an alternative proposal to the so-called compactified
models (like the ADD models ArkaniHamed1998a ). In such scenarios, the
curvature of space engenders a localization mechanism which admits a
suppression of the higher dimensional terms, thus recovering, within certain
limits, the Newton’s gravitation theory in the brane. Given that RS models
rely on thin branes, and the standard model fields are confined in a thin
slice of space, a more realistic framework suggests that the matter fields
ought be smeared over extra dimensions. Mostly driven by a system of dark
scalar fields, thick braneworlds hence replace the thin brane with some
topological defect, therefore admitting some lump-like (non-topological)
defect solution for the warp factor Bernardini2016 ; Almeida2009 ; Bazeia2004
; Dzhunushaliev2010 ; Bazeia2002 ; DeWolfe2000 ; Ahmed2013 ; Gremm2000a ;
Gremm2000 ; Kehagias2001 ; Kobayashi2002 ; Bronnikov2003 ; BAZEIA2009b ;
BarbosaCendejas2013 ; ZHANG2008 ; Melfo2003 ; Bazeia2009a ; Koley2005 ;
Bazeia2014 ; Chinaglia2016 ; Bernardini2013 . In this case, the non-
topological nature of curvature begets the same localization mechanism of the
RS models for gravitational and matter fields Bernardini2016 ; Almeida2009 ;
Bazeia2004 ; Dzhunushaliev2010 ; Bazeia2002 ; DeWolfe2000 ; Ahmed2013 ;
Gremm2000a ; Gremm2000 ; Kehagias2001 ; Kobayashi2002 ; Bronnikov2003 ;
BAZEIA2009b ; BarbosaCendejas2013 ; ZHANG2008 ; Melfo2003 ; Bazeia2009a ;
Koley2005 ; Bazeia2014 ; Chinaglia2016 ; Bernardini2013 ; HERRERAAGUILAR2010 ;
Guo2013 , which leads to straightforward astrophysical and cosmological
implications Hall1999 ; Ahmed2014 ; Csaki2000a ; Rubakov1983a ; Binetruy2000 ;
Binetruy2000a ; Cline1999 ; Csaki1999 ; Csaki2000b ; Flanagan2000 ; Kanti2000
; Kanti1999 ; Bazeia2008 ; George2009 ; Kadosh2012 ; Kim2000 ; Binetruy2000b ;
Bowcock2000 ; Guha2018 ; Mukohyama2000 ; Chung1999 ; Ida2000 ; Chung2000 ;
Mukohyama2000a ; MERSINI2001 ; Bernardini2014 ; Casadio2014 .

Besides the above mentioned five dimensional scenarios, six dimensional models
have also been considered Csaki2000 ; ArkaniHamed2000 ; PARAMESWARAN200754 ;
Liu2007 ; Dzhunushaliev2008 ; Koley2006 ; Singleton2004 ; Gherghetta2000 ;
Park2003 ; Dzhunushaliev2009 ; Multamaeki2002 ; Gregory2000 ; Cohen1999 and
stimulated the emergence of novel classes of six dimensional intersecting
braneworlds Gauy2022 . In the context of the background results that shall be
recovered along this work, six dimensional localized branes generated by two
intersecting scalar fields (where the warp factor and the components of the
metric are assumed as separable functions) have recently been categorized into
five different analytical models Gauy2022 , classified from $I$ to $V$. In
this case, a twofold-warp factor constructed from two intersecting warp
factors results into a bulk configuration where the brane localization is
driven by two crossing scalar fields.

Considering that any braneworld model should recover standard four dimensional
physics, at least within reasonable limits Antoniadis1998 , and assuming that
all physical fields must be localized in a brane-like region of space, it is
natural to suppose that such a two co-dimensional thick brane-world admits the
localization of gravity for all intersecting thick braneworld configurations,
from $I$ to $V$, in correspondence with the results from Ref. Gauy2022 .
Besides the inherent classical aspects, from systematic metric perturbations
over the braneworld RS-1 ; RS-2 , which renders a linearized formulation that
encompass non-linear aspects of gravity, the equations for the metric
fluctuations can be reduced to time independent Schrödinger-like equations
RS-1 ; RS-2 ; Csaki2000 ; Gremm2000a ; Gremm2000 , which are then subjected to
some separation of variables technique that results into the quantum
mechanical analogue problem.

The quantum mechanical aspects will then be further extended to the
interpretation of the associated wave functions, solutions of such
Schrödinger-like equation, which are not only relevant for the localization on
the brane, but also for identifying the Newtonian limit. Recovering the
Newtonian gravity in the brane, up to a sensitive limit, is indeed an
important test for braneworld models. One may determine the Newtonian limit by
assuming that perturbations of the metric are generated by a particle of mass
$\mathcal{M}$. As a first approach, the effective coupling between the massive
particle with gravity becomes dependent on the normalization of the graviton
modes and on its value at the position of the particle (which is generically
placed at the maximum of the wave functions of the zero mode). However, a more
realistic description would consider any particle to be smeared over the co-
dimensions Csaki2000 ; Brandhuber1999 , in contrast to a Dirac delta
localization. In this case, the gravitational constant would no longer depend
on the value of the graviton modes at the position of the particle, but would
be determined by an average value of the matter distribution with the graviton
modes. For the proposal considered here, the simplified stratagem of a Dirac
delta distribution is enough, since one can at least determine the correct
gravitational constant, even if failing to ascertain the correct gravitational
interaction111Generically, the Newtonian limit of some braneworld should
append some modifications to the gravitational interaction. To avoid
contradictions with experimental data one expects the corrections to be
insignificant above some scale, thus recovering Newtonian gravity..

Broadly speaking, the pertinence of considering six dimensional braneworlds
resides in obtaining Schrödinger-like linearized equations for a two (and not
only one) dimensional curved background. On one hand, intersecting thick
branes allow for a further separation of the co-dimensional variables for a
subset of our models (from $III$ to $V$). On the other hand, for another
subset (models $I$ and $II$), the variables are not separable unless the
graviton mass is null. Furthermore, for two of the models which shall be
considered here, $IV$ and $V$, Schrödinger-like equations solutions are
analytically defined for the entire spectrum of gravitons. In particular, this
provides a wide range for the mass spectrum. For models $I$ to $III$, however,
only zero modes can be analytically extracted. Regardless, some remarkable
features are still accomplished from the Newtonian limit for all these models,
from $I$ to $V$, as they shall be scrutinized along this paper.

The paper is thus organized as follows. In Sec. II, a brief review of the
solutions for $(5+1)$-dimensional intersecting thick braneworlds driven by two
scalar fields, in correspondence to models from $I$ to $V$ (cf. Ref. Gauy2022
) is presented. Sec. III is devoted to determine the equations for the
gravitational perturbations for a general background, which straightforwardly
maps the quantum mechanical analogue problem that describes the localization
and spectrum of the graviton modes. Finally, in Sec. IV, the Schrödinger
equation for each intersecting thick braneworld model is yielded and their
corresponding spectrum of gravitons as well as their Newtonian limit are
obtained. Our conclusions are drawn in Sec. V pointing out the most relevant
features and delineating comparisons between the proposed models.

##  II (5+1)-Dimensional Intersecting Brane-Worlds Preliminaries

Let the braneworld be a six dimensional manifold $\mathbb{E}^{6}$ that is, as
a set, equivalent to the product space $\mathbb{M}^{4}\times\mathbb{B}^{2}$,
where $\mathbb{M}^{4}$ and $\mathbb{B}^{2}$ are manifolds of four and two
dimensions, respectively. The following ansatz is assumed for the metric of
$\mathbb{E}^{6}$ Gauy2022 :

$\bm{g}=e^{-2A}\omega_{\mu\nu}\mathrm{d}x^{\nu}\mathrm{d}x^{\mu}+e^{-2f}\mathrm{d}u^{2}+e^{-2h}\mathrm{d}v^{2},$
(1)

where $A$ is the warp factor, $\bm{\omega}$ is the metric of the space-time
$\mathbb{M}^{4}$,
$\bm{\omega}:\mathbb{M}^{4}\rightarrow\mathcal{T}^{\left(0,2\right)}\mathbb{M}^{4}$,
and $e^{-2f}$ and $e^{-2h}$ are the components of metric $\bm{\sigma}$ of the
internal space, $\mathbb{B}^{2}$,
$\bm{\sigma}:\mathbb{B}^{2}\rightarrow\mathcal{T}^{\left(0,2\right)}\mathbb{B}^{2}$222To
clear up the notation, Greek indices ($\mu$, $\nu$,…) are valued in the set
$\\{0,1,2,3\\}$, uppercase Latin indices ($M$, $N$,…) in the set
$\\{0,1,2,3,4,5\\}$, lowercase Latin indices ($m$, $n$, $i$, $j$,…),
representing the bulk co-dimension, in the set $\\{4,5\\}$, and the labels
$x^{4}=u$ and $x^{5}=v$, representing the choice of coordinates for the co-
dimensions $(\mathbb{B}^{2})$. The use of notation $T_{45}\equiv T_{uv}$
whenever suited, indicates that $``4\leftrightarrow u^{\prime\prime}$ and
$``5\leftrightarrow v^{\prime\prime}$. Derivatives, whenever suited, will be
represented by a comma, i.e. $f_{,\mu}\leftrightarrow\partial f/\partial
x^{\mu}$. Finally, tensors when being referred to its (abstract) entirety will
be in boldface, as $\bm{g}$, but its components will be cast in regular font,
as $g_{\mu\nu}$.. Here $A:\mathbb{B}^{2}\rightarrow\mathbb{R}$, which means
that $A=A(u,v)$; $\omega_{\mu\nu}:\mathbb{M}^{4}\rightarrow\mathbb{R}$;
$f:\mathbb{B}^{2}\rightarrow\mathbb{R}$; and
$h:\mathbb{B}^{2}\rightarrow\mathbb{R}$.

The action of gravity is the usual Einstein-Hilbert action in six dimensions
minimally coupled to two canonical real scalar fields,

$S=\int\mathrm{d}^{6}x\sqrt{-\mathrm{g}}\left[2{M}^{4}\,R-\left(\frac{g^{MN}}{2}\phi_{,M}\phi_{,N}+\frac{g^{MN}}{2}\zeta_{,M}\zeta_{,N}+\mathcal{V}\right)\right],$
(2)

where $R$ is a six dimensional Ricci scalar,
$\phi:\mathbb{B}^{2}\rightarrow\mathbb{R}$ ($\phi{\equiv}\phi(u,v)$),
$\zeta:\mathbb{B}^{2}\rightarrow\mathbb{R}$ ($\zeta{\equiv}\zeta(u,v)$),
$\mathcal{V}$ is some function of $\phi$ and $\zeta$, and
$\mathrm{g}=\det{\left(g_{MN}\right)}$.

For intersecting braneworlds, one assumes that the scalar fields are functions
of different variables, i.e. $\phi_{,v}=0$ and $\zeta_{,u}=0$, and the metric
components are separable functions, i.e. $A=\hat{A}(v)+\tilde{A}(u)$,
$f=\hat{f}(v)+\tilde{f}(u)$ and $h=\hat{h}(v)+\tilde{h}(u)$. In this case, it
follows from Einstein equations that the metric must take one of the two forms
Gauy2022

$\bm{g}=e^{-2\tilde{A}}\omega_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+e^{-2\tilde{f}}\mathrm{d}u^{2}+e^{-2\tilde{h}}\mathrm{d}v^{2},$
(3)
$\bm{g}=e^{-2\hat{A}}e^{-2\tilde{A}}\omega_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+e^{-2p\hat{A}}e^{-2\tilde{f}}\mathrm{d}u^{2}+e^{-2\hat{h}}e^{-2\left(1-p\right)\tilde{A}}\mathrm{d}v^{2}.$
(4)

Metric Eq. (3) corresponds to a string like defect Koley2006 ; Gherghetta2000
; Park2003 ; PARAMESWARAN200754 ; Singleton2004 ; Multamaeki2002 and metric
Eq. (4), which shall be considered from this point, can be re-organized in
terms of the so-called models from $I$ to $V$ (cf. Gauy2022 ), as they are
described by

$\displaystyle\bm{g}^{I}$
$\displaystyle=\sqrt{\cosh\left(2c_{u}u\right)\left|\cos\left(2c_{v}v\right)\right|}\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\sqrt{\frac{\left|\cos\left(2c_{v}v\right)\right|^{p}}{\cosh^{p-1}\left(2c_{u}u\right)}}\left(\mathrm{d}u^{2}+\mathrm{d}v^{2}\right)\text{,
}p>3;$ (5) $\displaystyle\bm{g}^{II}$
$\displaystyle=\sqrt{\left|\cos\left(2c_{u}u\right)\cos\left(2c_{v}v\right)\right|}\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\sqrt{\frac{\left|\cos\left(2c_{v}v\right)\right|^{p}}{\left|\cos\left(2c_{u}u\right)\right|^{p-1}}}\left(\mathrm{d}u^{2}+\mathrm{d}v^{2}\right)\text{,
}\frac{1}{2}\leq p\leq 3;$ (6) $\displaystyle\bm{g}^{III}$
$\displaystyle=\sqrt{\left|\cos\left(2\sqrt{C}v\right)\right|}e^{-2\tilde{A}}\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+e^{-2\tilde{f}}\mathrm{d}u^{2}+e^{-2\tilde{A}}\mathrm{d}v^{2};$
(7) $\displaystyle\bm{g}^{IV}$
$\displaystyle=\frac{4\Lambda}{3C}\cos^{2}\left(\frac{\sqrt{C}v}{2}\right)e^{-2\tilde{A}}\omega^{+}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+e^{-2\tilde{f}}\mathrm{d}u^{2}+e^{-2\tilde{A}}\mathrm{d}v^{2};$
(8) $\displaystyle\bm{g}^{V}$
$\displaystyle=\cos^{2/3}\left(\sqrt{3\left|\Lambda\right|}v\right)e^{-2\tilde{A}}\omega^{-}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+e^{-2\tilde{f}}\mathrm{d}u^{2}+\cos^{2/3}\left(\sqrt{3\left|\Lambda\right|}v\right)e^{-2\tilde{A}}\mathrm{d}v^{2},$
(9)

where $c_{u}$, $c_{v}$, $C$ and $\Lambda$ are separation constants;
$\omega^{\pm}_{\mu\nu}$ represents solutions of the equation
$\mathcal{R}_{\mu\nu}=\Lambda\omega_{\mu\nu}$, with either positive (for
$\omega^{+}_{\mu\nu}$) or negative (for $\omega^{-}_{\mu\nu}$) values of
$\Lambda$; and $\mathcal{R}_{\mu\nu}$ represents the Ricci tensor compatible
with $\omega_{\mu\nu}$. The warp factor $\tilde{A}$ of models $III$, $IV$ and
$V$ are not strictly specified by Einstein equations and can be left free of
constraints. The localization of gravitons and the Newtonian limit can thus be
settled for the above set of geometries, from $I$ to $V$.

##  III Gravitational Fluctuations

A basic requirement of any braneworld model is that it reproduces, up to a
sensible limit, Newtonian gravity RS-1 ; RS-2 ; Csaki2000 ; Gremm2000a ;
Gremm2000 . The Newtonian limit of any braneworld model can be studied within
the framework of linearized gravity, where it is assumed that the spacetime
can be described by small fluctuations about a given background. Following
similar arguments as outlined by Csaki2000 333Extending this line of reasoning
in order to account for a curved four dimensional space Isaacson1968 .,
tensorial perturbations of (1) are written as

$\bar{\bm{g}}=e^{-2A}\left[{\omega}_{\mu\nu}+\varpi_{\mu\nu}\right]\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+\sigma_{ij}\mathrm{d}y^{i}\mathrm{d}y^{j},$
(10)

where $\varpi_{\mu\nu}$ are functions such that
$\varpi_{\mu\nu}:\mathbb{M}^{4}\times\mathbb{B}^{2}\rightarrow\mathbb{R}$
(i.e. $\varpi_{\mu\nu}(x^{\rho},u,v)$). The Einstein tensor ($\bar{G}_{MN}$)
compatible with $\bar{\bm{g}}$ is then expressed, up to the first order in
$\varpi_{\mu\nu}$, by

$\displaystyle\bar{G}_{MN}$ $\displaystyle={G}_{MN}+\delta G_{MN}$
$\displaystyle={G}_{MN}+\frac{1}{2}\hat{g}^{PK}\hat{\nabla}_{P}\hat{\nabla}_{N}{\varpi}_{MK}+\frac{1}{2}\hat{g}^{PK}\hat{\nabla}_{P}\hat{\nabla}_{M}{\varpi}_{KN}-\frac{1}{2}\hat{\square}{\varpi}_{MN}-\frac{1}{2}\hat{g}^{PK}\hat{\nabla}_{M}\hat{\nabla}_{N}{\varpi}_{PK}$
$\displaystyle\qquad\qquad-\frac{1}{2}\varpi_{MN}\hat{R}+\frac{1}{2}\hat{g}_{MN}\varpi_{PK}\hat{R}^{PK}-\frac{1}{2}\hat{g}_{MN}\hat{g}^{SD}\hat{g}^{PK}\hat{\nabla}_{P}\hat{\nabla}_{S}{\varpi}_{KD}+\frac{1}{2}\hat{g}_{MN}\hat{g}^{SD}\hat{\square}{\varpi}_{SD}$
$\displaystyle\qquad\qquad\qquad\qquad-2\,\hat{g}^{PK}\left(\hat{\nabla}_{N}{\varpi}_{MK}+\hat{\nabla}_{M}{\varpi}_{KN}-\hat{\nabla}_{K}{\varpi}_{MN}\right)A_{,P}+6\varpi_{MN}\hat{g}^{PS}\,A_{,S}A_{,P}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+2\hat{g}_{MN}\hat{g}^{PK}\hat{g}^{SD}\left(\hat{\nabla}_{K}{\varpi}_{PD}+\hat{\nabla}_{P}{\varpi}_{KD}-\hat{\nabla}_{D}{\varpi}_{PK}\right)A_{,S}-4\,\varpi_{MN}\hat{\square}A$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-6\hat{g}_{MN}\hat{g}^{PD}\hat{g}^{SJ}\varpi_{DJ}\,A_{,S}A_{,P}+4\,\hat{g}_{MN}\hat{g}^{PS}\hat{g}^{KD}\varpi_{SD}\hat{\nabla}_{P}\hat{\nabla}_{K}A,$
(11)

where $\hat{g}_{MN}=e^{2A}g_{MN}$ and $\hat{\nabla}$ is a covariant derivative
compatible with $\hat{\bm{g}}$. Introducing the assumed constraints,
$\varpi_{Mi}=A_{,\mu}=0$, fixing the gauge
$\hat{\nabla}^{M}\varpi_{MN}=\hat{g}^{MN}\varpi_{MN}=0$, noticing the
commutation properties of the covariant
derivatives444$\hat{g}^{PK}\hat{\nabla}_{P}\hat{\nabla}_{N}{\varpi}_{MK}=\hat{g}^{PK}\hat{\nabla}_{N}\hat{\nabla}_{P}{\varpi}_{MK}-\hat{g}^{SD}\hat{g}^{PK}\varpi_{SK}{\hat{R}}_{DMPN}+\hat{g}^{SD}\varpi_{MS}{\hat{R}}_{DN}$.
and considering that555Which is true whenever one assumes the gauge condition
$\hat{g}^{MN}\varpi_{MN}=0$. $\varpi_{\rho\kappa}\hat{R}^{\rho\kappa}=0$, one
obtains the first order perturbed Einstein tensor in the form of

$\delta
G_{MN}=\frac{1}{2}\hat{g}^{\zeta\delta}\varpi_{\zeta\left(M\right.}{\hat{R}}_{\left.N\right)\delta}-\varpi^{\delta\rho}{\hat{R}}_{\delta
M\rho
N}-\frac{1}{2}\hat{\square}{\varpi}_{MN}+2\,\hat{\sigma}^{ij}A_{,i}\hat{\nabla}_{j}{\varpi}_{MN}+\varpi_{MN}\left(6\hat{\sigma}^{ij}\,A_{,i}A_{,j}-4\,\hat{\square}A-\frac{1}{2}\hat{R}\right),$
(12)

where $t_{\left(MN\right)}=t_{MN}+t_{NM}$.

One also notices that the metric fluctuations induce perturbations in the
stress energy tensor. In particular, in the context of our analysis, only
stress energy tensors constructed out of scalar fields shall be considered.
One would also expect perturbations in the configuration of the scalar fields,
but the scalar and tensorial perturbations are completely decoupled one from
each other when linear perturbations are considered Csaki2000 (see Appendix A
for further details). For this reason, when dealing with the dynamics of the
gravitational field, one can disregard the perturbations of the scalar fields.

The stress energy tensor calculated out of metric $\bar{\bm{g}}$ is then given
by

$\bar{T}_{MN}={T}_{MN}-e^{-2A}\varpi_{MN}\left(\frac{{g}^{KS}\phi_{,S}\phi_{,K}}{2}+\frac{{g}^{KS}\zeta_{,S}\zeta_{,K}}{2}+V\right).$
(13)

Following the same stratagem from Ref. Csaki2000 , one finds that

$\varpi_{MP}{T^{P}}_{N}=-\varpi_{MN}\left(\frac{{g}^{KS}\phi_{,S}\phi_{,K}}{2}+\frac{{g}^{KS}\zeta_{,S}\zeta_{,K}}{2}+V\right),$
(14)

which can be used to derive the perturbations of the stress energy tensor,

$\delta T_{MN}=4M^{4}\hat{g}^{\rho\zeta}\varpi_{M\rho}\hat{R}_{\zeta
N}+4M^{4}\varpi_{MN}\left(6\hat{g}^{ij}A_{,i}A_{,j}-4\hat{\square}A-\frac{1}{2}\hat{R}\right).$
(15)

Finally, equalizing the first order contributions, $\delta G_{MN}$ and $\delta
T_{MN}$, from Eqs. (12) and (15), through the Einstein equation, one finds

$-\frac{1}{2}\hat{\square}{\varpi}_{MN}+2\,\hat{\sigma}^{ij}A_{,i}\hat{\nabla}_{j}{\varpi}_{MN}=\frac{1}{2}\hat{g}^{\zeta\delta}\varpi_{\zeta\left[M\right.}{\hat{R}}_{\left.N\right]\delta}+\varpi^{\delta\rho}{\hat{R}}_{\delta
M\rho N},$ (16)

where $t_{[MN]}=t_{MN}-t_{NM}$. The above result be simplified by noticing
that $\hat{\nabla}_{j}{\varpi}_{MN}={\varpi}_{MN,j}$, since
$\hat{\Gamma}^{N}_{jM}=0$ and that $\hat{\bm{g}}$ is factorable. One thus has

$\hat{\bm{R}}=\mathcal{R}^{\rho}_{\;\;\alpha\mu\nu}(x^{\kappa})\;\frac{\partial\;\;}{\partial
x^{\rho}}\otimes\mathrm{d}x^{\alpha}\otimes\mathrm{d}x^{\nu}\otimes\mathrm{d}x^{\nu}+\hat{\Sigma}^{j}_{\;\;klc}(x^{s})\;\frac{\partial\;\;}{\partial
x^{j}}\otimes\mathrm{d}x^{k}\otimes\mathrm{d}x^{l}\otimes\mathrm{d}x^{c},$
(17)

with $\mathcal{R}^{\rho}_{\;\;\alpha\mu\nu}$ and $\hat{\Sigma}^{j}_{\;\;klc}$
encoding the curvature of $\left(\mathbb{M}^{4},\bm{\omega}\right)$ and
$\left(\mathbb{B}^{2},\bm{\hat{\sigma}}\right)$, respectively (cf. Gauy2022 ).
To summarize, one can still write the equation for the perturbation of the
gravitational field for general bent branes,

$\hat{\square}{\varpi}_{\mu\nu}-4\,\hat{\sigma}^{ij}A_{,i}{\varpi}_{\mu\nu,j}=\omega^{\zeta\delta}\varpi_{\zeta\left[\nu\right.}{\mathcal{R}}_{\left.\mu\right]\delta}-2\varpi^{\delta\rho}{\mathcal{R}}_{\delta\mu\rho\nu},$
(18)

and notice that, for solutions (5)-(9), only the Ricci tensor is written as
$\mathcal{R}_{\mu\nu}=\Lambda\omega_{\mu\nu}$. For further simplifications, a
maximally symmetric spacetime $\left(\mathbb{M}^{4},\bm{\omega}\right)$ can be
assumed, i.e.
${\mathcal{R}}_{\delta\mu\rho\nu}=\left(\omega_{\delta\rho}\omega_{\mu\nu}-\omega_{\delta\nu}\omega_{\mu\rho}\right){\Lambda}/{3}$,
such that Eq. (18) is thus simplified into the form of

$\hat{\square}{\varpi}_{\mu\nu}-4\,\hat{\sigma}^{ij}A_{,i}{\varpi}_{\mu\nu,j}=\frac{2\Lambda}{3}\varpi_{\mu\nu},$
(19)

which is essential in discussing intersecting branes.

Finally, considering the perturbative approach applied to the action, one has

$\delta
S=2{M}^{4}\int\mathrm{d}^{6}xe^{-4A}\sqrt{-\hat{g}}\left[-\frac{1}{4}\hat{\nabla}^{K}\varpi^{\mu\nu}\hat{\nabla}_{K}\varpi_{\mu\nu}-\hat{g}^{ij}\varpi^{\mu\nu}\hat{\nabla}_{j}\varpi_{\mu\nu}A_{,i}+\varpi^{\mu\nu}\varpi_{\mu\nu}\left(2A^{,i}A_{,i}-\frac{\Lambda}{6}-\frac{1}{2}\hat{\square}A\right)\right],$
(20)

which will later select the localized solutions of Eq. (19), which namely
describe the four dimensional gravity.

###  III.1 The Quantum Mechanical Analogy

Eq. (19) can be readily refined by assuming a separation of variables
algorithm,

$\varpi_{\mu\nu}=\sum_{m\in\mathbb{I}}\Phi_{m}(u,v)\widetilde{\varpi}^{m}_{\mu\nu}(x^{\rho}),$
(21)

which implies into two equations,

$\left(\bm{\varDelta}-m^{2}\right)\widetilde{\varpi}^{m}_{\mu\nu}=0,$ (22)

and

${\hat{\triangle}}^{2}\Phi_{m}-4\hat{\sigma}^{ij}A_{,i}\Phi_{m,j}=\left(\frac{2\Lambda}{3}-m^{2}\right)\Phi_{m},$
(23)

where
$\bm{\varDelta}=\omega^{\alpha\beta}\hat{\nabla}_{\alpha}\hat{\nabla}_{\beta}$,
$\hat{\triangle}^{2}=\hat{\sigma}^{ij}\hat{\nabla}_{i}\hat{\nabla}_{j}$ and
$m^{2}$ is a separation constant.

A simpler form of Eq. (23) can be achieved by rescaling the field
$\Phi_{m}=e^{2A}\chi_{m}$, so as to give

$-{\hat{\triangle}}^{2}\chi_{m}+2\left(2\hat{\sigma}^{ij}A_{,i}A_{,j}-{\hat{\triangle}}^{2}A\right)\chi_{m}=\left(m^{2}-\frac{2\Lambda}{3}\right)\chi_{m},$
(24)

which shall be identified with a time independent Schrödinger-like equation in
curved space $\left(\mathbb{B}^{2},\hat{\bm{\sigma}}\right)$, with the energy
$E_{QM}=m^{2}-{2\Lambda}/{3}$ and the “quantum mechanical” potential

$V_{QM}(u,v)=2\left(2A^{,i}A_{,i}-{\hat{\triangle}}^{2}A\right)=2\left[2\hat{\sigma}^{ij}A_{,i}A_{,j}-\frac{1}{\sqrt{\hat{\sigma}}}\left(\sqrt{\hat{\sigma}}\hat{\sigma}^{ij}A_{,i}\right)_{,j}\right].$
(25)

A “flat” Schrödinger-like equation is indeed solely justified for a
conformally flat metric
$(\bar{f}=\bar{h}=0\iff\hat{\bm{\sigma}}=\bm{\gamma})$, with
$\gamma=\text{diag}\left({1,1}\right)$. Otherwise the curvature of
$\left(\mathbb{B}^{2},\hat{\bm{\sigma}}\right)$ is completely arbitrary, and
the curvature intricacies introduce the possibility of some additional
localization aspect.

Following the same algorithm from Eq. (21), and rescaling the scalar field by
$\Phi_{m}=e^{2A}\chi_{m}$, the action for the perturbations can be written as

$\delta
S=-\frac{{M}^{4}}{2}\sum_{m_{1},m_{2}\in\mathbb{I}}\int\mathrm{d}^{2}y\sqrt{\hat{\sigma}}\chi_{m_{2}}\chi_{m_{1}}\int\mathrm{d}^{4}x\sqrt{-{\omega}}\left(\hat{\nabla}^{\kappa}\widetilde{\varpi}_{m_{1}}^{\mu\nu}\hat{\nabla}_{\kappa}\widetilde{\varpi}^{m_{2}}_{\mu\nu}+{m_{1}}^{2}\widetilde{\varpi}_{m_{1}}^{\mu\nu}\widetilde{\varpi}^{m_{2}}_{\mu\nu}\right),$
(26)

which corresponds to the same problem driven by Eq. (22), if and only if the
gravitational modes are

  1. 1.

Normalizable, i.e.
$0<\displaystyle\int\mathrm{d}^{2}y\sqrt{\hat{\sigma}}\chi_{m}{}^{2}\neq\infty,\;\;\forall
m\in\mathbb{I}$;

  2. 2.

Orthogonal, i.e.
$\displaystyle\int\mathrm{d}^{2}y\sqrt{\hat{\sigma}}\chi_{m_{2}}\chi_{m_{1}}=0\text{
if }m_{1}\neq m_{2}$.

When the above conditions are satisfied, for operators compatible with
$\bm{\omega}$, the action reads

$\delta
S=-\frac{{M}^{4}}{2}\sum_{m\in\mathbb{I}}\int\mathrm{d}^{2}y\sqrt{\hat{\sigma}}{\chi_{m}}^{2}\int\mathrm{d}^{4}x\sqrt{-{\omega}}\left(\varDelta^{\kappa}\widetilde{\varpi}_{m}^{\mu\nu}\varDelta_{\kappa}\widetilde{\varpi}^{m}_{\mu\nu}+{m}^{2}\widetilde{\varpi}_{m}^{\mu\nu}\widetilde{\varpi}^{m}_{\mu\nu}\right),$
(27)

which does not provide a coupling between the different states
$\widetilde{\varpi}_{m_{i}}$. In this scope, variations of action (27) lead to
the Eq. (22). In particular, in higher dimensional theories there should be as
many massive gravitons as can be fit in the set $\mathbb{I}$, which contains
the normalizable states of the Schrödinger equation (24).

The requirement of normalizability again ratifies the quantum mechanical
analogy. The gravitational modes $\chi_{m}$ must satisfy a Schrödinger-like
equation and be normalized in the curved space
$(\mathbb{B}^{2},\hat{\bm{\sigma}})$, and the localization of gravity at the
vicinity of the brane now becomes contingent on the “quantum mechanical”
problem described by Eqs. (24) and (27). The problem of locality is then
reduced to solving the “quantum mechanical” problem described by Eq. (24)
according to the normalization condition from Eq. (27).

###  III.2 The Newtonian Limit

Rigorously, for thick braneworlds, matter fields should be smeared over the
bulk. But, for the sake of simplicity, the gravitational potential will be
generated by a point-like source of mass666This same strategy was employed by
Ref. Csaki2000 . $\mathcal{M}$. The action of a point-like particle, up to the
first order in $\varpi$, is written as (see Appendix B for further details)

$\delta
S_{p}=-\frac{\mathcal{M}}{2}\sum_{m\in\mathbb{I}}\int\mathrm{d}^{6}x\;\chi_{m}\delta(x^{Q}-x^{P})\widetilde{\varpi}^{m}_{\mu\nu}\mathsf{v}^{\mu}\mathsf{v}^{\nu}.$
(28)

where $\mathsf{v}$ represents the velocity of the particle in space-time and
$x^{P}$ is the position in $\mathbb{E}^{6}$. After a re-parameterization of
the proper time so to satisfy
$\omega_{\mu\nu}\mathsf{v}^{\mu}\mathsf{v}^{\nu}=-1$, for a particle at
$y^{i}_{0}$, the total action, to the leading order in $\varpi$, becomes

$\displaystyle\delta S=$
$\displaystyle-\sum_{m\in\mathbb{I}}\left[\frac{{M}^{4}}{2}\int\mathrm{d}^{2}y\,\sqrt{\hat{\sigma}}\,{\chi_{m}}^{2}\int\mathrm{d}^{4}x\sqrt{-{\omega}}\left(\varDelta^{\kappa}\widetilde{\varpi}_{m}^{\mu\nu}\varDelta_{\kappa}\widetilde{\varpi}^{m}_{\mu\nu}+{m}^{2}\widetilde{\varpi}_{m}^{\mu\nu}\widetilde{\varpi}^{m}_{\mu\nu}\right)\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+\frac{\mathcal{M}}{2}e^{2A(y_{0}^{i})}\chi_{m}(y_{0}^{i})\int\mathrm{d}x^{4}\delta(x^{\mu}-x^{\alpha})\widetilde{\varpi}^{m}_{\mu\nu}\mathsf{v}^{\mu}\mathsf{v}^{\nu}\right],$
(29)

where $x^{\alpha}$ and $y_{0}^{i}$ represent the position of the point-like
particle in $\mathbb{M}^{4}$ and $\mathbb{B}^{2}$, respectively. Varying with
respect to $\varpi$ leads to the equations of motion

$\left(\varDelta^{2}-m^{2}\right)\tilde{\varpi}^{m}_{\mu\nu}=\frac{\mathcal{M}e^{2A(y_{0}^{i})}\chi_{m}(y_{0}^{i})}{2M^{4}\displaystyle\int\mathrm{d}^{2}y\,\sqrt{\hat{\sigma}}\,{\chi_{m}}^{2}}\frac{\mathsf{v}_{\mu}\mathsf{v}_{\nu}\delta(x^{\mu}-x^{\alpha})}{\sqrt{-\omega}}.$
(30)

Hence, the Newtonian limit is obtained through the $00$ component of Eq. (30),
with $v^{1}=v^{2}=v^{3}=0$ and $v^{0}=1/\sqrt{-\omega_{00}}$, for a static
particle. Also supposing that the particle finds itself at the center of a
system of coordinates ($\mathsf{r}=0$), the configuration has been stabilized,
i.e. with $\widetilde{\varpi}$ independent of time, and the space
displacements for identifying the Newtonian potential are not of cosmological
scale ($\mathsf{r}\sim 1/\sqrt{\Lambda}$). It allows for setting the
approximation $\bm{\omega}\approx\bm{\eta}$ so to simplify the equations of
motion into the form of

$\left(\nabla^{2}-m^{2}\right)\tilde{\varpi}^{m}_{00}=\frac{\mathcal{M}e^{2A(y_{0}^{i})}\chi_{m}(y_{0}^{i})}{2M^{4}\displaystyle\int\mathrm{d}^{2}y\sqrt{\hat{\sigma}}{\chi_{m}}^{2}}\delta(\bm{\mathsf{r}}),$
(31)

with the direct solution777The solution is true for as long as
$m\in\mathbb{R}$, otherwise, i.e. $m\in\mathbb{C}$ ($Re(m)=0$), one would find
$\frac{\cos\left(m\mathsf{r}\right)}{\mathsf{r}}$ instead of
$\frac{e^{-m\mathsf{r}}}{\mathsf{r}}$.

$\tilde{\varpi}^{m}_{00}=\frac{\mathcal{M}e^{2A(y_{0}^{i})}\chi_{m}(y_{0}^{i})}{8\pi
M^{4}\displaystyle\int\mathrm{d}^{2}y\sqrt{\hat{\sigma}}{\chi_{m}}^{2}}\frac{e^{-m\mathsf{r}}}{\mathsf{r}},$
(32)

so to obtain the Newtonian potential,

$\phi_{N}(\mathsf{r})=\frac{\varpi_{00}(y_{0}^{i},\mathsf{r})}{4}=\sum_{m\in\mathbb{I}}\frac{\mathcal{M}e^{4A(y_{0}^{i})}\left[\chi_{m}(y_{0}^{i})\right]^{2}}{32\pi
M^{4}\displaystyle\int\mathrm{d}^{2}y\sqrt{\hat{\sigma}}{\chi_{m}}^{2}}\frac{e^{-m\mathsf{r}}}{\mathsf{r}}$
(33)

which can be considered in order to make the connection with the phenomenology
of braneworlds.

##  IV The Quantum Analogue Problem For Intersecting Branes

The prescription of the preceding subsections can be readily particularized to
intersecting thick braneworlds. The quantum problem, Eq. (24), for the
geometries described by metric (4) is characterized by the equation

$\displaystyle-e^{\left(p-1\right)\tilde{A}+\tilde{f}}\left[e^{\left(p-1\right)\tilde{A}+\tilde{f}}\chi_{m,u}\right]_{,u}-e^{-p\hat{A}+\hat{h}}\left(e^{-p\hat{A}+\hat{h}}\chi_{m,v}\right)_{,v}+2\Bigg{[}2e^{2\left(p-1\right)\tilde{A}+2\tilde{f}}\tilde{A}_{,u}{}^{2}+2e^{-2p\hat{A}+2\hat{h}}\hat{A}_{,v}{}^{2}\Bigg{.}$
$\displaystyle\qquad\qquad\qquad\Bigg{.}-e^{\left(p-1\right)\tilde{A}+\tilde{f}}\left[e^{\left(p-1\right)\tilde{A}+\tilde{f}}\tilde{A}_{,u}\right]_{,u}-e^{-p\hat{A}+\hat{h}}\left(e^{-p\hat{A}+\hat{h}}\hat{A}_{,v}\right)_{,v}\Bigg{]}\chi_{m}=e^{2p\tilde{A}}e^{-2\left(p-1\right)\hat{A}}\left(m^{2}-\frac{2\Lambda}{3}\right)\chi_{m}.$
(34)

A separation of variables technique, with
$\chi_{m}=\sum_{k\in\mathbb{K}}\tilde{\chi}_{k}(u)\hat{\chi}_{km}(v)$, can be
applied to Eq. (34) if:

  1. 1.

$m=\sqrt{2\Lambda/3}$ (i.e. the zero mode);

  2. 2.

$p=0$ (or $p=1$) (i.e. for models $III$, $IV$ and $V$).

If $p\neq 0$ the solutions of Eq. (34) are not separable unless
$m=\sqrt{2\Lambda/3}$, which can always be expressed by $\chi_{0}=be^{-2A}$
(see Csaki2000 ). On the other hand, whenever $p=0$ the variables are
separable and Eq. (34) is reduced to two equations

$-e^{\tilde{f}-\tilde{A}}\left[e^{\tilde{f}-\tilde{A}}\tilde{\chi}_{k,u}\right]_{,u}+\left[4e^{2\tilde{f}-2\tilde{A}}\tilde{A}_{,u}{}^{2}-2e^{\tilde{f}-\tilde{A}}\left(e^{\tilde{f}-\tilde{A}}\tilde{A}_{,u}\right)_{,u}+k\right]\tilde{\chi}_{k}=0,$
(35)

and

$-e^{\hat{h}}\left(e^{\hat{h}}\hat{\chi}_{mk,v}\right)_{,v}+\left[4e^{2\hat{h}}\hat{A}_{,v}{}^{2}-2e^{\hat{h}}\left(e^{\hat{h}}\hat{A}_{,v}\right)_{,v}-k\right]\hat{\chi}_{mk}=e^{2\hat{A}}\left(m^{2}-\frac{2\Lambda}{3}\right)\hat{\chi}_{mk}.$
(36)

The normalization of the gravitational modes, for the $p=0$ models, is fixed
by

$\int\sqrt{\hat{\sigma}}\chi_{m}{}^{2}\mathrm{d}u\mathrm{d}v=\sum_{k\in\mathbb{K}}\int\left(e^{\tilde{A}-\tilde{f}}\tilde{\chi}_{k}{}^{2}\right)\mathrm{d}u\,\int\left(e^{2\hat{A}-\hat{h}}\hat{\chi}_{mk}{}^{2}\right)\mathrm{d}v<\infty.$
(37)

Eq. (35) and the normalization condition for $\tilde{\chi}_{k}$ are precisely
the same as in five dimensions888Since coordinates can always be chosen such
that $\tilde{f}=\tilde{A}$.. However, Eq. (36) represents a distinct problem,
as it shall be discussed in the following.

###  IV.1 Models $I$ and $II$

As argued before, for the $p\neq 0$ constructions, i.e. models $I$ and $II$,
one can merely determine the zero modes Gauy2022 as, respectively,

$\chi^{I}_{0}=B^{I}\sqrt{\cosh\left(2c_{u}u\right)\left|\cos\left(2c_{v}v\right)\right|},$
(38)

and

$\chi^{II}_{0}=B^{II}\sqrt{\left|\cos\left(2c_{u}u\right)\cos\left(2c_{v}v\right)\right|},$
(39)

where $B^{I}$ and $B^{II}$ are normalization constants. Wave functions (38)
and (39) are normalizable, since

$\int\sqrt{\hat{\sigma}}\left(\chi^{I}_{0}\right)^{2}\mathrm{d}u\mathrm{d}v=\frac{\left({B^{I}}\right)^{2}}{4c_{u}c_{v}}\int^{\infty}_{-\infty}\operatorname{sech}^{\frac{p-3}{4}}\left(x\right)\mathrm{d}x\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\cos^{\frac{p+2}{4}}\left(\varphi\right)\mathrm{d}\varphi=\frac{\left({B^{I}}\right)^{2}}{4c_{u}c_{v}}\frac{\pi\Gamma\left(\frac{p-3}{8}\right)\Gamma\left(\frac{p+6}{8}\right)}{\Gamma\left(\frac{p+1}{8}\right)\Gamma\left(\frac{p+10}{8}\right)},$
(40)

and

$\int\sqrt{\hat{\sigma}}\left(\chi^{II}_{0}\right)^{2}\mathrm{d}u\mathrm{d}v=\frac{\left({B^{II}}\right)^{2}}{4c_{u}c_{v}}\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\sec^{\frac{p-3}{4}}\left(\theta\right)\mathrm{d}\theta\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}}\cos^{\frac{p+2}{4}}\left(\varphi\right)\mathrm{d}\varphi=\frac{\left({B^{II}}\right)^{2}}{4c_{u}c_{v}}\frac{\pi\Gamma\left(\frac{7-p}{8}\right)\Gamma\left(\frac{p+6}{8}\right)}{\Gamma\left(\frac{11-p}{8}\right)\Gamma\left(\frac{p+10}{8}\right)}.$
(41)

Additional massive modes are not straightforwardly obtained, thus the
corrections to the Newtonian gravity cannot be evaluated. Regardless, the
Newtonian gravitational constant can be settled from Eqs. (40) and (41):

$G^{I}_{N}=\frac{1}{2^{5}\pi\left(M^{I}_{pl}\right)^{2}}=\frac{c_{u}c_{v}\Gamma\left(\frac{p+1}{8}\right)\Gamma\left(\frac{p+10}{8}\right)}{2^{3}\pi^{2}M^{4}\Gamma\left(\frac{p-3}{8}\right)\Gamma\left(\frac{p+6}{8}\right)},$
(42)

and

$G^{II}_{N}=\frac{1}{2^{5}\pi\left(M^{II}_{pl}\right)^{2}}=\frac{c_{u}c_{v}\Gamma\left(\frac{11-p}{8}\right)\Gamma\left(\frac{p+10}{8}\right)}{2^{3}\pi^{2}M^{4}\Gamma\left(\frac{7-p}{8}\right)\Gamma\left(\frac{p+6}{8}\right)}.$
(43)

The gravitational constant $G_{N}$ and the associated Planck scale $M_{pl}$
are depicted in Fig. 1.

Ordinarily, for brane models, the gravitational scale of model $II$ is
controlled by the parameters $c_{u}$ and $c_{v}$. In contrast, the strength of
gravity for model $I$ can be set by choosing the parameter $p$, which can take
widely different values when $p\gtrsim 3$. This is relevant when addressing
the hierarchy problem, because a large Planck scale can be achieved for any
value of $c_{u}$ and $c_{v}$. Unfortunately, model $I$ is plagued by several
modeling issues: the stress energy tensor has singularities and the total
defect energy formation is infinite Gauy2022 .

(a)  (b)

Figure 1: (Color online) (a) The gravitational constant
$G^{*}_{N}=2^{3}\pi^{2}M^{4}G_{N}/c_{u}c_{v}$ as a function of $p$. (b) The
Planck Scale $M^{*}_{pl}=2M_{pl}\sqrt{c_{u}c_{v}}/M^{2}\sqrt{\pi}$ as a
function of $p$. The plots are for model $I$ (black line) and $II$ (red line).

Even though models $I$ and $II$ are constructed from the same solution
Gauy2022 , the transition from one to the other involves redefining $c_{u}$
from completely real, for model $I$, to completely imaginary, for model $II$.
This transition will imply in a discontinuous behavior for any physical
constant, as exhibited by the discontinuity depicted in Fig. 1 at the boundary
between models $I$ and $II$.

###  IV.2 Model $III$

Eq. (36) for model $III$ (represented by metric (7)) is expressed by

$-\hat{\chi}^{III}_{mk,vv}-\left[C+k+C\sec^{2}\left(2\sqrt{C}v\right)\right]\hat{\chi}^{III}_{mk}=\displaystyle\frac{m^{2}\hat{\chi}^{III}_{mk}}{\sqrt{\cos\left(2\sqrt{C}v\right)}}.$
(44)

For the general massive case, Eq. (44) cannot be straightforwardly integrated,
and the unique immediate solution is the zero mode, which is proportional to
the warp factor when $k=0$. For any $k$, the zero mode can be written as

$\hat{\chi}^{III}_{0k}=B^{III}_{0k}\sqrt{\cos\left(2\sqrt{C}v\right)}\,_{2}F_{1}\left[\frac{1}{4}\left(1-\sqrt{1+\frac{k}{C}}\right),\frac{1}{4}\left(1+\sqrt{1+\frac{k}{C}}\right);1;\cos^{2}\left(2\sqrt{C}v\right)\right],$
(45)

As it will be further argued in Subsec. IV.4, the allowed wave functions must
be continuously differentiable. The first derivative of (45) is necessarily
discontinuous at $v=0$, unless the first or second argument of the
hypergeometric function, ${}_{2}F_{1}$, is a non-positive integer $-j$,
$j\in\mathbb{N}$, or $k=8Cj(2j+1)$. The degeneracies of the zero mode, for
model $III$, are depicted in Fig. 2.

Figure 2: (Color online) The degeneracies of the zero mode of model $III$ as
functions of $\varphi=2\sqrt{C}v$ for $j=0$ (full line), $j=1$ (dashed line),
$j=2$ (dotted line) and $j=3$ (dot-dashed line).

Eq. (44) can be still more simplified if one makes $C=0$ and $v=r\varphi$,
leading to a trivial extension of five dimensional models Gauy2022 ,

$-\hat{\chi}^{III}_{mk,\varphi\varphi}=r^{2}\left(m^{2}+k\right)\hat{\chi}_{mk},$
(46)

which results into

$\hat{\chi}^{III}_{mk}=B\cos\left(n\varphi\right),$ (47)

where $m^{2}+k=n^{2}/r^{2}$, $B$ is a normalization constant and one assumes
boundary conditions999With the intent of discarding the
$\sin\left(n\varphi\right)$ terms. such that $\hat{\chi}^{III}_{mk}=0$ at
$\varphi=\pm\pi$. This trivial solution will be important for the sphere
models, where one will be able to compare trivial constructions with a more
sophisticated configuration, namely that one engendered by model $IV$.

###  IV.3 Model $IV$

To simplify the analysis that follows, it is more convenient to work with
coordinates such that

$v=\frac{2}{\sqrt{C}}\arcsin\left[\tanh\left(\frac{\sqrt{C}y}{2}\right)\right],$
(48)

which, from Eq. (8), implies into the metric $IV$ recasted as

$g^{IV}=\frac{4\Lambda}{3C}\operatorname{sech}^{2}\left(\frac{\sqrt{C}y}{2}\right)e^{-2\tilde{A}}\omega_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+e^{-2\tilde{A}}\mathrm{d}u^{2}+e^{-2\tilde{A}}\operatorname{sech}^{2}\left(\frac{\sqrt{C}y}{2}\right)\mathrm{d}y^{2}.$
(49)

After a rescaling of the wave function,
$\hat{\chi}^{IV}_{mk}=\operatorname{sech}^{1/2}\left({\sqrt{C}y}/{2}\right)\hat{\xi}^{IV}_{mk}$,
the Schrödinger-like equation (36) becomes a Pöschl-Teller equation
Fluegge1998 ,

$-\hat{\xi}^{IV}_{mk,yy}-\lambda\left(\lambda-1\right)\operatorname{sech}^{2}\left(z\right)\hat{\xi}^{IV}_{mk}=E\;\hat{\xi}^{IV}_{mk},$
(50)

where $\lambda=2\sqrt{1+\frac{k}{C}}+\frac{1}{2}$,
$E=\frac{3m^{2}}{\Lambda}-\frac{17}{4}$ and $z={\sqrt{C}y}/{2}$. The general
solution of Eq. (50) is

$\hat{\xi}^{IV}_{mk}=B_{1}\cosh^{\lambda}\left(z\right){}_{2}F_{1}\left[\alpha,\beta;\frac{1}{2};-\sinh^{2}\left(z\right)\right]+B_{2}\cosh^{\lambda}\left(z\right)\sinh\left(z\right){}_{2}F_{1}\left[\alpha+\frac{1}{2},\beta+\frac{1}{2};\frac{3}{2};-\sinh^{2}\left(z\right)\right],$
(51)

with the parameters $\alpha=\frac{1}{2}\left(\lambda-\sqrt{-E}\right)$ and
$\beta=\frac{1}{2}\left(\lambda+\sqrt{-E}\right)$. If $\lambda\leq 1$ (i.e.
$k\leq-15C/16$) or $E>0$ (i.e. $m^{2}>\frac{17}{12}\Lambda$) the above
solutions correspond to propagating modes. Otherwise, imposing unitary
boundary conditions Gremm2000 ; HERRERAAGUILAR2010 suppresses all propagating
modes. In this case, the unitary spectrum shall be composed uniquely of bound
states101010Because they do not generate any flux into the singularities at
the boundaries of space..

The normalization condition for $\hat{\xi}_{mk}$, in terms of conformal
coordinates (48), reads

$0<\int^{\infty}_{-\infty}\left(\hat{\xi}^{IV}_{mk}\right)^{2}\mathrm{d}y\neq\infty.$
(52)

Therefore, normalizable modes will exist for $\lambda>1$ and negative $E$,
where

$\hat{\xi}^{IV+}_{jk}=B^{+}_{jk}\cosh^{2\sqrt{1+\frac{k}{C}}+\frac{1}{2}}\left(z\right){}_{2}F_{1}\left[\frac{1}{2}+j,2\sqrt{1+\frac{k}{C}}-j;\frac{1}{2};-\sinh^{2}\left(z\right)\right]$
(53)

represents normalizable even eigenstates, with mass eigenvalues

$m^{IV+}_{jk}=\sqrt{\frac{\Lambda}{3}\left[\frac{17}{4}-\left(2\sqrt{1+\frac{k}{C}}-\frac{1}{2}-2j\right)^{2}\right]},\;\;0\leq
j<\sqrt{1+\frac{k}{C}}-\frac{1}{4};$ (54)

and

$\hat{\xi}^{IV-}_{lk}=B^{-}_{lk}\cosh^{2\sqrt{1+\frac{k}{C}}+\frac{1}{2}}\left(z\right)\sinh\left(z\right){}_{2}F_{1}\left[\frac{3}{2}+l,2\sqrt{1+\frac{k}{C}}-l;\frac{3}{2};-\sinh^{2}\left(z\right)\right],$
(55)

represents normalizable odd eigenstates, with mass eigenvalues

$m^{IV-}_{lk}=\sqrt{\frac{\Lambda}{3}\left[\frac{17}{4}-\left(2\sqrt{1+\frac{k}{C}}-\frac{3}{2}-2l\right)^{2}\right]},\;\;0\leq
l<\sqrt{1+\frac{k}{C}}-\frac{3}{4};$ (56)

where $j$ and $l$ are natural numbers, and $B^{\pm}_{jk}$ are normalization
constants. Some normalizable modes are depicted in Fig. 3. The masses, as
described by Eqs. (54) and (56), are strictly real valued as long as
$-15C/16<k\leq 0$, while for positive values of $k$ some tachyonic states are
allowed.

Figure 3: (Color online) Normalized gravitational wave functions
$\hat{\chi}^{IV}$ for model $IV$ as functions of $z=v\sqrt{C}/2$, for
$k/C=-7/16$ (black lines), $k/C=0$ (blue lines), $k/C=9/16$ (red lines) and
$k/C=33/16$ (brown lines). The full, dashed and dot-dashed lines represent
$j=0$, $l=0$ and $j=1$, respectively.

Interestingly, model $IV$ does not have a singular Ricci scalar111111The
singularities at the edges of space are associated with the Kretschmann
scalar., but it exhibits a mass gap between the zero and massive modes. This
is noteworthy because one would expect that a mass gap would imply in
singularities of the Ricci scalar, which always happens for five dimensional
models HERRERAAGUILAR2010 . Thus, one may conjecture that mass gaps and naked
singularities are connected, but the singularity might be related with scalar
invariants other than the Ricci scalar.

###  IV.4 Model $V$

Finally, once applied to model $V$, Eq. (36) is cast into the form of

$-\hat{\chi}^{V}_{mk,vv}-\sqrt{\frac{\left|\Lambda\right|}{3}}\tan\left(\sqrt{3\left|\Lambda\right|}v\right)\hat{\chi}^{V}_{mk,v}-\left[\frac{4\left|\Lambda\right|\sec^{2}\left(\sqrt{3\left|\Lambda\right|}v\right)}{3}+k\right]\hat{\chi}^{V}_{mk}=m^{2}\hat{\chi}^{V}_{mk}.$
(57)

Rescaling $\hat{\chi}^{V}_{mk}$ by
$\cos^{1/6}\left(\sqrt{3\left|\Lambda\right|}v\right)$ transforms the above
result from Eq. (57) into a trigonometric Pöschl-Teller equation, with the
straightforward solution identified by

$\hat{\chi}^{V}_{l}=B^{V}_{l}\cos^{\frac{2}{3}}\left(\sqrt{3\left|\Lambda\right|}v\right)\,_{2}F_{1}\left[l,\frac{1}{2}-l;1;\cos^{2}\left(\sqrt{3\left|\Lambda\right|}v\right)\right],$
(58)

with the eigenvalues identified by

$m^{V}_{kl}=\sqrt{\left[12l\left(l-\frac{1}{2}\right)-\frac{2}{3}\right]|\Lambda|-k},$
(59)

where $B^{V}_{l}$ is a normalization constant.

The hypergeometric component, ${}_{2}F_{1}\left(\alpha,\beta;\gamma;z\right)$,
have arguments such that $\gamma-\alpha-\beta=1/2$, thus a discontinuity of
the first derivatives, at $v=0$, is only avoidable if either $\alpha$ or
$\beta$ is a non-positive integer $-j$, $j\in\mathbb{N}$ (see Appendix C for
further details). Otherwise, an unphysical discontinuity is identified for the
stress energy tensor of the perturbations, since the latter depends on the
first derivatives of (58). Therefore, the allowed values of $l$ are given by
$l=-j$ or $l=j+{1}/{2}$, both implying into the same results.

The normalization condition for model $V$ reads

$\int\left(e^{2\hat{A}-\hat{h}}\hat{\chi}_{mk}{}^{2}\right)\mathrm{d}v=\frac{\left(B^{V}_{j}\right)^{2}}{\sqrt{3\left|\Lambda\right|}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left[\sum_{i=0}^{j}\left(-1\right)^{i}\frac{j!}{\left(i!\right)^{2}\left(j-i\right)!}\frac{\Gamma\left(\frac{1}{2}+j+i\right)}{\Gamma\left(\frac{1}{2}+j\right)}\cos^{2i}\left(x\right)\right]^{2}\cos\left(x\right)\mathrm{d}x,$
(60)

which is clearly finite, since the integrand is bounded at the entire domain
$\left(-\pi/2,\pi/2\right)$ for every $j\in\mathbb{N}$. The normalized profile
of the wave function, (58), is depicted in Fig. 4a and, for some particular
values of $k$, the graviton masses are depicted in Fig. 4b.

Notice that the constant $k$ is defined out of the eigenvalue spectrum of Eq.
(35), which is exactly the same Schrödinger-like equation, with energy $-k$,
of the well known five dimensional braneworld models RS-1 ; RS-2 ;
Bernardini2016 ; Almeida2009 ; Bazeia2004 ; Dzhunushaliev2010 ; Bazeia2002 ;
DeWolfe2000 ; Ahmed2013 ; Gremm2000a ; Gremm2000 ; Kehagias2001 ;
Kobayashi2002 ; Bronnikov2003 ; BAZEIA2009b ; BarbosaCendejas2013 ; ZHANG2008
; Melfo2003 ; Bazeia2009a ; Koley2005 ; Bazeia2014 ; Chinaglia2016 ;
Bernardini2013 .

(a)  (b)

Figure 4: (Color online) (a) Normalized wave functions $\hat{\chi}^{V}_{j}$ of
model $V$ as functions of $\varphi=v\sqrt{3\left|\Lambda\right|}$ for $j=0$
(full line), $j=1$ (dashed line), $j=2$ (dotted line) and $j=3$ (dot-dashed
line). (b) The graviton masses $m^{V}_{kj}$ of model V, with $\Lambda=-1$, for
$k=-2\times 10^{3}$ (black dots), $k=-10^{3}$ (blue dots), $k=0$ (red dots),
$k=10^{3}$ (brown dots) and $k=2\times 10^{3}$ (purple dots).

Before discussing sphere models, a short note about occurrence of
instabilities in the above discussed models is pertinent. As identified, the
scalar field perturbation is completely decoupled from tensorial ones, thus it
be addressed separately. If the potential for the scalar fields do not present
a local minima, occurrence of instabilities is not discarded. However, the
dynamics of the scalar field is not only governed by the potential, but also
by the curvature of space, i.e. $A$, $h$ and $f$. Thus, even if there not
being local minima from $\mathcal{V}$, stable configurations are possible
under particular conditions. As an example, consider models $III$, $IV$ and
$V$ Gauy2022 . The scalar field $\phi$ follows a similar structure to the ones
found in five dimensional models Gauy2022 ; Gremm2000 ; Afonso2006 ;
Sasakura2002 , thus should be stable by extension. Since model $IV$ only
presents one scalar field ($\phi$) it should be stable. On the other hand,
scalar field $\zeta$ could exhibit instabilities in models $III$ and $V$,
since potential $\mathcal{V}$ is independent on $\zeta$. But the existence of
curvature implies in the constraint equation,

$\square\left(\zeta+\delta\zeta\right)=0\implies\hat{\square}\left(\delta\zeta\right)-4\,\hat{\sigma}^{ij}A_{,i}\left(\delta\zeta\right)_{,j}=0.$
(61)

Therefore, the scalar field $\zeta$ satisfies the same stability equation
(with $\Lambda=0$) as the gravitational field (cf. Eq. (19)), and should also
be stable121212Similarly, the stability analysis for models $I$ and $II$ is
more intricate, since the potential depends on both $\phi$ and $\zeta$..

###  IV.5 The $\mathbb{S}^{2}$ Models

The sphere models based on models $III$ and $IV$ Gauy2022 are achieved by
assuming that $\tilde{A}=-\ln\left(\sin\theta\right)$, $\tilde{f}=0$,
$u=r\theta$ and $v=r\varphi$. Therefore, Eq. (35) can be written as

$-\frac{\left[\sin\left(\theta\right)\tilde{\chi}^{\zeta}_{k,\theta}\right]_{,\theta}}{\sin\left(\theta\right)}+\left[\frac{4+kr^{2}}{\sin^{2}\left(\theta\right)}-6\right]\tilde{\chi}^{\zeta}_{k}=0,$
(62)

which has the solution

$\tilde{\chi}^{\zeta}_{k}=B_{1k}P_{2}^{\sqrt{kr^{2}+4}}\left(\cos\left(\theta\right)\right)+B_{2k}Q_{2}^{\sqrt{kr^{2}+4}}\left(\cos\left(\theta\right)\right),$
(63)

where $B_{1k}$ and $B_{2k}$ are normalization constants.

To avoid singularities and trivial null solutions (which would be non-
normalizable), expression (63) restricts the possible values of $k$. More
precisely, normalizable modes exclusively achieved for $k={-3}/{r^{2}}$ or
$k=0$ are both depicted in Fig. 5. However, the constant $B_{2k}$ must be
fixed to zero, since $Q_{2}^{\sqrt{kr^{2}+4}}$ is non-normalizable in this
space for all $k$.

Figure 5: (Color online) Normalized gravitational wave functions
$\tilde{\chi}^{\zeta}_{k}$ for the sphere models, for $k=0$ (full black line)
and $k=-3/r^{2}$ (black dashed line).

####  IV.5.1 Model $III$ $(C=0)$

Let the sphere models be constructed out of model $III$, with $C=0$, which is
supported by the metric

$\bm{g}^{III}_{\zeta}=\sin^{2}\left(\theta\right)\left(\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+r^{2}\mathrm{d}\varphi^{2}\right)+r^{2}\mathrm{d}\theta^{2}.$
(64)

The gravitational massive modes are then identified by

$\chi^{III}_{\frac{\sqrt{n^{2}+3}}{r}}=B_{n1}\cos\left(n\varphi\right)\cos\left(\theta\right)\sin\left(\theta\right)\text{,
}n\neq 1;$ (65)
$\chi^{III}_{\frac{n}{r}}=B_{n2}\cos\left(n\varphi\right)\sin^{2}\left(\theta\right)\text{,
}n\neq 2;$ (66)
$\chi^{III}_{\frac{2}{r}}=B_{22}\cos\left(2\varphi\right)\sin^{2}\left(\theta\right)+B_{11}\cos\left(\varphi\right)\cos\left(\theta\right)\sin\left(\theta\right).$
(67)

In this case, a point-like source, placed at
$\left(\theta=\pi/2,\varphi=0\right)$, produces a Newtonian
potential131313Here one assumes that $B_{11}=0$ for simpler equations. of the
form

$\phi^{III}_{N}(\mathsf{r})=\frac{3\mathcal{M}}{2^{8}\pi^{2}r^{2}M^{4}}\frac{1}{\mathsf{r}}\left(1+\frac{2}{e^{\frac{\mathsf{r}}{r}}-1}\right)\approx\begin{cases}\displaystyle\frac{3\mathcal{M}}{2^{8}\pi^{2}r^{2}M^{4}}\frac{1}{\mathsf{r}}\text{,
if }&\mathsf{r}\gg r\\\ \hfill\\\
\displaystyle\frac{3\mathcal{M}}{2^{7}\pi^{2}rM^{4}}\frac{1}{\mathsf{r}^{2}}\text{,
if }&\mathsf{r}\ll r\end{cases}.$ (68)

Thus, in the region $\mathsf{r}\gg r$, the corrections to the Newtonian
potential are exponentially suppressed, and one recovers the ordinary
Newtonian theory. Likewise, whenever $\mathsf{r}\ll r$, the result from (68)
is dominated by the higher dimensional term, thus gravity behaves as if the
universe was five dimensional. In summary, the Newtonian potential behaves
exactly like in a five dimensional ADD model ArkaniHamed1998a . On the other
hand, the Newtonian gravitational constant is

$G^{III}_{N}=\frac{3}{2^{8}\pi^{2}r^{2}M^{4}},$ (69)

which is similar to the Newtonian constant as determined from a six
dimensional ADD model ArkaniHamed1998a .

####  IV.5.2 Model $IV$

Model $IV$ with a spherical internal space is characterized by the metric
Gauy2022 :

$\bm{g}^{IV_{n}}_{\zeta}=\frac{4r^{2}\Lambda}{3n^{2}}\cos^{2}\left(\frac{n\varphi}{2}\right)\sin^{2}\left(\theta\right)\omega^{+}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}+r^{2}\mathrm{d}\theta^{2}+r^{2}\sin^{2}\left(\theta\right)\mathrm{d}\varphi^{2},$
(70)

where $n\in\left\\{1,2\right\\}$ and represents two distinct configurations of
spherical models, which shall be labeled $IV_{1}$ and $IV_{2}$, for $n=1$ and
$n=2$, respectively.

Correspondently, two normalizable graviton modes are identified,

$\chi^{IV_{1}}_{\frac{2\Lambda}{3}}=B_{1}\cos^{2}\left(\frac{\varphi}{2}\right)\sin^{2}\left(\theta\right),$
(71)

and

$\chi^{IV_{1}}_{\frac{4\Lambda}{3}}=B_{2}\sin\left(\frac{\varphi}{2}\right)\cos^{2}\left(\frac{\varphi}{2}\right)\sin^{2}\left(\theta\right),$
(72)

where $B_{1}$ and $B_{2}$ are normalization constants, and
$\chi^{IV_{1}}_{\frac{2\Lambda}{3}}$ and $\chi^{IV_{1}}_{\frac{4\Lambda}{3}}$
represent gravitons of mass $m=\sqrt{2\Lambda/3}$ and $m=2\sqrt{\Lambda/3}$,
respectively. The Newtonian potential associated with model $IV_{1}$ is

$\phi^{IV_{1}}_{N}(\mathsf{r})=\frac{9\mathcal{M}}{2^{9}r^{4}\Lambda\,\pi^{2}M^{4}}\frac{e^{-\sqrt{\frac{2\Lambda}{3}}\mathsf{r}}}{\mathsf{r}}\approx\frac{9\mathcal{M}}{2^{9}r^{4}\Lambda\,\pi^{2}M^{4}}\frac{1}{\mathsf{r}},$
(73)

where the last approximation reinforces our choice of displacements much
smaller than the cosmological scale, i.e. $\mathsf{r}\ll 1/\sqrt{\Lambda}$.
Therefore, as far as the approximations here considered, model $IV_{1}$
implies, precisely, in Newton’s theory, with the gravitational constant

$G^{IV_{1}}_{N}=\frac{9}{2^{9}r^{4}\Lambda\,\pi^{2}M^{4}},$ (74)

which is dependent upon the cosmological constant $\Lambda$ and the radius $r$
of $\mathbb{S}^{2}$.

On the other hand, a spherical model with $n=2$ ($IV_{2}$) implies into two
normalizable gravitational modes:

${\chi}^{IV_{2}}_{\frac{2\Lambda}{3}}=B_{1}\cos^{2}\left(\varphi\right)\sin^{2}\left(\theta\right),$
(75)

and

${\chi}^{IV_{2}}_{\frac{4\Lambda}{3}}=B_{2}\sin\left(\varphi\right)\cos^{2}\left(\varphi\right)\sin^{2}\left(\theta\right)+B_{3}\cos\left(\varphi\right)\sin\left(\theta\right)\cos\left(\theta\right),$
(76)

where $B_{1}$, $B_{2}$ and $B_{3}$ are normalization constants.

The Newtonian potential for such a configuration is expressed by

$\phi^{IV_{2}}_{N}(\mathsf{r})=\frac{9\mathcal{M}}{2^{7}\pi^{2}M^{4}r^{4}\Lambda}\frac{e^{-\sqrt{\frac{2\Lambda}{3}}\mathsf{r}}}{\mathsf{r}}\approx\frac{9\mathcal{M}}{2^{7}\pi^{2}M^{4}r^{4}\Lambda}\frac{1}{\mathsf{r}},$
(77)

where, once again, displacement are not of cosmological scale. Therefore model
$IV_{2}$ also implies in Newtonian theory, with the gravitational constant

$G^{IV_{2}}_{N}=\frac{9}{2^{7}\pi^{2}M^{4}r^{4}\Lambda},$ (78)

which is a similar result to that one from model $IV_{1}$.

Had one proposed a more realistic model for matter141414Which is outside the
scope of the present paper., then Eqs. (72) and (76) would have significant
effects in both the gravitational constant and Newtonian potential, which
could imply into significant phenomenological differences between models
$IV_{1}$ and $IV_{2}$. The same is true for model $III$: in a more realistic
description, Eq. (65) would modify Eqs. (68) and (69).

##  V Conclusion

The gravitational fluctuations and phenomenological implications of some novel
braneworld models (cf. Ref. Gauy2022 ) have been explored. For the so-called
models $I$, $II$ and $III$, zero modes have been computed, and for models $IV$
and $V$, the entire spectrum of graviton modes were analytically identified.
In addition, the implications of spherical internal spaces have been
investigated for models $III$ and $IV$.

Generically, in contrast to ordinary five dimensional models, six dimensional
models rely on a more intricate quantum mechanical two dimensional analogue
problem, where the assessment of the localization of gravity is driven by the
curved space co-dimensional variables, $(\mathbb{B}^{2},\bm{\hat{\sigma}})$,
rendering a Schrödinger-like equation for each co-dimension, from which
localization modes have been identified.

Despite the reiterated physical appealing limitations of models $I$ and $II$
(cf. Ref. Gauy2022 ), the profile of the gravitational constant $G_{N}$ has
been obtained, at least, as a pedagogical exercise. Results from model $I$
attest that the gravitational strength may be as small as one wishes by
setting the parameter $p$ close to $3$, instead of reflecting some usual brane
“radius” (parameters $c_{u}$ and $c_{v}$) tuning effect. In opposition, model
$II$ behaves precisely as one would expect out of braneworlds: the
gravitational strength is driven by the parameters $c_{u}$ and $c_{v}$.

More problematically, model $III$ implied into a Schrödinger-like equation
whose massive states are not straightforwardly determined, with the exception
for a trivial extension of five dimensional models ($C=0$), for which massive
states are similar to flat compactified models.

The most physically appealing intersecting braneworld, model $IV$, revealed a
Schrödinger-like equation with a Pösch-Teller potential, admitting a finite
and discrete number of gravitons, whose masses are bounded from above by an
interplay between the separation constant $k$ and the cosmological
constant151515For instance, if $k$ is non-positive one can find at most two
massive gravitons. $\Lambda$. Any propagating mode was summarily discarded
after imposing unitary boundary conditions, thus rendering the singularities
at the boundaries of space harmless. It is also noteworthy that model $IV$
presents a mass gap between the zero and massive modes, all the while
retaining a smooth Ricci scalar but a singular Kretschmann scalar. This is in
contrast to five dimensional models, where mass gaps and a singular Ricci
scalar are directly connected HERRERAAGUILAR2010 . Thus we conjectured that
the existence of a mass gap in higher dimensional braneworlds, other than
five, may be connected to naked singularities of other scalar invariants
besides the Ricci scalar.

The unique model constructed out of an anti-de Sitter brane, model $V$, has
alluded to a Schrödinger-like equation with a trigonometric Pösch-Teller
potential and a discrete set of gravitational modes. Akin to the usual result
in braneworlds, model $V$ presents countably, but infinite, gravitons whose
masses are bounded from below by the value of the separation constant $k$.

Finally, concerning some technicalities pointed out by Ref. Gauy2022 , through
models $III$ and $IV$, one could also construct braneworld models whose
internal space are spheres. The spherical models constructed out of model
$III$, but with $C=0$, rely on a trivial extension of five dimensional
braneworlds. Remarkably, such trivial extensions imply into a gravitational
interaction that behaves like a five dimensional ADD model, while the
gravitational constant behaves like a six dimensional one. On the other hand,
sphere models constructed out of model $IV$ implied in two normalizable modes
and the usual Newtonian interaction, with the gravitational constant being
determined by a combination of the cosmological constant ($\Lambda$) and the
radius of $\mathbb{S}^{2}$ ($r$).

Just to conclude, considering that the grounds for the above discussed
intersecting braneworld models have been established, the localization of
scalar, gauge and spinor fields shall be in the core of our subsequent
investigations.

###### Acknowledgements.

HMG is grateful for the financial support provided by CNPq (Grant No.
141924/2019-5). The work of AEB is supported by the Brazilian Agencies FAPESP
(Grant No. 20/01976-5) and CNPq (Grant No. 301000/2019-0).

##  Appendix A On the decoupling between tensorial and scalar perturbations

In Sec. III, the perturbations were cast form of Eq. (10). Just as a
preliminary consideration, the general form from Eq. (10) should involve
perturbations of the warp factor, yet this can be disregarded because the
latter can always be encompassed by $\varpi_{\mu\nu}$, as follows,

$\displaystyle\bar{\bm{g}}=$ $\displaystyle
e^{-2\bar{A}}\left(\omega_{\mu\nu}+\varpi_{\mu\nu}\right)\mathsf{d}x^{\mu}\mathsf{d}x^{\nu}+\sigma_{ij}\mathsf{d}y^{i}\mathsf{d}y^{j}$
(79) $\displaystyle=$ $\displaystyle
e^{-2A}\left(\omega_{\mu\nu}+\varpi_{\mu\nu}-2\delta
A\,\omega_{\mu\nu}\right)\mathsf{d}x^{\mu}\mathsf{d}x^{\nu}+\sigma_{ij}\mathsf{d}y^{i}\mathsf{d}y^{j},$
(80)

where $\bar{A}=A+\delta A$ and $\delta A$ is the perturbation of the warp
factor. In which concerns the scalar perturbation related to $\delta A$, every
calculation of Sec. III follows straightforwardly from defining
$\tilde{\varpi}_{\mu\nu}=\varpi_{\mu\nu}-2\delta A\,\omega_{\mu\nu}$.

Of course, to clear up the above statement, from a more explicit approach,
once that one assumes the metric to be (79), the gravitational portion of the
action, if expanded up to the second order in the perturbative parameters, can
be written as

$\displaystyle\bar{S}_{g}=$ $\displaystyle
2{M}^{4}\int\mathrm{d}^{6}x\sqrt{-\bar{\mathsf{g}}}\;\bar{R}$ $\displaystyle=$
$\displaystyle
2{M}^{4}\int\mathrm{d}^{6}xe^{-2\bar{A}}\sqrt{-\omega\sigma}\left[\mathcal{R}+e^{-2\bar{A}}\left(\Sigma-2\triangle^{2}\bar{A}-20\bar{A}^{,i}\bar{A}_{,i}\right)+10\hat{\square}\bar{A}-20\bar{A}^{,\mu}\bar{A}_{,\mu}-\varpi_{\mu\nu}\mathcal{R}^{\mu\nu}-20\varpi^{\mu\nu}A_{,\mu}\bar{A}_{,\nu}\right]$
$\displaystyle+2{M}^{4}\int\mathrm{d}^{6}xe^{-2A}\sqrt{-\omega\sigma}\Bigg{[}\hat{\nabla}_{S}\left(\hat{g}^{SN}\varpi^{\mu\nu}\hat{\nabla}_{N}\varpi_{\mu\nu}-\hat{g}^{S\eta}\varpi^{\mu\nu}\hat{\nabla}_{\nu}\varpi_{\mu\eta}\right)-20{\varpi^{\mu}}_{\kappa}\varpi^{\kappa\nu}A_{,\mu}A_{,\nu}-\frac{1}{4}\varpi^{\mu\nu}\varpi_{\mu\nu}\hat{R}\Bigg{.}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+10{\varpi^{\mu}}_{\kappa}\varpi^{\kappa\nu}\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}A-\frac{1}{4}\hat{\nabla}^{K}\varpi^{\mu\nu}\hat{\nabla}_{K}\varpi_{\mu\nu}-\frac{5}{2}\varpi^{\mu\nu}\varpi_{\mu\nu}\hat{\square}A+5\varpi^{\mu\nu}\varpi_{\mu\nu}A^{,P}A_{,P}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad\,\,\Bigg{.}+\frac{1}{2}\hat{\nabla}^{\nu}\varpi^{\mu\kappa}\hat{\nabla}_{\kappa}\varpi_{\mu\nu}+{\varpi^{\mu}}_{\kappa}\varpi^{\kappa\nu}\mathcal{R}_{\mu\nu}+5\hat{g}^{SJ}\varpi^{\mu\nu}\left(2\hat{\nabla}_{\nu}\varpi_{\mu
J}-\hat{\nabla}_{J}\varpi_{\mu\nu}\right)A_{,S}\Bigg{]}.$ (81)

The only terms that can exhibit some coupling between the tensorial,
$\varpi_{\mu\nu}$, and warp factor, $\delta A$, perturbations are161616There
are also $\varpi^{MN}\hat{\nabla}_{M}\hat{\nabla}_{N}\delta A$ and
$\varpi_{MN}\hat{g}^{MN}\hat{g}^{AB}\hat{\nabla}_{A}\hat{\nabla}_{B}\delta A$,
which are either null
$\left(\varpi_{MN}\hat{g}^{MN}\hat{g}^{AB}\hat{\nabla}_{A}\hat{\nabla}_{B}\delta
A=0\right)$ or contribute only as a boundary term
$\left(\varpi^{MN}\hat{\nabla}_{M}\hat{\nabla}_{N}\delta
A=\hat{\nabla}_{M}\left(\varpi^{MN}\hat{\nabla}_{N}\delta A\right)\right)$ in
the transverse traceless gauge
($\hat{\nabla}^{M}\varpi_{MN}=\hat{g}^{MN}\varpi_{MN}=0$). Therefore, one has
a vanishing contribution from
$\left(\partial_{\mu}\partial_{\nu}-\frac{1}{4}\omega_{\mu\nu}\omega^{\alpha\beta}\partial_{\alpha}\partial_{\beta}\right)\delta
A$, i.e. the fluctuations of the scalar contributions are completely decoupled
from the transverse traceless gravitational fluctuations, which is consistent
with preliminary results in six dimensional models Csaki2000 .
$e^{-2\bar{A}}\varpi_{\mu\nu}\mathcal{R}^{\mu\nu}$ and
$e^{-2\bar{A}}\varpi^{\mu\nu}{A}_{,\mu}\bar{A}_{,\nu}$. By construction, these
terms are null, since $A_{,\mu}=\phi_{,\mu}=\zeta_{,\mu}=0$. Therefore, there
is no coupling between $\varpi_{\mu\nu}$ and $\delta A$, such that neither
$\varpi_{\mu\nu}$ nor $\delta A$ do affect one each other.

Likewise, the effects of the scalar fields ($\phi$ and $\zeta$) on the
localization of gravitational fields can also be evaluated. For the metric
described by Eq. (10), the action of the scalar fields is written as

$\displaystyle\bar{S}_{\bar{\phi},\bar{\zeta}}=$
$\displaystyle-\int\mathrm{d}x^{6}\sqrt{-\bar{g}}\left[\frac{\bar{g}^{MN}}{2}\bar{\phi}_{,M}\bar{\phi}_{,N}+\frac{\bar{g}^{MN}}{2}\bar{\zeta}_{,M}\bar{\zeta}_{,N}+\mathcal{V}\left(\bar{\phi},\bar{\zeta}\right)\right]$
$\displaystyle=$
$\displaystyle-\int\mathrm{d}x^{6}\sqrt{-g}\left(1-\frac{1}{4}\varpi^{MN}\varpi_{MN}\right)\left[\frac{\bar{g}^{MN}}{2}\bar{\phi}_{,M}\bar{\phi}_{,N}+\frac{\bar{g}^{MN}}{2}\bar{\zeta}_{,M}\bar{\zeta}_{,N}+\mathcal{V}\left(\bar{\phi},\bar{\zeta}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int\mathrm{d}x^{6}\sqrt{-g}\,\varpi^{MN}\varpi_{MN}\left[\frac{g^{MN}}{2}\phi_{,M}\phi_{,N}+\frac{g^{MN}}{2}\zeta_{,M}\zeta_{,N}+\mathcal{V}\left(\phi,\zeta\right)\right]$
$\displaystyle\qquad\qquad\qquad-\int\mathrm{d}x^{6}\sqrt{-g}\left[\frac{{g}^{MN}}{2}\bar{\phi}_{,M}\bar{\phi}_{,N}+\frac{{g}^{MN}}{2}\bar{\zeta}_{,M}\bar{\zeta}_{,N}+\frac{\varpi^{MN}}{2}\bar{\phi}_{,M}\bar{\phi}_{,N}+\frac{\varpi^{MN}}{2}\bar{\zeta}_{,M}\bar{\zeta}_{,N}+\mathcal{V}\left(\bar{\phi},\bar{\zeta}\right)\right],$
(82)

where $\bar{\phi}=\phi+\delta\phi$ and $\bar{\zeta}=\zeta+\delta\zeta$. The
only relevant terms for the coupling between tensorial and scalar
perturbations are $\varpi^{MN}\bar{\phi}_{,M}\bar{\phi}_{,N}$ and
$\varpi^{MN}\bar{\zeta}_{,M}\bar{\zeta}_{,N}$, which once again are evidently
null since $\varpi^{MN}\phi_{,M}=\varpi^{MN}\zeta_{,M}=0$.

Hence, to summarize, the tensorial fluctuations, $\varpi_{\mu\nu}$, are
completely decoupled from the scalar perturbations, i.e. $\delta A$,
$\delta\phi$ and $\delta\zeta$, and the action that drives the dynamics of
$\varpi_{\mu\nu}$ can be cast in the form of (20), as pointed out in Ref.
Csaki2000 . In the above context, one can disregard the perturbations of the
scalar fields when dealing with the dynamics of gravity.

##  Appendix B The point-like particle stress energy tensor

From a general standpoint, one may write the action of matter fields as

$S_{\mathcal{M}}=\int\mathrm{d}^{6}x\sqrt{-\bar{g}}\mathcal{L}_{\mathcal{M}}(\Phi,\bar{\bm{g}}),$
(83)

where $\Phi$ represents all matter fields. If
$\bar{\bm{g}}=\bm{g}+\bm{\varpi}$, where $\bm{\varpi}$ is a perturbation, then
the matter action to first order in $\varpi$ takes the form

$S_{\mathcal{M}}=\int\mathrm{d}^{6}x\sqrt{-g}\mathcal{L}_{\mathcal{M}}\left(\Phi,\bm{g}\right)+\sum_{m\in\mathbb{I}}\int\mathrm{d}^{6}xe^{-6A}\sqrt{-\hat{g}}\;\;\chi_{m}\left({\widetilde{\varpi}_{m}}\right)_{\mu\nu}\frac{{T}^{\mu\nu}}{2},$
(84)

where ${T}^{\mu\nu}$ is the stress energy tensor calculated out of
$\mathcal{L}_{\mathcal{M}}(\Phi,\bm{g})$. On the other hand, if the matter is
regarded as a point-like particle of mass $\mathcal{M}$, then the stress
energy tensor is,

$S_{p}=\mathcal{M}\int\mathrm{d}\tau\sqrt{-{g}_{MN}\mathsf{v}^{M}\mathsf{v}^{N}}\implies
T^{MN}(x^{Q})=\frac{\mathcal{M}\delta(x^{Q}-x^{P})\mathsf{v}^{M}(x^{P})\mathsf{v}^{N}(x^{P})}{\sqrt{-g}}.$
(85)

Substituting (85) into (84) reduces to (28).

##  Appendix C On the smoothness of hypergeometric functions

Assume a general wave function of the form

$\chi=\cos^{l}\left(y\right)\,{}_{2}F_{1}\left[\alpha,\beta;\gamma;\cos^{2}\left(y\right)\right],$
(86)

its first derivative is simply

$\chi_{,y}=-l\cos^{l-1}\left(y\right)\sin\left(y\right)\,{}_{2}F_{1}\left[\alpha,\beta;\gamma;\cos^{2}\left(y\right)\right]-2\sin\left(y\right)\cos^{l+1}\left(y\right)\,{}_{2}F_{1}^{\prime}\left[\alpha,\beta;\gamma;\cos^{2}\left(y\right)\right].$
(87)

The derivatives of hypergeometric functions follow the rule

${}_{2}F_{1}^{\prime}\left[\alpha,\beta;\gamma;z\right]=\frac{\mathrm{d}\;{}_{2}F_{1}\left[\alpha,\beta;\gamma;z\right]}{\mathrm{d}z}=\frac{\alpha\beta}{\gamma}\,{}_{2}F_{1}\left[\alpha+1,\beta+1;\gamma+1;z\right].$
(88)

Substituting into $\chi_{,y}$ one finds

$\chi_{,y}=\cos^{l-1}\left(y\right)\sin\left(y\right)\left\\{-l\,{}_{2}F_{1}\left[\alpha,\beta;\gamma;\cos^{2}\left(y\right)\right]-2\cos^{2}\left(y\right)\frac{\alpha\beta}{\gamma}\,{}_{2}F_{1}\left[\alpha+1,\beta+1;\gamma+1;\cos^{2}\left(y\right)\right]\right\\}.$
(89)

Let the parameters $\alpha$, $\beta$ and $\gamma$ satisfy
$\gamma-\alpha-\beta=1/2$, then
${}_{2}F_{1}\left[\alpha+1,\beta+1;\gamma+1;\cos^{2}\left(y\right)\right]$
will not converge at $\cos^{2}\left(y\right)=1$. To analyze this divergence
further one uses the following property of hypergeometric functions
Fluegge1998 :

$\displaystyle{}_{2}F_{1}\left(\alpha,\beta;\gamma;z\right)$
$\displaystyle=\frac{\Gamma(\gamma)\Gamma(\gamma-\alpha-\beta)}{\Gamma(\gamma-\alpha)\Gamma(\gamma-\beta)}{}_{2}F_{1}\left(\alpha,\beta;\alpha+\beta-\gamma+1;1-z\right)$
$\displaystyle\qquad\qquad+\frac{\Gamma(\gamma)\Gamma(\alpha+\beta-\gamma)}{\Gamma(\alpha)\Gamma(\beta)}\left(1-z\right)^{\gamma-\alpha-\beta}{}_{2}F_{1}\left(\gamma-\alpha,\gamma-\beta;\gamma-\alpha-\beta+1;1-z\right),$
(90)

thus the first derivative satisfy

$\displaystyle\chi_{,y}=$
$\displaystyle-l\cos^{l-1}\left(y\right)\sin\left(y\right)\,{}_{2}F_{1}\left[\alpha,\beta;\gamma;\cos^{2}\left(y\right)\right]$
$\displaystyle\qquad\qquad-2\cos^{l+1}\left(y\right)\sin\left(y\right)\frac{\alpha\beta}{\gamma}\frac{\Gamma(\gamma+1)\Gamma(-1/2)}{\Gamma(\gamma-\alpha)\Gamma(\gamma-\beta)}{}_{2}F_{1}\left[\alpha+1;\beta+1,3/2;\sin^{2}\left(y\right)\right]$
$\displaystyle\qquad\qquad\qquad\qquad-2\cos^{l+1}\left(y\right)\sin\left(y\right)\frac{\alpha\beta}{\gamma}\frac{\Gamma(\gamma+1)\Gamma(1/2)}{\Gamma(\alpha+1)\Gamma(\beta+1)}\left[\sin^{2}\left(y\right)\right]^{-1/2}{}_{2}F_{1}\left[\gamma-\alpha,\gamma-\beta;1/2;\sin^{2}\left(y\right)\right],$
(91)

and at the vicinity of $y=0$ it behaves like

$\chi_{,y}\Big{|}_{y\rightarrow
0}=-2\,\text{sign}\left[\sin\left(y\right)\right]\frac{\alpha\beta}{\gamma}\frac{\Gamma(\gamma+1)\Gamma(1/2)}{\Gamma(\alpha+1)\Gamma(\beta+1)},$
(92)

leading to a discontinuity unless $\alpha$ or $\beta$ is a non-positive
integer $-j$, $j\in\mathbb{N}$.

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