# A braneworld model in a massive gravity

Ke<EMAIL_ADDRESS>Shi-Fa<EMAIL_ADDRESS>Bao-Min
<EMAIL_ADDRESS>corresponding author aSchool of Physical Science and
Technology, Southwest University, Chongqing 400715, China
bCenter for Joint Quantum Studies and Department of Physics, School of
Science, Tianjin University, Tianjin 300350, China
cDepartment of Physics, Nanchang University, Nanchang 330031, China
dCenter for Relativistic Astrophysics and High Energy Physics, Nanchang
University, Nanchang 330031, China

###### Abstract

A Randall-Sundrum-like braneworld scenario is constructed in a 5D extension of
the Lorentz-violating massive gravity. The gauge hierarchy problem is solved
in current model. The linear perturbations are calculated, and it is found
that the tensor and vector perturbations are robust and free from the ghost
and tachyonic instabilities, however, the scalar perturbation is a ghost
filed. After Kaluza-Klein reduction, all the tensor, vector and scalar modes
are massive and possess the mass splitting of order of TeV in their respective
mass spectra. The massive ground states of tensor and scalar modes propagate
only along the brane, however, the vector ground state is absent in the mass
spectrum. By introducing the Goldberger-Wise mechanism to stabilize the extra
dimension, the 4D effective theory on the brane includes a nearly massless
graviton plus three towers of very massive spin-2, spin-1 and ghost spin-0
particles.

###### pacs:

04.50.+h, 04.50.Kd

##  I Introduction

Since Kaluza and Klein proposed the idea that our fundamental spacetime may be
more than four dimensions in 1920s Kaluza1921 ; Klein1926 , extra dimension
theories have received much attention from theoretical physicists. In Kaluza-
Klein (KK) theory, the extra dimension is compactified as a circle with the
radius of Planck length $l_{Pl}\sim 10^{-35}$m, and this strategy of
compactifying extra dimensions dominates higher-dimensional unified physics
Overduin1997 . Inspired by the string theory, the braneworld scenario provides
another way to hide the extra dimensions. In this scenario, our universe is a
brane embedded in a higher-dimensional bulk. All the Standard Model particles
are trapped on the brane and only the graviton can propagate into the extra
dimensions.

In 1999, Randall and Sundrum proposed a well-known braneworld model
Randall1999 , which is now dubbed RS1 model. In this model, the bulk is a
slice of a five-dimensional anti-de Sitter (AdS) spacetime, and there is a
brane at each boundary of the slice. In order to maintain a flat four-
dimensional spacetime on the brane, the fine-tuning condition requires that
the tension of the ultraviolet (UV) brane be positive while the tension of the
infrared (IR) brane be negative. Our universe lives on the IR brane, so owing
to the exponentially warped extra dimension, a fundamental electroweak scale
of the order of Planck scale ($M_{Pl}\sim 10^{16}$TeV) is redshifted to the
observed effective electroweak scale ($\sim$246GeV). The RS1 model provides a
natural way to solve the gauge hierarchy problem without introducing a new
large hierarchy, so it draws lots of attention in the last two decades, see,
e.g., Refs. Rubakov2001 ; Csaki2004 ; Maartens2010 ; Gherghetta2010 ;
Csaki1999 ; Cline1999 ; Goldberger1999a ; Rattazzi2001 ; Kisselev2016 ;
Blanke2018  for introduction and recent works.

The RS1 model is based on the 5D general relativity (GR), where the 5D
graviton is massless and possesses 5 degrees of freedom (DOF), therefore,
there is a massless 4D graviton, a massless scalar radion, and a tower of
massive 4D graviton in the effective 4D theory on the brane Randall1999 . In
the braneworld scenario, the effective 4D theory on the brane is related to
the DOF of the high-dimensional gravitational theory. So it is natural to ask
what is the difference between the 4D effective theories on the brane if the
higher-dimensional graviton possesses different numbers of DOF. One simple
exploration is to replace GR with the massive gravity theory, where the
higher-dimensional massive graviton possesses more numbers of DOF.

Massive gravity theory is a generalization of GR by endowing the graviton with
a nonzero mass. The first pioneering work was done by Fierz and Pauli in 1939
by adding non-derivative interaction terms, known as the Fierz-Pauli mass
term, into the linearized level of GR Fierz1939 . However, the theory does not
reduce to GR in the massless limit, so it suffers from a discontinuity
problem, known as van Dam-Veltman-Zakharov discontinuity Dam1970 ;
Zakharov1970 ; VanNieuwenhuizen1973 . This problem can be alleviated by
generalizing the linear Fierz-Pauli mass term to nonlinear terms
Vainshtein1972 . However, the nonlinear terms will generate a higher
derivative term and lead to the Boulware-Deser ghost instability Boulware1972
. A breakthrough was achieved by de Rham, Gabadadze, and Tolley about a decade
ago Rham2010a ; Rham2011 . They constructed a nonlinear massive gravity theory
with the field equation at most second order in time derivatives by getting
rid of ghosts for all nonlinear self-interactions of the longitudinal
component in the decoupling limit Hassan2012 ; Hassan2012a . This massive
gravity theory is now dubbed dRGT gravity. However, it is found that the dRGT
gravity may suffer from other pathologies. In Ref. Deser2013 , Deser and
Waldron argued that the theory admits second order superluminal shock wave
solutions, and the acausal characteristic can arise for any choice of
background. A detailed and correct analysis of characteristic equations was
performed by Izumi and Ong Izumi2013a , and they showed that the theory does
admit a well-posed Cauchy problem. Such acausality problem were further
discussed in Refs. Deser2013a ; Deser2014a ; Hassan2018 . The problem of
lacking the stable and realistic Friedmann-Lemaître-Robertson-Walker
cosmological solutions was discussed in Refs. DAmico2011 ; Gumrukcuoglu2011 ;
Gumrukcuoglu2012 ; DeFelice2012a ; DeFelice2013 .

On the other hand, although the Lorentz invariance is widely regarded as a
fundamental symmetry of nature, there are still some hints that the Lorentz
invariance may be broken at high energies in quantum gravity Kostelecky1989 ;
AmelinoCamelia2003 . Especially, the aforementioned standard problems of
Lorentz-invariant massive gravity theories may be avoided by switching to the
Lorentz-violating massive gravity theory, see Refs. Rubakov2004 ; Dubovsky2004
; Dubovsky2005 ; Blas2009 ; Lin2013  for examples. A simple example of
Lorentz-violating massive gravities was proposed by Lin in Ref. Lin2014 ,
where the theory is built with the Einstein-Hilbert action plus 3 canonical
massless scalar fields. The spatial condensation of the scalar fields breaks
spatial diffeomorphisms, which gives rise to 3 Goldstone excitations. After
“eating” the 3 Goldstone bosons in the unitary gauge, the graviton gets weight
and possesses 5 well-behaved DOF. It was found that the theory could remove
the infrared divergence of the inflationary loop diagram owing to the graviton
mass Lin2014 . For a recent review on massive gravity theories see Refs.
Hinterbichler2012 ; Rham2014 ; Rham2017  and the references therein.

The higher-dimensional scenario has a close relation with massive gravities,
as it naturally incorporates a tower of non-pathological massive spin-2
gravitons in the 4D graviton spectrum. Nevertheless, they are usually not a
purely massive gravity theories due to the presence of a massless graviton
associated with the unbroken 4D diffeomorphisms. A more interesting theory is
the Dvali-Gabadadze-Porrati model, whose graviton looks like a resonance and
effectively acquires a soft mass Dvali2000 ; Dvali2000a . In Ref. Chacko2004 ,
the authors considered the RS1 model with a localized Fierz-Pauli mass term
for the graviton on the IR brane, which reproduces GR for observers localized
on the UV brane at distances smaller than the IR scale but realizes a massive
gravity at longer distance with a ghost radion. The embedding of a 4D dRGT
massive gravity into a 5D dRGT massive gravity by resorting to the single-
brane RS2 model was studied by Gabadadze in Ref. Gabadadze2017 , where the
value of strong scale $\Lambda_{3}$ can be significantly increased. More
relevant works see Refs. Gabadadze2004 ; Kakushadze2014 ; Gabadadze2018 ;
Kaloper2019  for examples.

In this work, we are interested in generating the massive gravity with spatial
condensation into a 5D spacetime and studying the braneworld scenario in the
massive gravity. The background spacetime in the spatial condensation scenario
exhibits invariance under translation and 3D rotations. In order to achieve a
Minkowski flat brane model, we require that the background spacetime is
invariant under the 4D Poincaré transformation in the higher dimensional
generalization. The condensation of 4 background scalars spontaneously breaks
the 4D diffeomorphisms, and generates 4 Goldstone excitations associated with
the broken symmetries. Consequently, by “eating” the 4 Goldstone excitations
in the unitary gauge, the 5D massless spin-2 graviton with 5 DOF becomes
massive and possesses 9 DOF on the spectrum.

The layout of the paper is as follows: In Sect. II, the toy model is built in
the 5D massive theory. In Sect. III, the full linear perturbations have been
considered. In Sect. IV, the mass spectra of KK particles have been calculated
and the resolution of the gauge hierarchy problem has been discussed as an
application of the model. In Sect. V, the Goldberger-Wise (GW) mechanism has
been used to stabilize the radius of the extra dimension. Finally, brief
conclusions are presented. Throughout the paper, the small Latin letters
$a,b,\cdots=0,1,2,3,$ label the group indices of the internal metric of scalar
fields, and the capital Latin letters $A,B,\cdots=0,1,2,3,5,$ and Greek
letters $\mu,\nu,\cdots=0,1,2,3,$ label the 5D and 4D spacetime indices,
respectively.

##  II Model building

We start from a 5D extension of the Lorentz-violating massive gravity Lin2014
, and the action is given by a 5D Einstein-Hilbert term plus 4 canonical
massless scalar fields, i.e.,

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

where $M$ is the 5D fundamental gravity scale, $\Lambda$ is the 5D
cosmological constant, $V_{I}$ and $V_{II}$ represent the brane tensions. The
parameter $m$ is proportional to the mass of the 5D graviton, and this
relationship will become evident in Sect. IV. Here, the internal metric
$\eta_{ab}$ for the scalar fields is selected to be the Minkowski metric. This
choice guarantees the preservation of 4D Poincaré symmetry, allowing for the
attainment of a Minkowski flat brane model. However, it is evident that the
utilization of the Minkowski metric results in one of the bulk scalars, namely
$\phi^{0}$, having a kinetic term with the wrong sign. Consequently, this
gives rise to a ghost scalar mode in the linear perturbation theory, which
will be clearly observed in Sect. III.

The background solution spontaneously breaks the 5D Lorentz invariance when
the scalar fields are fixed to their vacuum values

$\langle\phi^{a}\rangle=\delta_{\mu}^{a}x^{\mu},$ (2)

with $x^{\mu}$ the coordinates of 4D spacetime. Thus, the condensation
spontaneously generates a preferred 4D frame.

In order to investigate a RS1 model-like braneworld solution, the metric
ansatz preserving the 4D Poincaré invariance reads

$ds^{2}_{5}=a^{2}(y){\eta}_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$ (3)

where $a(y)$ is the warp factor and $y\in[-y_{\pi},y_{\pi}]$ denotes an
$S^{1}/Z_{2}$ orbifold extra dimension. The UV brane with tension
$V_{\text{I}}$ locates at $y=0$ and the IR brane with tension $V_{\text{II}}$
locates at $y=y_{\pi}$.

The field equations are obtained by varying the action (1) with respect to the
metric $g^{MN}$,

$\displaystyle G_{MN}$ $\displaystyle=$ $\displaystyle
m^{2}\Big{(}\partial_{M}\phi^{a}\partial_{N}\phi^{a}-\frac{1}{2}g_{MN}\partial^{K}\phi^{a}\partial_{K}\phi^{a}\Big{)}-\Lambda
g_{MN}-\frac{\sqrt{-g_{\text{I}}}}{\sqrt{-g}}\frac{V_{\text{I}}}{M^{3}}g_{\text{I}\mu\nu}\delta^{\mu}_{M}\delta^{\nu}_{N}\delta\left(y\right)$
(4)
$\displaystyle-\frac{\sqrt{-g_{\text{II}}}}{\sqrt{-g}}\frac{V_{\text{II}}}{M^{3}}g_{\text{II}\mu\nu}\delta^{\mu}_{M}\delta^{\nu}_{N}\delta\left(y-y_{\pi}\right).$

Further, with the ansatzes of braneworld metric and scalar field condensation,
the field equations are written explicitly as

$\displaystyle 3\left(H^{\prime}+2H^{2}\right)\\!$ $\displaystyle=\\!$
$\displaystyle-\frac{m^{2}}{a^{2}}-\Lambda-\frac{V_{\text{I}}\delta\left(y\right)+V_{\text{II}}\delta\left(y\\!-\\!y_{\text{b}}\right)}{M^{3}},$
(5) $\displaystyle 6H^{2}\\!$ $\displaystyle=$
$\displaystyle\\!-\frac{2m^{2}}{a^{2}}-\Lambda,$ (6)

where $H\equiv a^{\prime}/a$ and the prime denotes the derivative with respect
to the extra dimension coordinate $y$.

The field equations simply give the following solution

$a(y)=e^{-k|y|}+\epsilon^{2}e^{k|y|},$ (7)

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. Since the minimum value of warp factor is $2\epsilon$, the
5D graviton has to be light enough, i.e., $\epsilon<a(y_{\pi})\sim 10^{-16}$,
to generate enough redshift for the fundamental electroweak scale. As
illustrated in Fig. 1, the warp factor is different from that of RS model only
far from the center for a tiny graviton mass, $m/k\ll 1$.

Figure 1: The profiles of the warp factor $a(y)$ with respect to different
mass parameters.

After matching the delta functions in (5), we have the fine-tuning conditions,

$\displaystyle V_{I}$ $\displaystyle=$ $\displaystyle
6kM^{3}\left[1-\frac{2\epsilon^{2}}{1+\epsilon^{2}}\right],$ (8)
$\displaystyle V_{II}$ $\displaystyle=$ $\displaystyle
6kM^{3}\left[1-\frac{2}{1+\epsilon^{2}e^{2ky_{\pi}}}\right].$ (9)

In the case of $e^{-ky_{\pi}}>\epsilon$, the exponential decay term of warp
factor (7) is dominant in the whole bulk. It is clear that $V_{I}>0$ and
$V_{II}<0$ in this case, namely, there is a positive tension brane located at
the origin and a negative tension brane at $y_{\pi}$. This is the model we
will focus on in the rest of the paper. Especially, in the limit $\epsilon
e^{ky_{\pi}}\ll 1$, the warp factor (7) approaches the exponential decay form
of RS model. Correspondingly, the fine-turning conditions reduce to the case
of RS model as well,

$V_{I}\approx-V_{II}\approx 6kM^{3},~{}~{}~{}\Lambda=-6k^{2}.$ (10)

Besides, there is another interesting case that both brane tensions are
positive in case of $e^{-k|y_{\pi}|}<\epsilon<1$. It would generate a new
brane configuration distinct from the RS1 model. Especially, the scenario that
our universe is confined on the positive tension brane may potentially provide
a possible model with better properties Csaki1999 ; Shiromizu2000 ; Yang2012a
. This model will be studied in detail in our another work.

##  III Linear perturbations

In order to investigate the stability of the model, we consider the full
linear perturbations against the background. The perturbed metric is written
as

$ds^{2}=\left(g_{MN}+h_{MN}\right)dx^{M}dx^{N},$ (11)

where $h_{MN}$ represents the perturbations against the background metric
$g_{MN}$ given in Eq. (3). Due to the 4D Lorentz symmetry of our background
spacetime, it is convenient to decompose the perturbed metric into the scalar,
transverse vector and transverse-traceless tensor modes, and to rewrite it by

$\displaystyle h_{55}\\!$ $\displaystyle=\\!$ $\displaystyle-2\xi,$ (12)
$\displaystyle h_{\mu 5}\\!$ $\displaystyle=\\!$
$\displaystyle-a\left(S_{\mu}+\partial_{\mu}\beta\right),$ (13) $\displaystyle
h_{\mu\nu}\\!$ $\displaystyle=$
$\displaystyle\\!a^{2}\\!\\!\left[\\!D_{\mu\nu}\\!+\\!2\eta_{\mu\nu}\psi\\!+\\!\frac{1}{2}\left(\partial_{\mu}F_{\nu}\\!+\\!\partial_{\nu}F_{\mu}\right)\\!+\\!2\partial_{\mu}\partial_{\nu}E\right]\\!\\!,$
(14)

where the vector modes satisfy the transverse condition
$\partial^{\mu}S_{\mu}=\partial^{\mu}F_{\mu}=0$, and the tensor mode satisfies
the transverse-traceless (TT) condition $\partial^{\mu}D_{\mu\nu}=0$.

Correspondingly, the perturbed scalar fields are

$\phi^{a}=x^{a}+\pi^{a},$ (15)

where $\pi^{a}=\delta^{a}_{\mu}\pi^{\mu}$ is the Goldstone excitation of the
condensation. In order to maintain the scalar condensation (2) under the
general coordinate transformation $x^{M}\to x^{M}+\epsilon^{M}$, the Goldstone
excitation has to transform opposite to the 4D coordinates simultaneously,
i.e., $\pi^{\mu}\to\pi^{\mu}-\epsilon^{\mu}$. This Stückelberg trick non-
linearly restores the general covariance of the theory. As a result, it
behaves like a vector field and can be decomposed as
$\pi^{\mu}=\eta^{\mu\nu}\left(\partial_{\nu}\varphi+A_{\nu}\right)$, with
$\varphi$ a scalar field and $A_{\mu}$ a transverse vector field.

Consequently, there are 1 TT tensor, 3 transverse vector and 5 scalar modes in
total. By decomposing
$\epsilon_{\mu}=a^{2}(\epsilon^{V}_{\mu}+\partial_{\mu}\epsilon^{s})$ with
$\epsilon^{V}_{\mu}$ a transverse vector and $\epsilon^{s}$ a scalar, these 9
perturbed modes transform as follows under the general coordinate
transformation,

$\displaystyle D_{\mu\nu}$ $\displaystyle\to$ $\displaystyle D_{\mu\nu},\quad
S_{\mu}\to S_{\mu}+a\epsilon^{V}_{\mu}{}^{\prime},\quad F_{\mu}\to
F_{\mu}-2\epsilon^{V}_{\mu},$ $\displaystyle A_{\mu}$ $\displaystyle\to$
$\displaystyle
A_{\mu}-{\epsilon}^{V}_{\mu},\quad\psi\to\psi-H\epsilon_{5},\quad E\to
E-\epsilon^{s},$ $\displaystyle\beta$ $\displaystyle\to$
$\displaystyle\beta+a{\epsilon}^{s}{}^{\prime}+\frac{\epsilon_{5}}{a},\quad\xi\to\xi+\epsilon^{\prime}_{5},\quad\varphi\to\varphi-{\epsilon}^{s}.$
(16)

Naively, we have 5 gauge freedoms to eliminate one vector and two scalar modes
by fixing ${\epsilon}^{V}_{\mu}$, ${\epsilon}^{s}$ and $\epsilon_{5}$. A
commonly used gauge choice is the so-called unitary gauge, in which the
Goldstone excitations $\pi^{a}$ of scalar fields are closed, i.e.,
$A_{\mu}=\varphi=0$. Moreover, we have another gauge freedom $\epsilon_{5}$ to
set $\psi=0$. Then from the fact that $Z_{\mu}-\pi_{\mu}$, with $Z_{\mu}\equiv
a^{2}\left(F_{\mu}/2+\partial_{\mu}E\right)$, is a gauge invariant quantity,
one observes that the 4 Goldstone excitations $\pi^{\mu}$ are “eaten” by the
5D graviton in the unitary gauge. Consequently, the 5D massless spin-2
graviton with 5 DOF gets weight and becomes massive, with 9 DOF on the
spectrum. After gauge fixing, we have 1 TT tensor, 2 transverse vector and 3
scalar modes left. However, not all modes among them are physical, since some
of them can be eliminated by some constraint equations, which can be easily
extracted by utilizing the Arnowitt-Deser-Misner decomposition.

The Arnowitt-Deser-Misner decomposition of the 5-dimensional spacetime metric
reads Manrique2011

$ds^{2}=N^{2}dy^{2}+\gamma_{\mu\nu}\left(dx^{\mu}+N^{\mu}dy\right)\left(dx^{\nu}+N^{\nu}dy\right),$
(17)

where $N$ is the lapse, $N^{\mu}$ the shift vector, and $\gamma_{\mu\nu}$ the
induced metric on the slices. Correspondingly, the bulk action of (1) is
rewritten as

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{M^{3}}{2}\int
d^{5}x\sqrt{-\gamma}\Big{[}N\left(R^{(4)}-2\Lambda-m^{2}\gamma^{\mu\nu}\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{a}\right)$
(18)
$\displaystyle-m^{2}N^{-1}\left(\phi^{a}{}^{\prime}-N^{\mu}\partial_{\mu}\phi^{a}\right)\left(\phi^{a}{}^{\prime}-N^{\nu}\partial_{\nu}\phi^{a}\right)-N^{-1}\left(E_{\mu\nu}E^{\mu\nu}-E^{2}\right)\Big{]},$

where
$E_{\mu\nu}=\frac{1}{2}\left(\gamma^{\prime}_{\mu\nu}-\nabla_{\nu}N_{\mu}-\nabla_{\mu}N_{\nu}\right)$.
The constraint equations are yielded by varying with respect to the lapse
function $N$ and shift vector $N_{\mu}$, i.e.,

$\displaystyle
R^{(4)}-2\Lambda-m^{2}\gamma^{\mu\nu}\partial_{\mu}\phi^{a}\partial_{\nu}\phi^{a}+N^{-2}\big{(}E_{\mu\nu}E^{\mu\nu}\\!$
$\displaystyle-$ $\displaystyle\\!E^{2}\big{)}$
$\displaystyle+m^{2}N^{-2}\left(\phi^{a}{}^{\prime}-N^{\mu}\partial_{\mu}\phi^{a}\right)\left(\phi^{a}{}^{\prime}-N^{\nu}\partial_{\nu}\phi^{a}\right)$
$\displaystyle=$ $\displaystyle 0,$ (19)
$\displaystyle\nabla_{\nu}\left(\frac{E^{\mu}_{\nu}-\delta^{\mu}_{\nu}E}{N}\right)-\frac{m^{2}}{N}\partial_{\mu}\phi^{a}\left(\phi^{a}{}^{\prime}-N^{\alpha}\partial_{\alpha}\phi^{a}\right)$
$\displaystyle=$ $\displaystyle 0.$ (20)

Further, by considering small perturbations of the metric functions, expressed
as $N=\bar{N}+\delta N$, $N^{\mu}=\bar{N}^{\mu}+\delta N^{\mu}$, and
$\gamma_{\mu\nu}=\bar{\gamma}_{\mu\nu}+\delta\gamma_{\mu\nu}$, where
$\bar{N}$, $\bar{N}^{\mu}$, and $\bar{\gamma}_{\mu\nu}$ represent the
background quantities with $\bar{N}=1$, $\bar{N}^{\mu}=0$, and
$\bar{\gamma}_{\mu\nu}=a^{2}(y)\eta_{\mu\nu}$, respectively. The perturbations
$\delta N$, $\delta N^{\mu}$, and $\delta\gamma_{\mu\nu}$ can be explicitly
extracted from Eqs. (12), (13), and (14). Consequently, one can obtain a
constraint equation for vector modes and two constraint equations for scalar
modes. Thus, one vector mode and two scalar modes can be worked out
algebraically, leaving us one vector and one scalar modes eventually.

After scalar-vector-tensor decomposition and expanding the action (1) to the
quadratic order of perturbations, the TT tensor, transverse vector and scalar
modes are decoupled with each other, so they can be treated separately.

###  III.1 Tensor mode

By including only the tensor perturbation in the perturbed metric and dropping
the boundary terms, the quadratic action for the tensor perturbation reads

$\displaystyle S^{(2)}_{T}$ $\displaystyle=$ $\displaystyle\frac{M^{3}}{8}\int
d^{4}xdya^{2}\left[-{a^{2}}D_{\alpha\beta}^{\prime}D{}^{\prime\alpha\beta}-{\partial_{\lambda}D_{\alpha\beta}}\partial^{\lambda}D^{\alpha\beta}-2m^{2}D_{\alpha\beta}D^{\alpha\beta}\right],$
(21)

where the indices are raised and lowered by the 4D Minkowski metric
$\eta_{\mu\nu}$. Further, after a coordinate transformation $dy=adz$ that
turns the background metric into a conformal form and a rescaling
$\tilde{D}={D}/{2}$, a canonical form is obtained as

$\displaystyle S^{(2)}_{T}$ $\displaystyle=$ $\displaystyle\frac{M^{3}}{2}\int
d^{4}xdza^{3}\left[-\dot{\tilde{D}}_{\alpha\beta}\dot{\tilde{D}}^{\alpha\beta}-\partial_{\lambda}\tilde{D}_{\alpha\beta}\partial^{\lambda}\tilde{D}^{\alpha\beta}-2m^{2}\tilde{D}_{\alpha\beta}\tilde{D}^{\alpha\beta}\right],$
(22)

where the dot denotes the derivative with respect to the extra dimension
coordinate $z$. The tensor mode is free from the ghost instability due to the
correct sign of kinetic term. Moreover, the action is expressed more concisely
in the momentum space, where the d’Alembert operator
$\partial^{\alpha}\partial_{\alpha}$ is replaced by
$-p^{2}\equiv-p^{\alpha}p_{\alpha}$, with $p^{\alpha}$ the four-momentum of
tensor mode. Then, it yields

$S^{(2)}_{T}\\!=\\!\frac{M^{3}}{2}\\!\\!\int\\!\\!d^{4}pdza^{3}\left[\\!-\dot{\tilde{D}}_{\alpha\beta}\dot{\tilde{D}}^{\alpha\beta}\\!-\\!\left({p^{2}}\\!+\\!{2m^{2}}\right)\tilde{D}_{\alpha\beta}\tilde{D}^{\alpha\beta}\right]\\!\\!.$
(23)

By variation with respect to $\tilde{D}^{\alpha\beta}$, we have the equation
of motion

$\ddot{\tilde{D}}_{\alpha\beta}+3H\dot{\tilde{D}}_{\alpha\beta}=(p^{2}+2m^{2}){\tilde{D}_{\alpha\beta}}.$
(24)

Note that here and in what follows, $H\equiv\dot{a}/a$ refers to the extra
dimension coordinate $z$. After redefining
$\tilde{D}_{\alpha\beta}=a^{-\frac{3}{2}}\mathcal{D}_{\alpha\beta}$, a
Schrödinger-like equation is obtained,

$-\ddot{\mathcal{D}}_{\alpha\beta}+\left(\frac{3}{2}\dot{H}+\frac{9}{4}H^{2}\right){\mathcal{D}}_{\alpha\beta}=M_{T}^{2}{\mathcal{D}}_{\alpha\beta},$
(25)

where $M_{T}^{2}\equiv-p^{2}-2m^{2}$. The Hamiltonian can be further
factorized into a supersymmetric quantum mechanics form, with
$H_{T}=A_{T}^{\dagger}A_{T}=\left(\partial_{z}+\frac{3}{2}H\right)\left(-\partial_{z}+\frac{3}{2}H\right)$.
Then, with the boundary condition
$\partial_{z}\tilde{D}_{\alpha\beta}|_{z=0,z_{b}}=0$, the self-adjoint
Hamiltonian gives non-negative eigenvalues Yang2017 , i.e., $M_{T}^{2}\geq 0$.
Consequently, with a positive 5D graviton mass $m$, the four-momentum of
tensor mode $p^{2}=-M_{T}^{2}-2m^{2}<0$, i.e., the tensor excitations are all
time-like particles. Thus, the model is also irrelevant to tachyonic
instability. If $m$ is set to zero, $M_{T}^{2}=-p^{2}\geq 0$ gives us the
well-known result that there is a massless graviton and a tower of massive
gravitons in RS1 model.

###  III.2 Vector modes

By including only the vector perturbations in the perturbed metric and
dropping the boundary terms, the quadratic action for vector perturbations
reads

$\displaystyle S_{V}^{(2)}\\!$ $\displaystyle=$
$\displaystyle\\!\frac{M^{3}}{16}\\!\int\\!d^{4}xdya^{2}\Big{[}\\!\\!-\\!a^{2}\partial_{\beta}F_{\alpha}{}^{\prime}\partial^{\beta}F^{\alpha}{}^{\prime}\\!-\\!2m^{2}\partial_{\beta}F_{\alpha}\partial^{\beta}F^{\alpha}-8a^{2}m^{2}A_{\alpha}{}^{\prime}A^{\alpha}{}^{\prime}-8m^{2}\partial_{\beta}A_{\alpha}\partial^{\beta}A^{\alpha}$
(26)
$\displaystyle-4\partial_{\beta}S_{\alpha}\partial^{\beta}S^{\alpha}-8m^{2}S_{\alpha}S^{\alpha}+8m^{2}\partial_{\beta}F_{\alpha}\partial^{\beta}A^{\alpha}-4a\partial_{\beta}F_{\alpha}{}^{\prime}\partial^{\beta}S^{\alpha}-16am^{2}S_{\alpha}A^{\alpha}{}^{\prime}\Big{]}.$

The constraint equation can be obtained by counting the first order
perturbations of Eq. (20), or simplify by varying the above quadratic action
with respect to $S^{\alpha}$, i.e.,

$2\partial_{\beta}{\partial^{\beta}S_{\alpha}}-4m^{2}S_{\alpha}+a\partial_{\beta}{\partial^{\beta}F_{\alpha}^{\prime}}-4am^{2}A_{\alpha}^{\prime}=0.$
(27)

Then, $S^{\alpha}$ can be worked out in momentum space, namely,

$S_{\alpha}=-\frac{ap^{2}F_{\alpha}^{\prime}+4am^{2}A_{\alpha}^{\prime}}{2(p^{2}+2m^{2})}.$
(28)

Working in the unitary gauge, $A_{\alpha}$ is gauged away. Then, substituting
the relation into the action (26) yields

$S_{V}^{(2)}\\!=\\!M^{3}\\!\int\\!d^{4}pdy\\!\left[-\frac{a^{4}p^{2}m^{2}F_{\alpha}{}^{\prime}F^{\alpha}{}^{\prime}}{8\left(p^{2}+2m^{2}\right)}\\!-\\!\frac{a^{2}p^{2}m^{2}}{8}F_{\alpha}F^{\alpha}\right]\\!.$
(29)

After performing a coordinate transformation into the $z$ coordinate, a
canonical normalized form can be obtained by redefining the vector
perturbation as

$\tilde{F}_{\alpha}=\sqrt{\frac{p^{2}m^{2}}{p^{2}+2m^{2}}}\frac{F_{\alpha}}{2},$
(30)

yielding,

$S^{(2)}_{V}\\!=\\!\frac{M^{3}}{2}\int
d^{4}pdza^{3}\left[-\dot{\tilde{F}}_{\alpha}\dot{\tilde{F}}^{\alpha}-\left({p^{2}}+{2m^{2}}\right)\tilde{F}_{\alpha}\tilde{F}^{\alpha}\right]\\!.$
(31)

The correct sign of the kinetic term ensures that the vector perturbation is
free from the ghost instability. Then the equation of motion reads

$\ddot{\tilde{F}}_{\alpha}+3H\dot{\tilde{F}}_{\alpha}=(p^{2}+2m^{2}){\tilde{F}_{\alpha}}.$
(32)

A Schrödinger-like equation is given by redefining $\tilde{F}_{\alpha}\to
a^{-\frac{3}{2}}\mathcal{F}_{\alpha}$,

$-\ddot{\mathcal{F}}_{\alpha}+\left(\frac{3}{2}\dot{H}+\frac{9}{4}H^{2}\right)\mathcal{F}_{\alpha}=M_{V}^{2}\mathcal{F}_{\alpha},$
(33)

where $M_{V}^{2}\equiv-p^{2}-2m^{2}$. The Hamiltonian can also be factorized
into a supersymmetric quantum mechanics form, with
$H_{V}=A_{V}^{\dagger}A_{V}=\left(\partial_{z}+\frac{3}{2}H\right)\left(-\partial_{z}+\frac{3}{2}H\right)$.
With the boundary condition $\partial_{z}\tilde{F}_{\alpha}|_{z=0,z_{b}}=0$,
the eigenvalues are non-negative $M_{V}^{2}\geq 0$, i.e., $p^{2}\leq-2m^{2}$.
Thus, the vector excitations are also time-like particles and irrelevant to
tachyonic instabilities.

For the cases of $p^{2}<-2m^{2}$, the formula (28) and the redefinition (30)
are valid. However, for the case of $M_{V}^{2}=-p_{0}^{2}-2m^{2}=0$, the
formula (28) is problematic, and the constraint equation (27) leads to

$F_{\alpha}^{\prime}=2A_{\alpha}^{\prime}.$ (34)

This implies that $F_{\alpha}-2A_{\alpha}\equiv f_{\alpha}(x)$ is a purely 4D
field. Especially, the field $f_{\alpha}(x)$ is a gauge invariant quantity
under the general coordinate transformation by the observation from (16).
Then, the quadratic action (26) reduces to

$\displaystyle S^{(2)}_{V0}=-\frac{M^{3}}{8}\int
d^{4}pdza^{3}m^{2}k_{0}^{2}\left(F_{\alpha}-2A_{\alpha}\right)\left(F^{\alpha}-2A^{\alpha}\right)=-\frac{M^{3}}{8}\int
d^{4}pdza^{3}m^{2}k_{0}^{2}f_{\alpha}f^{\alpha}.$ (35)

The equation of motion reads $p_{0}^{2}f_{\alpha}=0$. Since
$p_{0}^{2}=-2m^{2}$, it leads to $f_{\alpha}=0$. This implies that the
lightest vector mode does not exist in the mass spectrum. This is curial to
recover the mass spectrum of RS1 model in the massless limit $m=0$, where no
massless vector mode exists due to the lack of continuous isometries of the
higher dimension in the presence of 3-branes Randall1999 . Moreover, all the
massive vector modes are gauge dependent in the massless limit $m=0$,
therefore, they can be gauged away by gauge fixing.

###  III.3 Scalar modes

By including only the scalar perturbations in the perturbed metric and
dropping the boundary terms, the quadratic action for scalar perturbations
reads

$\displaystyle S^{(2)}_{S}\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!M^{3}\\!\\!\int\\!\\!d^{4}xdya^{2}\\!\Big{(}6a^{2}H^{2}\xi^{2}\\!+\\!4m^{2}\psi^{2}\\!-\\!4m^{2}\xi\psi\\!+\\!6a^{2}\psi^{\prime}{}^{2}+12a^{2}H\xi\psi^{\prime}\\!+\\!3\partial_{\alpha}\psi\partial^{\alpha}\psi\\!-\\!\frac{m^{2}}{2}\partial_{\alpha}\beta\partial^{\alpha}\beta\\!$
(36)
$\displaystyle\\!+\\!3\xi\partial_{\alpha}\partial^{\alpha}\psi+3aH\xi\partial_{\alpha}\partial^{\alpha}\beta+m^{2}\xi\partial_{\alpha}\partial^{\alpha}\varphi-m^{2}\xi\partial_{\alpha}\partial^{\alpha}E-2m^{2}\psi\partial_{\alpha}\partial^{\alpha}\varphi+2m^{2}\psi\partial_{\alpha}\partial^{\alpha}E$
$\displaystyle\\!+3a^{2}H\xi\partial_{\alpha}\partial^{\alpha}E^{\prime}+3a\beta\partial_{\alpha}\partial^{\alpha}\psi^{\prime}+m^{2}a\beta\partial_{\alpha}\partial^{\alpha}\varphi^{\prime}-\frac{m^{2}}{2}a^{2}\partial_{\alpha}\varphi^{\prime}\partial^{\alpha}\varphi^{\prime}+3a^{2}\psi^{\prime}\partial_{\alpha}\partial^{\alpha}E^{\prime}$
$\displaystyle\\!-\frac{m^{2}}{2}\partial_{\alpha}\partial^{\alpha}E\partial_{\lambda}\partial^{\lambda}E-\frac{m^{2}}{2}\partial_{\alpha}\partial^{\alpha}\varphi\partial_{\lambda}\partial^{\lambda}\varphi+m^{2}\partial_{\alpha}\partial^{\alpha}\varphi\partial_{\lambda}\partial^{\lambda}E\Big{)}.$

From the first order perturbation of Eqs. (19) and (20) or simply varying the
above action respect to the modes $\beta$ and $\xi$ respectively, the
constraint equations are obtained as

$\displaystyle
12a^{2}H^{2}\xi-4m^{2}\psi+12a^{2}H\psi^{\prime}+3\partial_{\alpha}\partial^{\alpha}\psi+3aH\\!\\!\\!\\!\\!$
$\displaystyle\\!\\!\\!\partial_{\alpha}\partial^{\alpha}\beta$
$\displaystyle+m^{2}\partial_{\alpha}\partial^{\alpha}\varphi-m^{2}\partial_{\alpha}\partial^{\alpha}E+3a^{2}H\partial_{\alpha}\partial^{\alpha}E^{\prime}$
$\displaystyle=$ $\displaystyle 0,$ (37) $\displaystyle
m^{2}\partial_{\alpha}\beta+3aH\partial_{\alpha}\xi+3a\partial_{\alpha}\psi^{\prime}+m^{2}a\partial_{\alpha}\varphi^{\prime}$
$\displaystyle=$ $\displaystyle 0.$ (38)

After closing the scalar perturbations $\varphi$ and $\psi$ in the unitary
gauge, we only have to take the remaining perturbations $\xi$, $\beta$ and $E$
into account in the quadratic action, i.e.,

$\displaystyle S^{(2)}_{S}$ $\displaystyle=$ $\displaystyle M^{3}\int
d^{4}xdya^{2}\Big{(}6a^{2}H^{2}\xi^{2}-\frac{m^{2}}{2}\partial_{\alpha}\beta\partial^{\alpha}\beta+3aH\xi\partial_{\alpha}\partial^{\alpha}\beta-m^{2}\xi\partial_{\alpha}\partial^{\alpha}E$
(39)
$\displaystyle+3a^{2}H\xi\partial_{\alpha}\partial^{\alpha}E^{\prime}-\frac{m^{2}}{2}\partial_{\alpha}\partial^{\alpha}E\partial_{\lambda}\partial^{\lambda}E\Big{)}.$

Correspondingly, the constraint equations are rewritten as

$\displaystyle
12a^{2}\\!H^{2}\xi\\!+\\!3aH\partial_{\alpha}\partial^{\alpha}\\!\beta\\!-\\!m^{2}\partial_{\alpha}\partial^{\alpha}\\!E\\!+\\!3a^{2}\\!H\partial_{\alpha}\partial^{\alpha}\\!E^{\prime}\\!$
$\displaystyle\\!=\\!$ $\displaystyle 0,$ (40) $\displaystyle
m^{2}\partial_{\alpha}\beta+3aH\partial_{\alpha}\xi\\!$ $\displaystyle\\!=\\!$
$\displaystyle 0.$ (41)

In momentum space, $\beta$ and $\xi$ can be worked out from the constraint
equations as

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle-\frac{3aH}{m^{2}}\xi,$
(42) $\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{p^{2}m^{2}(3a^{2}HE^{\prime}-m^{2}E)}{3a^{2}H^{2}\left(3p^{2}+4m^{2}\right)}.$
(43)

Thus, the perturbations $\beta$ and $\xi$ can be eliminated by substituting
these relations into the action (39), then a quadratic action for scalar
perturbation $E$ is achieved as

$S^{(2)}_{S}\\!\supset\\!M^{3}\\!\\!\int\\!\\!d^{4}pdy\frac{3a^{2}p^{4}m^{2}}{6p^{2}\\!+\\!8m^{2}}\left[-a^{2}E^{\prime}E^{\prime}\\!-\\!\left(p^{2}\\!+\\!2m^{2}\right)\\!E^{2}\right]\\!.$
(44)

Further, after redefining the scalar perturbation as

$\tilde{E}=\sqrt{\frac{-3p^{4}m^{2}}{3p^{2}+4m^{2}}}E,$ (45)

the above action can be rewritten as a canonical normalized form in the
conformal coordinate $z$, i.e.,

$S^{(2)}_{S}\supset\frac{M^{3}}{2}\int
d^{4}pdza^{3}\left[\dot{\tilde{E}}\dot{\tilde{E}}+\left({p^{2}}+{2m^{2}}\right)\tilde{E}^{2}\right].$
(46)

It is clear that there is an overall wrong-sign in the Lagrangian, which
implies that the scalar mode $\tilde{E}$ is a ghost field. However, the
equation of motion is not affected by the overall sign of the Lagrangian, so
it has a similar form to the tensor and vector modes, given by

$\ddot{\tilde{E}}+3H\dot{\tilde{E}}=(p^{2}+2m^{2}){\tilde{E}}.$ (47)

By redefining $\tilde{E}=a^{-\frac{3}{2}}{\varepsilon}$, it can be rewritten
as a Schrödinger-like equation,

$-\ddot{{\varepsilon}}+\left(\frac{9}{4}H^{2}+\frac{3}{2}\dot{H}\right){\varepsilon}=M_{S}^{2}{\varepsilon},$
(48)

where $M_{S}^{2}\equiv-p^{2}-2m^{2}$. The Hamiltonian can be factorized as a
supersymmetric quantum mechanics form as well,
$H_{S}=A_{S}^{\dagger}A_{S}=\left(\partial_{z}+\frac{3}{2}H\right)\left(-\partial_{z}+\frac{3}{2}H\right)$,
so with the boundary condition $\partial_{z}\tilde{E}|_{z=0,z_{b}}=0$, the
eigenvalues are non-negative $M_{S}^{2}\geq 0$, i.e., $p^{2}\leq-2m^{2}$.
Thus, the redefinition (45) is well-defined. In this case, all the KK
excitations of the scalar mode are time-like particles.

In RS1 model, the presence of IR brane abruptly ends AdS space, so it
spontaneously breaks the conformal invariance of AdS bulk in the IR. The
massless radion is just the Goldstone boson associated with the broken
dilatation invariance Arkani-Hamed2001a ; Rattazzi2001 . For the current
model, the scalar curvature reads
$R=-20k^{2}+\frac{4m^{2}e^{2ky}}{1+\epsilon^{2}e^{2ky}}$ in the bulk. Thus,
the presence of scalar condensation induces a slight deformation in the bulk
geometry, causing it to deviate from a pure AdS space. Consequently, the
radion acquires a tiny mass due to this explicit symmetry breaking. In the
massless limit $m=0$, the quadratic action (36) will reduce to that of RS
model, with only a massless radion in the mass spectrum Goldberger1999a ;
Charmousis2000 ; Callin2005 .

##  IV Mass Spectra and Hierarchy Resolution

For the lightest tensor and scalar modes corresponding to $M_{T,S}=0$, their
wave functions can be easily solved from Eqs. (25) and (48). After returning
to the coordinate space, and utilizing the KK decompositions,
$\mathcal{D}_{\alpha\beta}(x,z)=d_{\alpha\beta}(x)\Psi(z)$ and
$\varepsilon(x,z)=e(x)\Psi(z)$, a Schrödinger-like equation

$-\ddot{\Psi}+\left(\frac{3}{2}\dot{H}+\frac{9}{4}H^{2}\right)\Psi=M_{T,S}^{2}\Psi,$
(49)

with $M_{T,S}^{2}=m_{T,S}^{2}-2m^{2}$, can be achieved from Eqs. (25) and
(48). Here, $m_{T,S}$ is the effective mass of KK particles in 4D point of
view, satisfying the 4D Klein-Gordon equations
$\Box^{(4)}d_{\mu\nu}=m_{T}^{2}d_{\mu\nu}$ and $\Box^{(4)}e(x)=m_{S}^{2}e(x)$.

The wave function of ground states is easily achieved by setting $M_{T,S}=0$
or $m_{T,S}=\sqrt{2}m$ in above Schrödinger-like equation, i.e.,

$\Psi^{(0)}(z)=N_{0}a(z)^{\frac{3}{2}},$ (50)

where $N_{0}$ is the normalization factor. The boundary conditions
$\partial_{z}\tilde{D}_{\alpha\beta}|_{z=0,z_{b}}=0$ and
$\partial_{z}\tilde{E}|_{z=0,z_{b}}=0$ lead to
$\left.\left(\Psi-\frac{3}{2}H\right)\right|_{z=0,z_{b}}=0$. Obviously, the
wave function $\Psi^{(0)}(z)$ satisfies the boundary condition. All KK
particles are massive in this model, which is a significant difference from
the RS1 model, where the lightest KK particles are massless spin-2 graviton
and spin-0 radion. Especially, from the redefinitions
$\tilde{D}_{\alpha\beta}=a^{-\frac{3}{2}}\mathcal{D}_{\alpha\beta}$ and
$\tilde{E}=a^{-\frac{3}{2}}{\varepsilon}$, their canonical normalized field
configurations are given by
$\tilde{D}^{(0)}_{\alpha\beta}=a^{-\frac{3}{2}}\mathcal{D}^{(0)}_{\alpha\beta}=d^{(0)}_{\alpha\beta}(x)$
and $\tilde{E}^{(0)}=a^{-\frac{3}{2}}{\varepsilon}^{(0)}=e^{(0)}(x)$.
Therefore, the lightest graviton and radion propagate only on the brane.

However, the effective mass of 4D graviton is severely constrained by the
gravitational experiments Rham2017 . For instance, the detection of
gravitational waves has provided a bound of the graviton mass, yielding
$m_{g}\leq 4.7\times 10^{-23}$eV Abbott2019 . Thus, the parameter $m$ must be
tinier than the experimental constraints, i.e., $m<3.3\times 10^{-23}$eV.
Since the lightweight radion has the same mass as the lightest graviton,
exchanging such nearly massless scalar particle would cause a fifth force and
violate experimental observations. Therefore, similar to the RS1 model, the
radion must gain weight to meet the experimental expectations, which is
realized through GW mechanism Goldberger1999a  in the next section.

With the normalization condition $\int^{z_{b}}_{-z_{b}}\Psi_{0}^{2}dz=1$, the
normalization factor is worked out as

$\displaystyle
N_{0}^{-2}=\frac{1}{k}\left[1-e^{-2ky_{\pi}}+4\epsilon^{2}ky_{\pi}-\epsilon^{4}(1-e^{2ky_{\pi}})\right].$
(51)

By shutting down the 5D graviton mass $m$, one recovers the result of RS1
model. However, if we remove the visible brane, i.e., $y_{\pi}\to\infty$, the
quasi-massless graviton is no longer normalizable. This means that the
effective 4D gravity theory cannot be recovered on the brane. Therefore, the
RS2-like non-compact single brane model Randall1999a  is not a physically
available one in current massive gravity.

The braneworld scenario provides a natural way to solve the gauge hierarchy
problem, which is a crucial motivation of the well-known ADD model Arkani-
Hamed1998  and RS1 model Randall1999 . In our toy model, the low-energy
effective theory is obtained by including the nearly-massless gravitons, i.e.,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle
a^{2}(y)\bar{g}_{\mu\nu}(x)dx^{\mu}dx^{\nu}+dy^{2}$ (52) $\displaystyle=$
$\displaystyle
a^{2}(y)\left[\eta_{\mu\nu}+\gamma_{\mu\nu}(x)\right]dx^{\mu}dx^{\nu}+dy^{2},$

then the 4D effective gravitational mass scale $M_{\text{eff}}$ is read from
the curvature term in the action (1),

$M_{\text{eff}}^{2}=M^{3}\int^{y_{\pi}}_{-y_{\pi}}a^{2}dy={N_{0}^{-2}}{M^{3}}.$
(53)

On the other hand, in order to produce a large hierarchy between the Planck
scale and the electroweak scale, our Universe should be confined to the IR
brane located at $y_{\pi}$. Then, the Higgs field action on the brane is non-
canonically normalized,

$S_{\mathcal{H}}\supset\\!\int{d^{4}x\sqrt{|g_{\text{II}}|}\left[-{g}_{\text{II}}^{\mu\nu}D_{\mu}\mathcal{H}^{{\dagger}}D_{\nu}\mathcal{H}-\lambda(\mathcal{H}^{{\dagger}}\mathcal{H}-v_{0}^{2})^{2}\right]},$
(54)

where $g_{\text{II}\mu\nu}=a(y_{\pi})\tilde{g}_{\mu\nu}$ is the induced metric
on the brane at $y_{\pi}$, and $v_{0}$ the fundamental Higgs vacuum
expectation value (VEV). Writting the warp factor explicitly, one has

$\displaystyle S_{\mathcal{H}}$ $\displaystyle\supset$ $\displaystyle\int
d^{4}x\sqrt{|\tilde{g}|}\left[-a(y_{\pi})^{2}\tilde{g}^{\mu\nu}D_{\mu}\mathcal{H}^{{\dagger}}D_{\nu}\mathcal{H}-a(y_{\pi})^{4}\lambda(\mathcal{H}^{{\dagger}}\mathcal{H}-v_{0}^{2})^{2}\right],$
(55)

With a field renormalization, $\mathcal{H}\to\tilde{\mathcal{H}}/a(y_{\pi})$,
the effective action of Higgs on the brane is

$S_{\mathcal{H}}\\!\supset\\!\int{d^{4}x\sqrt{|\tilde{g}|}\left[-\tilde{g}^{\mu\nu}D_{\mu}\tilde{\mathcal{H}}^{{\dagger}}D_{\nu}\mathcal{H}-\lambda(\tilde{\mathcal{H}}^{{\dagger}}\mathcal{H}-v_{\text{eff}}^{2})^{2}\right]},$
(56)

where the effective Higgs VEV, $v_{\text{eff}}=a(y_{\pi})v_{0}$, sets the
electroweak scale on the brane.

Therefore, in order to solve the gauge hierarchy problem, the fundamental
parameters $M$, $k$, $v_{0}$ are simply chosen to be of the same order as the
Planck scale, $M_{\text{Pl}}$. Then, the warp factor has to provide enough
redshift to recover a TeV scale of Higgs VEV on the brane, namely,
$a(y_{\pi})\sim 10^{-16}$. In the condition $e^{-k|y_{\pi}|}>\epsilon$ and
$m<3.3\times 10^{-23}$eV, it leads to $e^{-ky_{\pi}}\approx
a(y_{\pi})-\frac{\epsilon^{2}}{a(y_{\pi})}\sim 10^{-16}$. Thus, it requires
$y_{\pi}\approx 37/k$.

Since the mass splitting scale of massive KK modes is inversely proportional
to the conformal size of extra dimension $z_{b}$, i.e., $\Delta m_{T}\propto
1/z_{b}$. From the coordinate transformation $dy=adz$, one has

$z_{\pi}=\frac{1}{k\epsilon}\left[\text{arctan}\left(\epsilon
e^{ky_{\pi}}\right)-\text{arctan}\left(\epsilon\right)\right],$ (57)

where the integral constant has been chosen so that $z(y=0)=0$. Since
$e^{-ky_{\pi}}\sim 10^{-16}$, $m<3.3\times 10^{-23}$eV and $k\sim
M_{\text{Pl}}$, one has $\epsilon e^{ky_{\pi}}\ll 1$. Thus, the formula can be
rewritten approximately as
$z_{\pi}\approx\frac{e^{ky_{\pi}}}{k}\left(1+\mathcal{O}(\epsilon^{2})\right)$.
So the mass splitting scale is similar to the RS1 model, i.e., $\Delta
m_{T}\sim ke^{-ky_{\pi}}\sim\mathcal{O}(\text{TeV})$.

##  V Radius stabilization

In RS1 model, the position of IR brane is put by hand, namely, the radius is
not dynamically fixed. It leads to the presence of a massless radion in the
effective theory, which corresponds to the fluctuations of the radius of
compact extra dimensions. However, the massless radion is phenomenologically
unacceptable since it would contribute to Newton’s law and cause a fifth
force. It is well-known that the GW mechanism Goldberger1999a  can be
introduced to stabilize the size of the extra dimension and to increase the
mass of radion. In current model, the position of IR brane is put by hand as
well, we can also utilize the GW mechanism here to stabilize the brane
position and increase the radion mass.

By adding a bulk scalar field $\Phi$ into the model, which has interaction
with the two branes, the action is given by

$\displaystyle S_{\Phi}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{4}x\int^{\pi}_{-\pi}d\theta\sqrt{-g}\left(-g^{MN}\partial_{M}\Phi\partial_{N}\Phi-
m_{\Phi}^{2}\Phi^{2}\right)-\int{}d^{4}x\sqrt{-g_{\text{I}}}\lambda_{\text{I}}(\Phi^{2}-v_{\text{I}}^{2})^{2}$
(58)
$\displaystyle-\int{}d^{4}x\sqrt{-g_{\text{II}}}\lambda_{\text{II}}(\Phi^{2}-v_{\text{II}}^{2})^{2},$

where $g_{MN}$ is the background metric (3) with the radius $r_{c}$ of
compactified extra dimension given by $y=r_{c}\theta$, $\lambda_{\text{I/II}}$
is the coupling parameters, and
$V_{\text{I/II}}=\lambda_{\text{I/II}}v_{\text{I/II}}^{4}$.

The equation of motion of the scalar filed $\Phi$ is achieved by varying the
action (58) with respect to $\Phi$,

$\displaystyle\frac{1}{r_{c}}{\partial_{\theta}}\left(a^{4}{\partial_{\theta}\Phi}\right)-a^{4}m_{\Phi}^{2}\Phi-{4a^{4}\lambda_{\text{I}}\Phi\left(\Phi^{2}-v_{\text{I}}^{2}\right)}\delta(\theta)-4a^{4}\lambda_{\text{II}}\left(\Phi^{2}-v_{\text{II}}^{2}\right)\Phi\delta(\theta-\pi)=0.$
(59)

Away from the two branes at $\theta=0,\pi$, the general solution of this
equation reads

$\displaystyle\Phi(\theta)=e^{2\sigma}\left[A{}_{2}\text{F}_{1}\left(2,\nu+2,\nu+1,-\epsilon^{2}e^{2\sigma}\right)e^{\nu\sigma}+B{}_{2}\text{F}_{1}\left(2,2-\nu,1-\nu,-\epsilon^{2}e^{2\sigma}\right)e^{-\nu\sigma}\right],$
(60)

where $\sigma(\theta)=kr_{c}\theta$, $\nu=\sqrt{4+m_{\Phi}^{2}/k^{2}}$,
${}_{2}\text{F}_{1}$ is the hypergeometric function, $A$ and $B$ are
integration constants. Especially, under the condition $\epsilon e^{\sigma}\ll
1$, up to the first order of correction, the solution can be approximately
written as

$\displaystyle\Phi(\theta)\simeq
e^{2\sigma}\left[A\left(1-2\epsilon^{2}\frac{\nu+2}{\nu+1}e^{2\sigma}\right)e^{\nu\sigma}+B\left(1-2\epsilon^{2}\frac{\nu-2}{\nu-1}e^{2\sigma}\right)e^{-\nu\sigma}\right].$
(61)

If we close the mass of 5D graviton, i.e., $\epsilon=0$, the solution reduces
to the one in GR Goldberger1999a . The boundary conditions on the branes can
be obtained by inserting the approximate solution into the equation of motion
(59) and matching the delta functions, given by

$\displaystyle
k\\!\left[(\nu\\!+\\!2)\\!\left(1\\!+\\!2\epsilon^{2}\frac{\nu-2}{\nu+1}\right)\\!A\\!-\\!(\nu\\!-\\!2)\\!\left(1\\!+\\!2\epsilon^{2}\frac{\nu+2}{\nu-1}\right)\\!B\right]$
$\displaystyle-2\lambda_{\text{I}}\left(1+4\epsilon^{2}\right)\Phi(0)\left(\Phi(0)^{2}-v_{\text{I}}^{2}\right)=0,~{}~{}~{}$
(62) $\displaystyle
ke^{2\sigma(\pi)}\left[(\nu+2)e^{\nu\sigma(\pi)}\left(1+2\epsilon^{2}\frac{\nu-2}{\nu+1}e^{2\sigma(\pi)}\right)A-(\nu-2)e^{-\nu\sigma(\pi)}\left(1+2\epsilon^{2}\frac{\nu+2}{\nu-1}e^{2\sigma(\pi)}\right)B\right]$
$\displaystyle+2\lambda_{v}\left(1+4\epsilon^{2}e^{2\sigma(\pi)}\right)\Phi(\pi)\left(\Phi(\pi)^{2}-v_{v}^{2}\right)=0.~{}~{}~{}$
(63)

The integration constants $A$ and $B$ can be determined from these boundary
conditions.

However, rather than solving the above equations directly, we employ the trick
of Ref. Goldberger1999a  to simplify the calculation. By substituting the
approximate solution back into the action and integrating over $\theta$, it
yields the effective potential of compactified radius $r_{c}$,

$\displaystyle V_{\Phi}(r_{c})$ $\displaystyle\approx$ $\displaystyle
kA^{2}(\nu+2)e^{2\nu kr_{c}\pi}\left(1-\frac{8\epsilon^{2}e^{2\pi
kr_{c}}}{\nu+1}\right)+e^{-4kr_{c}\phi}\lambda_{\text{II}}\left(1+4\epsilon^{2}e^{2\pi
kr_{c}}\right)\left(\Phi(\pi)^{2}-v_{\text{II}}^{2}\right)^{2}$ (64)
$\displaystyle+\lambda_{\text{I}}\left(\Phi(0)^{2}-v_{\text{I}}^{2}\right)^{2},$

where the limit of $e^{kr_{c}\pi}\gg 1$ has been used in the calculation.
Assuming that the interaction parameters $\lambda_{\text{I}}$ and
$\lambda_{\text{II}}$ are very large Goldberger1999a , this effective
potential implies the solution $\Phi(0)=v_{\text{I}}$ and
$\Phi(\pi)=v_{\text{II}}$. Then, the integration constants $A$ and $B$ can be
solved approximately, yielding

$\displaystyle A$ $\displaystyle\approx$ $\displaystyle
v_{\text{II}}e^{-(\nu+2)\sigma(\pi)}\left[1-2\epsilon^{2}\frac{\nu+2}{\nu+1}e^{2\sigma(\pi)}\right]-v_{\text{I}}e^{-2\nu\sigma(\pi)}\left[1+4\epsilon^{2}\frac{\nu}{\nu^{2}-1}e^{2\sigma(\pi)}\right],$
(65) $\displaystyle B$ $\displaystyle\approx$ $\displaystyle
v_{\text{I}}\left[1+2\epsilon^{2}\frac{\nu+2}{\nu+1}\left(e^{\nu\sigma(\pi)}-e^{2\sigma(\pi)}\right)\right]-v_{\text{II}}e^{-(\nu+2)\sigma(\pi)}\left[1+2\epsilon^{2}\frac{\nu+2}{\nu+1}e^{\nu\sigma(\pi)}\right].$
(66)

Further, assuming that the mass of the scalar filed is a small quantity, i.e.,
${m_{\Phi}}/{k}\ll 1$, one has $\nu=\sqrt{4+\frac{m_{\Phi}^{2}}{k^{2}}}\simeq
2+\delta$, where $\delta=\frac{m_{\Phi}^{2}}{4k^{2}}$ is a small quantity.
With this approximation, the effective potential can be rewritten as

$\displaystyle V_{\Phi}(r_{c})$ $\displaystyle\approx$
$\displaystyle(4+\delta)ke^{-4kr_{c}\pi}\left(1-\frac{8\epsilon^{2}e^{2kr_{c}\pi}}{\delta+3}\right)\left[\left(1+\frac{4(\delta+2)\epsilon^{2}e^{2kr_{c}\pi}}{\delta^{2}+4\delta+3}\right)e^{-\delta
kr_{c}\pi}v_{\text{I}}\right.$ (67)
$\displaystyle-\left.\left(1-\frac{2(\delta+4)\epsilon^{2}e^{2kr_{c}\pi}}{\delta+3}\right)v_{\text{II}}\right]^{2}.$

As a result, the effective potential has a minimum at

$r_{c}\approx\frac{1}{\pi
k\delta}\ln\left(\frac{v_{h}}{v_{v}}\right)-\frac{4\epsilon^{2}}{3\pi
k}\left(\frac{v_{h}}{v_{v}}\right)^{\frac{2}{\delta}}\left(1+\frac{v_{v}}{v_{h}}\right).$
(68)

The first leading term is just the result of GW mechanism in GR
Goldberger1999a , and the second term proportional to $\epsilon^{2}$ is a tiny
correction stemming from the 5D graviton mass. The mechanism provides a
dynamical way to stabilize the compactified radius of the extra dimension
without introducing another large hierarchy. For instance, if $m_{\Phi}/k=0.2$
and ${v_{h}}/{v_{v}}=1.45$, one obtains $ky_{\pi}=kr_{c}\pi\approx 37$ to
generate a proper hierarchy. After the radius stabilization, the radion
acquires a mass roughly $\mathcal{O}(\delta)$TeV Goldberger2000 , which is
still smaller than the TeV scale.

##  VI Conclusions

In this work, we generalized the RS1 model in a 5D extension of the Lorentz-
violating massive gravity. It is found that the theory supports two distinct
brane configurations. The configuration possessing both positive and negative
tension branes is similar to the RS1 model, while the other possessing only
two positive tension branes is distinct from RS1 model. The gauge hierarchy
problem was also solved for the RS1-like brane configuration as an
application. In order to analyze the mass spectra and stability, we considered
full linear perturbations against the background metric. It is found that the
tensor and vector modes are free from the ghost and tachyonic instabilities,
however, the scalar mode is a ghost filed.

Figure 2: The mass spectra of RS1 model and current model, where the blue
lines refer to the tensor mode, red lines to vector mode, and green lines to
ghost scalar mode.

As shown in Fig. 2, all the KK particles are massive. The tensor and scalar
modes have similar mass spectra which start from $\sqrt{2}m$ with a mass
splitting of TeV scale, nevertheless, the lightest vector mode does not exist
in the mass spectrum. Furthermore, the ground states of tensor and scalar
modes propagate only along the brane. The graviton mass $\sqrt{2}m$ has to be
tiny enough to fit the experimental constraints. Consequently, the low-energy
effective theory on the brane is a massive gravity plus a ghost scalar field.
Although the ghost scalar filed would lead to quantum instability, it is
interesting to note that the gravity coupled to a real ghost scalar field with
a negative sign for the kinetic and potential terms can support non-singular,
static solutions for topologically non-trivial wormhole-like geometry
Kodama1978 ; Kodama1979 .

Further, by imposing the GW mechanism to stabilize the size of the extra
dimension, the light weight ghost radion would get weight to be the order of
$\mathcal{O}(\delta)$ TeV. After the radius stabilization, the 4D effective
theory on the brane includes a nearly massless graviton plus three towers of
very massive spin-2, spin-1 and ghost spin-0 particles.

In order to achieve a Minkowski flat brane model, the 4D Poincaré invariance
is required to be preserved on the brane. Therefore, the internal metric of
the scalar fields is chosen to be the Minkowski metric in the action (1).
However, the non-positive definite Minkowski metric leads to the bulk scalar
$\phi^{0}$ having a kinetic term with the wrong sign. Since there is no
further constraint to eliminate this ghost scalar mode, it results in the
presence of the ghost in the linear theory. In order to get rid of the ghost
scalar mode, we can choose the internal metric to be $\delta_{ij}$ with
$i,j=1,2,3$, which is positive definite and removes the problematic mode
$\phi^{0}$. However, in this scenario, the scalar condensation
$\langle\phi^{a}\rangle=\delta_{i}^{a}x^{i}$ breaks the 4D Lorentz symmetry
and preserves the $SO(3)$ symmetry on the brane. As a result, we cannot
achieve a Minkowski flat brane model anymore. Instead, the brane needs to be
bent, for example, de Sitter brane. The attempt to establish such a stable
bent brane model is left for our another work.

## ACKNOWLEDGMENTS

We would like to especially thank Prof. Yu-Xiao Liu for a very helpful
discussion of our paper. This work was supported by the National Natural
Science Foundation of China under Grant Nos. 12005174 and 12165013. K. Yang
acknowledges the support of Natural Science Foundation of Chongqing, China
under Grant No. cstc2020jcyj-msxmX0370. B.-M. Gu is also supported by Jiangxi
Provincial Natural Science Foundation under grant No. 20224BAB211026.

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