aainstitutetext: School of Physics and Electronics Science, Hunan University,
Changsha 410082, Chinabbinstitutetext: Lanzhou Center for Theoretical Physics,
Key Laboratory of Theoretical Physics of Gansu Province, and Key Laboratory of
Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou, Gansu
730000, Chinaccinstitutetext: College of Physics, Nanjing University of
Aeronautics and Astronautics, Nanjing 211106, China

# Localization of q-form fields on a de Sitter brane in chameleon gravity

Yi Zhong c,* Tao-Tao Sui ***Corresponding authors<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recently, it was found that the vector field can be naturally localized on the
thick brane in chameleon gravity. In this work, we extend this study to
encompass de Sitter brane scenario. We focus on the localization of q-form
fields. The scalar and vector fields can be localized on the de Sitter brane,
while the KR field cannot be localized. The condition for localization of the
scalar and vector fields is obtained. Furthermore, we investigate the
localization characteristics in two examples with given conformal factor
$b(\phi)$. In the first case, the effective potentials and KK modes of the
matter fields are obtained asymmetric even though the de Sitter brane has
$Z_{2}$ symmetry. In the second case, volcano-like effective potentials are
generated in the de Sitter brane model.

##  1 Introduction

The idea that there may exist an extra spatial dimension has been an
interesting topic ever since the proposal of the Kaluza-Klein (KK) theory in
the 1920s. An intriguing theory within this field is the Randall-Sundrum (RS)
model, which offers a potential solution to a perplexing puzzle in physics
known as the hierarchy problem Randall1999 ; Randall1999a . This problem
revolves around the question of why gravity seems weaker compared to the other
fundamental forces. In the RS model, the brane is a four-dimensional
hypersurface embedded in a five-dimensional spacetime known as the ”bulk.”
However, due to the geometrically thin nature of the brane, the curvature
becomes divergent on the brane. Hence, the concept of the thick brane or
domain wall is introduced, wherein the domain wall is a topological defect of
the background scalar field in the bulk Gremm2000b ; Gass1999 ; Gremm2000 ;
Afonso2006 . In the braneworld scenario, gravity is free to propagate in the
bulk, while the matter fields are confined on the brane. As a result, the
essential role of the localization mechanism for matter fields emerges within
the braneworld theory’s framework. This mechanism not only yields a four-
dimensional effective theory at low energy scales but also provides potential
effects of the extra dimension Bajc2000 ; Oda2000 ; Melfo2006 ; Liu2009 ;
Almeida2009 ; Liang2009 ; Vaquera-Araujo2015 ; Salvio2007 . In other words,
the action of matter fields in five-dimensional spacetime can be dimensionally
reduced to a zero-mass four-dimensional action along with a series of massive
four-dimensional actions. The zero-mass four-dimensional action corresponds to
particles in the Standard Model, while the massive four-dimensional actions
correspond to excited KK particles at high energy scales. The free scalar
fields can be localized on the branes , while the localization of fermion
fields requires the Yukawa couplings with background scalar field Liu2018b .
The localization of the q-form fields has been investigated in Refs.Fu2012 ;
Fu2014 ; Fu2016 . However, the localization mechanism of the vector field
remains an open question. Some researchers have sought to address this by
introducing couplings between the vector field and the background scalar
field, coupling between the vector field and spacetime curvature to tackle the
issue of vector field localization Vaquera-Araujo2015 ; Chumbes2012a ;
Belchior2023b .

On the other hand, general relativity has been found to encounter several
challenges, encompassing both phenomenological and theoretical issues. Among
these challenges are the dark energy problem, and the singularity problem. In
order to address these intricate issues, numerous alternative theories of
gravity have been put forth. And the braneworld theory has also been extended
to the framework of modified gravities and new features have been found
Liu2018c . For example, it was found that thick branes can be generated by
pure geometry in Weyl gravity and $f(R)$ gravity Barbosa-Cendejas2006b ;
Zhong2016 , and may have inner structure in mimetic gravity and $f(T)$ gravity
Zhong2018 ; Yang2012 .

Recently, it was found that the vector field can be naturally localized on the
thick brane in the chameleon gravity Zhong2023 . The chameleon gravity was
proposed to explain the dark energy problem Khoury2004 ; Khoury2004a . In this
theory, matter fields couple to a physical metric $\widetilde{g}_{\mu\nu}$,
and $\widetilde{g}_{\mu\nu}$ is related to the metric $g_{\mu\nu}$ which
describe the spacetime geometry by a conformal transformation with a function
of the chameleon scalar field $\phi$, i.e.
$\widetilde{g}_{\mu\nu}=b^{2}(\phi)g_{\mu\nu}$. The chameleon scalar field
varies in mass depending on the density of surrounding matter. This
environmental nature of the screening means that such a scalar field may be
relevant on other scales Burrage2018 . The highly-nonlinear nature of
screening mechanisms means that they evade classical fifth-force searches, and
there has been an intense effort towards designing new and novel tests to
probe them, both in the laboratory and using astrophysical objects, and by
reinterpreting existing datasets Burrage2018 . A similar screen mechanism has
been studied in $f(R)$ gravity, and this type of $f(R)$ gravity is
mathematically equivalent to chameleon gravity, where the function $f(R)$
corresponds to the conformal factor $b(\phi)$ in chameleon gravity
DeFelice2010 ; Burrage2018 .

In chameleon gravity, various matter fields couple to the chameleon scalar
field $\phi$ through the physical metric $\widetilde{g}_{\mu\nu}$, and this
provides a potential mechanism to localize the vector field. Brane cosmology
in chameleon gravity was investigated in Refs. Saaidi2012 ; Bisabr2017 . In
Ref. Zhong2023 , localization of various matter fields on a flat brane was
investigated and new features were found. Considering the accelerated
expansion of our universe, there is a compelling incentive to extend this
study to encompass the de Sitter brane scenario. See Refs. Liu2009a ; Liu2009b
; Guerrero2019 ; Yang2020 ; Guo2013  for works on the localization of matter
fields on a de Sitter brane. In this work, we will focus on the localization
of q-form fields on a de Sitter brane in chameleon gravity.

The layout of the paper is as follows: In Section 2, we construct a de Sitter
brane in the framework of chameleon gravity. In Section 3, we investigate the
localization of q-form fields on the de Sitter brane. In Section 4, we provide
the specific form of the conformal factor $b(\phi)$ and investigate two cases
with novel localization properties. Throughout this paper, $M,N,\cdots$ denote
the indices of the five-dimensional coordinates, and $\mu,\nu,\cdots$ denote
the ones on the brane.

##  2 De Sitter brane in chameleon gravity

In this study, we begin with the following five-dimensional action which
describes a canonical scalar field minimally coupled to gravity Burrage2018a :

$S=\int
d^{5}x\sqrt{-g}\left[\frac{1}{2}R-\frac{1}{2}\nabla_{M}\phi\nabla^{M}\phi-V(\phi)\right]+S_{m}[\tilde{g}_{MN},\psi_{i}],$
(1)

where the matter fields $\psi_{i}$ coupled to the Jordan frame metric
$\tilde{g}_{MN}=b^{2}(\phi)g_{MN}$. We will not specify the specific form of
the conformal factor initially; instead, we will engage in a general
discussion. Later, in Section 4, we will discuss its specific form.

The metric of a de-Sitter brane is assumed as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\text{e}^{2A(z)}\left(-dt^{2}+\text{e}^{2Ht}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2})+dz^{2}\right)$
(2) $\displaystyle=$
$\displaystyle\text{e}^{2A(z)}\left(\gamma_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}\right),$

where $z$ stands for the extra coordina, $H$ is the Hubble parameter, and
$H>0$. $a(z)=\text{e}^{A(z)}$ is the warp factor,
$\gamma_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+\text{e}^{2Ht}(dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2})$.
Since the brane is maximally symmetric, the scalar field can be assumed as
$\phi=\phi(z)$. When constructing the braneworld background solution, we
neglect the influence of matter fields on the brane, i.e., we set $S_{m}=0$.
With the metric assumption of (2), and varying the action (1) with respect to
the Einstein frame metric $g_{MN}$ and the scalar field $\phi$, we obtain the
following equations

$\displaystyle 3A^{\prime 2}-\phi^{\prime 2}-3A^{\prime\prime}-3H^{2}$
$\displaystyle=$ $\displaystyle 0,$ (3) $\displaystyle
2\text{e}^{2A}V(\phi)+9(A^{\prime 2}-H^{2})+3A^{\prime\prime}$
$\displaystyle=$ $\displaystyle 0,$ (4) $\displaystyle
4A^{\prime}\left(\phi^{\prime
2}+6H^{2}\right)-\text{e}^{2A}\phi^{\prime}V^{\prime}(\phi)$ $\displaystyle=$
$\displaystyle 0,$ (5)

where the primes denote the derivatives with respect to $z$. Note that there
are only two independent equations in Eqs. (3)-(5), We assume the following
warp factor of a de sitter brane Kobayashi2002a ,

$\displaystyle\text{e}^{2A(z)}=\text{sech}^{p}(kz),$ (6)

with the scalar field solved as

$\displaystyle\phi(z)=\phi_{0}\text{arcsin}\left(\tanh(kz)\right),$ (7)

where $\phi_{0}=\left[3p(1-p)\right]^{\frac{1}{2}}$, $k=\frac{H}{p}$ and
$0<p<1$. The corresponding potential of the scalar field is

$\displaystyle
V(\phi)=\frac{1}{2}k^{2}p\left(9p+3\right)\cos^{2(1-p)}\left(\frac{\phi}{\phi_{0}}\right).$
(8)

(a) The warp factor (b) The scalar field

Figure 1: The shapes of the warp factor $a(z)$ and the scalar field $\phi(z)$
in the de Sitter brane model.

For convenience, we express the Jordan frame metric as follows:

$\tilde{ds}^{2}=\tilde{g}_{MN}dx^{M}dx^{N}=\tilde{a}^{2}(z)\left[\gamma_{\mu\nu}dx^{\mu}dx^{\nu}+dz^{2}\right],$
(9)

where we introduce an effective warped factor
$\tilde{a}(z)=\text{e}^{\tilde{A}(z)}=\text{e}^{B(\phi)+A(z)}$, and
$b(\phi)=\text{e}^{B(\phi)}$. All functions derived from $\tilde{g}_{MN}$ will
be denoted with a tilde in the following. Notably, the braneworld solution in
this setup remains the same as the standard case. Moreover, since the four-
dimensional gravity is only related to the tensor mode of the perturbation of
the metric, the localization of gravity in our model is exactly the same with
that in the standard case in general relativity, and it is not affected by the
conformal factor $b(\phi)$. However, we anticipate novel characteristics
concerning the localization of various matter fields since the matter fields
couple to the Jordan frame metric $\tilde{g}_{MN}$ instead of the Einstein
frame metric $g_{MN}$.

##  3 Localization of the q-form fields

Firstly, we focus on localizing the $q$-form field $X_{M_{1}M_{2}\ldots
M_{q}}$ which corresponds to a scalar field when $q=0$, a vector field when
$q=1$, and a Kalb-Ramond (KR) field when $q=2$ on a de-Sitter brane. Our
approach involves investigating a general $q$-form field first and then
delving into the specific cases of scalar, vector, and Kalb-Ramond fields. The
research regarding the localization of $q$-form fields on a flat brane in
general relativity can be found in Refs. Fu2012 ; Fu2014 ; Fu2016 . Similar
derivations of the formulas of a q-form field in a de sitter brane model can
be found in Ref. Yang2020 . The action of a massless $q$-form field in five-
dimension is given by

$S_{q}=-\frac{1}{2(q+1)!}\int d^{5}x\sqrt{-\tilde{g}}Y^{M_{1}M_{2}\ldots
M_{q+1}}Y_{M_{1}M_{2}\ldots M_{q+1}},$ (10)

where $\tilde{g}$ represents the determinant of the Jordan frame metric
$\tilde{g}_{MN}$. The indices of the matter fields are raised and lowered
using $\tilde{g}_{MN}$, and the strength of the $q$-form field is defined as
$Y_{M_{1}M_{2}\ldots M_{q+1}}=\partial_{[M_{1}}X_{M_{2}M_{3}\ldots M_{q+1}]}$.
By varying action (10) with respect to $X_{M_{1}M_{2}\ldots M_{q}}$, the
equations of motion of the $q$-form field can be obtained,

$\displaystyle\partial_{M_{1}}\left(\sqrt{-\tilde{g}}Y^{M_{1}M_{2}\ldots
M_{q+1}}\right)=0.$ (11)

By substituting the Jordan frame metric (9), we can get the equations of
motion as

$\displaystyle\partial_{\mu_{1}}\left(\sqrt{-\tilde{g}}Y^{\mu_{1}\mu_{2}\ldots\mu_{q+1}}\right)+\partial_{z}\left(\sqrt{-\tilde{g}}Y^{z\mu_{2}\ldots\mu_{q+1}}\right)$
$\displaystyle=$ $\displaystyle 0,$ (12)
$\displaystyle\partial_{\mu_{1}}\left(\sqrt{-\tilde{g}}Y^{\mu_{1}\ldots\mu_{q}z}\right)$
$\displaystyle=$ $\displaystyle 0.$ (13)

We should note that the five-dimension massless $q$-form field action (10) is
invariant under the gauge transformation $X_{M_{1}M_{2}\ldots
M_{q}}\rightarrow\tilde{X}_{M_{1}M_{2}\ldots M_{q}}=X_{M_{1}M_{2}\ldots
M_{q}}+\partial_{[M_{1}}\Lambda_{M_{2}M_{3}\ldots M_{q}]}$, where
$\Lambda_{M_{2}M_{3}\ldots M_{q}}$ can be considered as an arbitrary
antisymmetric tensor field. In order to remove the normative degrees of
freedom, we can choose the gauge $X_{\mu_{1}\ldots\mu_{q-1}z}=0$. Then, the
rest components of the $q$-form field with the KK decomposition can be
expressed as

$\displaystyle
X_{\mu_{1}\mu_{2}\ldots\mu_{q}}(x,z)=\sum_{n}\hat{X}_{\mu_{1}\mu_{2}\ldots\mu_{q}}^{(n)}(x)\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}.$
(14)

Here, ‘n’ can be interpreted as representing different KK modes, and as a
result, we can obtain the following KK decomposition for the strength

$\displaystyle Y_{\mu_{1}\mu_{2}\ldots\mu_{q+1}}(x,z)$ $\displaystyle=$
$\displaystyle\sum_{n}\hat{Y}_{\mu_{1}\mu_{2}\ldots\mu_{q+1}}^{(n)}(x)\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}},$
$\displaystyle Y_{\mu_{1}\mu_{2}\ldots\mu_{q}z}(x,z)$ $\displaystyle=$
$\displaystyle\frac{1}{q+1}\sum_{n}\hat{X}_{\mu_{1}\ldots\mu_{q}}(x)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}$
(15) $\displaystyle\times$
$\displaystyle\left(\Upsilon^{\prime}_{n}(z)+\frac{(2q-3)}{2}\tilde{A}^{\prime}\Upsilon_{n}(z)\right),$

where $\hat{Y}_{\mu_{1}\mu_{2}\ldots\mu_{q+1}}^{(n)}(x)=\partial_{[\mu
1}\hat{X}_{\mu_{2}\ldots\mu_{q+1}]}(x)$ is the field strength on the brane.
With the above KK decomposition, and substituting the relations Eqs. (3) into
the equations of motion (12), we can reduce the equations of motion (12) into

$\displaystyle\sum_{n}\Big{\\{}\partial_{\mu_{1}}\big{(}\sqrt{-\tilde{g}}\hat{Y}^{\mu_{1}\mu_{2}\ldots\mu_{q+1}}_{(n)}(x)\big{)}\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}$
(16)
$\displaystyle+\frac{1}{q+1}\partial_{z}\Big{(}\sqrt{-\tilde{g}}\partial_{z}\big{(}\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}\big{)}\Big{)}\hat{X}^{\mu_{2}\mu_{3}\ldots\mu_{q+1}}_{(n)}(x)\Big{\\}}=0.$

For different KK modes, we can see that they are independent from each other.
In other words, we can discuss the different KK modes separately. So, for the
‘n-th’ KK mode, we divide the left and right sides of above equation by
$\hat{X}^{\mu_{2}\mu_{3}\ldots\mu_{q+1}}_{(n)}(x)\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}\sqrt{-\gamma}$
at the same time, and note that
$\sqrt{-\tilde{g}}=\text{e}^{5\tilde{A}}\sqrt{-\gamma}$, then we obtain

$\displaystyle\frac{\partial_{\mu_{1}}\big{(}\sqrt{-\gamma}\hat{Y}^{\mu_{1}\mu_{2}\ldots\mu_{q+1}}_{(n)}(x)\big{)}}{\sqrt{-\gamma}\hat{X}^{\mu_{2}\mu_{3}\ldots\mu_{q+1}}_{(n)}(x)}+\frac{1}{q+1}\frac{\partial_{z}\Big{(}\text{e}^{5\tilde{A}}\partial_{z}\big{(}\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}\big{)}\Big{)}}{\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}}=0.$
(17)

We can see that equation (17) is made up of two parts, the first term is a
function of the four-dimensional coordinates, and the second term is of the
extra dimension. Therefore, both the two terms must vanish. The first term of
the four-dimensional part $\hat{Y}_{\mu_{1}\mu_{2}\ldots\mu_{q+1}}^{(n)}(x)$

$\displaystyle\partial_{\mu_{1}}\left(\sqrt{-\gamma}\hat{Y}^{\mu_{1}\mu_{2}\ldots\mu_{q+1}}_{(n)}(x)\right)=\frac{1}{q+1}m_{n}^{2}X^{\mu_{2}\ldots\mu_{q+1}}\sqrt{-\gamma},$
(18)

where $m_{n}$ is the mass of the KK mode, and second part yields

$\displaystyle\partial_{z}\Big{(}\text{e}^{5\tilde{A}}\partial_{z}\big{(}\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}\big{)}\Big{)}=m_{n}^{2}\Upsilon_{n}(z)\text{e}^{\frac{(2q-3)\tilde{A}}{2}}.$
(19)

extra dimensional which can be considered as a Schrödinger-like equation of
the KK modes $\Upsilon_{n}(z)$,

$\displaystyle\left[-\partial^{2}_{z}+V_{q}(z)\right]\Upsilon_{n}(z)=m_{n}^{2}\Upsilon_{n}(z),$
(20)

where the $m_{n}$ can be considered as the effective mass of the four-
dimensional q-form field, and the effective potential $V_{q}(z)$ is

$\displaystyle
V_{q}(z)=\frac{3-2q}{2}\tilde{A}^{\prime\prime}+\frac{(3-2q)^{2}}{4}\tilde{A}^{\prime
2}.$ (21)

The derivation of the Schrödinger-like equation above is an extension of the
derivation process of the Schrödinger-like equation in Ref. Zhong2023 , where
the flat four-dimensional metric is replaced with an de Sitter metric
$\gamma_{\mu\nu}$ that is independent of $z$. In order to ensure that the
five-dimensional action (10) reduces to a four-dimensional effective action on
the brane, the integral of the extra part should be convergent, which can be
realized by introducing the orthonormal condition

$\displaystyle\int^{+\infty}_{-\infty}\Upsilon_{m}(z)\Upsilon_{n}(z)dz=\delta_{mn}.$
(22)

The action (10) can be reduced to

$\displaystyle S_{q}=$ $\displaystyle-\frac{1}{2(q+1)}\sum_{n}\int
d^{4}x\sqrt{-\gamma}\Big{(}\hat{Y}_{\mu_{1}\mu_{2}\ldots\mu_{q+1}}^{(n)}\hat{Y}^{(n)\mu_{1}\mu_{2}\ldots\mu_{q+1}}\\!$
(23)
$\displaystyle+\frac{m_{n}^{2}}{q+1}\hat{X}_{\mu_{1}\mu_{2}\ldots\mu_{q}}^{(n)}\hat{X}^{(n)\mu_{1}\mu_{2}\ldots\mu_{q}}\Big{)},$

where the indices are raised and lowered by the metric $\gamma_{\mu\nu}$. The
effective action describes a massless $q$-form field ($n=0$) and a series of
massive $q$-form fields ($n\geq 1$) on the de Sitter brane.

The Kaluza-Klein reduction procedure can also be applied to the gravity part
of the action (1). We can consider the tensor mode of the perturbation of the
metric $g_{MN}$. By implementing the KK decomposition and integrating with
respect to the extra dimension, the first term $\int
d^{5}x\sqrt{-g}\frac{1}{2}R$ reduces to $S_{eff}\propto\int
d^{4}x\sqrt{-\gamma}\frac{1}{2}R^{(4)}$, where $R^{(4)}$ represents the
effective four-dimensional Ricci scalar on the brane. As for the second term
$\int
d^{5}x\sqrt{-g}\left[-\frac{1}{2}\nabla_{M}\phi\nabla^{M}\phi-V(\phi)\right]$,
which describes the chameleon scalar field, it becomes a constant.

To recover the four-dimensional effective theory at low energy scale, we have
to ensure that the zero modes with $m_{n}=0$ can be localized on the brane.
The zero modes can be solved from (20) by the factorizing method:

$\displaystyle\Upsilon_{0}(z)=N_{0}\text{e}^{\frac{(3-2q)\tilde{A}}{2}},$ (24)

where $N_{0}$ is a normalization constant. Therefore, the condition for
localization of zero mode is reduced to

$\displaystyle\int^{+\infty}_{-\infty}\Upsilon_{0}^{2}(z)dz=1,$ (25)

or equivalently,

$\displaystyle\int^{+\infty}_{-\infty}\left[a(z)b(\phi)\right]^{3-2q}dz<\infty.$
(26)

Obviously, the condition for localization requires
$\left[a(z)b(\phi)\right]^{3-2q}_{z\rightarrow\pm\infty}\rightarrow 0$. Since
the exponentiation $3-2q$ for the scalar and vector fields have opposite signs
with that of KR field, which means that when the zero mode of the scalar and
vector field localize on the dS brane, the KR filed can not be localized on
the dS brane at the same time. As KR particles have not been discovered yet on
the brane, it is reasonable for the vector field to be localized but KR field
to not be localized on the brane.

In summary, in chameleon gravity, the scalar and vector fields can be
localized on the de Sitter brane, while the KR field can not be localized at
the same time. The condition to localize both the scalar and vector fields is

$\displaystyle\int^{+\infty}_{-\infty}\left[a(z)b(\phi)\right]dz<\infty.$ (27)

In addition, since $a(z)$ is an even function and $\phi(z)$ is odd, to ensure
that the action does not vanishes, $b(\phi)$ can not be an odd function of
$\phi$.

On the condition that the above condition is satisfied, the effective
potentials and the four-dimensional effective actions of the scalar and vector
fields can be obtained.

###  3.1 Scalar field

For the scale field $\Phi(x^{\mu},z)$, the KK decomposition (14) turns to

$\displaystyle\Phi(x^{\mu},z)=\sum_{n}\varphi^{(n)}(x^{\mu})\chi_{n}(z)\text{e}^{-\frac{3}{2}\tilde{A}}.$
(28)

and the effective potential for the scalar field reads

$\displaystyle
V_{\text{s}}(z)=\frac{3}{2}\tilde{A}^{\prime\prime}+\frac{9}{4}\tilde{A}^{\prime
2}.$ (29)

Therefore, the massless scalar zero mode reads

$\displaystyle\chi_{0}(z)\propto\text{e}^{\frac{3\tilde{A}}{2}}.$ (30)

The four-dimensional effective action for the scalar field becomes

$\displaystyle S_{\text{s}}=-\sum_{n\geq 1}\int
d^{4}x\sqrt{-\gamma}\left(\frac{1}{2}\partial^{\mu}\varphi^{(0)}\partial_{\mu}\varphi^{(0)}+\frac{1}{2}\partial^{\mu}\varphi^{(n)}\partial_{\mu}\varphi^{(n)}+m_{(n)}^{2}\varphi^{(n)}\varphi^{(n)}\right).$
(31)

which describes a massless and a series of massive scalar fields,

###  3.2 Vector field

Then we consider the vector field $A_{\mu}(x,z)$ with the gauge $A_{5}=0$. The
KK decomposition is

$\displaystyle
A_{\mu}(x,z)=\sum_{n}a^{(n)}_{\mu}(x)\alpha_{n}(z)\text{e}^{-\frac{\tilde{A}}{2}}.$
(32)

The effective potential $V_{\text{v}}(z)$ for the vector field is given by

$\displaystyle
V_{\text{v}}(z)=\frac{1}{2}\tilde{A}^{\prime\prime}+\frac{1}{4}\tilde{A}^{\prime
2},$ (33)

and the massless vector zero mode reads

$\displaystyle\alpha_{0}(z)\propto\text{e}^{\frac{\tilde{A}}{2}}.$ (34)

Then, the four-dimensional effective action for the scalar field reads

$\displaystyle S_{\text{v}}=-\sum_{n\geq 1}\int
d^{4}x\sqrt{-\gamma}\left(\frac{1}{4}f^{(0)\mu\nu}f^{(0)}_{\mu\nu}+\frac{1}{4}f^{(n)\mu\nu}f^{(n)}_{\mu\nu}+\frac{1}{2}m_{n}^{2}a^{(n)\mu}a^{(n)}_{\mu}\right),$
(35)

which describes a massless vector field and a series of massive vector fields.

##  4 Setting up $b(\phi)$

In the last section, we demonstrated that the zero modes of both scalar and
vector fields can be localized on a de sitter brane generated by a chameleon
scalar field. With the provision of $b(\phi)$, we can further explore
additional localized characteristics. In this section, we will study two
examples that exhibit novel properties.

###  4.1 Case I: Asymmetric effective potentials

For the first case, we consider the assumption that
$b(\phi)=\text{e}^{\beta\phi}$, which constitutes the initial form of the
conformal factor $b(\phi)$ Waterhouse2006 . This particular form may stem from
processes like dimensional reduction and was widely used in string theory, see
Refs. Damour1994 ; Fradkin1985 ; Callan1985 ; Callan1986  for instance. The
gravity dual of Little String Theory also have similar action Antoniadis2012 .
In this assumption, since function $b(\phi)$ is asymmetric, while the warp
factor $a(z)$ is an even function, the effective potentials and zero modes
become asymmetric, which are plotted in Figs. 2 and 3. When $\beta=0$, our
model everts to the standard case in general relativity Guo2013 .

Therefore, we can obtain asymmetric effective potentials and the KK modes in a
thick brane with $Z_{2}$ symmetry in chameleon gravity. It was found in
Refs.Liu2009a ; Du2013  that the asymmetry of the effective potential may
affects the spectrum of KK modes.

(a) The scalar field (b) The vector field

Figure 2: The effective potential of the scalar and vector fields. The
parameter $p$ is sect as $p=0.6$.

(a) Scalar field (b) Vector field

Figure 3: The zero modes of the scalar and vector fields. The parameter $p$ is
sect as $p=0.6$.

###  4.2 Case II: Volcano-like effective potentials

In a de Sitter brane in general relativity, the effective potential of matters
field is usually Pöchl–Teller (PT) potentials, and the zero modes of the
scalar field and the vector field are localized on the brane Liu2009a ;
Guo2013 . Moreover, it was found that the excited KK mode with a mass of
$m_{s}=\sqrt{2}H\sim 10^{-33}\text{eV}$ for the scalar field is also a bound
state Liu2009a ; Guo2013 . Since the mass $m_{s}$ is small, this excited KK
mode of the scalar field may have observable effects and could be inconsistent
with experimental observations.

In this subsection, we will show that in chameleon gravity, the effective
potential could be a volcano-like potential so that all the massive KK modes
are not localized. We can assume that
$b(\phi)=\frac{\cos^{2}\left(\frac{\phi}{\phi_{0}}\right){}^{-p/2}}{\left(\tanh^{-1}\left(\sin\left(\frac{\phi}{\phi_{0}}\right)\right){}^{2}+1\right){}^{3/2}}$.
With this assumption, the effective potential of the scalar and vector fields
becomes

$\displaystyle V_{\text{s}}(z)$ $\displaystyle=$
$\displaystyle\frac{9k^{2}\left(11k^{2}z^{2}-2\right)}{4\left(k^{2}z^{2}+1\right)^{2}},$
(36) $\displaystyle V_{\text{v}}(z)$ $\displaystyle=$
$\displaystyle\frac{3k^{2}\left(5k^{2}z^{2}-2\right)}{4\left(k^{2}z^{2}+1\right)^{2}},$
(37)

which are volcano-like potentials. And both the scalar field and vector field
KK modes have only one bound state, which is the zero mode. The shapes of the
effective potentials and zero modes are plotted in Fig. 4.

(a) Scalar field (b) Vector field

Figure 4: The zero modes and effective potentials of the scalar and vector
fields.

##  5 Conclusion

In this work, we investigated the localization of $q$-form fields on a de
Sitter brane construct by a chameleon scalar field. It was found that the de
Sitter brane solution in chameleon gravity is the same as the one in general
relativity. The scalar and vector fields can be localized on the de Sitter
brane, while the KR field can not be localized at the same time. The condition
for localization of the scalar and vector fields is obtained. Furthermore, we
investigated the localization characteristics in two examples of given
conformal factor $b(\phi)$. In the first case, we assume that
$b(\phi)=\text{e}^{\beta\phi}$ , which is inspired by the string theory. it
was shown that the effective potentials and KK modes of the matter fields
becomes asymmetric even though the de Sitter brane has $Z_{2}$ symmetry. In
the second case, we generated volcano-like effective potentials in the de
Sitter brane by assuming that
$b(\phi)=\frac{\cos^{2}\left(\frac{\phi}{\phi_{0}}\right){}^{-p/2}}{\left(\tanh^{-1}\left(\sin\left(\frac{\phi}{\phi_{0}}\right)\right){}^{2}+1\right){}^{3/2}}$.
In this case, only the zero mode of the scalar and vector fields are localized
on the brane.

###### Acknowledgements.

This work was supported by the Natural Science Foundation of Hunan Province,
China (Grant No. 2022JJ40033), and the National Natural Science Foundation of
China (Grant No. 12305061 and No. 12247101).

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