# Braneworld sum rules and positive tension branes in a massive gravity

Ke Yang<EMAIL_ADDRESS>Bao-Min Gu<EMAIL_ADDRESS>Yi Zhong
<EMAIL_ADDRESS>School of Physical Science and Technology, Southwest
University, Chongqing 400715, China  Department of Physics, Nanchang
University, Nanchang 330031, China  Center for Relativistic Astrophysics and
High Energy Physics, Nanchang University, Nanchang 330031, China  School of
Physics and Electronics Science, Hunan University, Changsha 410082, China
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, China

###### Abstract

By taking advantage of the braneworld sum rules, we explore the feasibility of
constructing a flat 3-brane scenario consisting solely of positive tension
branes in a 5D extension of the Lorentz-violating massive gravity. It is found
that the theory supports three distinct brane configurations, one of which is
exactly what we expected, consisting solely of two positive tension branes.
The cosmological problem of Randall-Sundrum-1 model and the gauge hierarchy
problem can be solved in this model simultaneously. Furthermore, the analysis
of linear perturbations reveals that the tensor, vector and scalar modes are
all massive and share the same mass spectrum, except that the ground state of
vector mode is absent. Moreover, the tensor and vector modes are robust, but
the scalar mode is ghost-like. Interestingly, even though the Kaluza-Klein
gravitons have an extremely small mass splitting scale, an estimation of the
effective gravitational potential and production of these gravitons on the
brane indicates that the phenomenology of the present model is equivalent to
that of the 6D ADD model.

###### keywords:

Extra dimensions , Randall-Sundrum model , Braneworld sum rules

††journal: Physics Letters B

##  1 Introduction

The possibility that our spacetime may have more dimensions other than four
has always been concerned by theoretical physicists since the proposal of
Klein-Kaluza (KK) theory in 1920s [1, 2]. Inspired by the string theory, the
braneworld scenario is an available mechanism for hiding the undiscovered
extra dimensions, where our visible universe is a 3-brane embedded in a
higher-dimensional bulk and all Standard Model particles are confined on the
brane. In 1999, Randall and Sundrum proposed a well-known braneworld model,
which provides a natural mechanism to solve the long-standing gauge hierarchy
problem of particle physics [3]. In the Randall-Sundrum-1 (RS1) model, there
is a 3-brane located at each boundary of the orbifold extra dimension, where
the one with negative tension is the infrared (IR) brane (or visible brane)
our universe confined on, and the other with positive tension is the
ultraviolet (UV) brane (or hidden brane).

It is well known that the overall sign of the source terms in the induced
Friedmann-like equation on the brane depends on the sign of the brane tension
[4, 5, 6]. Since we live on a negative tension brane in RS1 model, it would
lead to a “wrong-signed” Friedmann-like equation and hence our observed
expanding universe cannot be recovered on the brane. Moreover, it was found
that the Standard Model fields can be localized on some positive tension
branes, such as D-branes and NS-branes [7]. Hence, it is more reasonable to
place our universe on a positive tension brane. One example of such a scenario
is the Randall-Sundrum-2 (RS2) model, which features a single positive tension
brane that we live on, but the gauge hierarchy problem is left [8]. In order
to solve the gauge hierarchy problem and cosmological problem simultaneously,
the authors considered some generations of RS1 model in modified gravitational
theories [9, 10, 11], where the massless 4D graviton is localized on the
negative tension brane, so our world should move onto the positive tension
brane in order to solve the gauge hierarchy problem.

However, the negative tension brane is a potentially unstable object [12], so
it would be a better way to construct the braneworld scenario with only
positive tension branes. In Ref. [7], the authors added a probe brane with a
small positive tension into the RS2 model to build a hierarchy resolved
configuration. For the exact brane solution, it is easy to show that the
constraints require the presence of negative tension brane in a 5D
compactification scheme in general relativity by resorting to the technique of
braneworld sum rules [13]. However, it can be evaded for a higher-dimensional
model such as the 6D anti-de Sitter soliton [14]. It is also found that the
constraints can be relaxed in some modified gravitational theories even in 5D
spacetime, such as in scalar-tensor gravity [15, 16] and $f(R)$ gravity [17,
18]. It gives us a hint that an extended RS1-like model with only positive
tension branes could be realized in some modified gravitational theories.

In this work, we are interested in building braneworld model with only
positive tension branes in a massive gravity, which is a generalization of
general relativity by endowing the graviton with a nonzero mass. See Refs.
[19, 20, 21] and the references therein for an introduction on massive gravity
theories. Specifically, under the help of braneworld sum rules, we focus on
generating the RS1-like scenario in a 5D extension of the Lorentz-violating
massive gravity [22]. In order to obtain a flat 3-brane configuration, 4D
Poincaré invariance has to be preserved in this theory. Therefore, we assume
that the background spacetime is invariant under 4D Poincaré transformation in
the 5D extension of the Lorentz-violating massive gravity. The 5D
diffeomorphisms are spontaneously broken due to the condensation of four
background scalars. Then, the condensation generates four Goldstone
excitations associated with the broken symmetries. Consequently, the 5D
massless spin-2 graviton with five degrees of freedom gets weight and
possesses nine degrees of freedom on the spectrum by “eating” the four
Goldstone excitations in the unitary gauge. Other works related to braneworld
scenario in massive gravities can be found in Refs. [23, 24, 25, 26, 27, 28,
29, 30].

The layout of the paper is as follows: In Sect. 2, the constraint from the
braneworld sum rules is discussed in the 5D extension of the Lorentz-violating
massive gravity. In Sect. 3, a hierarchy-resolving toy model is built. The
corresponding mass spectra of KK particles are discussed in Sect. 4, and some
low-energy phenomenology of the model is investigated in Sect. 5. Finally,
brief conclusions and discussions are presented. Throughout the paper, the
small Latin letters $(a,b,\cdots=0,1,2,3)$ are used to label the group indices
of the internal metric of scalar fields, while the capital Latin letters
$(A,B,\cdots=0,1,2,3,5)$ and Greek letters $(\mu,\nu,\cdots=0,1,2,3)$ are used
to label the 5D and 4D spacetime indices, respectively.

##  2 Braneworld sum rules

To obtain a flat 3-brane configuration, we start from the most general metric
ansatz keeping the four-dimensional Poincaré invariance, given by

$ds^{2}=g_{MN}dx^{M}dx^{N}=a^{2}(y)\eta_{\mu\nu}(x)dx^{\mu}dx^{\nu}+dy^{2},$
(1)

where $a(y)$ is the warp factor, and $y\in[-y_{\pi},y_{\pi}]$ denotes a
compact $S^{1}/Z_{2}$ orbifold extra dimension. Correspondingly, the 5D Ricci
tensor can be written as

$\displaystyle
R_{\mu\nu}=-a^{2}\eta_{\mu\nu}\left(H^{\prime}+4H^{2}\right),\quad
R_{55}=-4\left(H^{\prime}+H^{2}\right),$ (2)

where the prime denotes the derivative with respect to $y$, and $H\equiv
a^{\prime}/a$. By tracing the above equations respectively, one obtains

$\displaystyle R^{\mu}_{\mu}=-4\left(H^{\prime}+4H^{2}\right),\quad
R^{5}_{5}=-4\left(H^{\prime}+H^{2}\right).$ (3)

Further, with the relation

$\left(a^{\alpha+1}H\right)^{\prime}=a^{\alpha+1}\left[H^{\prime}+(\alpha+1)H^{2}\right],$
(4)

where $\alpha$ is an arbitrary constant, one obtains a useful relation by
combining Eqs. (3), i.e.,

$\displaystyle\left(a^{\alpha+1}H\right)^{\prime}=\frac{a^{\alpha+1}}{4}\left[\left(\alpha-3\right){R^{5}_{5}}-\alpha
R^{\mu}_{\mu}\right].$ (5)

Here, we would like to consider a theory with a 5D Einstein-Hilbert term plus
four canonical scalar fields, whose action is given by

$\displaystyle S$
$\displaystyle=M_{*}^{3}\int{d^{5}x}\sqrt{-g}\left[\frac{R}{2}-\frac{1}{2}m^{2}g^{MN}\partial_{M}\phi^{a}\partial_{N}\phi^{a}-V(\phi^{a}\phi^{a})\right]$
$\displaystyle-\int d^{4}x\sqrt{-g_{\text{I}}}V_{\text{I}}-\int
d^{4}x\sqrt{-g_{\text{II}}}V_{\text{II}},$ (6)

where $M_{*}$ is the 5D fundamental gravity scale, $m$ is a parameter
proportional to the mass of 5D graviton, $V(\phi^{a}\phi^{a})$ is the self-
interaction potential, and $V_{\text{I}}$ and $V_{\text{II}}$ represent the
brane tensions at $y=0$ and $y=y_{\pi}$ respectively. In oder to achieve a
Minkowski flat 3-brane model, the internal metric of the scalar fields has to
be chosen as the Minkowski metric $\eta_{ab}$.

The corresponding field equations are obtained by varying the action (6) with
respect to the metric $g^{MN}$,

$\displaystyle
R_{MN}-\frac{R}{2}g_{MN}=m^{2}\Big{(}\partial_{M}\phi^{a}\partial_{N}\phi^{a}-\frac{1}{2}g_{MN}\partial^{K}\phi^{a}\partial_{K}\phi^{a}\Big{)}-Vg_{MN}$
$\displaystyle-\frac{V_{\text{I}}}{M_{*}^{3}}g_{\text{I}\mu\nu}\delta^{\mu}_{M}\delta^{\nu}_{N}\delta\left(y\right)-\frac{V_{\text{II}}}{M_{*}^{3}}g_{\text{II}\mu\nu}\delta^{\mu}_{M}\delta^{\nu}_{N}\delta\left(y-y_{\pi}\right).$
(7)

Then, $R^{\mu}_{\mu}$ and $R^{5}_{5}$ can be expressed explicitly as

$\displaystyle R^{\mu}_{\mu}$
$\displaystyle=m^{2}\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{a}+\frac{8V}{3}+\frac{4V_{\text{I}}}{3M_{*}^{3}}\delta\left(y\right)+\frac{4V_{\text{II}}}{3M_{*}^{3}}\delta\left(y-y_{\pi}\right),$
(8) $\displaystyle R^{5}_{5}$
$\displaystyle=m^{2}({\phi^{a}}^{\prime})^{2}+\frac{2V}{3}+\frac{4V_{\text{I}}}{3M_{*}^{3}}\delta\left(y\right)+\frac{4V_{\text{II}}}{3M_{*}^{3}}\delta\left(y-y_{\pi}\right).\
$ (9)

After inserting above Eqs. (8) and (9) into Eq. (5), one has

$\displaystyle\left(a^{\alpha+1}H\right)^{\prime}=-\frac{a^{\alpha+1}}{4}\Big{[}\alpha
m^{2}\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{a}-(\alpha-3)m^{2}({\phi^{a}}^{\prime})^{2}$
$\displaystyle+2(\alpha+1)V+4\frac{V_{\text{I}}}{M_{*}^{3}}\delta(y)+4\frac{V_{\text{II}}}{M_{*}^{3}}\delta\left(y-y_{\pi}\right)\Big{]}.$
(10)

Since the $S^{1}/Z_{2}$ orbifold is periodic and compact, the integral of left
hand side of the above equation (10) vanishes [13]. Especially, for
$\alpha=-1$, one obtains a useful constraint,

$\displaystyle{M_{*}^{3}}\oint\Big{(}m^{2}\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{a}-4m^{2}({\phi^{a}}^{\prime})^{2}\Big{)}dy={4\left(V_{\text{I}}+V_{\text{II}}\right)}.$
(11)

If the scalar fields further depend only on the extra dimension, which is
typically the case considered in braneworld models, the constraint (11)
reduces to

$-m^{2}{M_{*}^{3}}\oint({\phi^{a}}^{\prime})^{2}dy=V_{\text{I}}+V_{\text{II}}.$
(12)

So in this case, the theory cannot support a model with only positive tension
branes. Instead, if the scalar fields depend only on the brane coordinates,
$\phi^{a}=\phi^{a}(x)$, the constraint (11) becomes

$m^{2}{M_{*}^{3}}\oint\partial^{\mu}\phi^{a}\partial_{\mu}\phi^{a}dy=4(V_{\text{I}}+V_{\text{II}}).$
(13)

It implies that the theory may support a model with only positive tension
branes in this case.

Specifically, here we consider a 5D extension of the Lorentz-violating massive
gravity [22], where the background solution spontaneously breaks the 5D
Lorentz invariance. The breaking of 5D Lorentz invariance stems from the
condensation of scalar fields via

$\langle\phi^{a}\rangle=\delta_{\mu}^{a}x^{\mu},$ (14)

with $x^{\mu}$ the brane coordinates. Thus, the condensation spontaneously
generates a preferred 4D frame. Moreover, the scalar potential
$V(\phi^{a}\phi^{a})$ takes its vacuum value, i.e., the 5D cosmological
constant $\Lambda$. In this case, the constraint simplifies to

$V_{\text{I}}+V_{\text{II}}=m^{2}{M_{*}^{3}}\oint a^{-2}dy.$ (15)

##  3 Model building and hierarchy resolution

With the ansatzes of the flat braneworld metric (1) and scalar field
condensation (14), the field equations (7) are written explicitly as

$\displaystyle 3\left(H^{\prime}+2H^{2}\right)$
$\displaystyle=-\frac{m^{2}}{a^{2}}-\Lambda-\frac{V_{\text{I}}\delta\left(y\right)+V_{\text{II}}\delta\left(y-y_{\text{b}}\right)}{M_{*}^{3}},$
(16) $\displaystyle 6H^{2}$ $\displaystyle=-\frac{2m^{2}}{a^{2}}-\Lambda.$
(17)

From the field equations, we can easily obtain the solution

$a(y)=e^{-k|y|}+\epsilon^{2}e^{k|y|},$ (18)

where $\epsilon\equiv\frac{m}{2\sqrt{3}k}$, $k^{2}\equiv-\Lambda/6$, and $|y|$
represents the absolute value of $y$ in order to be consistent with the
$Z_{2}$ symmetry. There are two branches in the solution of the warp factor,
one is exponential growth and the other is exponential decay. Its minimum
appears at $y_{\text{m}}=\log(1/\epsilon)/k$ with the value
$a(y_{\text{m}})=2\epsilon$.

By matching the delta functions in (16), we have the fine-tuning conditions,

$V_{\text{I}}=6kM_{*}^{3}\left[1-\frac{2\epsilon^{2}}{1+\epsilon^{2}}\right],~{}~{}V_{\text{II}}=6kM_{*}^{3}\left[1-\frac{2}{1+\epsilon^{2}e^{2ky_{\pi}}}\right].$
(19)

It is straightforward to verify that the sum of brane tensions satisfies the
constraint (15) from braneworld sum rules.

From the fine-tuning conditions (19), we observe that there are three
different brane configurations depending on the size of extra dimension.
Especially, the IR brane tension vanishes in the fine tuning conditions (19)
for $y_{\pi}=y_{\text{m}}$, so a single brane configuration is obtained in
this case. Nevertheless, we are more interested in two brane configurations in
this work, in which the gauge hierarchy problem may be solved. So we prefer to
leave a brief discussion on it for the last section. Here we focus on the two
brane configurations as illustrated in Fig. 1.

(a) Configuration 1

(b) Configuration 2

Figure 1: Diagrams of two distinct brane configurations.

As shown in Fig. 1(a), the first configuration corresponds to the case of
$e^{-ky_{\pi}}>\epsilon^{2}e^{ky_{\pi}}$, i.e., the exponential decay branch
of warp factor is dominant in the bulk. Then, the fine-tuning conditions (19)
yield $V_{I}>0$ and $V_{II}<0$. Especially, in the limit $\epsilon
e^{ky_{\pi}}\ll 1$, the exponential growth branch of the warp factor can be
neglected compared to the exponential decay branch. Now the brane
configuration reduces to that of RS1 model, with the fine-turning conditions,

$V_{I}\approx-V_{II}\approx 6kM_{*}^{3},~{}~{}~{}\Lambda=-6k^{2}.$ (20)

This brane configuration has been investigated in detail in the previous work
by some of our authors [31].

As shown in Fig. 1(b), the second configuration corresponds to the case of
$e^{-ky_{\pi}}<\epsilon^{2}e^{ky_{\pi}}<1$, where the exponential growth
branch of warp factor is dominant near the IR brane. By observing from the
fine-tuning conditions (19), it is interesting that both the brane tensions
are positive in this case. This is the brane configuration that we have
expected from the braneworld sum rules. Now since our universe is confined on
the positive tension brane at $y_{\pi}$, a “correct-signed” induced Friedmann-
like equation can be obtained on the brane.

As a crucial motivation of the braneworld, the gauge hierarchy problem can be
solved in a natural way in this scenario. By confining the Higgs field on the
IR brane at $y_{\pi}$, the fundamental Higgs vacuum expectation value (VEV)
$v_{0}$ is redshifted by the warped factor, and therefore, the effective Higgs
VEV measured by the observers on the brane is
$v_{\text{eff}}=a(y_{\pi})v_{0}$, which sets the electroweak scale of the
Standard Model [3]. If all the fundamental parameters $M_{*}$, $k$, $v_{0}$
are set to be the order of Planck scale $M_{\text{Pl}}\sim 10^{16}$TeV, there
is no fundamental hierarchy between them. If the 5D graviton mass which is
proportional to $m$ is light enough, i.e., $m/k<10^{-16}$, the minimum of warp
factor $a(y_{\text{m}})=2\epsilon<10^{-16}$. Then, an effective TeV
electroweak scale could be generated by just placing the IR brane at the place
where $a(y_{\pi})\sim 10^{-16}$.

We will see in the next section that the constraint on the 4D graviton mass
implies $m<3.3\times 10^{-23}$eV, and as a result, the minimum of warp factor
$a(y_{\text{m}})=2\epsilon\leq 10^{-51}$. Thus, the exponential growth branch
of warp factor is completely dominant at IR brane, i.e.,
$a(y_{\pi})\approx\epsilon^{2}e^{ky_{\pi}}\sim 10^{-16}$. Then, the size of
the extra dimension is approximated given by
$y_{\pi}\approx\log\left[{a(y_{\pi})}/{\epsilon^{2}}\right]/k$. For instance,
$\epsilon\sim 10^{-51}$ yields $y_{\pi}\approx 198/k$ and $\epsilon\sim
10^{-60}$ yields $y_{\pi}\approx 239/k$. Therefore, the size of extra
dimension is roughly two orders of magnitude larger than the Planck length. As
a comparison, the size of extra dimension is $y_{\pi}\approx 37/k$ in RS1
model.

##  4 Mass spectra of KK states

In order to investigate the mass spectra of KK excitations, we consider the
linear perturbations against the background,

$ds^{2}=\left(g_{MN}+h_{MN}\right)dx^{M}dx^{N},$ (21)

where $g_{MN}$ is the background metric (1) and $h_{MN}$ represents the linear
perturbations. Due to the 4D Lorentz invariance of the background spacetime,
it is convenient to decompose the perturbations $h_{MN}$ into the scalar,
transverse vector and transverse-traceless tensor modes, as

$\displaystyle h_{55}$ $\displaystyle=-2\xi,$ (22) $\displaystyle h_{\mu 5}$
$\displaystyle=-a\left(S_{\mu}+\partial_{\mu}\beta\right),$ (23)
$\displaystyle h_{\mu\nu}$
$\displaystyle=a^{2}\left[D_{\mu\nu}+2\eta_{\mu\nu}\psi+\frac{1}{2}\left(\partial_{\mu}F_{\nu}+\partial_{\nu}F_{\mu}\right)+2\partial_{\mu}\partial_{\nu}E\right],$
(24)

where the transverse-traceless tensor $D_{\mu\nu}$ satisfies the condition
$\eta^{\mu\rho}\partial_{\rho}D_{\mu\nu}=0$ and the transverse vector modes
$S_{\mu}$ and $F_{\mu}$ satisfy the condition
$\eta^{\mu\rho}\partial_{\rho}S_{\mu}=\eta^{\mu\rho}\partial_{\rho}F_{\mu}=0$.

The perturbed scalar fields are $\phi^{a}=x^{a}+\pi^{a}$, with
$\pi^{a}=\delta^{a}_{\mu}\pi^{\mu}$ the Goldstone excitation of the
condensation. The Goldstone excitation transforms like a vector field under
the general coordinate transformation in order to maintain the scalar
condensation (14) unchanged, i.e., $\pi^{\mu}\to\pi^{\mu}-\epsilon^{\mu}$.
Correspondingly, it can be decomposed as
$\pi^{\mu}=\eta^{\mu\nu}\left(\partial_{\nu}\varphi+A_{\nu}\right)$, where
$\varphi$ is a scalar field and $A_{\mu}$ a transverse vector field satisfying
$\eta^{\mu\rho}\partial_{\rho}A_{\mu}=0$.

Note that the quantity $Z_{\mu}-\pi_{\mu}$ is a gauge invariant quantity,
where $Z_{\mu}\equiv a^{2}\left(F_{\mu}/2+\partial_{\mu}E\right)$ does not
contain physical degrees of freedom in general relativity. In unitary gauge,
the Goldstone excitations $\pi^{\mu}$ of scalar fields vanish. Then, it is
clear that the four Goldstone excitations $\pi^{\mu}$ are “eaten” by
$Z_{\mu}$, which survives in the linear perturbation theory and becomes
physical degrees of freedom. Consequently, the 5D massless spin-2 graviton
with five degrees of freedom gets weight and possesses nine degrees of freedom
on the spectrum after “eating” the four Goldstone excitations.

By substituting the full perturbed metric (21) into the action (6) and
expanding the action to the quadratic order of perturbations, the transverse-
traceless tensor, transverse vector and scalar modes are decoupled with each
other, so they can be treated separately. Working in the unitary gauge
$A_{\alpha}=\varphi=\psi=0$, the quadratic actions for tensor, vector and
scalar modes are finally obtained respectively [31],

$\displaystyle S^{(2)}_{\text{T}}\\!$
$\displaystyle=\\!-\frac{M_{*}^{3}}{2}\\!\int\\!d^{4}xdza^{3}\Big{[}\dot{\tilde{D}}_{\alpha\beta}\dot{\tilde{D}}^{\alpha\beta}\\!+\\!\partial^{\alpha}\tilde{D}_{\alpha\beta}\partial_{\alpha}\tilde{D}^{\alpha\beta}\\!+\\!{2m^{2}}\tilde{D}_{\alpha\beta}\tilde{D}^{\alpha\beta}\Big{]},$
(25) $\displaystyle S^{(2)}_{\text{V}}\\!$
$\displaystyle\\!=-\frac{M_{*}^{3}}{2}\int
d^{4}xdza^{3}\Big{[}\dot{\tilde{F}}_{\alpha}\dot{\tilde{F}}^{\alpha}\\!+\\!\partial^{\alpha}\tilde{F}_{\alpha}\partial_{\alpha}\tilde{F}^{\alpha}\\!+\\!{2m^{2}}\tilde{F}_{\alpha}\tilde{F}^{\alpha}\Big{]},$
(26) $\displaystyle S^{(2)}_{\text{S}}\\!$
$\displaystyle=\\!\frac{M_{*}^{3}}{2}\int
d^{4}xdza^{3}\left[\dot{\tilde{E}}\dot{\tilde{E}}+\partial^{\alpha}\tilde{E}\partial_{\alpha}\tilde{E}+{2m^{2}}\tilde{E}\tilde{E}\right],$
(27)

where the indices are raised and lowered by the 4D Minkowski metric
$\eta_{\mu\nu}$, $z$ is the conformal coordinate obtained through a coordinate
transformation $dy=adz$, and in order to canonically normalize these actions,
the modes have been rescaled as
$\tilde{D}_{\alpha\beta}=\frac{D_{\alpha\beta}}{2}$,
$\tilde{F}_{\alpha}=\sqrt{\frac{k^{2}m^{2}}{k^{2}+2m^{2}}}\frac{F_{\alpha}}{2}$,
and $\tilde{E}=\sqrt{\frac{-3k^{4}m^{2}}{3k^{2}+4m^{2}}}E$, with $k^{\alpha}$
the four-momentum of various modes.

The forms of the three actions are similar, except that there is an overall
wrong-sign in the action (27) of scalar mode. Thus, the scalar perturbation
$\tilde{E}$ is a ghost field. Further, by varying the actions (25), (26), and
(27) with respect to various modes respectively, we have the equation of
motion

$\partial^{\alpha}\partial_{\alpha}\Upsilon+\ddot{\Upsilon}+3H\dot{\Upsilon}=2m^{2}{\Upsilon},$
(28)

where $\Upsilon$ represents $\tilde{D}_{\alpha\beta}$, $\tilde{F}_{\alpha}$,
and $\tilde{E}$. With the KK decomposition
$\Upsilon=\upsilon(x)a^{-\frac{3}{2}}(z)\Psi(z)$, the equation of motion (28)
reduces to a 4D Klein-Gordon equations
$\Box^{(4)}\upsilon(x)=M^{2}\upsilon(x)$ and a Schrödinger-like equation,

$-\ddot{\Psi}+\left(\frac{3}{2}\dot{H}+\frac{9}{4}H^{2}\right){\Psi}=\mathcal{M}^{2}{\Psi},$
(29)

where $\mathcal{M}^{2}\equiv M^{2}-2m^{2}$, and $M$ is the effective mass of
various KK states observed on the brane.

The Hamiltonian can be factorized as
$H_{T}=A_{T}^{\dagger}A_{T}=\left(\partial_{z}+\frac{3}{2}H\right)\left(-\partial_{z}+\frac{3}{2}H\right)$,
which is self-adjoint. With the Neumann boundary condition
$\partial_{z}\tilde{D}_{\alpha\beta}|_{z=0,z_{b}}=0$, it is easy to show that
all the eigenvalues $\mathcal{M}^{2}$ are non-negative [32]. Thus, it leads to
$M^{2}\geq 2m^{2}$, namely, all the KK particles of various modes are massive.

Further, by setting $\mathcal{M}^{2}=0$ or $M=\sqrt{2}m$, the ground state of
the Schrödinger-like equation is obtained as

$\Psi_{0}(z)=N_{0}a(z)^{\frac{3}{2}},$ (30)

where the normalization factor $N_{0}$ can be worked out from the
normalization condition $\int^{z_{b}}_{-z_{b}}\Psi_{0}^{2}dz=1$, yielding

$\displaystyle
N_{0}^{-2}=\frac{1}{k}\left[1-e^{-2ky_{\pi}}+4\epsilon^{2}ky_{\pi}-\epsilon^{4}(1-e^{2ky_{\pi}})\right].$
(31)

Since the exponential growth branch of warp factor is completely dominant at
the IR brane, the normalization factor is approximated as
$N_{0}\approx\sqrt{k}$.

It is noted that for the ground state of vector mode with the mass
$M=\sqrt{2}m$, the rescaling
$\tilde{F}_{\alpha}=\sqrt{\frac{k^{2}m^{2}}{k^{2}+2m^{2}}}\frac{F_{\alpha}}{2}$
is ill-defined. Through a careful analysis, it is shown that the ground state
of the vector mode does not exist in the mass spectra [31]. This is curial for
recovering the mass spectra of RS1 model when the 5D graviton mass is turned
off, as there is no massless vector mode in RS1 model due to the lack of
continuous isometries of the bulk in the presence of 3-branes [3]. Therefore,
there exist only ground states of tensor and scalar modes in the mass spectra.

From the KK decomposition $\Upsilon=\upsilon(x)a^{-\frac{3}{2}}(z)\Psi(z)$,
the canonical normalized field configuration is given by
$\Upsilon_{0}=\upsilon_{0}(x)$. Therefore, the lightest tensor and scalar
modes propagate only on the brane, corresponding to the massive 4D graviton
and massive radion respectively. However, the mass of 4D graviton is severely
constrained by the gravitational experiments [21]. For example, the detection
of gravitational waves constrains the bound of the graviton mass to be
$m_{\text{g}}\leq 4.7\times 10^{-23}$eV [33]. It leads to $m\leq 3.3\times
10^{-23}$eV in our model. Under the constraint, the radion is also quasi-
massless. However, the radion can gain weight through Goldberger-Wise
mechanism and decoupled from the low-energy mass spectra [34].

By including only the contribution of the quasi-massless graviton in the
action (6), the 4D effective gravitational mass scale $M_{\text{eff}}$ reads,

$M_{\text{eff}}^{2}=M_{*}^{3}\int^{y_{\pi}}_{-y_{\pi}}a^{2}dy={N_{0}^{-2}}{M_{*}^{3}}.$
(32)

Therefore, the 4D effective gravitational mass scale is given by
$M_{\text{eff}}^{2}\simeq M_{*}^{3}/k$, which is the order of Planck scale as
expected.

For the excited KK states, their wave functions $\Psi_{M}$ can be worked out
by solving the Schrödinger-like equation (29). However, due to the complicated
form of the effective potential, the Schrödinger-like equation cannot be
solved directly in $z$ coordinate. By noting that the warp factor
$a^{\text{L}}(y)=e^{-ky}$ is dominated in the region $0<ky<\log(1/\epsilon)$,
while $a^{\text{R}}(y)=\epsilon^{2}e^{ky}$ is dominated in
$\log(1/\epsilon)<ky<ky_{\pi}$, the Schrödinger-like equation can be solved
approximately in these two regions respectively, yielding

$\displaystyle\Psi^{\text{L}}_{M}(y)$
$\displaystyle=e^{\frac{ky}{2}}\left[N_{n}J_{2}\left(\frac{M}{k}e^{ky}\right)+C_{1}Y_{2}\left(\frac{M}{k}e^{ky}\right)\right],$
(33) $\displaystyle\Psi^{\text{R}}_{M}(y)$
$\displaystyle=e^{-\frac{ky}{2}}\left[C_{2}J_{2}\left(\frac{M}{k\epsilon^{2}}e^{-ky}\right)+C_{3}Y_{2}\left(\frac{M}{k\epsilon^{2}}e^{-ky}\right)\right],$
(34)

where $N_{n}$ is the normalization factor, and $J_{2}$ and $Y_{2}$ are Bessel
functions of order 2. After imposing the boundary condition
$\partial_{y}\tilde{D}_{\alpha\beta}|_{y=0,y_{\pi}}=0$, i.e.,
${\Psi^{\text{L}}_{M}}^{\prime}-\frac{3}{2}\frac{a^{\text{L}}{}^{\prime}}{a^{\text{L}}}\Psi^{\text{L}}_{M}|_{y=0}=0$
and
$\Psi^{\text{R}}_{M}{}^{\prime}-\frac{3}{2}\frac{a^{\text{R}}{}^{\prime}}{a^{\text{R}}}\Psi^{\text{R}}_{M}|_{y=y_{\pi}}=0$,
one has

$\displaystyle
C_{1}=-\frac{J_{1}\left(\frac{M}{k}\right)}{Y_{1}\left(\frac{M}{k}\right)}N_{n},\quad
C_{3}=-\frac{J_{1}\left(\frac{M}{k\epsilon^{2}}e^{-ky_{\pi}}\right)}{Y_{1}\left(\frac{M}{k\epsilon^{2}}e^{-ky_{\pi}}\right)}C_{2}.$
(35)

Since the terms of $J_{2}$ dominate near $ky\sim\log(1/\epsilon)$ in both
$\Psi^{\text{L}}_{M}(y)$ and $\Psi^{\text{R}}_{M}(y)$, joining the two
functions together at $ky_{\text{m}}=\log(1/\epsilon)$ requires that
$C_{2}=N_{n}/\epsilon$ and $J_{2}\left(\frac{M}{k\epsilon}\right)=0$. This
condition yields a discrete eigenvalue spectrum as
$\mathcal{M}_{n}=x_{n}k\epsilon=\frac{x_{n}m}{2\sqrt{3}}$, where $x_{n}$
satisfies $J_{2}\left(x_{n}\right)=0$, e.g. $x_{1}=5.136$, $x_{2}=8.417$ and
$x_{3}=11.620$. Thus, the mass spectrum of excited KK states is given by

$M_{n}=m\sqrt{2+\frac{x_{n}^{2}}{12}}.$ (36)

By utilizing the approximate formula of the zero point of $J_{2}$,
$x_{n}\approx\left(n+\frac{3}{4}\right)\pi$, the mass spectrum (36) can be
approximated as $M_{n}\approx\frac{x_{n}m}{2\sqrt{3}}\approx\frac{n\pi
m}{2\sqrt{3}}$, for $n\gg 1$. Thus, the mass splitting reads $\Delta
M_{n}\approx\frac{\pi m}{2\sqrt{3}}$.

On the other hand, the mass splitting scale of excited KK states can also be
estimated from the fact that it is approximately quantized in units of inverse
size of conformal extra dimension, i.e., $\Delta M_{n}\sim{1}/{z_{\pi}}$. From
the coordinate transformation $dy=adz$, one has

$z_{\pi}=\frac{1}{k\epsilon}\left[\text{arctan}\left(\epsilon
e^{ky_{\pi}}\right)-\text{arctan}\left(\epsilon\right)\right],$ (37)

where the integral constant has been chosen so that $z(y=0)=0$. With
$\epsilon\leq 10^{-51}$ and $ky_{\pi}\sim\mathcal{O}(200)$, the formula can be
rewritten approximately as $z_{\pi}\approx\frac{\pi}{2\epsilon
k}=\frac{\sqrt{3}\pi}{m}$. So the mass splitting scale approximately reads
$\Delta M_{n}\sim\frac{m}{\sqrt{3}\pi}$, which is the same magnitude as the
previous result. Since $\Delta M_{n}\sim m<10^{-23}$eV, these massive KK
excited states are extremely light in this model.

##  5 Phenomenology on the brane

Since the mass splitting scale of KK gravitons is extremely small, an enormous
amount of KK gravitons could be easily produced in accelerators, which may
cause unacceptable large experimental signals. Therefore, it is necessary to
check some low-energy phenomenology on the IR brane.

First, we consider the correction to the Newtonian gravitational potential on
the brane. Due to the extremely small mass splitting scale, the mass spectrum
of KK gravitons can be approximated as continuous. The effective gravitational
potential between two point-like sources of mass $m_{1}$ and $m_{2}$ separated
by a distance $r$ on the brane takes the form [35, 36]

$U(r)=-G_{4}\frac{m_{1}m_{2}}{r}\left(1+\int dM\rho(M)e^{-Mr}\right),$ (38)

where $\rho(M)$ is the relative density of states on the IR brane for excited
KK states, defined by [36],
$\rho(M)\equiv\frac{\left|\Psi_{M}(y_{\pi})\right|^{2}}{\left|\Psi_{0}(y_{\pi})\right|^{2}}$.

Since
$\sqrt{x}J_{2}(x)\approx\sqrt{\frac{2}{\pi}}\cos\left(x-\frac{5}{4}\pi\right)$
for a large $x$, the dominant terms $J_{2}$ in $\Psi^{\text{L}}_{M}(y)$ and
$\Psi^{\text{R}}_{M}(y)$ have the plane wave behavior around
$y_{\text{m}}=\log(1/\epsilon)/k$. So after normalizing these KK states as
plane waves such that the physical quantities always involve an integration
over $M$ for which the proper measure is $dM$ [7, 36], the normalization
factor is given by $N_{n}=\sqrt{{M}/{k}}$. Thus, from Eq. (34), we have

$\Psi_{M}(y_{\pi})=\Psi^{\text{R}}_{M}(y_{\pi})\approx-\left(\frac{M}{k\epsilon^{2}e^{ky_{\pi}}}\right)^{\frac{1}{2}}.$
(39)

With
$\Psi_{(0)}(y_{\pi})\approx\sqrt{k}\left(\epsilon^{2}e^{ky_{\pi}}\right)^{3/2}$,
the relative density of states is obtained finally

$\rho(M)\approx\frac{M}{k^{2}\left(\epsilon^{2}e^{ky_{\pi}}\right)^{4}}.$ (40)

As a result, the effective gravitational potential on the IR brane reads

$\displaystyle U(r)$
$\displaystyle=-G_{4}\frac{m_{1}m_{2}}{r}\left(1+\frac{1}{k^{2}r^{2}\left(\epsilon^{2}e^{ky_{\pi}}\right)^{4}}\right)$
$\displaystyle=-G_{4}\frac{m_{1}m_{2}}{r}\left[1+\frac{1}{\left(10^{-4}\text{eV}\right)^{2}r^{2}}\right].$
(41)

This deviation is the same as that of ADD model with two extra dimensions
[37].

Another widely considered process is the real emission of the KK gravitons,
which could be observed as missing energy in the accelerators. The total cross
section for the production of these on-shell massive gravitons in a typical
process $e^{+}e^{-}\to\gamma+$ KK gravitons can be roughly estimated to be
[36]

$\sigma(e^{+}e^{-}\to\gamma+\not{E}_{\text{KK}})\sim\frac{\alpha}{M^{2}_{\text{Pl}}}\int_{0}^{E_{c}}dM\rho(M),$
(42)

where $E_{c}$ is the center of mass energy for the process and
$\not{E}_{\text{KK}}$ is the missing energy carried away by KK gravitons.
After some simple algebra, the final result for total cross section is of
order

$\sigma(e^{+}e^{-}\to\gamma+\not{E}_{\text{KK}})\sim\frac{\alpha
E_{c}^{2}}{k^{4}\left(\epsilon^{2}e^{ky_{\pi}}\right)^{4}}\sim\frac{\alpha
E_{c}^{2}}{\text{TeV}^{4}}.$ (43)

This result is consistent with that of 6D ADD model as well [38].

The reason why the phenomenology generated by only one extra dimension in
present model are similar to those of 6D ADD model can be seen from the
couplings of the excited KK gravitons to matter [39, 9, 11],

$\displaystyle{\zeta}_{n}\sim\frac{a^{-\frac{3}{2}}(y_{\pi})\tilde{\Psi}_{M}(y_{\pi})}{M_{*}^{\frac{3}{2}}}\sim\frac{\sqrt{mM_{n}}}{\sqrt{k}M_{\text{Pl}}^{3/2}\left(\epsilon^{2}e^{ky_{\pi}}\right)^{2}}\sim\frac{\sqrt{n}m}{\text{TeV}^{2}}.$
(44)

Here, $\tilde{\Psi}_{M}(y_{\pi})\equiv\Psi_{M}(y_{\pi})/\sqrt{z_{\pi}}$, in
which, the factor $1/\sqrt{z_{\pi}}$ is included in order to restore an
appropriate dimension when returning from continuous integration to discrete
summation [8]. It is clear that the coupling between excited KK gravitons to
matter in present model
${\zeta}_{n}\sim\frac{m}{10^{-4}\text{eV}}\frac{\sqrt{n}}{M_{\text{Pl}}}$ is
much smaller than that of ADD model in which ${\zeta}_{n}\sim
1/M_{\text{Pl}}$. Nevertheless, in present model, the number of species of KK
gravitons with masses below $E_{c}$ is
$n\sim(E_{c}z_{\pi})\sim\frac{E_{c}}{\Delta M_{n}}\sim\frac{E_{c}}{m}$, which
is much more than the number of 6D ADD model
$n\sim(E_{c}R)^{2}\sim(\frac{E_{c}}{10^{-4}\text{eV}})^{2}$. Therefore, from
$\sigma(e^{+}e^{-}\to\gamma+\not{E}_{\text{KK}})\sim\sum_{n}\alpha{\zeta}_{n}^{2}\sim\alpha
n{\zeta}_{n}^{2}\sim\frac{\alpha E_{c}^{2}}{\text{TeV}^{4}}$, the total cross
sections of the two model are ultimately the same.

##  6 Conclusions and discussions

In RS1 model, the fact that our world resides on the negative tension brane
results in a “wrong-signed” Friedmann-like equation, making it unable to
describe an expanding universe on the brane. To overcome this issue, one
possible solution is to construct a braneworld scenario using only positive
tension branes. By taking advantage of braneworld sum rules, it was found that
this brane configuration may be supported in the 5D extension of the Lorentz-
violating massive gravity. Therefore, we generalized RS1-like model in this
gravity and found that the theory supports three distinct brane
configurations, among which the configuration with only two positive tension
branes is exactly what we expected. By confining our world on the positive
tension IR brane, a “correct-signed” 4D Friedmann-like equation can be
recovered in this model and the gauge hierarchy problem can be solved as well.

By considering the full linear perturbations against the background metric, it
is found that the tensor and vector modes are robust, but the scalar mode is
ghost-like. By solving the quadratic actions of various perturbed modes, it
was found that they share the same mass spectrum, which start from $\sqrt{2}m$
and have a mass splitting scale roughly $\Delta M_{n}\sim m$, except that the
ground state of vector mode is absent in the mass spectrum. Since the graviton
mass is severely constrained by experimental observations, the mass splitting
of KK gravitons has to be extremely small. By further studying the interaction
between KK gravitons and matter fields on the brane, it was found that the
large amount of KK gravitons leads to the same phenomenology as that of the
ADD model with two extra dimensions.

If we remove the IR brane, i.e., $y_{\pi}\to\infty$, the ground state would no
longer be normalizable. In this case, the effective 4D gravity cannot be
recovered on the brane. So the RS2-like single brane model with non-compact
extra dimension is not viable in present theory. However, it is interesting
that a single brane model with compact extra dimension can be obtained by
setting $y_{\pi}=y_{\text{m}}=\log(1/\epsilon)/k$ to vanish the IR brane
tension in the fine tuning conditions (19). Now our world should move to the
only remaining UV brane and hence the gauge hierarchy problem is left. The
mass splitting scale approximates $\Delta
M_{n}\sim{1}/{z_{\pi}}\sim\frac{2m}{\sqrt{3}\pi}$, which is the same order as
that of above model with positive tensions. However, due to the wave functions
of KK particles are heavily suppressed by the potential barrier of order of
$M_{\text{Pl}}$ at $y=0$, the total cross section of the process
$e^{+}e^{-}\to\gamma+$ KK gravitons is estimated as
$\sigma\sim{E_{c}^{2}}/{M_{\text{Pl}}^{4}}$, and the gravitational potential
reads $U(r)=-G_{4}\frac{m_{1}m_{2}}{r}\left(1+\frac{1}{k^{2}r^{2}}\right)$,
which is in fact identical to those in the RS2 model.

In order to obtain a Minkowski flat brane model, the internal metric has been
chosen as the Minkowski metric in the action (6). However, due to the unbroken
4D Lorentz invariance and the non-Fierz-Pauli form of the mass term, the ghost
scalar mode appears at the linear perturbation theory. The ghost mode could be
removed by breaking further the 4D Lorentz invariance on the brane, such as
leaving a residual SO(3) symmetry. However, since the 4D Lorentz invariance is
necessary to build the flat 3-brane configuration, there would be no flat
brane solution anymore in a 4D Lorentz symmetry breaking theory. Therefore,
one can only expect (anti)-de Sitter 3-brane solutions in such a theory with
residual SO(3) symmetry. These models are left for our further investigation.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China
under Grant Nos. 12005174, 12165013, and 12247101. K. Yang acknowledges the
support of Natural Science Foundation of Chongqing, China (Grant No.
cstc2020jcyj-msxmX0370). B.-M. Gu acknowledges the support of Jiangxi
Provincial Natural Science Foundation (Grant No. 20224BAB211026). Y. Zhong
acknowledges the support of the Fundamental Research Funds for the Central
Universities (Grant No. 531118010195) and the Natural Science Foundation of
Hunan Province, China (Grant No. 2022JJ40033).

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