# Spin-2 Kaluza-Klein Scattering in a Stabilized Warped Background

R. Sekhar Chivukulaa Dennis Forena Kirtimaan A. Mohanb Dipan Senguptaa,c
Elizabeth H. Simmonsa a Department of Physics and Astronomy, 9500 Gilman
Drive, University of California, San Diego  b Department of Physics and
Astronomy, 567 Wilson Road, Michigan State University, East Lansing  cARC
Centre of Excellence for Dark Matter Particle Physics, Department of Physics,
University of Adelaide, South Australia 5005, Australia
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Scattering amplitudes involving massive spin-2 particles typically grow
rapidly with energy. In this paper we demonstrate that the anomalous high-
energy growth of the scattering amplitudes cancel for the massive spin-2
Kaluza-Klein modes arising from compactified five-dimensional gravity in a
stabilized warped geometry. Generalizing previous work, we show that the two
sum rules which enforce the cancellations between the contributions to the
scattering amplitudes coming from the exchange of the (massive) radion and
those from the exchange of the tower of Goldberger-Wise scalar states
(admixtures of the original gravitational and scalar fields of the theory)
still persist in the case of the warping which would be required to produce
the hierarchy between the weak and Planck scales in a Randall-Sundrum model.
We provide an analytic proof of one combination of these generalized scalar
sum rules, and show how the sum rule depends on the Einstein equations
determining the background geometry and the mode-equations and normalization
of the tower of physical scalar states. Finally, we provide a consistent and
self-contained derivation of the equations governing the physical scalar modes
and we list, in appendices, the full set of sum rules ensuring proper high-
energy growth of all $2\to 2$ massive spin-2 scattering amplitudes.

††preprint: MSUHEP-22-021, ADP-22-20/T1191

###### Contents

  1. I Introduction
  2. II The Lagrangian
  3. III Kaluza-Klein Mode Expansions
    1. III.1 Spin-2 Sector
    2. III.2 Spin-0 Sector
  4. IV Massive spin-2 couplings, scattering amplitudes, and Sum Rules
    1. IV.1 Scattering Amplitudes and Couplings
    2. IV.2 Sum Rules Insuring Consistency of Scattering Amplitudes
    3. IV.3 Sum-Rule Summary
  5. V Perturbative Analysis of a Warped Stabilized Model
    1. V.1 The DFGK Model
    2. V.2 The Perturbative DFGK Model
      1. V.2.1 Spin-2 wavefunctions and masses in perturbation theory
      2. V.2.2 Spin-0 wavefunctions and masses in perturbation theory
    3. V.3 Numerical verification of sum rules
      1. V.3.1 Spin-2 Sum-Rules and Completeness
      2. V.3.2 The Spin-0 Sum Rule
  6. VI Conclusion
  7. A The Derivation of the Scalar Kaluza-Klein Modes
    1. A.1 The Lagrangian
    2. A.2 The Lagrangian to Quadratic Order
    3. A.3 Equations of Motion
      1. A.3.1 Background Equations of Motion
      2. A.3.2 Lagrangian at Quadratic Order: Mode Equations
      3. A.3.3 The Gauge Condition
    4. A.4 Kaluza-Klein Decomposition of the Scalar Field
    5. A.5 The Canonical Scalar Mode Expansion
  8. B The Inelastic Sum Rules Relating Couplings and Masses
    1. B.1 Definitions
    2. B.2 Applications of Completeness
    3. B.3 B-to-A Formulas
    4. B.4 The $\mu_{j}^{4}$ Sum Rule
    5. B.5 The $\mu_{j}^{6}$ Sum Rule
      1. B.5.1 Elastic
      2. B.5.2 Inelastic
    6. B.6 Summary of Proven Sum Rules (Inelastic)
    7. B.7 Summary of Proven Sum Rules (Elastic)
    8. B.8 Unproven Rules
  9. C Eigenvalues and Eigenfunctions of Spin-2 and Spin-0 Modes
    1. C.1 Perturbation theory and a general Sturm-Liouville Problem
      1. C.1.1 Mass Corrections
      2. C.1.2 Solving the In-homogeneous Differential Equation using Variation of Parameters
      3. C.1.3 Wave functions in Rayleigh-Schrödinger perturbation theory
    2. C.2 Wavefunction and Masses of KK modes in the DFGK model
      1. C.2.1 Spin-2 mass and wavefunction corrections
      2. C.2.2 The massless graviton to order $\epsilon^{2}$
      3. C.2.3 Massive spin-2 modes to order $\epsilon^{2}$
    3. C.3 Spin-0 mass and wave functions in the DFGK model
      1. C.3.1 Radion wavefunction and mass to order $\epsilon^{2}$
      2. C.3.2 GW-scalar mass and wavefunction to leading order

##  I Introduction

Historically, extra-dimensional theories of gravity were introduced soon after
Einstein’s discovery of the general theory of relativity. In the original form
extra dimensions were introduced by Kaluza and Klein (KK) to unify
electromagnetism with gravity, the only two fundamental forces known at the
time Kaluza:1921tu ; Klein:1926tv . Extra-dimensional models have continued to
evolve since the late 1970’s, thanks in large part to the development of
string theory. Over the last three decades, low-energy realizations of extra-
dimensional models gained prevalence as well-motivated scenarios of physics
beyond the standard model. One of the most popular and phenomenologically-
viable models of extra dimensions is the Randall-Sundrum model Randall:1999ee
; Randall:1999vf , wherein a compact extra dimension in Anti-de Sitter space
is used to generate relative exponential factors; this factor allows particles
fixed to a brane to interact at electroweak strength while also ensuring bulk-
propagating gravity is weak in the observed extended four-dimensional space,
thus providing a geometric solution for the hierarchy problem of the standard
model.

Low-energy four-dimensional effective field theories arising from compactified
theories of gravity involve towers of interacting spin-0 and spin-2 fields
(and potentially spin-1 fields as well, though these are often eliminated by
imposing an orbifold symmetry on the compact extra dimension). The massive
spin-2 resonances – sometimes called Kaluza-Klein (KK) gravitons – are
particularly interesting. The existence of self interactions between these KK
gravitons is problematic because typically scattering amplitudes between
massive spin-2 particles grow far too rapidly with energy to keep unitarity
constraints satisfied much beyond the mass of lightest massive spin-2 state
involved. For example, theories of massive gravity that extend 4D general
relativity by adding a Fierz-Pauli mass term Fierz:1939ix result in 2-to-2
scattering amplitudes for the helicity-zero channel (the channel whose
amplitude has the highest energy-growth) that grow like
$s^{5}/(m_{\text{FP}}^{8}M_{\text{Pl}}^{2})$ ArkaniHamed:2002sp , where
$m_{\text{FP}}$ is the mass of the graviton, $s$ the squared center-of-mass
energy, and $M_{\text{Pl}}$ the reduced Planck mass. Adding carefully-chosen
potential terms to the model Hinterbichler:2011tt ; deRham:2014zqa can soften
matrix element growth down to ${\cal O}(s^{3})$. However, compactified
theories of extra-dimensional gravity should – due to higher-dimensional
diffeomorphism invariance, requiring a smooth $m_{\text{KK}}\to 0$ limit –
grow at most as ${\cal O}(s)$.

As we have demonstrated in earlier work Chivukula:2019rij ; Chivukula:2019zkt
; Chivukula:2020hvi ; Foren:2020egq , while the individual contributions to
massive helicity-zero spin-2 scattering amplitudes do each grow like ${\cal
O}(s^{5})$ in compactified gravity theories, cancellations occur between these
contributions ultimately ensure the overall 2-to-2 scattering amplitudes grow
no faster than ${\cal O}(s)$.111See also Bonifacio:2019ioc ; Hang:2021fmp .
These cancellations require that the couplings and masses of the Kaluza-Klein
spin-2 and spin-0 modes be constrained to satisfy various sum-rule relations,
which we have shown to be satisfied in both flat and warped compactifications.

In the original formulation of the RS1 (Randall-Sundrum I) model, extra-
dimensional gravity generates a massless radion in the effective 4D
theory222Along with the usual massless 4D graviton.. A massless radion couples
to the trace of the stress-energy tensor, yielding a Brans-Dicke like theory,
in odds with predictions of the general theory of relativity. Additionally,
this massless radion sources a Casimir force Hofmann:2000cj , thereby
destabilizing the extra dimension and leading to its collapse. A naive
computation of scattering amplitudes of massive spin-2 KK particles in a
compactified theory of gravity, with a radion mass introduced by hand, reveals
that scattering amplitudes grow like
$m^{2}_{\text{radion}}s^{2}/(m_{\text{KK}}^{4}M_{\text{Pl}}^{2})$, i.e. ${\cal
O}(s^{2})$ instead of ${\cal O}(s)$. In order to stabilize the extra dimension
in a way that retains validity of the theory up to the Planck scale, the
radion must not only be made massive, but its mass must be generated
dynamically. These dynamics must also provide additional contributions
guaranteeing that the overall scattering amplitude grows no faster than ${\cal
O}(s)$ – as we have explicitly shown in Chivukula:2021xod and explore here in
detail.

In the same year that Randall and Sundrum published their model, Goldberger
and Wise published a dynamical mechanism for stabilizing the model’s extra
dimension Goldberger:1999uk ; Goldberger:1999un . Their mechanism shares
conceptual similarities with the standard technique of generating massive
gauge bosons by spontaneously breaking the associated underlying gauge
symmetry, wherein dynamics producing a non-zero vacuum expectation value (VEV)
for a gauge-variant operator induces mixing between the longitudinal
components of the gauge bosons with Goldstone bosons. In the Goldberger-Wise
(GW) mechanism, a new 5D bulk scalar field $\hat{\Phi}(x,y)$ is appended to
the RS1 model and included in new potential terms that also involve (via
standard gravitational factors) the usual RS1 metric fields. That bulk scalar
field then spontaneously acquires a background profile $\phi_{0}(y)$ with
nonconstant dependence on the extra-dimensional coordinate $y$, causing mixing
between background fluctuations $\hat{f}$ of the bulk scalar field and scalar
fluctuations $\hat{r}$ of the RS1 metric. While this could in principle yield
two physically-relevant superpositions of the bulk scalar field and scalar
metric fluctuations, one combination is automatically forced to vanish in
unitary gauge, leaving just one physical 5D scalar field in the spectrum.

Quantitatively, the new mixed scalar sector decomposes into a tower of spin-0
modes, all described by a single Sturm-Liouville equation with non-trivial
Robin boundary conditions involving delta function contributions at the
boundaries of the RS1 geometry. In this way, the GW mechanism ultimately
generates an infinite tower of physical massive spin-0 states. But what
happened to the radion? If the background profile of the bulk scalar field
were, instead, made constant in the extra-dimensional coordinate, no extra-
dimensional symmetries would be spontaneously broken and all but the lowest
state in the spin-0 tower would cease to mix with the gravitational sector of
the theory. In this limit, the lowest state would become massless, and –
indeed – its couplings would exactly match those of the original unstabilized
RS1 radion. Hence, in the GW-stabilized RS1 model we are considering, the
radion should be associated with the lightest massive state among an entire
tower of massive spin-0 states.

The assessment of the validity of the GW-stabilized RS1 effective field theory
proceeds as for the corresponding unstabilized case, by calculating $2\to 2$
massive KK graviton scattering, now with an extended scalar sector, as
compared to only the radion in the unstabilized case. An account of this
calculation was provided in Chivukula:2021xod , where we formulated an
extended set of sum-rules required to ensure that scattering amplitudes were
well behaved in a stabilized theory of extra dimension without reference to
any explicit GW model. In addition, we proposed a simple model of a
stabilized-but-approximately-flat extra dimension, the “flat stabilized”
model, and we demonstrated that the revised sum-rules were satisfied in this
model.

In this work we extend prior results into a new domain by computing the
couplings and masses of the scalar and spin-2 states in a Randall-Sundrum
model with a Goldberger-Wise stabilization mechanism in the phenomenologically
interesting case in which the warping reproduces the hierarchy between the
weak and Planck scales. We provide a self-contained derivation of the
equations governing the physical scalar modes. We show how one combination of
the generalized sum-rules in particular relies explicitly on the equations
determining the background bulk scalar field and metric, including the scalar
mode wavefunctions and their normalization conditions. We introduce a model in
which the Goldberger-Wise dynamics are a small perturbation away from the
unstabilized warped RS1 model. We then demonstrate numerically that all of the
sum rules needed to ensure that the anomalous growth of the scattering
amplitudes cancel are satisfied to leading non-trivial order in perturbation
theory.

The computations performed in the pure gravity sector in this paper and the
preceding ones Chivukula:2019rij ; Chivukula:2019zkt ; Chivukula:2020hvi ;
Chivukula:2021xod , demonstrating that amplitudes in compactified extra
dimensional theories grow no faster than $\mathcal{O}(s)$, have important
phenomenological consequences. The underlying higher-dimensional
diffeomorphism invariance that ensures that scattering amplitudes are well
behaved also guarantees that scattering amplitudes involving matter particles
should also be compliant with the same principle. For example, in calculations
for cosmological observables like relic abundances of dark matter with KK
graviton portals (both for freeze-in and freeze-out), the velocity averaged
cross sections must be properly estimated at large $\sqrt{s}$. An erroneous
estimate with anomalously growing cross sections would lead to inaccurate
predictions for cosmological observables within the scope of these models. We
will illustrate these issues in detail in future work.

The rest of the paper is organized as follows. In Sec. II we describe the
Lagrangian of the Goldberger-Wise Randall-Sundrum model and set notation for
the background geometry. In Sec. III we describe the spin-0 and spin-2 mode
expansions. Our analysis of the Kaluza-Klein expansions for this system
follows the computations of Csaki:2000zn ; Kofman:2004tk ; Boos:2005dc ;
Boos:2012zz , generalized to de Donder gauge, and is presented in detail for
completeness and clarity in Appendix A. A review of Kaluza-Klein mode
scattering and couplings, and description of the version of the sum rules of
Chivukula:2021xod used here, as well as a description of the analytic proof of
one combination of the sum rules involving the scalar sector is given in Sec.
IV. Details of the analytic proof of the sum rule is given in Appendix B.5.1.
We also provide, in the totality of Appendix B, a complete list of the sum-
rule relations which must be satisfied for all $2\to 2$ massive spin-2
scattering amplitudes to grow no faster than ${\cal O}(s)$ – completing the
analyses begun in Chivukula:2019rij ; Chivukula:2019zkt ; Chivukula:2020hvi ;
Foren:2020egq . A description and the analysis of the perturbative warped-
stabilized model is given in Sec. V. In particular, our numerical checks of
the sum rules in this model are illustrated in Fig. 3 and Fig. 4 of Sec. V.3.
Our perturbative analysis requires a slight generalization of Rayleigh-
Schrödinger perturbation theory to account for perturbations in the weight-
function of the corresponding Sturm-Liouville problem, and this formalism is
described in Appendix C. Our conclusions are given in Sec. VI. Mathematica
Mathematica files giving the expressions for all the spin-2 and spin-0
perturbative wavefunctions can be found on
GitHub.333https://github.com/kirtimaan/Stabilized-Extra-Dimenion

##  II The Lagrangian

In this section we outline schematically how a canonical 4D effective
Lagrangian is derived from a 5D RS1 model stabilized by the Goldberger-Wise
mechanism. We provide a self-contained discussion of the details of this
derivation in Appendix A utilizing arguments similar to those found in refs.
Kofman:2004tk ; Boos:2005dc ; Boos:2012zz , generalized to de Donder gauge to
enable consistent scattering amplitude computations for processes involving
the (massless) graviton. In this section we specify our notation and outline
the results needed to present our computations.

We begin by writing down the Lagrangian which consists of the following terms.

$\displaystyle\mathcal{L}_{\text{5D}}\equiv\mathcal{L}_{\text{EH}}+\mathcal{L}_{\Phi\Phi}+\mathcal{L}_{\text{pot}}+\mathcal{L}_{\text{GHY}}+\Delta\mathcal{L}~{}.$
(1)

Here $\mathcal{L}_{\text{EH}}$ comes from the usual Einstein-Hilbert action,
$\mathcal{L}_{\Phi\Phi}$ and $\mathcal{L}_{\text{pot}}$ are the kinetic and
potential terms respectively of a bulk scalar field $\hat{\Phi}(x,y)$,
$\mathcal{L}_{\text{GHY}}$ is the Gibbons-Hawking-York (GHY) boundary term
York:1972sj ; Gibbons:1976ue , and $\Delta\mathcal{L}$ is a useful total
derivative we define in Appendix A. The combination of
$\mathcal{L}_{\text{GHY}}$ and $\Delta\mathcal{L}$ is required to have a well-
posed variational principle for the gravitational action Dyer:2008hb . This
Lagrangian is a function of the 5D metric $G$, which we parameterize in terms
a 4D metric perturbation $g_{\mu\nu}$ and a scalar metric perturbation
$\hat{r}$ as Charmousis:1999rg

$\displaystyle[G_{MN}]=\left(\begin{tabular}[]{ c c
}${\,\,\,g_{\mu\nu}\,\,\exp{\left[-2\left(A(y)+\frac{e^{2A(y)}}{2\sqrt{6}}\kappa\,\hat{r}(x,y)\right)\right]}}$&${0}$\\\
${0}$&${-\left(1+\frac{e^{2A(y)}}{\sqrt{6}}\kappa\,\hat{r}(x,y)\right)^{2}\,\,\,}$\end{tabular}\right)\
.$ (4)

in terms of coordinates $x^{M}\equiv(x^{\mu},y)$, where $y\in(-\pi r_{c},+\pi
r_{c}]$ parameterizes an orbifolded extra-dimension444That is, the extra-
dimension is a circle in which the points with coordinates $y$ and $-y$ are
identified. As we will see, this view of the extra dimension (as opposed to
treating it as a line-segment) allows us to motivate and use the boundary
conditions of the Kaluza-Klein mode equations at the orbifold fixed-points at
$y=0$ and $y=\pi$ more easily.; for convenience we define $\varphi\equiv
y/r_{c}$, and use $y$ or $\varphi$ interchangeably as the coordinate of the
fifth dimension. Note that this form of the metric was used in our previous
works Chivukula:2019rij ; Chivukula:2019zkt ; Chivukula:2020hvi for the
unstabilized RS1 metric, following Charmousis:1999rg to bring the quadratic
Lagrangian to a canonical form from the outset. In the case of the stabilized
model, the situation is further complicated by a nontrivial mixing between the
bulk scalar field and the scalar metric fluctuations. As we will explain below
and demonstrate explicitly in Appendix A, the total derivative
$\Delta\mathcal{L}$ helps us bring the Lagrangian into a canonical form.

The warp factor $A(y)$ encodes the warped background geometry. In RS1
Randall:1999ee , in which the extra dimension is unstabilized, $A(y)=k|y|$,
where $k$ is related to the spacetime curvature. The Goldberger-Wise mechanism
Goldberger:1999uk ; Goldberger:1999un complicates the background geometry such
that the specific form of $A(y)$ becomes dependent on the details of the
mechanism’s bulk scalar interactions. Crucially, the scalar (bulk and
boundary) potential terms in $\mathcal{L}_{\text{pot}}$ are chosen such that
the scalar field gains a $y$-dependent background field value, and the trade-
off between the contributions to the action from bulk kinetic energy terms
$\mathcal{L}_{\Phi\Phi}$ and the scalar potential(s) stabilize the size of the
extra-dimension.

The Lagrangian thus far is written in terms of the metric perturbations
$\\{g_{\mu\nu},\hat{r}\\}$ and the bulk scalar field $\hat{\Phi}$. We next
expand $g_{\mu\nu}$ and $\hat{\Phi}$ about their background values:

$\displaystyle g_{\mu\nu}(x,y)$
$\displaystyle\equiv\eta_{\mu\nu}+\kappa\,\hat{h}_{\mu\nu}(x,y)$
$\displaystyle\hat{\Phi}(x,y)$
$\displaystyle\equiv\frac{1}{\kappa}\hat{\phi}\equiv\frac{1}{\kappa}\left[\phi_{0}(y)+\hat{f}(x,y)\right]~{}.$
(5)

The background metric $\eta_{\mu\nu}=\text{Diag}(+1,-1,-1,-1)$ of $g_{\mu\nu}$
is determined by demanding Lorentz-invariance along the extended dimensions,
while the background value $\phi_{0}/\kappa$ of $\hat{\Phi}$ must be found by
solving the theory’s field equations. We normalize the Lagrangian such that
the 5D gravitational coupling $\kappa$ is related to the 5D Planck mass
$M_{\text{Pl,5D}}$ according to $\kappa^{2}=4/M_{\text{Pl,5D}}^{3}$. Note that
the factors of $\kappa$ (units: energy-3/2) included in Eq. (5) are such that
$\phi_{0}$ and $\hat{f}$ are unitless in natural units. Following Kaluza-Klein
(KK) decomposition, the 5D tensor field $\hat{h}_{\mu\nu}$, the 5D scalar
field $\hat{r}$, and 5D scalar fluctuation field $\hat{f}$ give rise to an
infinite tower of 4D states.

Perturbatively expanding the Lagrangian Eq. (1) order-by-order in $\kappa$
yields terms containing various powers of $h_{\mu\nu}$, $\hat{f}$, and
$\hat{r}$. In particular, at quadratic order in the fields, we find a
complicated expression, c.f. Eqs. (117)-(122). Thankfully, there are residual
five-dimensional diffeomorphism transformations which leave the form of Eq.
(4) invariant - these transformations allow us to reorganize how the physical
content is embedded in the fields and thereby attain explicitly canonical
quadratic Lagrangians. This process will also mix the 5D fields $\hat{r}$ and
$\hat{f}$ (and their constituent 4D states) together in a process that
eventually leaves a single scalar tower of physical states. In particular, in
order to ultimately bring the quadratic 5D Lagrangian into a form suitable for
generating canonical 4D Lagrangians, we impose the gauge-fixing constraint555A
demonstration that one can always impose this gauge constraint can be found in
Refs. Kofman:2004tk ; Boos:2012zz .

$\displaystyle(\partial_{\varphi}\phi_{0})\,\hat{f}(x,y)\equiv\sqrt{6}\,e^{2A(\varphi)}\,\big{(}\partial_{\varphi}\hat{r}\big{)}~{},$
(6)

to eliminate the field $\hat{f}$ in terms of $\hat{r}$.666Note that in the
limit in which there is no non-trivial scalar background,
$\phi^{\prime}_{0}\to 0$, this constraint leaves only the constant
($\varphi$-independent) mode of $\hat{r}$ in the theory - a mode corresponding
to the massless radion of the unstabilized theory.  In this gauge the 5D
theory’s independent field degrees of freedom consist only of the 5D scalar
field $\hat{r}$ and the 5D tensor field $\hat{h}_{\mu\nu}$. To yield a 4D
effective theory, each of these 5D fields is subsequently decomposed into a
tower of 4D Kaluza-Klein (KK) modes. We emphasize here that bringing the
Lagrangian to a canonical form is a non-trivial task, and it is of paramount
importance to figure out all the interactions of both the gravitational and
the scalar sector that will eventually determine the structure of the matrix
elements and the couplings.

In order to calculate the desired matrix elements, we require the cubic and
quartic self-interactions of the 5D tensor field $\hat{h}_{\mu\nu}$ as well as
the $\hat{h}\hat{h}\hat{r}$ cubic interaction. The $\hat{h}$ self-interactions
(and their 4D effective equivalents) are changed from our previous works
Chivukula:2019zkt ; Chivukula:2019rij ; Foren:2020egq only in the specific
choice of $A(y)$. Following integration-by-parts and the elimination of total
derivatives, we find that the $\hat{h}\hat{h}\hat{r}$ interaction is similarly
identical to the unstabilized case Chivukula:2020hvi :

$\displaystyle\mathcal{L}_{hhr}$
$\displaystyle=-\dfrac{\kappa}{2r_{c}^{2}}\sqrt{\dfrac{3}{2}}e^{-2A(\varphi)}\Big{[}(\hat{h}^{\,\prime})^{2}-\hat{h}^{\,\prime}_{\mu\nu}(\hat{h}^{\mu\nu})^{\,\prime}\Big{]}\hat{r}~{}.$
(7)

Thus, the primary difference between the stabilized and unstabilized cases as
far as $\hat{h}\hat{h}\hat{r}$ is concerned regards the KK decomposition of
the 5D field $\hat{r}(x,y)$. In the unstabilized case, $\hat{r}$ generates
only a single massless scalar state $\hat{r}(x)$ (see footnote 6)— the usual
RS1 radion. In the stabilized case, $\hat{r}$ has nontrivial $y$-dependence
and instead generates an infinite tower of massive scalars
$\\{\hat{r}^{(i)}(x)\\}$, wherein the lightest of these scalars (with KK
number $i=0$) is identified as the massive radion and the heavier states are
called Goldberger-Wise (GW) scalars. 777Note that, having chosen to express
the physical degrees of freedom in terms of the scalar field $\hat{r}$ in Eq.
(6), the form of the couplings between the massive spin-2 states and the tower
of GW states is precisely the same as the form of the radion coupling in RS1.
However, as we will see, the mode equation and normalization conditions for
the physical GW scalars lead to additional complications. From here on, we
will drop the argument $\varphi$ in the warp factor $A(\varphi)$ for
convenience.

##  III Kaluza-Klein Mode Expansions

Upon Kaluza-Klein (KK) decomposition, the scalar and tensor modes come from
extra-dimensional wavefunctions which satisfy one-dimensional Sturm-Liouville
(SL) problems. For completeness and notational consistency, we provide details
of the derivation of the SL problems in the tensor and scalar sectors in
Appendix A \- following the procedures originally described in Boos:2005dc ;
Boos:2012zz , (see also Ref. Chivukula:2021xod ). We report the results here
for the convenience of the reader and to highlight the differences that arise
when solving these problems in the stabilized RS1 model.

###  III.1 Spin-2 Sector

The tensor field $h_{\mu\nu}(x,y)$ is decomposed into a tower of 4D KK states
$\hat{h}^{(n)}_{\mu\nu}(x)$ in the usual way as follows: recalling
$\varphi\equiv y/r_{c}$,

$\displaystyle\hat{h}_{\mu\nu}(x,y)=\dfrac{1}{\sqrt{\pi
r_{c}}}\sum_{n=0}^{+\infty}\hat{h}^{(n)}_{\mu\nu}(x)\,\psi_{n}(\varphi)~{}.$
(8)

Here $r_{c}$ is the radius of the extra dimension and $\psi_{n}(\varphi)$ is
the 5D wavefunction of the $n^{\text{th}}$ mode that satisfies the following
SL differential equation:

$\displaystyle\partial_{\varphi}[e^{-4A}\,\partial_{\varphi}\psi_{n}]=-\mu_{n}^{2}e^{-2A}\psi_{n}$
(9)

where the wavefunctions satisfy the boundary conditions where
$(\partial_{\varphi}\psi_{n})=0$ at $\varphi\in\\{0,\pi\\}$. The eigenvalues
$\mu_{n}=m_{n}r_{c}$ are the masses $m_{n}$ of the $n^{\text{th}}$ spin-2 KK
mode. The wavefunctions are normalized as follows

$\displaystyle\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
10.0pte^{-2A}\,\psi_{m}\,\psi_{n}$ $\displaystyle=\delta_{m,n}\ ,$ (10)

and satisfy the completeness relation

$\displaystyle\delta(\varphi_{2}-\varphi_{1})=\dfrac{1}{\pi}\,e^{-2A}\sum_{j=0}^{+\infty}\psi_{j}(\varphi_{1})\,\psi_{j}(\varphi_{2})~{}.$
(11)

The form of this SL problem is identical to the unstabilized case and differs
only in how the new background geometry influences the warp factor $A$. In the
unstabilized case, the warp factor is simply $kr_{c}|\varphi|$. In the
stabilized case, the bulk scalar potential modifies the background geometry
such that $A(\varphi)$ satisfies the (Einstein) equation

$\displaystyle
A^{\prime\prime}=\dfrac{1}{12}\bigg{[}(\phi_{0}^{\prime})^{2}+4\sum_{i=1,2}V_{i}r_{c}\,\delta_{i}\bigg{]}$
(12)

in terms of the background scalar field $\phi_{0}$ and the brane-localized
potentials $V_{1,2}$ at $\varphi=0$ and $\varphi=\pi$ respectively (refer to
Appendix A for additional details).

###  III.2 Spin-0 Sector

The spin-0 sector of the model arises from two sources. The first is the
scalar metric fluctuation (where even the lightest mode will be $y$-dependent
in the stabilized model), and the second is the new bulk scalar field. The two
sectors mix via the gauge condition noted in Eq. (6). Consequently, we attain
a single physically-relevant 5D physical scalar perturbation $\hat{r}(x,y)$.
Similar to the tensor perturbation, the KK decomposition of the 5D scalar
field $\hat{r}(x,y)$ into a tower of spin-0 KK modes proceeds by introducing
extra-dimensional wavefunctions $\gamma_{i}(\varphi)$ and a tower of 4D scalar
fields $\hat{r}^{(i)}(x)$ parameterized as follows:

$\displaystyle\hat{r}(x,y)=\dfrac{1}{\sqrt{\pi
r_{c}}}\sum_{i=0}^{+\infty}\hat{r}^{(i)}(x)\,\gamma_{i}(\varphi)~{}.$ (13)

The mode equation that brings the 5D scalar Lagrangian to canonical form,
however, is quite different from the tensor case and involves nontrivial
boundary terms

$\displaystyle\partial_{\varphi}\bigg{[}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}(\partial_{\varphi}\gamma_{i})\bigg{]}-\dfrac{e^{2A}}{6}\gamma_{i}=-\mu_{(i)}^{2}\,\dfrac{e^{4A}}{(\phi_{0}^{\prime})^{2}}\,\gamma_{i}\,\Bigg{\\{}1+\frac{2\,\delta(\varphi)}{\Big{[}2\ddot{V}_{1}r_{c}-\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}+\frac{2\,\delta(\varphi-\pi)}{\Big{[}2\ddot{V}_{2}r_{c}+\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}\Bigg{\\}}~{},$
(14)

where $\phi^{\prime}_{0}\equiv(\partial_{\varphi}\phi_{0})$ and the
eigenvalues $\mu_{(n)}=m_{(n)}r_{c}$ are the masses $m_{(n)}$ of the
$n^{\text{th}}$ scalar KK mode. The Dirac deltas enforce the following
(orbifold) boundary conditions:

$\displaystyle(\partial_{\varphi}\gamma_{i})\bigg{|}_{\varphi=0}$
$\displaystyle=-\bigg{[}2\ddot{V}_{1}r_{c}-\dfrac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\bigg{]}^{-1}\,\mu_{(i)}^{2}\,e^{2A}\,\gamma_{i}\bigg{|}_{\varphi=0}~{},$
$\displaystyle(\partial_{\varphi}\gamma_{i})\bigg{|}_{\varphi=\pi}$
$\displaystyle=+\bigg{[}2\ddot{V}_{2}\,r_{c}+\dfrac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\bigg{]}^{-1}\,\mu_{(i)}^{2}\,e^{2A}\,\gamma_{i}\bigg{|}_{\varphi=\pi}~{},$
(15)

where $\ddot{V}_{1,2}$ are second functional derivatives of the brane
potentials evaluated at the background-field configuration. Note that these
boundary conditions reduce to Neumann form in the “stiff-wall” limit,
$\ddot{V}_{1,2}\to+\infty$, a limit which will be useful to us during our
numerical work in Sec. V. In the form of Eq. (14), the Sturm-Liouville nature
of the problem is manifest, and the orthogonality and completeness of the
wavefunctions follow immediately Boos:2005dc ; Kofman:2004tk ; fulton1977two ;
binding1994sturm .888The completeness of the solutions to the scalar Sturm-
Liouville problem in Eq. (14) and the positivity of the scalar mass-squared
eigenvalues $\mu_{(i)}^{2}$ are only assured if the coefficients of the
$\delta$-function terms are non-negative fulton1977two ; binding1994sturm . We
will assume that the brane-potentials and background field are such as to
satisfy this condition (as they do in the stiff-wall limit we use later).
Physically this constraint is more easily understood in the analogous case of
the modes of a string: in that case the $\delta$-functions can be understood
as representing point masses which can freely move at the boundary of the
string, and the coefficients are proportional to these masses and must
therefore be positive for stability.

Due to the mixing between the gravitational and bulk scalar sectors in Eq.
(6), however, an unconventional normalization of the scalar wavefunctions is
required to bring the scalar kinetic energy terms to canonical form999Because
$\mu^{2}_{(i)}>0$ in the GW model, this normalization choice (albeit unusual)
is consistent. Taking the unstabilized limit $\phi^{\prime}_{0}\to 0$ (in
which $\mu_{(0)}\to 0$ and all other scalar states decouple; see Eq (6))
however requires care. This limit is discussed in the context of the “flat-
stabilized” model in Chivukula:2021xod .

$\displaystyle\delta_{mn}$
$\displaystyle=\dfrac{6\mu^{2}_{(n)}}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\gamma_{m}\gamma_{n}\dfrac{e^{4A}}{(\phi_{0}^{\prime})^{2}}\Bigg{\\{}1+\frac{2\,\delta(\varphi)}{\Big{[}2\ddot{V}_{1}r_{c}-\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}+\frac{2\,\delta(\varphi-\pi)}{\Big{[}2\ddot{V}_{2}r_{c}+\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}\Bigg{\\}}$
(16) $\displaystyle=\dfrac{6}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{[}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}\gamma_{m}^{\,\prime}\,\gamma_{n}^{\,\prime}+\dfrac{e^{2A}}{6}\gamma_{m}\gamma_{n}\bigg{]}~{},$
(17)

where the second line follows by applying the differential equation (14) and
integration-by-parts on the periodic doubled orbifold. The scalar wavefunction
completeness relation follows from Eq. (16):

$\delta(\varphi_{2}-\varphi_{1})=\dfrac{6}{\pi}\dfrac{e^{4A(\varphi_{1})}}{(\phi_{0}^{\prime}(\varphi_{1}))^{2}}\,\Bigg{\\{}1+\frac{2\,\delta(\varphi_{1})}{\Big{[}2\ddot{V}_{1}r_{c}-\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}+\frac{2\,\delta(\varphi_{1}-\pi)}{\Big{[}2\ddot{V}_{2}r_{c}+\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}\Bigg{\\}}~{}\sum_{j=0}^{+\infty}\mu_{(j)}^{2}\gamma_{j}(\varphi_{1})\,\gamma_{j}(\varphi_{2})~{}.$
(18)

The second form of the normalization condition in Eq. (17) is useful in
computational work, since it does not rely on knowledge of the eigenvalues.
Details are given in Appendix A.

Because of the Neumann boundary conditions $(\partial_{\varphi}\psi_{n})=0$
and the simplicity of Eq. (9), there is always a massless spin-2 mode (with a
wavefunction constant in $\varphi$) in the tensor tower. This is not the case
for the scalar tower. Due to the non-constant potential of the background
scalar, along with its vacuum expectation value, the lightest spin-0 state
(identified as the radion with a wavefunction $\gamma_{0}$) acquires a mass
$\mu_{(0)}$.

##  IV Massive spin-2 couplings, scattering amplitudes, and Sum Rules

Figure 1: Matrix element diagrams contributing to $n,n\to n,n$ massive spin-2
KK boson scattering. Here $n$ refers to the KK mode number of the external
state. The intermediate states $x$ include a massive radion, the graviton, a
tower of massive spin-2 KK bosons, and a tower of GW scalars.

As discussed extensively in the literature (see, for example,
Hinterbichler:2011tt ; deRham:2014zqa , and references therein),
phenomenological calculations incorporating massive spin-2 states often
generate matrix element diagrams which exhibit anomalous high-energy behavior.
Extra-dimensional models of gravity possess an underlying 5D diffeomorphism
invariance that ensures their amplitudes are well behaved. That is, any
overall bad high-energy growth necessarily signals the omission of additional
important physics. Such omissions can produce erroneous phenomenological
results.101010Following our previous computations Chivukula:2019rij ;
Chivukula:2019zkt ; Chivukula:2020hvi it was shown in deGiorgi:2021xvm that in
freeze-out computations with KK-graviton portal dark matter scenarios exhibit
amplitudes that grow no faster than $\mathcal{O}(s)$ in an unstabilized model.
The stabilized and phenomenologically relevant RS1 model is significantly more
difficult to calculate, and will be presented in a future work.

In our previous work we analyzed the diagrams which contribute to
$2\rightarrow 2$ massive spin-2 mode scattering for several variants of the
Randall-Sundrum I model; within each of these analyses, we found there exist
individual diagrams which diverge as fast as $\mathcal{O}(s^{5})$ at high
energies ($s$ being the usual Mandelstam parameter) and that cancellations
between diagrams ensure the total matrix element only grows as fast as
$\mathcal{O}(s)$ Chivukula:2019rij ; Chivukula:2019zkt ; Chivukula:2020hvi ;
Foren:2020egq . This genuine $\mathcal{O}(s)$ growth is important because it
insures the 4D effective theory breaks down at an energy scale consistent with
the physics of the underlying extra-dimensional theory. The central purpose of
this paper is to verify the various coupling relations and sum rules
Chivukula:2021xod required to ensure these cancellations in a general
Goldberger-Wise-stabilized Randall-Sundrum I model, report how most of these
rules may be proved in full generality, and (in the next section) numerically
demonstrate leading $\mathcal{O}(s)$ growth of the matrix element to second-
order in a solvable warped stabilized model.

In this section we review the definitions of the KK mode-couplings relevant to
the scattering computations, IV.1, describe the sum-rules which apply to these
couplings and show how they are related to the physics of the Goldberger-Wise
model, IV.2, and provide a brief summary of the sum-rules we numerically
verify in Sec. IV.3.

###  IV.1 Scattering Amplitudes and Couplings

The tree-level diagrams relevant to the aforementioned $2\rightarrow 2$ matrix
element are shown in Fig. 1; details can be found in Chivukula:2019rij ;
Chivukula:2019zkt ; Foren:2020egq ; Chivukula:2021xod ; Chivukula:2020hvi .
The general strategy employed in this paper follows that of the unstabilized
case. Briefly, the total matrix element $\mathcal{M}$ involves a contact
diagram $(\mathcal{M}_{c})$ as well as infinite sums over the diagrams
($\mathcal{M}_{j,X}$, $\mathcal{M}_{(i),X}$) describing $X=s$, $t$, and
$u$-channel exchanges of intermediate spin-2 states and spin-0 states:

$\displaystyle\mathcal{M}=\mathcal{M}_{c}+\sum_{X\in\\{s,t,u\\}}\bigg{[}\sum_{j=0}^{+\infty}\mathcal{M}_{j,X}+\sum_{i=0}^{+\infty}\mathcal{M}_{(i),X}\bigg{]}$

where in general KK numbers within parentheses (the $(i)$) refer to those of
the spin-0 states, while those without parentheses (the $j$) reference the
spin-2 states. We expand the matrix element $\mathcal{M}$ in a Taylor series
in $s$ as Chivukula:2020hvi ,

$\mathcal{M}(s,\theta)=\sum_{\sigma\in\tfrac{1}{2}\mathbb{Z}}\overline{\mathcal{M}}^{(\sigma)}(\theta)\cdot
s^{\sigma}$ (19)

and isolate the kinematic factors and the couplings. Generally, 5D
diffeomorphism demands that all coefficients of $s^{\sigma}$ with $\sigma>1$
in this expansion must vanish. At present, because we consider the process
involving helicity-zero external states, half-integer values of $\sigma$
automatically vanish. Demanding $\mathcal{O}(s)$ growth at most in this matrix
element necessitates relationships between various masses and coupling
structures in the theory.

The couplings present in each of these diagrams come from wavefunction overlap
integrals attained following Kaluza-Klein decomposition of the fields in the
Lagrangian; this procedure of attaining a 4D effective theory from a
Lagrangian such as Eq. 1 via KK decomposition is explained in detail in Refs.
Chivukula:2020hvi ; Foren:2020egq .

  * •

The contact diagram $\mathcal{M}_{c}$ involves the 4-point massive spin-2
vertex and contributes the following wavefunction overlap integrals:

$\displaystyle a_{klmn}$
$\displaystyle\equiv\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
10.0pte^{-2A}\,\psi_{k}\,\psi_{l}\,\psi_{m}\,\psi_{n}~{},$ $\displaystyle
b_{k^{\prime}l^{\prime}mn}$
$\displaystyle\equiv\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
10.0pte^{-4A}\,(\partial_{\varphi}\psi_{k})(\partial_{\varphi}\psi_{l})\psi_{m}\,\psi_{n}~{}.$

For helicity-zero, $\mathcal{M}_{c}$ diverges like $\mathcal{O}(s^{5})$.

  * •

The spin-2 mediated diagrams $\mathcal{M}_{j,X}$ involve 3-point spin-2
vertices and contribute the following wavefunction overlap integrals:

$\displaystyle a_{lmn}$
$\displaystyle\equiv\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
10.0pte^{-2A}\,\psi_{l}\,\psi_{m}\,\psi_{n}~{},$ $\displaystyle
b_{l^{\prime}m^{\prime}n}$
$\displaystyle\equiv\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
10.0pte^{-4A}\,(\partial_{\varphi}\psi_{l})(\partial_{\varphi}\psi_{m})\psi_{n}~{}.$

For helicity-zero, each $\mathcal{M}_{j,X}$ diverges like $\mathcal{O}(s^{5})$

  * •

The spin-0 mediated diagrams $\mathcal{M}_{(i),X}$ involve $3$-point
scalar-(spin-2)-(spin-2) couplings and contribute the following wavefunction
overlap integrals:

$\displaystyle a_{l^{\prime}m^{\prime}(n)}$
$\displaystyle\equiv\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
10.0pte^{-2A}\,(\partial_{\varphi}\psi_{l})(\partial_{\varphi}\psi_{m})\gamma_{n}~{}.$

Due to the structure of the Lagrangian, no corresponding
“$b_{l^{\prime}m^{\prime}(n)}$” or “$b_{lm^{\prime}(n^{\prime})}$” is
generated. For helicity-zero, each $\mathcal{M}_{(i),X}$ diverges like
$\mathcal{O}(s^{3})$.

In general, (i) couplings labeled with an “$a$” have an $e^{-2A}$ weight
factor whereas those labeled with a “$b$” involve $e^{-4A}$, (ii) the
subscript KK indices indicate the relevant wavefunctions to include in each
integral (remembering that parentheses indicate scalar modes), and (iii) a
subscript KK index with a prime denotes that the corresponding mode number’s
wavefunction should be differentiated with respect to the extra-dimensional
coordinate $\varphi$.

We can reduce the number of coupling integrals present by using the properties
of the KK wavefunctions; namely, the mode equation and completeness relations,
Eqs. (9) and (11). For example, in prior work Chivukula:2020hvi ;
Foren:2020egq we showed how the spin-2 mode equation Eq. (9) and the
corresponding completeness relation Eq. (11) relate some of the $a$ and $b$
couplings:

$\displaystyle
b_{l^{\prime}m^{\prime}n}=\dfrac{1}{2}\Big{[}\mu_{l}^{2}+\mu_{m}^{2}-\mu_{n}^{2}\Big{]}a_{lmn}\hskip
35.0ptb_{n^{\prime}n^{\prime}nn}=\dfrac{\mu^{2}_{n}}{3}a_{nnnn}~{}.$ (20)

We use these relations and eliminate all $b$-type overlap integrals in favor
of $a$-type integrals. Doing so, we may write the sum rules entirely using
$a$-type overlap integrals.

###  IV.2 Sum Rules Insuring Consistency of Scattering Amplitudes

By requiring the scattering amplitude to grow no faster than $\mathcal{O}(s)$
in the GW model Chivukula:2021xod , we previously determined the following
general sum rules should be satisfied:

$\displaystyle\sum_{j=0}a_{nnj}^{2}$ $\displaystyle=a_{nnnn}~{},$ (21)
$\displaystyle\sum_{j=0}\mu_{j}^{2}a_{nnj}^{2}$
$\displaystyle=\dfrac{4}{3}\mu_{n}^{2}a_{nnnn}~{},$ (22)
$\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{4}a^{2}_{nnj}$
$\displaystyle=\dfrac{4}{15}\mu_{n}^{4}(4\,a_{nnnn}-3\,a^{2}_{nn0})+\dfrac{36}{5}\sum_{i=0}^{+\infty}a^{2}_{n^{\prime}n^{\prime}(i)}~{},$
(23) $\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{6}a^{2}_{nnj}$
$\displaystyle=-4\mu_{n}^{6}a^{2}_{nn0}+9\sum_{i=0}^{+\infty}(4\mu_{n}^{2}-\mu_{(i)}^{2})a^{2}_{n^{\prime}n^{\prime}(i)}~{},$
(24)

The first two sum rules, Eqs. (21) and (22), insure that the contributions to
the scattering amplitudes growing like ${\cal O}(s^{5})$ and ${\cal O}(s^{4})$
vanish. They follow directly from the Sturm-Liouville form of the spin-2 KK
mode equation, Eq. (9) and the corresponding completeness relation Eq. (11)-
so the proofs given in Chivukula:2019zkt ; Chivukula:2020hvi ; Foren:2020egq
apply to any model producing a geometry defined by a warp function $A(y)$.
However, the sum rules in Eqs. (23) and (24), which insure cancellation of the
contributions to the amplitude growing like ${\cal O}(s^{3})$ and ${\cal
O}(s^{2})$, involve the scalar tower present in the GW model.

By combining the last two sum rules, Eqs. (23) and (24), to eliminate the
common sum $\sum_{i}a_{n^{\prime}n^{\prime}(i)}^{2}$, we find a mixed rule:

$\displaystyle\sum_{j=0}^{+\infty}\bigg{[}5\mu_{n}^{2}-\mu_{j}^{2}\bigg{]}\mu_{j}^{4}a^{2}_{nnj}$
$\displaystyle=\dfrac{16}{3}\mu_{n}^{6}a_{nnnn}+9\sum_{i=0}^{+\infty}\mu_{(i)}^{2}a_{n^{\prime}n^{\prime}(i)}^{2}~{}.$
(25)

The only scalar tower sum
$\sum_{i=0}^{+\infty}\mu_{(i)}^{2}a^{2}_{n^{\prime}n^{\prime}(i)}$ remaining
in this particular combination of the $\mathcal{O}(s^{3})$ and
$\mathcal{O}(s^{2})$ sum rules can be eliminated using the spin-0 completeness
relation Eq. (18). Since the spin-2 wavefunctions satisfy Neumann boundary
conditions, $(\partial_{\varphi}\psi_{n})=0$ at $\varphi=0$ and $\pi$, we find

$\displaystyle\sum_{i=0}^{+\infty}\mu_{(i)}^{2}a_{n^{\prime}n^{\prime}(i)}^{2}$
$\displaystyle=\dfrac{1}{6}\bigg{\\{}\int d\varphi\hskip
5.0pt(\partial_{\varphi}\phi_{0})^{2}e^{-8A}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}~{}.$
(26)

Hence the combination of sum-rules in Eq. (25) does not depend on the explicit
form of the scalar wavefunctions $\gamma_{i}(\varphi)$, but only on the spin-2
wavefunctions $\psi_{n}(\varphi)$, the exponentiated warp-factor
$e^{A(\varphi)}$, and (the derivative of) the background scalar field
$\phi_{0}$.

In Appendix B.5.1 we show, by applying only the spin-2 mode equation (9),
Neumann boundary conditions, and completeness relations (11), that

$\displaystyle\sum_{j=0}^{+\infty}\bigg{[}5\mu_{n}^{2}-\mu_{j}^{2}\bigg{]}\mu_{j}^{4}a^{2}_{nnj}$
$\displaystyle=\dfrac{16}{3}\mu_{n}^{6}a_{nnnn}+18\bigg{\\{}\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt(\partial_{\varphi}^{2}A)e^{-8A}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}~{}.$
(27)

In the GW model we have the Einstein equation (via Eq. (12))
$(\partial_{\varphi}^{2}A)=(\partial_{\varphi}\phi_{0})^{2}/12+\sum_{i}V_{i}r_{c}\,\delta_{i}(\varphi)/3$.
The Dirac delta terms vanish because $(\partial_{\varphi}\psi_{n})=0$ at the
boundaries, and hence

$\displaystyle\sum_{j=0}^{+\infty}\bigg{[}5\mu_{n}^{2}-\mu_{j}^{2}\bigg{]}\mu_{j}^{4}a^{2}_{nnj}$
$\displaystyle=\dfrac{16}{3}\mu_{n}^{6}a_{nnnn}+\dfrac{3}{2}\bigg{\\{}\int
d\varphi\hskip
5.0pt(\partial_{\varphi}\phi_{0})^{2}e^{-8A}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}~{}.$
(28)

Applying Eq. (26) we immediately obtain Eq. (25). Hence Eq. (25) depends non-
trivially on the dynamics of the GW model - in particular on the Einstein
equations for the warp-factor and on the scalar completeness relation which
follows from the mode equation and the scalar mode normalization condition.

Using the spin-2 completeness relations and the relations between the $a$ and
$b$ couplings in Eq. (20) and the sum-rule in Eq. (21), we find

$\sum_{j=0}^{+\infty}\mu^{4}_{j}a^{2}_{nnj}=4\,c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\,\dfrac{4}{3}\mu_{n}^{4}a_{nnnn}~{}.$
(29)

Here we have defined the quantity

$\displaystyle
c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}=\sum_{j=0}^{+\infty}b^{2}_{n^{\prime}n^{\prime}j}\equiv\dfrac{1}{\pi}\int
d\varphi\hskip 5.0pte^{-6A}(\partial_{\varphi}\psi_{n})^{4}~{},$ (30)

which depends only on the spin-2 wavefunctions. Plugging this into Eq. (28) we
find

$\sum_{j=0}^{+\infty}\mu_{j}^{6}a^{2}_{nnj}=\dfrac{4}{3}\mu_{n}^{6}a_{nnnn}+20\mu_{n}^{2}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}-\dfrac{3}{2}\bigg{\\{}\int
d\varphi\hskip
5.0pt(\partial_{\varphi}\phi_{0})^{2}e^{-8A}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}~{},$
(31)

which depends only on the spin-2 wavefunctions and the background scalar-field
configuration $\phi_{0}$.

Finally, having demonstrated that one linear combination of Eqs. (23) and (24)
is determined entirely by the spin-2 sector of the GW model, we can also use
Eq. (29) to isolate the contribution from the GW scalar sector

$\displaystyle\sum_{i=0}^{+\infty}a^{2}_{n^{\prime}n^{\prime}(i)}=\dfrac{5}{9}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\dfrac{1}{9}\mu_{n}^{4}a^{2}_{nn0}+\dfrac{1}{27}\mu^{4}_{n}a_{nnnn}~{},$
(32)

which succinctly summarizes the necessary (but unproven) relationship between
the scalar and spin-2 couplings (and hence wavefunctions) which must be
satisfied in order for the spin-2 scattering amplitudes to grow no faster than
${\cal O}(s)$.

###  IV.3 Sum-Rule Summary

In summary, the spin-2 coupling relations

$\displaystyle\sum_{j=0}a_{nnj}^{2}$ $\displaystyle=a_{nnnn}~{},$ (21
revisited) $\displaystyle\sum_{j=0}\mu_{j}^{2}a_{nnj}^{2}$
$\displaystyle=\dfrac{4}{3}\mu_{n}^{2}a_{nnnn}~{},$ (22 revisited)
$\displaystyle\sum_{j=0}^{+\infty}\mu^{4}_{j}a^{2}_{nnj}$
$\displaystyle=4c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\,\dfrac{4}{3}\mu_{n}^{4}a_{nnnn}~{},$
(29 revisited) $\displaystyle\sum_{j=0}^{+\infty}\mu_{j}^{6}a^{2}_{nnj}$
$\displaystyle=\dfrac{4}{3}\mu_{n}^{6}a_{nnnn}+20\mu_{n}^{2}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}-\dfrac{3}{2}\bigg{\\{}\int
d\varphi\hskip
5.0pt(\partial_{\varphi}\phi_{0})^{2}e^{-8A}(\partial_{\varphi}\psi_{n})^{4}\bigg{\\}}~{},$
(31 revisited)

follow from the form of the spin-2 mode equations, spin-2 wavefunction
completeness, the Einstein equations for the warp factor, and the scalar
completeness relation which follows from the scalar mode equation and the
scalar mode normalization condition. Analytic derivations of all of these
relations are given above or in Appendix B.5.1. With respect to guaranteeing
that the massive spin-2 scattering amplitudes grow no faster than ${\cal
O}(s)$, these relations show that the first-two sum-rules Eqs. (21)-(22) and
one combination of the sum-rules in Eqs. (23) and (24) are always satisfied.

Separately, the sum-rule

$\displaystyle\sum_{i=0}^{+\infty}a^{2}_{n^{\prime}n^{\prime}(i)}=\dfrac{5}{9}c_{n^{\prime}n^{\prime}n^{\prime}n^{\prime}}+\dfrac{1}{9}\mu_{n}^{4}a^{2}_{nn0}+\dfrac{1}{27}\mu^{4}_{n}a_{nnnn}~{},$
(32 revisited)

which depends on the spin-0 GW scalar couplings must also be satisfied in
order for the spin-2 scattering amplitudes to grow no faster than ${\cal
O}(s)$. With the methods discussed here, we have been unable to prove
analytically that this scalar sum-rule is satisfied in
general.111111Generalizations of the proven rules (and their proofs) as well
as additional unproven rules necessary for $\mathcal{O}(s)$ growth of the
inelastic amplitude $(k,l)\rightarrow(m,n)$ are provided in Appendix B.

In Chivukula:2021xod , we demonstrated that these sum-rules in Eqs. (21)-(24)
were satisfied in the “flat-stabilized” model - a slight deformation of a flat
extra-dimensional model in which the radion is massive and the size of the
extra dimension is stable. We now demonstrate numerically that these relations
are satisfied in the presence of the warping required to produce the hierarchy
between the weak and Planck scales in the Randall-Sundrum model.

##  V Perturbative Analysis of a Warped Stabilized Model

In the original formulation of the RS1 model Randall:1999ee ; Randall:1999vf ,
Randall and Sundrum constructed a consistent solution to the Einstein field
equations by choosing a warp factor $A(\varphi)=kr_{c}|\varphi|$ sourced by
brane and bulk cosmological constants. Once the Goldberger-Wise mechanism is
implemented, such a simple functional form becomes unavailable: the background
geometry is augmented, the Einstein field equations are changed, and the warp
factor $A(\varphi)$ is made more complicated. The background Einstein field
equations of the stabilized model (Eqs. (85), (86), (88), and (89)) are
coupled non-linear equations with respect to $A(\varphi)$ that depend on the
derivative of the scalar background $\phi_{0}^{\prime}(\varphi)$ and are
generally difficult to solve. DeWolfe, Freedman, Gubser, and Karch have
constructed a specific class of exactly solvable potentials (the DFGK model)
DeWolfe:1999cp which make calculations in the stabilized RS1 model feasible.

In this section, we review the DFGK class of solutions to set notation, and we
detail a specific DFGK model which enables perturbative expansion around the
(solved) warped unstabilized RS1 model Chivukula:2021xod . Subsequently, for
the physical spin-0 and spin-2 towers, we perturbatively compute the
wavefunctions and masses using the Sturm-Liouville equations described in
section III. Finally, we demonstrate that the sum rules defined in section IV
are numerically satisfied at second order in the expansion parameter, the
lowest order necessary to generate a nonzero radion mass.

###  V.1 The DFGK Model

A key strategy employed in the DFGK model DeWolfe:1999cp is the introduction
of a superpotential-inspired function $W[\hat{\phi}]$ which is used to
simplify the stabilized model’s background field equations (Eqs. (85), (86),
(88), and (89)). In particular, it is assumed that the scalar bulk and brane
potentials may be parameterized as:

$\displaystyle
Vr_{c}^{2}=\dfrac{1}{8}\bigg{(}\dfrac{dW}{d\hat{\phi}}\bigg{)}^{2}-\dfrac{W^{2}}{24}~{},$
(33) $\displaystyle
V_{1}r_{c}=+\dfrac{W}{2}+\beta_{1}^{2}\bigg{[}\hat{\phi}(\varphi)-\phi_{1}\bigg{]}^{2}~{},\hskip
20.0pt$ $\displaystyle\hskip
20.0ptV_{2}r_{c}=-\dfrac{W}{2}+\beta_{2}^{2}\bigg{[}\hat{\phi}(\varphi)-\phi_{2}\bigg{]}^{2}~{},$
(34)

In this case, the background scalar and Einstein equations are solved if

$\displaystyle(\partial_{\varphi}A)=\frac{W}{12}\bigg{|}_{\hat{\phi}=\phi_{0}}\text{
sign}(\varphi)~{},\hskip 20.0pt$ $\displaystyle\hskip
20.0pt(\partial_{\varphi}\phi_{0})=\dfrac{dW}{d\hat{\phi}}\bigg{|}_{\hat{\phi}=\phi_{0}}\text{
sign}(\varphi)~{},$ (35)

where $\phi_{1}\equiv\hat{\phi}(0)$ and $\phi_{2}\equiv\hat{\phi}(\pi)$. Ref.
DeWolfe:1999cp introduces a convenient $W[\hat{\phi}]$ with the following
specific form:121212Here $u$ is a parameter, and not the $\hat{u}$ field of
Eq. 62.

$\displaystyle
W[\hat{\phi}(\varphi)]=12kr_{c}-\dfrac{1}{2}\hat{\phi}(\varphi)^{2}\,ur_{c}~{}.$
(36)

Plugging this into Eq. 35 we find solutions for the bulk scalar vacuum and the
warp factor:

$\displaystyle\phi_{0}(\varphi)$
$\displaystyle=\phi_{1}e^{-ur_{c}|\varphi|}~{},$ (37) $\displaystyle
A(\varphi)$
$\displaystyle=kr_{c}|\varphi|+\dfrac{1}{48}\phi_{1}^{2}\bigg{[}e^{-2ur_{c}|\varphi|}-1\bigg{]}\
.$ (38)

In the limit that the parameter $u$ vanishes, this reduces to the usual
unstabilized RS1, with the bulk field acquiring a constant vacuum expectation
value. The parameters $u$, $\phi_{1}$, and $\phi_{2}$ are related according to

$ur_{c}=\dfrac{1}{\pi}\log\dfrac{\phi_{1}}{\phi_{2}}~{}.$ (39)

We next define the small-$u$ limit carefully to facilitate a perturbative
analysis.

###  V.2 The Perturbative DFGK Model

The forms of $\phi_{0}(\varphi)$ and $A(\varphi)$ in Eqs. (37)-(38) are useful
for solving the background equations, but it remains difficult to find general
analytic solutions (i.e. the KK wavefunctions and masses) to the differential
equations defined in section III. Therefore, we define a limit of the model so
that we may perturbatively expand $\phi_{0}$ and $A$ around the unstabilized
background (for which analytic solutions are well known), take the stiff wall
limit (with $\ddot{V}_{1,2}\to\infty$ so that the scalar boundary conditions
in Eq. (15) reduce to Neumann conditions at $\varphi=0,\,\pi$), and develop
solutions for the Sturm-Liouville problems order-by-order in perturbation
theory. Details of the perturbation theory can be found in Appendix C. Here,
we now proceed to introduce the effective warp parameter $\tilde{k}$ and
perturbation parameter $\epsilon$ Chivukula:2021xod .

Suppose we series expand $A(\varphi)$ with respect to the unitless quantity
$ur_{c}$. Expanding around $ur_{c}=0$ yields

$\displaystyle A(\varphi)$
$\displaystyle=kr_{c}|\varphi|-\bigg{[}\dfrac{\phi_{1}^{2}}{24}|\varphi|\bigg{]}(ur_{c})+\bigg{[}\dfrac{\phi_{1}^{2}}{24}|\varphi|^{2}\bigg{]}(ur_{c})^{2}+\mathcal{O}\Big{(}(ur_{c})^{3}\Big{)}$
(40)
$\displaystyle=\bigg{[}k-\dfrac{\phi_{1}^{2}u}{24}\bigg{]}r_{c}|\varphi|+\bigg{[}\dfrac{\phi_{1}^{2}(ur_{c})^{2}}{24}\bigg{]}|\varphi|^{2}+\mathcal{O}\Big{(}(ur_{c})^{3}\Big{)}~{}.$
(41)

The first term in the second line demonstrates that, when $ur_{c}$ is
sufficiently small, the stabilized model is a small deformation of an
unstabilized Randall-Sundrum I model Randall:1999ee ; Randall:1999vf . Because
we intend to work in the $ur_{c}\rightarrow 0$ limit, we will eliminate the
actual warp parameter $k$ in favor of the effective warp parameter,

$\tilde{k}\equiv k-\phi_{1}^{2}u/24~{},$ (42)

that applies in that limit.131313In Chivukula:2021xod we discussed the
properties of the “flat-stabilized model” with $\tilde{k}=0$.

In order to simplify various factors that would otherwise be present in
multiple equations, we will also replace $ur_{c}$ (and its role as our
expansion parameter) with the rescaled dimensionless perturbative parameter
$\epsilon\equiv\phi_{1}(ur_{c})/\sqrt{24}$. This definition of $\epsilon$
simplifies $A(\varphi)$ at $\mathcal{O}(\epsilon^{2})$

$\displaystyle A(\varphi)$
$\displaystyle=\tilde{k}r_{c}|\varphi|+\dfrac{\phi_{1}^{2}}{48}\bigg{[}\exp\bigg{(}-\dfrac{4\sqrt{6}}{\phi_{1}}\epsilon|\varphi|\bigg{)}-1\bigg{]}+\dfrac{\phi_{1}}{2\sqrt{6}}\epsilon|\varphi|~{},$
(43)
$\displaystyle=\tilde{k}r_{c}|\varphi|+\epsilon^{2}|\varphi|^{2}+\mathcal{O}(\epsilon^{3})~{}.$
(44)

and yields, to all orders in $\epsilon$,

$\displaystyle W[\hat{\phi}(\varphi)]$
$\displaystyle=12\tilde{k}r_{c}+\dfrac{\sqrt{6}}{\phi_{1}}\bigg{[}\phi_{1}^{2}-\hat{\phi}(\varphi)^{2}\bigg{]}\epsilon~{},$
(45) $\displaystyle\phi_{0}(\varphi)$
$\displaystyle=\phi_{1}\exp\bigg{(}-\dfrac{2\sqrt{6}}{\phi_{1}}\epsilon|\varphi|\bigg{)}~{}=\rm\phi_{1}exp(-\alpha\epsilon|\varphi|).$
(46)

where $\alpha=\dfrac{2\sqrt{6}}{\phi_{1}}$. It is the form of the warp-factor
shown in Eq. (43) that we use in subsequent perturbative computations.

####  V.2.1 Spin-2 wavefunctions and masses in perturbation theory

Figure 2: We show for each spin-2 KK mode $n$, the ratio of mass correction to
leading order mass times $\epsilon^{2}$ (dots). The mass correction
$\delta\mu_{n}^{2}$ is calculated in the large $\tilde{k}r_{c}$ limit to order
$\epsilon^{2}$, as shown in Eq. (268). The line represents the asymptotic form
of the expression given in Eq. (51), which is valid for large spin-2 KK
number.

To order ${\cal O}(\epsilon^{2})$, the spin-2 mode equation Eq. (9)
becomes141414Here, and in subsequent equations related to the perturbative
model in this paper, we expand $A(\varphi)$ to ${\cal O}(\epsilon^{2})$ using
Eq. (43), but we do not record the expansion of the exponents in powers of
$\epsilon$ to retain succinctness of expressions when writing. However, these
exponents are actually expanded during our calculations, e.g. when we define
the perturbed differential equations in (225)-(226).

$\partial_{\varphi}\left[e^{-4\tilde{k}r_{c}\varphi-4\epsilon^{2}\varphi^{2}}\,\partial_{\varphi}\psi_{n}\right]=-\mu^{2}_{n}e^{-2\tilde{k}r_{c}\varphi-2\epsilon^{2}\varphi^{2}}\psi_{n}~{}.$
(47)

The form of this equation, along with the Neumann boundary conditions
$\partial_{\varphi}\psi_{n}=0$ at $\varphi=0,\,\pi$, insures that the zero
mode (the graviton) is massless and has a wavefunction which is constant in
$\varphi$. Expanding this equation in powers of $\epsilon^{2}$, we solve Eq.
(9) using perturbation theory as described in Appendix C. The perturbative
expressions for the spin-2 wavefunctions and masses are quite lengthy and in
order to simplify them, we restrict ourselves to the limit when
$\tilde{k}r_{c}$ is large. The expressions for the spin-2 mass and
wavefunctions are given to order $\epsilon^{2}$ in the large-$\tilde{k}r_{c}$
limit in Appendix C.2.

In order to illustrate the effects of the geometry on the spin-2 masses, we
calculate the masses for Eq. (9) using the WKB approximation. The asymptotic
formula for the masses is given by151515There are O(1) corrections to
$\mu^{2}_{n}\ell^{2}$, that do not grow with $n$, due to the Neumann boundary
conditions. These effects can be included, but do not effect the analysis
given here.

$\displaystyle\mu_{n}=m_{n}r_{c}=\frac{n\pi}{l}~{},\hskip
35.0ptl=\int_{0}^{\pi}d\varphi\ e^{A}\ .$ (48)

The equations above show how the eigenvalues are positive and form an infinite
tower of states. Using the form of $A(\varphi)$ from Eq. (38), we find the
mass of spin-2 KK modes for large mode number to be

$\displaystyle\mu_{n}\Big{|}_{n\gg
1}=\frac{(n\pi)\tilde{k}r_{c}}{e^{\pi\tilde{k}r_{c}}-1}-(n\pi)\frac{e^{\pi\tilde{k}r_{c}}[\pi\tilde{k}r_{c}(\pi\tilde{k}r_{c}-2)+2]-2}{\tilde{k}r_{c}(e^{\pi\tilde{k}r_{c}}-1)^{2}}\epsilon^{2}+O\left(\epsilon^{3}\right)$
(49)

In the limit when $\tilde{k}r_{c}\gg 1$, which is the phenomenologically
interesting limit, this expression further simplifies to

$\mu_{n}\Big{|}_{n\gg
1}\simeq(n\pi)\tilde{k}r_{c}e^{-\pi\tilde{k}r_{c}}\bigg{\\{}1-\frac{[\pi\tilde{k}r_{c}(\pi\tilde{k}r_{c}-2)+2]}{\tilde{k}^{2}r_{c}^{2}}\epsilon^{2}\bigg{\\}}\
.$ (50)

We can see how the effect of the back reaction on the geometry from the bulk
scalar field reduces the masses of the massive spin-2 modes. Further, the
square of the ratio of the correction to the mass ($\delta\mu_{n}$) to the
leading order mass $(\mu_{n}^{(0)})$ is

$\left.\frac{\delta\mu_{n}^{2}}{\left(\mu_{n}^{(0)}\right)^{2}}\right|_{n\gg
1;\tilde{k}r_{c}\gg
1}\simeq\left[-2\pi^{2}+\frac{4\pi}{\tilde{k}r_{c}}-\frac{4}{\tilde{k}^{2}r_{c}^{2}}+\ldots\right]\epsilon^{2}\,.$
(51)

We can therefore conclude that the perturbation theory we use is valid when
$|\epsilon|\ll\ 1/(\sqrt{2}\ \pi)$. In Fig. 2, we compare the full expression
for the mass corrections (given in Eq. (268) of Appendix C) to the asymptotic
form shown here. Here we see that the full form of the mass, represented by
the blue dots approaches the asymptotic value, represented by the bold red
line.

While the asymptotic formula is simple and provides a convenient cross-check
of our calculations, it is insufficient for our present purposes. In order to
demonstrate cancellations and verify sum rules, we need to be able to evaluate
the ${\cal O}(\epsilon^{2})$ wavefunctions and masses without approximation.
We provide exact expressions to order $\epsilon^{2}$ in the perturbation
theory within Appendix C.2. These expressions are consistent with the large
$\tilde{k}r_{c}$ limit results. Full expressions (which are valid for
arbitrary values of $\tilde{k}r_{c}$) are provided as supplementary
Mathematica files.

####  V.2.2 Spin-0 wavefunctions and masses in perturbation theory

Eq. (46) implies that

$(\phi^{\prime}_{0})^{2}=24\epsilon^{2}e^{-2\alpha\epsilon\varphi}~{}$ (52)

in the bulk. To simplify our analysis, we consider the perturbative solution
for the scalar tower in the “stiff-wall” limit, $\ddot{V}_{1,2}\to+\infty$, so
that Eq. (15) reduces to Neumann conditions

$\partial_{\varphi}\gamma_{i}\big{|}_{\varphi=0,~{}\pi}=0~{},$ (53)

Using this and the expansion of $A(\varphi)$ to order ${\cal O}(\epsilon^{2})$
from Eq. (43), the spin-0 mode equation Eq. (14) becomes (after multiplying
through by $24\epsilon^{2}$)

$\partial_{\varphi}\left[e^{(2\tilde{k}r_{c}\varphi+2\alpha\epsilon\varphi+2\epsilon^{2}\varphi^{2})}\partial_{\varphi}\gamma_{i}\right]-4\epsilon^{2}e^{(2\tilde{k}r_{c}\varphi+2\epsilon^{2}\varphi^{2})}\gamma_{i}=-\mu^{2}_{i}e^{(4\tilde{k}r_{c}\varphi+2\alpha\epsilon\varphi+4\epsilon^{2}\varphi^{2})}\gamma_{i}~{},$
(54)

while the normalization conditions to this order are,

$\delta_{mn}=\dfrac{1}{4\pi\epsilon^{2}}\int_{-\pi}^{+\pi}d\varphi\left[e^{(2\tilde{k}r_{c}\varphi+2\alpha\epsilon\varphi+2\epsilon^{2}\varphi^{2})}\gamma^{\prime}_{m}\gamma^{\prime}_{n}+4\epsilon^{2}e^{(2\tilde{k}r_{c}+2\epsilon^{2}\varphi^{2})\varphi}\gamma_{m}\gamma_{n}\right]~{}.$
(55)

Expanding the spin-0 mode equation Eq. (54) in powers of $\epsilon$, we solve
the differential Eq. (14) using perturbation theory described in Appendix C.
In particular, using Eq. (233), we find that the radion (identified as the
zero mode of the KK expansion in Eq. (13)) acquires a mass-squared at order
$\epsilon^{2}$ in perturbation theory

$\mu^{2}_{(0)}=\dfrac{8\epsilon^{2}}{1+e^{2\pi\tilde{k}r_{c}}}+\mathcal{O}(\epsilon^{3})~{},$
(56)

while the radion wavefunction evaluated to order $\epsilon^{2}$ is

$\gamma_{(0)}=\sqrt{\frac{\tilde{k}r_{c}\pi}{e^{2\tilde{k}r_{c}\pi}-1}}+C(\tilde{k}r_{c})\bigg{\\{}2\tilde{k}r_{c}\varphi+\text{sech}(\pi\tilde{k}r_{c})\sinh[\tilde{k}r_{c}(\pi-2\varphi)]-\tanh(\pi\tilde{k}r_{c})\bigg{\\}}\epsilon^{2}+\mathcal{O}(\epsilon^{3})\
.$ (57)

Here $C(\tilde{k}r_{c})$ is determined through normalization; the normalized
radion wave function is provided in the appendix in Eq. (273). In the limit
where $\epsilon$ vanishes, the gravitational degrees of freedom and the bulk
scalar cease mixing. This results in a massless radion, an unstabilized extra
dimension, and an entirely separate tower of scalar states. Note that the
leading order radion wavefunction is flat, signalling that it is massless at
that order.

The GW scalar wavefunction to order $\epsilon$ is given in Eq. (275). As
expected, the wavefunctions are comprised of Bessel functions. Due to the
normalization condition in Eq. (55), the massive GW scalar mode wavefunctions
have no $\epsilon^{0}$ terms and start at order $\epsilon$. Since the sum
rules in Eq. (21) through Eq. (25) have only products of GW scalar
wavefunctions, when trying to verify them to order $\epsilon^{2}$, we only
need expressions of the wavefunctions to order $\epsilon$. We therefore, only
provide expressions for the GW scalar wavefunction and masses to leading
order.161616Note that the limit $\tilde{k}r_{c}=0$, dubbed as the flat
stabilized model was studied previously in our work Chivukula:2021xod , where
we show that the sum rules required for the scattering amplitudes to grow only
as $\mathcal{O}(s)$ were satisfied to ${\cal O}(\epsilon^{2})$.

###  V.3 Numerical verification of sum rules

Figure 3: Verification of sum rules (Eqs. (21),(22),(29) and (31), reproduced
at the bottom of the diagram) for elastic scattering of KK mode number one
($n=1$). The index $j$, shown on the $x$-axis, is the number of spin-2 modes
included in the sum of the left hand side of the sum rules. The $y$-axis
indicates the log of relative error
$(\log_{10}(1-\Delta(\text{LHS})/\Delta(\text{RHS})))$ between the ${\cal
O}(\epsilon^{2})$ corrections to LHS and RHS of the the relevant equations.
This has been evaluated in the large $\tilde{k}r_{c}$ limit.

We now verify the sum rules summarized in section IV.3 for the warped-
stabilized model using our perturbative computations. We can substitute
expression for wavefunctions and masses calculated in the DFGK model and
provided in Appendix C.2 and evaluate the overlap integrals numerically. The
sum rules in Eq. (21),(22),(29) and (31) and Eq. (32) can thereby be evaluated
order-by-order in $\epsilon$. We know that the sum rules are satisfied for the
unstabilized RS model Chivukula:2020hvi , and therefore these expressions
agree to leading-order ($\epsilon^{0}$). Using our perturbative experessions,
we verify here that the sum-rules are satisfied to leading non-trivial order,
${\cal O}(\epsilon^{2})$. Equivalently, we show that the ${\cal
O}(\epsilon^{2})$ contributions on the left- and right-hand sides of Eq.
(21),(22),(29) and (31) and Eq. (32) agree.

Note that the left-hand sides of these expressions are given as infinite sums
over different overlap integrals. It is therefore not possible to perform the
entire sum. Instead, we perform the sum up a “cutoff” KK-number and show that
the relative error in the ${\cal O}(\epsilon^{2})$ contributions to the left
hand side (LHS) converge to the ${\cal O}(\epsilon^{2})$ contributions to
right hand side (RHS) of each expression as the number of KK modes included in
the sum increases.

####  V.3.1 Spin-2 Sum-Rules and Completeness

We begin with Eq. (21), (22), (29), and (31). The result of this exercise is
shown in Fig. 3, in the case $n=1$ (e.g. for elastic scattering of spin-2
modes at KK level 1) and for $\tilde{k}r_{c}=40/\pi=12.73$. We see that each
of the series converge nicely with the relative error reducing with the
addition of terms to the sum on the LHS of the equation. As described in
detail in Section IV, Eqs. (21), (22), and (29) can be proven directly using
the completeness properties of the solutions of the spin-2 mode equation Eq.
(9). Their numerical verification demonstrates that accuracy of our
perturbative analysis, The first two of these equations demonstrate that the
${\cal O}(s^{5})$ and ${\cal O}(s^{4})$ contributions to helicity-0 spin-2
elastic scattering vanish to this order in perturbation theory.

We also see that the sum-rule in Eq. (31) converges as well. As discussed
above, Eq. (31) depends non-trivially on the dynamics of the Goldberger-Wise
model implemented here - in particular on the Einstein equation coupling the
scalar-potential to the curvature of the extra dimension Eq. (12) and on the
completeness conditions of the spin-0 modes Eq. (18). As discussed in Section
IV.3, this verifies that one linear-combination of Eqs. (23) and (24) are also
satified.

Finally, in all cases we see that the series only converges rapidly once we
have included the $j=2$ term. This is because the overlap integral defining
the coupling between two spin-2 level-1 states and a spin-2 state at level $j$
is largest for $j=2$ – which can be understood as a remnant of the “discrete”
KK momentum-conservation which would be present in a flat extra-dimension.

####  V.3.2 The Spin-0 Sum Rule

Figure 4: Verification of the scalar sum rule in Eq. (32) for elastic
scattering of spin-2 KK mode number $n=1$ (red), $n=5$ (blue) and $n=11$
(green). Here $n$ represents the KK mode number of the spin-2 particles in the
external legs. On the $x$-axis we show the number of GW-scalar modes included
in the sum ($i$) of the left hand side of the sum rules in Eq. (32) versus the
log of the relative error
$(\log_{10}(1-\Delta(\text{LHS})/\Delta(\text{RHS})))$ between the ${\cal
O}(\epsilon^{2})$ contributions to both sides of the relevant equation. This
has been evaluated in the large $\tilde{k}r_{c}$ limit.

Finally, we examine the sum-rule in Eq. (32) for which we have no analytic
proof, and which depends on the couplings of the individual spin-0 states to
the massive spin-2 KK modes. The result of this exercise is shown in Fig. 4,
in the case $n=1,\,5,\,11$ (e.g. for scalar-exchange contributions to elastic
scattering of spin-2 modes at KK level 1, 5, and 11) and for
$\tilde{k}r_{c}=12.73$. Again, what is plotted here is the agreement between
the ${\cal O}(\epsilon^{2})$ contributions to the left- and right-hand sides
of Eq. (32) – which, at this order, depends explicitly on the forms of the
scalar wavefunctions that solve Eq. (14) and which are subject to the
normalization conditions of Eq. (17).

The non-zero difference between the left- and right-hand sides of the ${\cal
O}(\epsilon^{2})$ corrections to Eq. (32) at $i=1$ demonstrate the need to
include the tower of Goldberger-Wise scalar states for the spin-2 scattering
amplitudes to have the proper high-energy behavior. Again we see that the
largest single contribution to the ${\cal O}(\epsilon^{2})$ corrections come
from the exchange of the GW scalar state whose mode-number is twice that of
the incoming particles – $i=2,10$ and 20, respectively, for incoming mode 1,
5, and 10 spin-2 states. However, the continued convergence when adding
additional states is also clear and, formally, the entire tower is needed for
the sum-rule to be satisfied.

##  VI Conclusion

In this paper we have presented a thorough analysis of the scattering of
massive spin-2 Kaluza-Klein excitations in phenomenologically realistic models
based on a warped geometry Randall:1999ee ; Randall:1999vf stabilized via the
Goldberger-Wise Goldberger:1999uk ; Goldberger:1999un mechanism. These results
significantly extend the work presented in Chivukula:2019rij ;
Chivukula:2019zkt ; Chivukula:2020hvi on the unstabilized RS1 model and the
results in Chivukula:2021xod on the “flat-stabilized” model
($\tilde{k}r_{c}=0$). We briefly recap our findings here:

  * •

We provided a complete and self-contained derivation of the mode expansions
for the spin-2 and spin-0 states and their interactions. Generalizing the
presentations in Kofman:2004tk ; Boos:2005dc ; Boos:2012zz , our computations
are given in de-Donder gauge for massless gravitons – allowing us to
consistently compute scattering amplitudes involving intermediate off-shell
states.

  * •

In previous work Chivukula:2021xod we had demonstrated the extended sum-rule
relationships between spin-2 and spin-0 modes, and their masses and couplings,
which must be satisfied in order for elastic massive spin-2 KK scattering to
grow no faster than ${\cal O}(s)$. Here, we have provided an analytic proof
for one combination of these sum rules, and showed its relation both to the
Einstein and scalar background field equations implementing the Goldberger-
Wise dynamics and also to the properties of the mode-equations for the
physical scalar fields (fields which are admixtures of bulk scalar and
gravitational modes in the original theory).

  * •

We have provided, in an appendix to this work, a complete list of the sum-rule
relations which must be satisfied if all $2\to 2$ massive spin-2 scattering
amplitudes, elastic or inelastic, are to grow no faster than ${\cal O}(s)$ –
completing the analyses begun in Chivukula:2019rij ; Chivukula:2019zkt ;
Chivukula:2020hvi ; Foren:2020egq .

  * •

Finally, using a version of the DFGK model DeWolfe:1999cp in which the
Goldberger-Wise dynamics can be treated perturbatively Chivukula:2021xod , we
have checked numerically that the sum rules which enforce the proper high-
energy behavior of massive spin-2 scattering continue to be satisfied in the
case of the large warping that would be required to produce the hierarchy
between the weak and Planck scales. These numerical computations demonstrate
that, in models with a massive radion, proper cancellation is achieved only
after including the contributions from the tower of scalar states present in
the Goldberger-Wise model.

In subsequent work we will take up the issues of providing an analytic proof
for the one remaining elastic sum-rule relationship for which we presently
have only numerical confirmation (Eq. (32)) Xing . In future work we will also
explore the phenomenological consequences of the fact that all spin-2
scattering amplitudes in models of compactified gravity can grow no faster
than ${\cal O}(s)$; specifically, we will study the implications for the
computation of relic abundances of dark matter particles in KK graviton-portal
theories and related theories.

Acknowledgements: This material is based upon work supported by the National
Science Foundation under Grant No. PHY- 1915147 .The work of DS for this paper
was supported in part by the University of Adelaide and the Australian
Research Council through the Centre of Excellence for Dark Matter Particle
Physics (CE200100008).

##  Appendix A The Derivation of the Scalar Kaluza-Klein Modes

The canonical quadratic Lagrangian for the Kaluza-Klein modes in a Goldberger-
Wise model has been derived in Refs. Kofman:2004tk ; Boos:2005dc ; Boos:2012zz
. Here, for completeness and as a guide to the interested reader, we present
our derivation of the canonical quadratic Lagrangian. In addition, since our
computation includes diagrams with off-shell gravitons, we are careful to
derive our results in de Donder gauge.171717If one is only concerned with
external gravitons, and not doing scattering computations, one can impose the
transverse-traceless conditions on all of the fields to simplify the
computations of the interactions. We include all the necessary details needed
in order to reproduce our results: deriving the background equations of motion
for a Goldberger-Wise-stabilized Randall-Sundrum I model, showing how the 5D
scalar field $\hat{r}$ and 5D tensor field $\hat{h}$ decouple, motivating the
gauge condition relating the fluctuation field $\hat{f}$ and the 5D scalar
field $\hat{r}$, describing how the linear equations of motion inspire the
Kaluza-Klein decomposition of the 5D scalar field $\hat{r}$, and demonstrating
that the 5D scalar field $\hat{r}(x,y)$ generates a tower of canonical
spin-$0$ fields $\hat{r}^{(i)}(x)$ with masses $m_{(i)}\equiv\mu_{(i)}/r_{c}$.

###  A.1 The Lagrangian

The Goldberger-Wise-stabilized Randall-Sundrum I Lagrangian is constructed
from several elements, including the spacetime metric. Focusing our attention
on the spin-2 ($\hat{h}_{\mu\nu}$) and scalar ($\hat{r}$) fluctuations about a
geometry determined by the warp-factor $A(y)$, we use the following
parameterization of the metric $G_{MN}$ and its 4D projection
$\overline{G}_{MN}$:

$\displaystyle[G_{MN}]=\left(\begin{tabular}[]{ c c
}${w\,g_{\mu\nu}}$&${0}$\\\ ${0}$&${-v^{2}}$\end{tabular}\right)\hskip
35.0pt[\overline{G}_{\mu\nu}]=w\,g_{\mu\nu}$ (60)

where (taking our parameterization from Charmousis:1999rg ),

$\displaystyle w=e^{-2[A(y)+\hat{u}(x,y)]}~{},\hskip 35.0pt$ $\displaystyle
v=1+2\hat{u}(x,y)~{},$ (61) $\displaystyle
g_{\mu\nu}=\eta_{\mu\nu}+\kappa\,\hat{h}_{\mu\nu}(x,y)~{},\hskip 35.0pt$
$\displaystyle\hat{u}=\dfrac{e^{2A(y)}}{2\sqrt{6}}\kappa\,\hat{r}(x,y)~{}.$
(62)

The compact extra dimension is parameterized on a circle of radius $r_{c}$
such that $y\in(-\pi r_{c},+\pi r_{c}]$ and the 5D coordinates
$X^{M}=(x^{\mu},y)$ define a 4D Minkowski spacetime “slice” at each fixed
value of $y$. We impose an orbifold invariance $y\leftrightarrow-y$ on the
infinitesimal spacetime interval $G_{MN}dX^{M}dX^{N}$, and identify $y=0$ and
$y=\pi r_{c}$ as orbifold fixed points. For our spacetime signatures, we use
the mostly-minus convention, i.e.
$\eta_{\mu\nu}\equiv\text{Diag}(+1,-1,-1,-1)$. Lowercase Greek letters (e.g.
$\mu$, $\nu$, $\rho$, $\sigma$) denote 4D indices, uppercase Latin letters
(e.g. $M$, $N$, $R$, $S$) denote 5D indices, and lowercase Latin letters (e.g.
$m$, $n$, $i$, $j$) denote Kaluza-Klein mode numbers.

We then define the Lagrangian as

$\displaystyle\mathcal{L}_{\text{5D}}\equiv\mathcal{L}_{\text{EH}}+\mathcal{L}_{\Phi\Phi}+\mathcal{L}_{\text{pot}}+\mathcal{L}_{\text{GHY}}+\Delta\mathcal{L}~{},$
(63)

where the Einstein-Hilbert (EH), Gibbons-Hawking-York (GHY), scalar kinetic
terms ($\Phi\Phi$), and scalar potential terms (pot) are defined as

$\displaystyle\mathcal{L}_{\text{EH}}$
$\displaystyle\equiv-\dfrac{2}{\kappa^{2}}\,\sqrt{G}\,R\ ,$ (64)
$\displaystyle\mathcal{L}_{\text{GHY}}$
$\displaystyle\equiv-\dfrac{4}{\kappa^{2}}\,\partial_{y}\Big{[}\sqrt{\overline{G}}\,K\Big{]}=\dfrac{2}{\kappa^{2}}\partial_{y}\left[\dfrac{w^{2}}{v}\sqrt{-\text{det}g}\left(\tilde{g}^{\mu\nu}\partial_{y}(g_{\mu\nu})+4\frac{\partial_{y}(w)}{w}\right)\right]\
,$ (65) $\displaystyle\mathcal{L}_{\Phi\Phi}$
$\displaystyle\equiv\sqrt{G}\bigg{[}\dfrac{1}{2}\tilde{G}^{MN}(\partial_{M}\hat{\Phi})(\partial_{N}\hat{\Phi})\bigg{]}\
,$ (66) $\displaystyle\mathcal{L}_{\text{pot}}$
$\displaystyle\equiv-\dfrac{4}{\kappa^{2}}\bigg{[}\sqrt{G}\,V[\hat{\Phi}]+\sum_{i=1,2}\sqrt{\overline{G}}\,V_{i}[\hat{\Phi}]\,\delta_{i}(\varphi)\bigg{]}\
.$ (67)

respectively. Here $R$ is the Ricci scalar and $K$ is the extrinsic curvature
at the boundaries. $G$ and $\bar{G}$ are the determinants of metric and the
induced metric respectively. The Dirac deltas on the branes are defined in the
limit as we approach the branes from within the $y\in[0,\pi r_{c}]$ half of
the bulk: $\delta_{1}(\varphi)\equiv\delta(\varphi-0^{+})$ and
$\delta_{2}(\varphi)\equiv\delta(\varphi-\pi^{-})$, where $\varphi=y/r_{c}$.

Note that no dynamics are given to describe the physics of the branes
themselves, which are assumed to arise through unspecified dynamics at some
higher scale (e.g. string physics) – and, in particular, no modes arising from
that physics (fluctuations of the branes themselves) are included. More on
this below, when we discuss the effect of taking the so-called “stiff-wall”
limit on the scalar mode-expansions.

Meanwhile, the contribution $\Delta\mathcal{L}$ in $\mathcal{L}_{\text{5D}}$
is a total derivative we add for convenience which generalizes a total
derivative from our unstabilized analysis Chivukula:2020hvi ; Foren:2020egq .
It cancels terms in the Lagrangian at linear order, eliminates mixing between
tensor and scalar 5D fields at quadratic order, and simplifies the vertices
relevant to this paper.181818If such a term were not introduced here, we would
recover its effects (at least to quadratic order) as additional total
derivative terms needed to make the Lagrangian explicitly canonical. This
$\Delta\mathcal{L}$ naively generalized the $\Delta\mathcal{L}$ introduced in
refs. Chivukula:2020hvi ; Foren:2020egq , and (unlike the latter) does not
eliminate Dirac deltas or twice-differentiated quantities to all orders.
Explicitly, we define it as

$\displaystyle\Delta\mathcal{L}\equiv\dfrac{2}{\kappa^{2}}\partial_{y}\left[\dfrac{w^{2}}{v}\sqrt{-\text{det}g}\left(-3\frac{\partial_{y}(w)}{w}\right)\right]~{}.$
(68)

The perturbative expansion of the gravitational contributions as series in
$\kappa$ proceeds as usual. For convenience, we rewrite the bulk scalar field
$\hat{\Phi}(x,y)$ such that it is perturbed about the background
$\phi_{0}(\varphi)/\kappa$ by an amount $\hat{f}/\kappa$, i.e.
$\hat{\Phi}\equiv\hat{\phi}/\kappa\equiv(\phi_{0}+\hat{f})/\kappa$. This
rewrite allows us to expand the bulk and brane potentials in
$\mathcal{L}_{\text{pot}}$ about $\hat{\phi}=\phi_{0}$ with respect to the
dimensionless scalar fluctuation field $\hat{f}$ like so:

$\displaystyle V[\hat{\Phi}]$
$\displaystyle=V+\dot{V}\,\hat{f}+\dfrac{1}{2}\ddot{V}\,(\hat{f})^{2}+\mathcal{O}\Big{(}(\hat{f})^{3}\Big{)}~{},$
(69) $\displaystyle V_{i}[\hat{\Phi}]$
$\displaystyle=V_{i}+\dot{V}_{i}\,\hat{f}+\dfrac{1}{2}\ddot{V}_{i}\,(\hat{f})^{2}+\mathcal{O}\Big{(}(\hat{f})^{3}\Big{)}~{},$
(70)

where dots denote $\hat{\phi}$ functional derivatives, and $V$ and $V_{i}$
(and their $\hat{\phi}$ derivatives) are set to
$\hat{\Phi}=\phi_{0}(y)/\kappa$ when their functional arguments are
unspecified.

The path forward is, in principle, clear: find the appropriate solutions for
the background fields $A(y)$ and $\phi_{0}(y)$, compute the Lagrangian that
describes the dynamics of fluctuations about these background fields, and then
diagonalize the quadratic terms in this Lagrangian to establish the
(canonically normalized) physical modes and their interactions. In practice
this is difficult because of the complicated algebraic structures involved in
the mixing between the scalar components of the metric and Goldberger-Wise
scalar field in the presence of a non-trivial scalar background. Some
simplification results from the fact that diffeomorphism invariance implies
that only one linear combination of the fields $\hat{r}$ and $\hat{f}$ are
physical, and we can set the gauge of the calculation such that these fields
are related according to:

$\displaystyle\phi^{\prime}_{0}\,\hat{f}=\sqrt{6}\,e^{2A}\,\hat{r}^{\,\prime}~{}.$
(71)

We begin our analysis in general, without imposing this gauge condition and
only use the gauge constraint Eq. (71) to identify the physical scalar modes
after deriving the background equations, which we turn to now.

###  A.2 The Lagrangian to Quadratic Order

After weak field expanding the Lagrangian Eq. (63) without applying the gauge
condition Eq. (71) or any background field equations, we obtain

$\displaystyle\mathcal{L}_{\text{5D}}$
$\displaystyle\equiv\mathcal{L}_{\text{5D},\text{bkgd}}+\mathcal{L}_{\text{5D},h}+\mathcal{L}^{*}_{\text{5D},r}+\mathcal{L}^{*}_{\text{5D},f}+\mathcal{L}^{*}_{\text{5D},hr}+\mathcal{L}^{*}_{\text{5D},hf}+\mathcal{L}^{*}_{\text{5D},rf}$
$\displaystyle\hskip
10.0pt+\mathcal{L}_{\text{5D},hh}+\mathcal{L}^{*}_{\text{5D},rr}+\mathcal{L}^{*}_{\text{5D},ff}+\mathcal{O}(\kappa)~{},$
(72)

to all orders in the background fields and up to quadratic order in the
fluctuations. The background-only terms in the Lagrangian are

$\displaystyle\mathcal{L}_{\text{5D},\text{bkgd}}$
$\displaystyle\equiv\dfrac{e^{-4A}}{2\,r_{c}^{2}\kappa^{2}}\bigg{[}24A^{\prime\prime}-48(A^{\prime})^{2}-(\phi_{0}^{\prime})^{2}-8\bigg{(}Vr_{c}^{2}+\sum_{i=1,2}V_{i}r_{c}\,\delta_{i}\bigg{)}\bigg{]}~{}.$
(73)

The terms linear in the fluctuations are

$\displaystyle\mathcal{L}_{\text{5D},h}$
$\displaystyle\equiv\dfrac{e^{-4A}}{4\,r_{c}^{2}\kappa}\bigg{[}24A^{\prime\prime}-48(A^{\prime})^{2}-(\phi_{0}^{\prime})^{2}-8\bigg{(}Vr_{c}^{2}+\sum_{i=1,2}V_{i}r_{c}\,\delta_{i}\bigg{)}\bigg{]}\hat{h}~{},$
(74) $\displaystyle\mathcal{L}_{\text{5D},r}^{*}$
$\displaystyle\equiv\dfrac{\sqrt{6}\,e^{-2A}}{r_{c}^{2}\kappa}\bigg{[}\hat{r}^{\,\prime\prime}-2A^{\prime}\,\hat{r}^{\,\prime}\bigg{]}$
$\displaystyle\hskip
10.0pt+\dfrac{e^{-2A}}{2\sqrt{6}\,r_{c}^{2}\kappa}\bigg{[}-48A^{\prime\prime}+48(A^{\prime})^{2}+3(\phi_{0}^{\prime})^{2}+8\bigg{(}Vr_{c}^{2}+2\sum_{i=1,2}V_{i}r_{c}\,\delta_{i}\bigg{)}\bigg{]}\hat{r}~{},$
(75) $\displaystyle\mathcal{L}_{\text{5D},f}^{*}$
$\displaystyle\equiv-\dfrac{e^{-4A}}{r_{c}^{2}\kappa}\phi_{0}^{\prime}\,\hat{f}^{\,\prime}-\dfrac{4\,e^{-4A}}{r_{c}^{2}\kappa}\bigg{[}\dot{V}r_{c}^{2}+\sum_{i=1,2}\dot{V}_{i}r_{c}\,\delta_{i}\bigg{]}\hat{f}~{},$
(76)

where $\hat{h}=\eta^{\mu\nu}\hat{h}_{\mu\nu}$. At quadratic order in the
fluctuations, “off-diagonal” (mode-mixing) quadratic terms are

$\displaystyle\mathcal{L}_{\text{5D},hr}^{*}$
$\displaystyle\equiv\dfrac{\kappa}{2}\,\mathcal{L}_{\text{5D},r}^{*}\,\hat{h}~{},$
(77) $\displaystyle\mathcal{L}_{\text{5D},hf}^{*}$
$\displaystyle\equiv\dfrac{\kappa}{2}\,\mathcal{L}_{\text{5D},f}^{*}\,\hat{h}~{},$
(78) $\displaystyle\mathcal{L}_{\text{5D},rf}^{*}$
$\displaystyle\equiv\sqrt{\dfrac{3}{2}}\dfrac{e^{-2A}}{r_{c}^{2}}\phi_{0}^{\prime}\,\hat{r}\,\hat{f}^{\,\prime}+\sqrt{\dfrac{8}{3}}\dfrac{e^{-2A}}{r_{c}^{2}}\bigg{[}\dot{V}r_{c}^{2}+2\sum_{i=1,2}\dot{V}_{i}r_{c}\,\delta_{i}\bigg{]}\hat{r}\,\hat{f}~{},$
(79)

and the “on-diagonal” quadratic terms are given by

$\displaystyle\mathcal{L}_{\text{5D},hh}$ $\displaystyle\equiv
e^{-2A}\bigg{[}(\partial^{\nu}\hat{h}_{\mu\nu})\,(\partial^{\mu}\hat{h})-(\partial^{\nu}\hat{h}_{\mu\nu})^{2}+\dfrac{1}{2}(\partial_{\mu}\hat{h}_{\nu\rho})^{2}-\dfrac{1}{2}(\partial_{\mu}\hat{h})^{2}\bigg{]}+\dfrac{e^{-4A}}{2\,r_{c}^{2}}\bigg{[}(\hat{h}^{\prime})^{2}-(\hat{h}^{\prime}_{\mu\nu})^{2}\bigg{]}$
$\displaystyle\hskip
10.0pt+\dfrac{e^{-4A}}{16\,r_{c}^{2}}\bigg{[}24A^{\prime\prime}-48(A^{\prime})^{2}-(\phi_{0}^{\prime})^{2}-8\bigg{(}Vr_{c}^{2}+\sum_{i=1,2}V_{i}r_{c}\,\delta_{i}\bigg{)}\bigg{]}\bigg{[}(\hat{h})^{2}-2(\hat{h}_{\mu\nu})^{2}\bigg{]}~{},$
(80) $\displaystyle\mathcal{L}_{\text{5D},rr}^{*}$
$\displaystyle\equiv\dfrac{1}{2}\,e^{+2A}\,(\partial_{\mu}\hat{r})^{2}-\dfrac{1}{r_{c}^{2}}\bigg{[}2(\hat{r}^{\,\prime})^{2}+3\,\hat{r}\,\hat{r}^{\,\prime\prime}\bigg{]}$
$\displaystyle\hskip
10.0pt-\dfrac{1}{12\,r_{c}^{2}}\bigg{[}-48A^{\prime\prime}+5(\phi_{0}^{\prime})^{2}+16\sum_{i=1,2}V_{i}r_{c}\,\delta_{i}\bigg{]}(\hat{r})^{2}~{},$
(81) $\displaystyle\mathcal{L}_{\text{5D},ff}^{*}$
$\displaystyle\equiv\dfrac{1}{2}\,e^{-2A}\,(\partial_{\mu}\hat{f})^{2}-\dfrac{e^{-4A}}{2\,r_{c}^{2}}(\hat{f}^{\,\prime})^{2}-\dfrac{2e^{-4A}}{r_{c}^{2}}\bigg{[}\ddot{V}r_{c}^{2}+\sum_{i=1,2}\ddot{V}_{i}r_{c}\,\delta_{i}\bigg{]}(\hat{f})^{2}~{}.$
(82)

Here we use an asterisk to denote that we have not yet applied a gauge
condition relating $\hat{r}$ and $\hat{f}$.

The first line of terms in Eq. (80) will yield the usual canonical spin-2
Lagrangians after Kaluza-Klein decomposition. As we will soon demonstrate, the
other terms in Eq. (80) will be cancelled when the background fields satisfy
their equations of motion. However, as close as $\mathcal{L}_{\text{5D},hh}$
is to the desired spin-2 result, the quadratic analysis overall is complicated
by the presence of the mixing terms $\mathcal{L}^{*}_{\text{5D},hr}$ and
$\mathcal{L}^{*}_{\text{5D},hf}$, which seemingly imply kinetic mixing between
the tensor field $\hat{h}$ and the scalar fields $\hat{f}$ and $\hat{r}$. In
order to eliminate these mixing terms, we must derive the equations of motion
for the background fields and for the fluctuations, which we discuss next.

###  A.3 Equations of Motion

The Einstein field equations derived from $\mathcal{L}_{\text{5D}}$ equal

$\displaystyle\mathcal{G}_{MN}-V[\hat{\Phi}]G_{MN}-\bigg{[}V_{1}[\hat{\Phi}]\dfrac{\delta_{1}(\varphi)}{r_{c}}+V_{2}[\hat{\Phi}]\dfrac{\delta_{2}(\varphi)}{r_{c}}\bigg{]}\dfrac{\sqrt{\overline{G}}}{\sqrt{G}}\,\overline{G}_{MN}=\dfrac{\kappa^{2}}{4}T_{MN}~{},$
(83)

where $\mathcal{G}_{MN}=R_{MN}-\tfrac{1}{2}G_{MN}\,R$ is the Einstein tensor,
$R_{MN}$ and $R=\tilde{G}^{AB}R_{AB}$ are the Ricci tensor and Ricci scalar
respectively, and the stress-energy tensor equals

$\displaystyle T_{MN}$
$\displaystyle=2\dfrac{\delta\mathcal{L}_{\Phi\Phi}}{\delta\tilde{G}^{MN}}-G_{MN}\mathcal{L}_{\Phi\Phi}~{},$
$\displaystyle=(\partial_{M}\hat{\Phi})(\partial_{N}\hat{\Phi})-G_{MN}\bigg{[}\dfrac{1}{2}\tilde{G}^{AB}(\partial_{A}\hat{\Phi})\,(\partial_{B}\hat{\Phi})\bigg{]}~{}.$
(84)

Recall that
$\hat{\Phi}\equiv\hat{\phi}/\kappa\equiv(\phi_{0}+\hat{f})/\kappa$. We will
discuss the Einstein field equations in terms of their decomposition as
$(M,N)=\left\\{(\mu,\nu),(\mu,5),(5,5)\right\\}$, to the first two orders in
$\kappa$.

####  A.3.1 Background Equations of Motion

To lowest order in $\kappa$, in which no fluctuation fields are present, only
the $(\mu,\nu)$ and $(5,5)$ Einstein field equations are nontrivial (because
of the Lorentz-invariance of the constant-$y$ subspaces, the $(\mu,\nu)$
components of the curvature are proportional to $\eta_{\mu\nu}$), and they
imply

$\displaystyle A^{\prime\prime}$
$\displaystyle=2(A^{\prime})^{2}+\dfrac{1}{24}(\phi_{0}^{\prime})^{2}+\dfrac{1}{3}\bigg{[}Vr_{c}^{2}+\sum_{i=1,2}V_{i}r_{c}\,\delta_{i}\bigg{]}~{},$
(85) $\displaystyle Vr_{c}^{2}$
$\displaystyle=-6(A^{\prime})^{2}+\dfrac{1}{8}(\phi_{0}^{\prime})^{2}~{},$

respectively for the background fields. The first of these equations implies
the boundary conditions (integrating over the endpoints using an $S^{1}/Z_{2}$
orbifold construction where $A(y)$ is assumed to be “even” under orbifold
reflection):

$\displaystyle V_{1}r_{c}\,\delta_{1}=+6A^{\prime}\,\delta_{1}\,~{},\hskip
35.0ptV_{2}r_{c}\,\delta_{2}=-6A^{\prime}\,\delta_{2}~{}.$ (86)

By combining the equations of (85), we may also write

$\displaystyle
A^{\prime\prime}=\dfrac{1}{12}\bigg{[}(\phi_{0}^{\prime})^{2}+4\sum_{i=1,2}V_{i}r_{c}\,\delta_{i}\bigg{]}$
((12) revisited)

Note that Eq. (85) ensures $\mathcal{L}_{\text{5D},\text{bkgd}}$ and
$\mathcal{L}_{\text{5D},h}$ from Eqs. (73)-(74) vanish, and ensures
$\mathcal{L}_{\text{5D},hh}$ yields canonical spin-2 Lagrangians after Kaluza-
Klein decomposition. Eq. (85) also simplifies the various pieces of the
Lagrangian, including the linear $\hat{r}$ terms:

$\displaystyle\mathcal{L}_{\text{5D},r}^{*}$
$\displaystyle=\dfrac{\sqrt{6}}{r_{c}^{2}\kappa}\partial_{\varphi}\bigg{[}e^{-2A}\hat{r}^{\,\prime}\bigg{]}~{}.$
(87)

While the mixing terms $\mathcal{L}^{*}_{\text{5D},hr}$ and
$\mathcal{L}^{*}_{\text{5D},hf}$ remain at this point, these will vanish once
we have analyzed the scalar sector, which we discuss now.

We obtain another background equation by considering the Euler-Lagrange
equation of the scalar field. The terms independent of the fluctuations yield

$\displaystyle\phi_{0}^{\prime\prime}$
$\displaystyle=4A^{\prime}\phi_{0}^{\prime}+4\dot{V}r_{c}^{2}+4\sum_{i=1,2}\dot{V}_{i}r_{c}\,\delta_{i}~{},$
(88)

which implies its own boundary conditions (again, assuming the background
scalar field configuration is even under the orbifold projection)

$\displaystyle\dot{V}_{1}r_{c}\,\delta_{1}=+\dfrac{1}{2}\phi_{0}^{\prime}\,\delta_{1}~{},\hskip
35.0pt\dot{V}_{2}r_{c}\,\delta_{2}=-\dfrac{1}{2}\phi_{0}^{\prime}\,\delta_{2}~{}.$
(89)

This simplifies $\mathcal{L}_{\text{5D},f}^{*}$, such that

$\displaystyle\mathcal{L}_{\text{5D},f}^{*}=-\dfrac{1}{r_{c}^{2}\kappa}\partial_{\varphi}\bigg{[}e^{-4A}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}~{}.$
(90)

This completes our derivation of background equations.

Recall that whenever we write a quantity multiplying $\delta_{1}(\varphi)$ or
$\delta_{2}(\varphi)$, it is understood that the quantity is evaluated in the
limit as $\varphi$ approaches the appropriate orbifold fixed point from inside
the $[0,\pi r_{c}]$ half of the bulk. This implies, for example, via Eq. (88),

$\displaystyle\phi_{0}^{\prime\prime}\,\delta_{i}=(\phi_{0}^{\prime\prime})_{\text{bulk}}\,\delta_{i}\equiv\bigg{[}4A^{\prime}\phi_{0}^{\prime}+4\,\dot{V}r_{c}^{2}\bigg{]}\delta_{i}~{}.$
(91)

This also ensures quantities like $A^{\prime}(\varphi)\,\delta_{i}(\varphi)$
in Eq. (86) and Eq. (91) are written unambiguously, despite
$A^{\prime}(\varphi)$ being orbifold odd across the orbifold fixed points.

####  A.3.2 Lagrangian at Quadratic Order: Mode Equations

Next, we examine the equations of motion derived from considering terms in the
Lagrangian that are quadratic or lower in the fluctuations. These will give us
the equations which will define the mode expansions – the Kaluza-Klein
decomposition – for the fluctuating fields. As mentioned in the previous
subsubsection, we will be ignoring the spin-2 fields – they will ultimately
decouple from the scalar fields after having performed the correct scalar-
field mode expansions.

We begin with the scalar fields in the metric. Simplifying the expressions
using the background equations Eqs. (85)-(88), the $(\mu,\nu)$, $(\mu,5)$, and
$(5,5)$ Einstein field equations at $\mathcal{O}(\kappa)$ are satisfied only
if, respectively,

$\displaystyle 0$
$\displaystyle=\bigg{[}\partial_{\varphi}-4A^{\prime}\bigg{]}\bigg{[}\sqrt{6}\,e^{2A}\,\hat{r}^{\,\prime}-\phi_{0}^{\prime}\,\hat{f}\bigg{]}~{},$
(92) $\displaystyle 0$
$\displaystyle=\partial_{\mu}\bigg{[}\sqrt{6}\,e^{2A}\,\hat{r}^{\,\prime}-\phi_{0}^{\prime}\,\hat{f}\bigg{]}~{},$
(93)
$\displaystyle\partial_{\varphi}\bigg{[}\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}$
$\displaystyle=e^{2A}\,r_{c}^{2}\,(\square\hat{r})+2A^{\prime}\bigg{\\{}2\,\hat{r}^{\,\prime}+\bigg{[}\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}\bigg{\\}}$
$\displaystyle\hskip
15.0pt+\dfrac{8\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}\bigg{[}\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}+\dfrac{1}{6}(\phi_{0}^{\prime})^{2}\,\hat{r}+2\,(\delta_{1}-\delta_{2})\bigg{[}\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}~{},$
(94)

where the final equation has also utilized the jump conditions of Eq. (89). As
noted in Ref. Csaki:2000zn , integrating Eq. (94), we end up with a tautology
and end up with boundary terms that provide no additional physical
information. By moving the Dirac deltas of Eq. (94) to the LHS and evaluating
the derivatives, we derive an alternative form of the equation which lacks
Dirac deltas (implicit or explicit):

$\displaystyle\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}^{\,\prime}=e^{2A}\,r_{c}^{2}\,(\square\hat{r})+4A^{\prime}\,\hat{r}^{\,\prime}+\dfrac{1}{6}(\phi_{0}^{\prime})^{2}\,\hat{r}+\sqrt{\dfrac{8}{3}}\,e^{-2A}\,\dot{V}r_{c}^{2}\,\hat{f}~{}.$
(95)

We may also consider the Euler-Lagrangian equation of the fluctuation field
$\hat{f}$ at this order, which yields

$\displaystyle\hat{f}^{\,\prime\prime}$
$\displaystyle=e^{2A}\,r_{c}^{2}\,(\square\hat{f})+4A^{\prime}\,\hat{f}^{\prime}+4\ddot{V}r_{c}^{2}\,\hat{f}+\sqrt{\dfrac{3}{2}}\,e^{2A}\,\phi_{0}^{\prime}\,\hat{r}^{\prime}+\sqrt{\dfrac{2}{3}}\,e^{2A}\,\bigg{[}4\dot{V}r_{c}^{2}+3A^{\prime}\phi_{0}^{\prime}\bigg{]}\hat{r}$
$\displaystyle\hskip
15.0pt+\bigg{[}4\ddot{V}_{1}r_{c}\,\hat{f}+\sqrt{\dfrac{2}{3}}\,e^{2A}\,\phi_{0}^{\prime}\,\hat{r}\bigg{]}\delta_{1}+\bigg{[}4\ddot{V}_{2}r_{c}\,\hat{f}-\sqrt{\dfrac{2}{3}}\,e^{2A}\,\phi_{0}^{\prime}\,\hat{r}\bigg{]}\delta_{2}~{}.$
(96)

This equation implies, via the orbifold construction, the boundary conditions:

$\displaystyle\hat{f}^{\,\prime}\delta_{1}=\bigg{[}2\ddot{V}_{1}r_{c}\,\hat{f}+\dfrac{1}{\sqrt{6}}\,e^{2A}\,\phi_{0}^{\prime}\,\hat{r}\bigg{]}\delta_{1}~{},\hskip
35.0pt\hat{f}^{\,\prime}\delta_{2}=-\bigg{[}2\ddot{V}_{2}r_{c}\,\hat{f}-\dfrac{1}{\sqrt{6}}\,e^{2A}\,\phi_{0}^{\prime}\,\hat{r}\bigg{]}\delta_{2}~{}.$
(97)

Multiply these jump conditions by $e^{-2A}\phi_{0}^{\prime}/\sqrt{6}$ and use
Eq. (95) to get

$\displaystyle\bigg{\\{}e^{2A}\,r_{c}^{2}\,(\square\hat{r})+4A^{\prime}\,\hat{r}^{\,\prime}+\bigg{[}\dfrac{4\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}-2\ddot{V}_{1}r_{c}\bigg{]}\bigg{[}\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}\bigg{\\}}\delta_{1}$
$\displaystyle=0~{},$ (98)
$\displaystyle\bigg{\\{}e^{2A}\,r_{c}^{2}\,(\square\hat{r})+4A^{\prime}\,\hat{r}^{\,\prime}+\bigg{[}\dfrac{4\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}+2\ddot{V}_{2}r_{c}\bigg{]}\bigg{[}\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}\bigg{\\}}\delta_{2}$
$\displaystyle=0~{}.$ (99)

Using Eq. (91), we may instead write the jump conditions Eq. (97) as

$\displaystyle\bigg{\\{}e^{2A}\,r_{c}^{2}\,(\square\hat{r})+4A^{\prime}\,\hat{r}^{\,\prime}+\bigg{[}\dfrac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}-4A^{\prime}-2\ddot{V}_{1}r_{c}\bigg{]}\bigg{[}\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}\bigg{\\}}\delta_{1}$
$\displaystyle=0~{},$
$\displaystyle\bigg{\\{}e^{2A}\,r_{c}^{2}\,(\square\hat{r})+4A^{\prime}\,\hat{r}^{\,\prime}+\bigg{[}\dfrac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}-4A^{\prime}+2\ddot{V}_{2}r_{c}\bigg{]}\bigg{[}\dfrac{e^{-2A}}{\sqrt{6}}\,\phi_{0}^{\prime}\,\hat{f}\bigg{]}\bigg{\\}}\delta_{2}$
$\displaystyle=0~{}.$ (100)

This form is more common in the literature.

The linear field equations Eqs. (92)-(94) and Eq. (96) describe the scalar
modes of the theory. Note, in particular, the recurring quantity
$\sqrt{6}\,e^{2A}\,\hat{r}^{\,\prime}-\phi_{0}^{\prime}\,\hat{f}$. This will
vanish once we impose the gauge condition (71), which is our next focus.

####  A.3.3 The Gauge Condition

The form of the metric specified by (60) - (62) does not completely fix the
“gauge” for this calculation: we have access to various five-dimensional
diffeomorphism transformations which maintain the form of the metric and with
which we can choose to simplify our computations. In particular, as shown in
Boos:2005dc , we can always perform a change of coordinate to impose the gauge
condition introduced previously

$\displaystyle\sqrt{6}\,e^{2A}\,\hat{r}^{\,\prime}=\phi_{0}^{\prime}\,\hat{f}~{}.$
(71)

One immediate consequence of this gauge choice is that the sum of the mixing
terms $\mathcal{L}^{*}_{\text{5D},hr}$ $\mathcal{L}^{*}_{\text{5D},hf}$
vanishes, eliminating (as promised) any problematic mixing between the scalar
and spin-2 mode sectors.

The physical implication of the gauge condition (71) is that one combination
of the scalar fields is a gauge-artifact, and does not correspond to a
propagating degree of freedom.191919The precise combination of Lagrangian
fields which is physical and the corresponding form of its interactions depend
on the gauge choice – although all physical amplitudes are gauge-invariant.
Note that the “mixing” of the scalar degree of freedom in the five-dimensional
metric $\hat{r}$ with the bulk scalar field $\hat{f}$ only occurs in the
presence of a $y$-dependent scalar background field configuration
($\phi^{\prime}_{0}\neq 0$). It is precisely this mixing between the two
sectors that enables the dynamics which stabilize the size of the extra-
dimension in the Goldberger-Wise mechanism Goldberger:1999uk ;
Goldberger:1999un and which simultaneously give rise to a “radion” mass. One
advantage of working in this “unitary” gauge and eliminating the fluctuations
of the scalar field $\hat{f}$ in favor of scalar fluctuations of the metric
$\hat{r}$ is that all couplings linear in the physical scalar fields have the
same algebraic form as couplings linear in the (massless) radion within the
unstabilized model – simplifying the required coupling computations.

Having imposed this gauge condition, the $(\mu,\nu)$ and $(\mu,5)$ linear
Einstein field equations Eqs. (92)-(93) vanish, the $(5,5)$ linear Einstein
field equation Eq. (94) simplifies to

$\displaystyle\hat{r}^{\,\prime\prime}$
$\displaystyle=e^{2A}\,r_{c}^{2}\,(\square\hat{r})+\bigg{[}6A^{\prime}+\dfrac{8\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}\bigg{]}\hat{r}^{\,\prime}+\dfrac{1}{6}(\phi_{0}^{\prime})^{2}\,\hat{r}+2\,(\delta_{1}-\delta_{2})\,\hat{r}^{\,\prime}~{},$
(101)

and the jump conditions Eq. (100) reduce to

$\displaystyle\bigg{\\{}e^{2A}\,r_{c}^{2}\,(\square\hat{r})-\bigg{[}2\ddot{V}_{1}r_{c}-\dfrac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\bigg{]}\hat{r}^{\,\prime}\bigg{\\}}\delta_{1}=\bigg{\\{}e^{2A}\,r_{c}^{2}\,(\square\hat{r})+\bigg{[}2\ddot{V}_{2}r_{c}+\dfrac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\bigg{]}\hat{r}^{\,\prime}\bigg{\\}}\delta_{2}=0~{}.$
(102)

These equations of motion for the field $\hat{r}$ at quadratic order will
define the Kaluza-Klein decomposition of the $\hat{r}$ field. Note that, being
careful about Dirac deltas,202020Refer to the discussion after Eq. (124) for
more details. In short, the quantity $1/(\phi_{0}^{\prime})^{2}$ cannot
generate Dirac deltas upon differentiation.

$\displaystyle\partial_{\varphi}\bigg{[}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}\hat{r}^{\,\prime}\bigg{]}$
$\displaystyle=\dfrac{2\,e^{2A}}{(\phi_{0}^{\prime})^{2}}A^{\prime}\,\hat{r}^{\,\prime}-\dfrac{2\,e^{2A}}{(\phi_{0}^{\prime})^{3}}(\phi_{0}^{\prime\prime})_{\text{bulk}}\,\hat{r}^{\,\prime}+\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}\hat{r}^{\,\prime\prime}~{},$
$\displaystyle=\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}\bigg{\\{}-\Big{[}6A^{\prime}+\dfrac{8\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}\Big{]}\hat{r}^{\,\prime}+\hat{r}^{\,\prime\prime}\bigg{\\}}~{},$
(103)

such that Eq. (101) may also be written in a more conventional form:

$\displaystyle\partial_{\varphi}\bigg{[}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}\hat{r}^{\,\prime}\bigg{]}-\dfrac{e^{2A}}{6}\hat{r}+2\Big{[}\delta_{2}(\varphi)-\delta_{1}(\varphi)\Big{]}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}\hat{r}^{\,\prime}=\dfrac{e^{4A}}{(\phi_{0}^{\prime})^{2}}r_{c}^{2}(\square\hat{r})$
(104)

In the next subsection, we use Eqs. (104) and (102) to define the Kaluza-Klein
decomposition of the 5D scalar field. (Note again that the jump conditions of
the field $\hat{r}$ are trivial in this form of the equation, and boundary
conditions in Eq. (102) are required.)

###  A.4 Kaluza-Klein Decomposition of the Scalar Field

Next, we assume we can decompose the 5D scalar field $\hat{r}(x,y)$ into a
tower of 4D fields $\hat{r}_{i}(x)$ and extra-dimensional wavefunctions
$\gamma_{i}(\varphi)$:

$\displaystyle\hat{r}(x,y)$ $\displaystyle=\dfrac{1}{\sqrt{\pi
r_{c}}}\sum_{i=0}^{+\infty}\hat{r}^{(i)}(x)\,\gamma_{i}(\varphi)~{},$ (105)

where the states are arranged in order of increasing mass and
$\varphi=y/r_{c}$. We will show that if the $\gamma_{i}$ satisfies the Sturm-
Liouville-like equation (compare to Eq. (104)):

$\displaystyle\partial_{\varphi}\bigg{[}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}(\partial_{\varphi}\gamma_{i})\bigg{]}-\dfrac{e^{2A}}{6}\gamma_{i}+2\Big{[}\delta_{2}(\varphi)-\delta_{1}(\varphi)\Big{]}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}(\partial_{\varphi}\gamma_{i})=-\mu_{(i)}^{2}\,\dfrac{e^{4A}}{(\phi_{0}^{\prime})^{2}}\,\gamma_{i}~{},$
(106)

with boundary conditions (compare to Eq. (102)):

$\displaystyle(\partial_{\varphi}\gamma_{i})\bigg{|}_{\varphi=0+}$
$\displaystyle=-\bigg{[}2\ddot{V}_{1}r_{c}-\dfrac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\bigg{]}^{-1}\,\mu_{(i)}^{2}\,e^{2A}\,\gamma_{i}\bigg{|}_{\varphi=0+}~{},$
$\displaystyle(\partial_{\varphi}\gamma_{i})\bigg{|}_{\varphi=\pi-}$
$\displaystyle=+\bigg{[}2\ddot{V}_{2}\,r_{c}+\dfrac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\bigg{]}^{-1}\,\mu_{(i)}^{2}\,e^{2A}\,\gamma_{i}\bigg{|}_{\varphi=\pi-}~{},$
(107)

the $\hat{r}^{(i)}(x)$ are the properly normalized scalar Kaluza-Klein fields.

These scalar boundary conditions can alternatively be enforced (recalling that
$\hat{r}$ and hence $\gamma_{i}$ are orbifold-even) using the equation
introduced in the body of the paper:

$\displaystyle\partial_{\varphi}\bigg{[}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}(\partial_{\varphi}\gamma_{i})\bigg{]}-\dfrac{e^{2A}}{6}\gamma_{i}=-\mu_{(i)}^{2}\,\dfrac{e^{4A}}{(\phi_{0}^{\prime})^{2}}\,\gamma_{i}\,\Bigg{\\{}1+\frac{2\,\delta(\varphi)}{\Big{[}2\ddot{V}_{1}r_{c}-\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}+\frac{2\,\delta(\varphi-\pi)}{\Big{[}2\ddot{V}_{2}r_{c}+\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}\Bigg{\\}}~{}.$
(14)

In this form, the Sturm-Liouville nature of the problem is manifest
Boos:2005dc ; Kofman:2004tk ; fulton1977two ; binding1994sturm . We will
choose to normalize the wavefunctions such that

$\displaystyle\delta_{m,n}$
$\displaystyle=\dfrac{6\mu^{2}_{n}}{\pi}\int_{-\pi}^{+\pi}d\varphi\,\gamma_{m}\gamma_{n}\dfrac{e^{4A}}{(\phi_{0}^{\prime})^{2}}\Bigg{\\{}1+\frac{2\,\delta(\varphi)}{\Big{[}2\ddot{V}_{1}r_{c}-\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}+\frac{2\,\delta(\varphi-\pi)}{\Big{[}2\ddot{V}_{2}r_{c}+\frac{\phi_{0}^{\prime\prime}}{\phi_{0}^{\prime}}\Big{]}}\Bigg{\\}}$
(108) $\displaystyle=\dfrac{6}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{[}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}\gamma_{m}^{\,\prime}\,\gamma_{n}^{\,\prime}+\dfrac{e^{2A}}{6}\gamma_{m}\gamma_{n}\bigg{]}~{},$
(109)

where the second line follows by applying the differential equation (14) and
integration by parts on the periodic doubled “orbifold”. We will show that
this normalization will yield properly normalized scalar Kaluza-Klein
modes.212121Note that this choice is consistent since we have not massless
physical scalar modes in this model. For our numerical investigations, we
consider the “stiff-wall” limit $\ddot{V}_{1,2}\to\infty$, in which case the
eigenmodes $\gamma_{i}$ satisfy Neumann boundary conditions. While the stiff-
wall limit is (ultimately) unphysical, it is consistent with the
simplification we made in ignoring the dynamics of the brane itself - and we
can expect the results of our analysis correctly describe low-energy
properties of the system. Outside of numerical investigations, we do not take
the stiff-wall limit.

To facilitate manipulations at the 5D level, define the following useful
auxillary field:

$\displaystyle\hat{z}$ $\displaystyle\equiv\dfrac{1}{\sqrt{\pi
r_{c}}}\sum_{i=0}^{+\infty}\mu_{(i)}^{2}\,\hat{r}^{(i)}(x)\,\gamma_{i}(\varphi)~{}.$
(110)

Using $\hat{z}$ and the decomposition in (105), the wavefunction differential
equation becomes

$\displaystyle\hat{r}^{\,\prime\prime}$
$\displaystyle=-e^{2A}\,\hat{z}+\bigg{[}6A^{\prime}+\dfrac{8\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}\bigg{]}\hat{r}^{\,\prime}+\dfrac{1}{6}(\phi_{0}^{\prime})^{2}\,\hat{r}+2\,(\delta_{1}-\delta_{2})\,\hat{r}^{\,\prime}~{},$
(111)

such that

$\displaystyle(\hat{r}^{\prime\prime})_{\text{bulk}}$
$\displaystyle\equiv-e^{2A}\,\hat{z}+\bigg{[}6A^{\prime}+\dfrac{8\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}\bigg{]}\hat{r}^{\,\prime}+\dfrac{1}{6}(\phi_{0}^{\prime})^{2}\,\hat{r}~{},$
(112)

and the wavefunction boundary conditions imply

$\displaystyle\bigg{\\{}-e^{2A}\,\hat{z}+\bigg{[}4A^{\prime}+\dfrac{4\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}\bigg{]}\hat{r}^{\,\prime}\bigg{\\}}\delta_{1}$
$\displaystyle=\bigg{\\{}+2\ddot{V}_{1}r_{c}\,\hat{r}^{\,\prime}\bigg{\\}}\delta_{1}~{},$
$\displaystyle\bigg{\\{}-e^{2A}\,\hat{z}+\bigg{[}4A^{\prime}+\dfrac{4\dot{V}r_{c}^{2}}{\phi_{0}^{\prime}}\bigg{]}\hat{r}^{\,\prime}\bigg{\\}}\delta_{2}$
$\displaystyle=\bigg{\\{}-2\ddot{V}_{2}r_{c}\,\hat{r}^{\,\prime}\bigg{\\}}\delta_{2}~{}.$
(113)

These boundary conditions are written in such a way to most easily replace
away $\ddot{V}_{i}$ for future convenience. Let us now return to the
Lagrangian.

###  A.5 The Canonical Scalar Mode Expansion

After applying the background equations of motion Eqs. (85), (86), (88), and
(89), as well as the gauge condition (71), we find the collections of
Lagrangian terms Eqs. (73)-(82) are simplified. Most contributions now
explicitly vanish:

$\displaystyle\mathcal{L}_{\text{5D},\text{bkgd}}\hskip 8.0pt=\hskip
8.0pt\mathcal{L}_{\text{5D},h}\hskip 8.0pt=\hskip
8.0pt\mathcal{L}_{\text{5D},r}^{*}+\mathcal{L}_{\text{5D},f}^{*}\hskip
8.0pt=\hskip
8.0pt\mathcal{L}_{\text{5D},hr}^{*}+\mathcal{L}_{\text{5D},hf}^{*}\hskip
8.0pt=\hskip 8.0pt0~{}.$ (114)

The tensor quadratic Lagrangian is now of the desired form to yield a tower of
canonical spin-2 states after Kaluza-Klein decomposition:

$\displaystyle\mathcal{L}_{\text{5D},hh}$ $\displaystyle\equiv
e^{-2A}\bigg{[}(\partial^{\nu}\hat{h}_{\mu\nu})\,(\partial^{\mu}\hat{h})-(\partial^{\nu}\hat{h}_{\mu\nu})^{2}+\dfrac{1}{2}(\partial_{\mu}\hat{h}_{\nu\rho})^{2}-\dfrac{1}{2}(\partial_{\mu}\hat{h})^{2}\bigg{]}+\dfrac{e^{-4A}}{2\,r_{c}^{2}}\bigg{[}(\hat{h}^{\prime})^{2}-(\hat{h}^{\prime}_{\mu\nu})^{2}\bigg{]}$
(115)

The scalar quadratic Lagrangian, however, remains quite complicated. We
organize the terms from each part of the quadratic scalar Lagrangian as
follows:

$\displaystyle\mathcal{L}_{\text{5D},rr}$
$\displaystyle=\mathcal{L}_{\text{5D},rr}^{*}+\mathcal{L}_{\text{5D},rf}^{*}+\mathcal{L}_{\text{5D},ff}^{*}$
(116)
$\displaystyle=\mathcal{L}_{\text{EH},rr}+\mathcal{L}_{\text{GHY},rr}+\mathcal{L}_{\Phi\Phi,rr}+\mathcal{L}_{\text{pot},rr}+\Delta\mathcal{L}_{rr}$
(117)

where

$\displaystyle\mathcal{L}_{\text{EH},rr}$
$\displaystyle=-\dfrac{1}{6}\,e^{2A}\,(\partial_{\mu}\hat{r})^{2}-\dfrac{2}{3}\,e^{2A}\,\hat{r}\,(\square\hat{r})-\dfrac{3}{r_{c}^{2}}\,(\hat{r}^{\,\prime})^{2}-\dfrac{4}{r_{c}^{2}}\,\hat{r}\,\hat{r}^{\,\prime\prime}+\dfrac{8}{3\,r_{c}^{2}}A^{\prime}\,\hat{r}\,\hat{r}^{\,\prime}+\dfrac{16}{3\,r_{c}^{2}}\,A^{\prime\prime}\,\hat{r}^{\,2}$
(118) $\displaystyle\mathcal{L}_{\text{GHY},rr}$
$\displaystyle=\dfrac{4}{r_{c}^{2}}\,(\hat{r}^{\,\prime})^{2}+\dfrac{4}{r_{c}^{2}}\,\hat{r}\,\hat{r}^{\,\prime\prime}-\dfrac{32}{3\,r_{c}^{2}}A^{\prime}\,\hat{r}\,\hat{r}^{\,\prime}-\dfrac{16}{3\,r_{c}^{2}}\,A^{\prime\prime}\,\hat{r}^{\,2}$
(119) $\displaystyle\mathcal{L}_{\Phi\Phi,rr}$
$\displaystyle=\dfrac{3}{(\phi_{0}^{\prime})^{2}}\,e^{2A}\,(\partial_{\mu}\hat{r}^{\,\prime})^{2}-\dfrac{3}{r_{c}^{2}(\phi_{0}^{\prime})^{4}}\Big{[}\phi_{0}^{\prime}\,\hat{r}^{\,\prime\prime}-\phi_{0}^{\prime\prime}\,\hat{r}^{\,\prime}+2\,A^{\prime}\,(\phi_{0}^{\prime})\,\hat{r}^{\,\prime}-\dfrac{1}{2}(\phi_{0}^{\prime})^{3}\,\hat{r}\Big{]}^{2}+\dfrac{(\phi_{0}^{\prime})^{2}}{3\,r_{c}^{2}}\,\hat{r}^{\,2}$
(120) $\displaystyle\mathcal{L}_{\text{pot},rr}$
$\displaystyle=\dfrac{4\,\dot{V}}{\phi_{0}^{\prime}}\,\hat{r}\,\hat{r}^{\,\prime}-\dfrac{12\,\ddot{V}}{(\phi_{0}^{\prime})^{2}}\,(\hat{r}^{\,\prime})^{2}-\sum_{i=1,2}\bigg{[}-\dfrac{4\,V_{i}}{3\,r_{c}}\,\hat{r}^{\,2}+\dfrac{8\,\dot{V}_{i}}{r_{c}\,\phi_{0}^{\prime}}\,\hat{r}\,\hat{r}^{\,\prime}-\dfrac{12\,\ddot{V}_{i}}{r_{c}\,(\phi_{0}^{\prime})^{2}}\,(\hat{r}^{\,\prime})^{2}\bigg{]}\delta_{i}$
(121) $\displaystyle\Delta\mathcal{L}_{rr}$
$\displaystyle=-\dfrac{3}{r_{c}^{2}}(\hat{r}^{\,\prime})^{2}-\dfrac{3}{r_{c}^{2}}\hat{r}\,\hat{r}^{\,\prime\prime}+\dfrac{8}{r_{c}^{2}}A^{\prime}\,\hat{r}\,\hat{r}^{\,\prime}+\dfrac{4}{r_{c}^{2}}A^{\prime\prime}\,\hat{r}^{\,2}$
(122)

For ease of comparison, we present these results without yet applying the
background equations of motion or integration-by-parts. Note that the squared
quantity in $\mathcal{L}_{\Phi\Phi,rr}$ is not singular because the delta-
functions in
$\phi^{\prime}_{0}\,\hat{r}^{\,\prime\prime}-\phi^{\prime\prime}_{0}\,\hat{r}^{\,\prime}$
cancel.

Rather than consider this quadratic scalar Lagrangian directly, we first add a
convenient total derivative (which we determined through trial-and-error).
Generally, adding a total derivative to a Lagrangian reorganizes how
information is stored in the fields, but ultimately does not change the
physics described by the Lagrangian; a classic example of this is the Gibbons-
Hawking-York total derivative, which is used to make the Einstein-Hilbert
Lagrangian of 4D gravity into a Lagrangian which only depends on fields and
their first derivatives Gibbons:1976ue . The total derivative we add to
$\mathcal{L}_{\text{5D},rr}$ (in addition to the total derivative
$\Delta\mathcal{L}$ defined in Eq. (68), which is already folded into
$\mathcal{L}_{\text{5D},rr}$) is

$\displaystyle\overline{\Delta}\mathcal{L}_{rr}$
$\displaystyle=\dfrac{1}{r_{c}^{2}}\partial_{\varphi}\bigg{\\{}\hat{r}\,\hat{r}^{\,\prime}-\dfrac{3\,e^{2A}}{(\phi_{0}^{\prime})^{2}}\hat{z}\,\hat{r}^{\,\prime}+\dfrac{12}{(\phi_{0}^{\prime})^{4}}\Big{[}A^{\prime}(\phi_{0}^{\prime})^{2}+V^{\prime}r_{c}^{2}\Big{]}(\hat{r}^{\,\prime})^{2}\bigg{\\}}$
(123)

such that we consider the combination

$\displaystyle\mathcal{L}_{\text{5D},rr}+\overline{\Delta}\mathcal{L}_{rr}~{}.$
(124)

The $\varphi$-derivative in $\overline{\Delta}\mathcal{L}_{rr}$ must be
evaluated with care, lest we generate spurious singularities. In particular,
Dirac delta-function singularities will be generated whenever
$\varphi$-differentiating a discontinuity in a function’s slope. In the
present calculation, such a discontinuity only ever happens at the orbifold
fixed points. The standard example of this from Randall-Sundrum models is the
twice-differentiated quantity
$|\varphi|^{\prime\prime}=\partial_{\varphi}(|\varphi|^{\prime})=\partial_{\varphi}(\text{sign
}\varphi)$, which equals $2(\delta_{1}-\delta_{2})$ in our scheme. If we are
not careful when taking $\varphi$-derivatives more generally, we can
accidentally generate spurious Dirac deltas which contradict our scheme.
Consider $\varphi$-differentiating a quantity which is a square of a
$\varphi$-differentiated quantity, such as $1/(\phi_{0}^{\prime})^{2}$.
Naively, we attain $-2\,\phi_{0}^{\prime\prime}/(\phi_{0}^{\prime})^{3}$,
which generates nonzero Dirac deltas through the $\phi_{0}^{\prime\prime}$.
These Dirac deltas are spurious. First, note that $\phi_{0}$ is a function of
$|\varphi|$, which means $\phi_{0}^{\prime}$ is proportional to
$|\varphi|^{\prime}=\text{sign}(\varphi)$. Thus
$1/(\phi_{0}^{\prime})^{2}\propto 1/(\text{sign }\varphi)^{2}=1$ and
$1/(\phi_{0})^{2}$ lacks the overall factor of $|\varphi|^{\prime}$ necessary
to generate Dirac deltas upon $\varphi$-differentiation. While naive
differentiation yields $-2\,\phi_{0}^{\prime\prime}/(\phi_{0}^{\prime})^{3}$,
careful analysis reveals the $\varphi$-derivative of
$1/(\phi_{0}^{\prime})^{2}$ is actually the Dirac delta-free quantity
$-2\,(\phi_{0}^{\prime\prime})_{\text{bulk}}/(\phi_{0}^{\prime})^{3}$.

For these reasons, evaluation of the $\varphi$-derivative present in the total
derivative $\overline{\Delta}\mathcal{L}_{rr}$ yields fewer Dirac deltas than
naively expected. Namely, they are only generated upon differentiating
$\hat{r}^{\prime}$ in the first two terms and $A^{\prime}$ & $V^{\prime}$ in
the third term. Explicitly, we thus calculate222222Technically these same
considerations are important when calculating, for example,
$\Delta\mathcal{L}$; however,
$\Delta\mathcal{L}_{rr}=\partial_{\varphi}[(4A^{\prime}\hat{r}-3\hat{r}^{\,\prime})\hat{r}]/r_{c}^{2}$,
and naive differentiation yields the correct result.

$\displaystyle\overline{\Delta}\mathcal{L}_{rr}$
$\displaystyle=\dfrac{1}{r_{c}^{2}}\bigg{\\{}(\hat{r}^{\,\prime})^{2}+\hat{r}\,\hat{r}^{\,\prime\prime}+\dfrac{6\,e^{2A}}{(\phi_{0}^{\prime})^{3}}\,(\phi_{0}^{\prime\prime})_{\text{bulk}}\,\hat{z}\,\hat{r}^{\,\prime}-\dfrac{6\,e^{2A}}{(\phi_{0}^{\prime})^{2}}\,A^{\prime}\,\hat{z}^{\,\prime}\,\hat{r}^{\,\prime}-\dfrac{3\,e^{2A}}{(\phi_{0}^{\prime})^{2}}\hat{z}\,\hat{r}^{\,\prime\prime}$
$\displaystyle\hskip
35.0pt-48\dfrac{(\phi_{0}^{\prime\prime})_{\text{bulk}}}{(\phi_{0}^{\prime})^{5}}\Big{[}A^{\prime}(\phi_{0}^{\prime})^{2}+V^{\prime}r_{c}^{2}\Big{]}(\hat{r}^{\,\prime})^{2}+\dfrac{24}{(\phi_{0}^{\prime})^{4}}\Big{[}A^{\prime}(\phi_{0}^{\prime})^{2}+V^{\prime}r_{c}^{2}\Big{]}\hat{r}^{\,\prime}\,(\hat{r}^{\,\prime\prime})_{\text{bulk}}$
$\displaystyle\hskip
35.0pt+\dfrac{12}{(\phi_{0}^{\prime})^{4}}\Big{[}A^{\prime\prime}(\phi_{0}^{\prime})^{2}+2A^{\prime}\phi_{0}^{\prime}\,(\phi_{0}^{\prime\prime})_{\text{bulk}}+V^{\prime\prime}r_{c}^{2}\Big{]}(\hat{r}^{\,\prime})^{2}\bigg{\\}}$
(125)

Having evaluated $\overline{\Delta}\mathcal{L}_{rr}$ as above, we next
consider the quadratic scalar terms
$\mathcal{L}_{\text{5D},rr}+\overline{\Delta}\mathcal{L}_{rr}$ after
performing the following sequence of manipulations:

  1. 1.

Use 4D integration-by-parts to eliminate any 4D d’Alembertian operators
$\square=\partial_{t}^{2}-\vec{\nabla}^{\,2}$, e.g. taking
$\hat{r}(\square\hat{r})$ to $-(\partial_{\mu}\hat{r})^{2}$.

  2. 2.

Eliminate all instances of $\hat{r}^{\,\prime\prime}$, $A^{\prime\prime}$,
$V$, and $\phi_{0}^{\prime\prime}$ (and their Dirac delta-free bulk forms) via
Eqs. (111), (85), and (88) respectively. Having done so, all Dirac deltas in
the original weak field expanded Lagrangian have been made explicit.

  3. 3.

Eliminate $V_{i}$ and $\dot{V}_{i}$ via the background equations Eqs. (86) and
(89) respectively.

  4. 4.

Eliminate $\ddot{V}_{i}$ (which always multiplies an $\hat{r}^{\,\prime}$) via
the boundary conditions, Eq. (113).

  5. 5.

Eliminate all instances of $\ddot{V}$, $V^{\prime}$, and $V^{\prime\prime}$,
and in favor of $\dot{V}$ and $\dot{V}^{\prime}$ via chain rule relations,
i.e.

$\displaystyle
V^{\prime\prime}=\dot{V}^{\prime}\,\phi_{0}^{\prime}+\dot{V}\,\phi_{0}^{\prime\prime}\hskip
55.0ptV^{\prime}=\dot{V}\,\phi_{0}^{\prime}\hskip
55.0pt\ddot{V}=\dfrac{\dot{V}^{\prime}}{\phi_{0}^{\prime}}$ (126)

where $\phi_{0}^{\prime\prime}$ is then replaced by Eq. (88), as done earlier.
With this, all Dirac deltas in $\overline{\Delta}\mathcal{L}$ are also
explicit.

After performing these replacements, we find all $\dot{V}$ and
$\dot{V}^{\prime}$ terms cancel, all Dirac deltas cancel, and we are left with
very few terms232323An alternate way of deriving the canonical quadratic
Lagrangian is to start with the expression on the right hand side of Eq.
(127), and substituting $\hat{z}$ from the (5,5) Einstein equation (95), as
well as a similar expression for $\hat{z}^{\prime}$ derived from the Euler-
Lagrange equation (96). The resulting expression can be shown to be equal to
the combination
$\mathcal{L}_{\text{5D},rr}+\overline{\Delta}\mathcal{L}_{rr}$. It is useful,
when performing these manipulations, to remove explicit Dirac delta terms by
using the background equations of motion given in Eq. (85) and Eq. (88).

$\displaystyle\mathcal{L}_{\text{5D},rr}+\Delta\mathcal{L}_{rr}$
$\displaystyle=\dfrac{e^{2A}}{2}\bigg{[}(\partial_{\mu}\hat{r})^{2}-\dfrac{\hat{z}\,\hat{r}}{r_{c}^{2}}\bigg{]}+\dfrac{3\,e^{2A}}{(\phi_{0}^{\prime})^{2}}\bigg{[}(\partial_{\mu}\hat{r}^{\,\prime})^{2}-\dfrac{\hat{z}^{\,\prime}\,\hat{r}^{\,\prime}}{r_{c}^{2}}\bigg{]}~{}.$
(127)

Upon Kaluza-Klein decomposition via Eqs. (105) and (110),
$\mathcal{L}_{\text{5D},rr}+\Delta\mathcal{L}_{rr}$ immediately yields:

$\displaystyle\sum_{m,n=0}^{+\infty}\dfrac{1}{2}\bigg{[}(\partial_{\mu}\hat{r}^{(m)})(\partial^{\mu}\hat{r}^{(n)})-\mu_{(m)}^{2}\,\hat{r}^{(m)}\,\hat{r}^{(n)}\bigg{]}\cdot\dfrac{6}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
5.0pt\bigg{[}\dfrac{e^{2A}}{(\phi_{0}^{\prime})^{2}}\gamma_{m}^{\,\prime}\,\gamma_{n}^{\,\prime}+\dfrac{e^{2A}}{6}\gamma_{m}\gamma_{n}\bigg{]}~{}.$
(128)

Recall that the scalar state wavefunctions are normalized according to Eq.
(109), such that the integral on the right above (including the $6/\pi$)
equals $\delta_{m,n}$. Consequently, we finally achieve our desired result:

$\displaystyle\mathcal{L}_{\text{5D},rr}+\Delta\mathcal{L}_{rr}$
$\displaystyle=\sum_{n=0}^{+\infty}\bigg{\\{}\dfrac{1}{2}(\partial^{\mu}\hat{r}^{(n)})^{2}-\dfrac{1}{2}\mu_{(n)}^{2}\,(\hat{r}^{(n)})^{2}\bigg{\\}}~{}.$
(129)

That is, the 5D field $\hat{r}(x,y)$ generates a scalar tower of canonical 4D
scalar states $\\{\hat{r}^{(n)}(x)\\}$, each having mass
$m_{(n)}\equiv\mu_{(n)}/r_{c}$, where $\mu_{(n)}$ is determined by solving the
differential equation problem for the wavefunctions $\\{\gamma_{n}\\}$ laid
out between Eqs. (106) and (107).

##  Appendix B The Inelastic Sum Rules Relating Couplings and Masses

This section derives & summarizes relationships between couplings and mass
spectra that are relevant to ensuring at-most $\mathcal{O}(s)$ growth of tree-
level inelastic 2-to-2 helicity-zero massive spin-2 KK mode scattering
amplitudes (i.e. the process $(k,l)\rightarrow(m,n)$) in the Goldberger-Wise-
stabilized Randall-Sundrum I model. We briefly consider the implications of
completeness before deriving a means of expressing all cubic and quartic
(spin-2 exclusive) B-type couplings in terms of A-type couplings and special
objects $B_{(kl)(mn)}$. These B-to-A formulas reduce the problem of finding
amplitude-relevant formulas to the problem of simplifying sums of the form
$\sum_{j}\mu_{j}^{2p}\,a_{klj}\,a_{mnj}$. The relevant (inelastic and elastic)
sum rules are derived and then summarized in their own subsections. The final
subsection describes the remaining set of (unproven) sum rules necessary for
at-most $\mathcal{O}(s)$ growth in the fully inelastic process.

This appendix is written as a stand-alone report of the sum-rule relationships
needed to insure that all inelastic scattering amplitudes (all $2\to 2$
scattering amplitudes with massive spin-2 fields of arbitrary mode number in
the external states) grow no faster than ${\cal O}(s)$ and report which we
have succeeded in proving – completing the program begin in Chivukula:2019rij
; Chivukula:2019zkt ; Chivukula:2020hvi ; Foren:2020egq . Section B.5.1
derives relationships used in Sec. IV.2 of the main body of this paper and can
be read independently.

###  B.1 Definitions

It is convenient to define generalized “couplings” to be overlap integrals of
spin-2 and spin-0 wavefunctions of the form

$\displaystyle x^{(p)}_{(k^{\prime}\cdots l)\cdots m^{\prime}\cdots n}$
$\displaystyle\equiv\dfrac{1}{\pi}\int_{-\pi}^{+\pi}d\varphi\hskip
10.0pt\varepsilon^{p}\,(\partial_{\varphi}\gamma_{k})\cdots\gamma_{l}\cdots(\partial_{\varphi}\psi_{m})\cdots\psi_{n}$
(130)

where $A(\varphi)$ is the warp-factor, $\varepsilon\equiv\exp(-A)$, and we add
an additional factor of $(\partial_{\varphi}A)/kr_{c}$ to the integrand if
only an odd number of differentiated wavefunctions are present in the
integrand otherwise. The most common integrands appearing in the 4D effective
Lagrangian carry powers $\varepsilon^{-2}$ and $\varepsilon^{-4}$, and are
given special symbols:

$\displaystyle a_{(k^{\prime}\cdots l)\cdots m^{\prime}\cdots n}\equiv
x^{(-2)}_{(k^{\prime}\cdots l)\cdots m^{\prime}\cdots n}\hskip
35.0ptb_{(k^{\prime}\cdots l)\cdots m^{\prime}\cdots
n}=x^{(-4)}_{(k^{\prime}\cdots l)\cdots m^{\prime}\cdots n}$ (131)

We will also encounter the label “$c$”, which is associated with $p=-6$. In
particular, we encounter this integral often:

$\displaystyle c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$
$\displaystyle\equiv
x^{(-6)}_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}=\dfrac{1}{\pi}\int
d\varphi\hskip
5.0pt\varepsilon^{-6}(\partial_{\varphi}\psi_{k})(\partial_{\varphi}\psi_{l})(\partial_{\varphi}\psi_{m})(\partial_{\varphi}\psi_{n})$
(132)

Another object that will be useful throughout the rest of this document is the
symbol $\mathcal{D}\equiv\varepsilon^{-4}\partial_{\varphi}$, which is a
combination of quantities that is often present as a result of the spin-2
Sturm-Liouville equation. When desperate for space, we will nest the notation
even further, utilizing
$\mathcal{D}_{n}\equiv\varepsilon^{-4}(\partial_{\varphi}\psi_{n})$.

We will ultimately derive sum rules that allow us to rewrite certain useful
sums of intermediate masses and couplings in terms of just the quartic A-type
coupling $a_{klmn}$, three $B_{(kl)(mn)}$ objects (of which any two fix the
value of the third), and integrals
$c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$ and
$x^{(-8)}_{\phi^{\prime}_{0}\phi^{\prime}_{0}n^{\prime}n^{\prime}n^{\prime}n^{\prime}}$.

###  B.2 Applications of Completeness

The spin-2 mode completeness relation is

$\displaystyle\delta(\varphi_{2}-\varphi_{1})=\dfrac{1}{\pi}\,\varepsilon(\varphi_{1})^{-2}\sum_{j=0}^{+\infty}\psi_{j}(\varphi_{1})\,\psi_{j}(\varphi_{2})$
(133)

where $\varepsilon(\varphi)\equiv e^{A(\varphi)}$ and the wavefunctions
$\psi_{n}$ satisfy
$\partial_{\varphi}\mathcal{D}\psi_{n}=\partial_{\varphi}[\varepsilon^{-4}(\partial_{\varphi}\psi_{n})]=-\mu_{n}^{2}\varepsilon^{-2}\psi_{n}$.
Spin-2 mode completeness allows us to collapse certain sums of cubic coupling
products into a single quartic coupling, e.g.

$\displaystyle a_{klmn}$
$\displaystyle=\sum_{j}a_{klj}\,a_{mnj}=\sum_{j}a_{kmj}\,a_{lnj}=\sum_{j}a_{knj}\,a_{lmj}$
(134) $\displaystyle b_{k^{\prime}l^{\prime}mn}$
$\displaystyle=\sum_{j}b_{k^{\prime}l^{\prime}j}\,a_{mnj}$ (135)

Furthermore, by combining cubic B-type couplings in this same way, we arrive
at

$\displaystyle c_{k^{\prime}l^{\prime}m^{\prime}n^{\prime}}$
$\displaystyle=\sum_{j}b_{k^{\prime}l^{\prime}j}\,b_{m^{\prime}n^{\prime}j}=\sum_{j}b_{k^{\prime}m^{\prime}j}\,b_{l^{\prime}n^{\prime}j}=\sum_{j}b_{k^{\prime}n^{\prime}j}\,b_{l^{\prime}m^{\prime}j}$

This is as far as direct applications of completeness can get us for now.

###  B.3 B-to-A Formulas

This subsection details how to eliminate all B-type couplings (e.g.
$b_{l^{\prime}m^{\prime}n}$ and $b_{k^{\prime}l^{\prime}mn}$) in favor of
A-type couplings (e.g. $a_{lmn}$ and $a_{klmn}$) and new structures
$B_{(kl)(mn)}$. To begin, we note we can absorb a factor of $\mu^{2}$ into
A-type couplings with help from the Sturm-Liouville equation. A standard
application of this technique proceeds as follows:

$\displaystyle\mu_{n}^{2}\,a_{lmn}$ $\displaystyle=\dfrac{1}{\pi}\int
d\varphi\hskip
5.0pt\varepsilon^{-2}\psi_{l}\psi_{m}\left[\mu_{n}^{2}\psi_{n}\right]$ (136)
$\displaystyle=\dfrac{1}{\pi}\int d\varphi\hskip
5.0pt\varepsilon^{-2}\psi_{l}\psi_{m}\bigg{[}-\varepsilon^{+2}\partial_{\varphi}(\mathcal{D}\psi_{n})\bigg{]}$
(137) $\displaystyle=\dfrac{1}{\pi}\int d\varphi\hskip
5.0pt\partial_{\varphi}\left[\psi_{l}\psi_{m}\right](\mathcal{D}\psi_{n})$
(138) $\displaystyle=\dfrac{1}{\pi}\int d\varphi\hskip
5.0pt\varepsilon^{-4}(\partial_{\varphi}\psi_{l})\psi_{m}(\partial_{\varphi}\psi_{n})+\dfrac{1}{\pi}\int
d\varphi\hskip
5.0pt\varepsilon^{-4}\psi_{l}(\partial_{\varphi}\psi_{m})(\partial_{\varphi}\psi_{n})$
(139) $\displaystyle=b_{l^{\prime}mn^{\prime}}+b_{lm^{\prime}n^{\prime}}$
(140)

where integration by parts was utilized between Eqs. (137) and (138); because
$(\mathcal{D}\psi_{n})$ vanishes on the boundaries, there is no surface term.
This and the equivalent calculation with the quartic A-type coupling yield

$\displaystyle\mu_{n}^{2}\,a_{lmn}$
$\displaystyle=b_{l^{\prime}mn^{\prime}}+b_{lm^{\prime}n^{\prime}}$ (141)
$\displaystyle\mu_{n}^{2}\,a_{klmn}$
$\displaystyle=b_{k^{\prime}lmn^{\prime}}+b_{kl^{\prime}mn^{\prime}}+b_{klm^{\prime}n^{\prime}}$
(142)

By considering different permutations of KK indices, each of these equations
corresponds to three and four unique constraints respectively. Because there
are only three unique cubic B-type couplings with KK indices $l$, $m$, and $n$
(specifically, $b_{l^{\prime}m^{\prime}n}$, $b_{l^{\prime}mn^{\prime}}$, and
$b_{lm^{\prime}n^{\prime}}$), Eq. (141) can be inverted to yield

$\displaystyle\boxed{b_{l^{\prime}m^{\prime}n}=\dfrac{1}{2}\left[\mu_{l}^{2}+\mu_{m}^{2}-\mu_{n}^{2}\right]a_{lmn}}$
(143)

with which we can eliminate all cubic B-type couplings in favor of the cubic
A-type coupling.

There are six unique quartic B-type couplings with KK indices $k$, $l$, $m$,
and $n$. We first halve this set by rewriting each quartic B-type coupling
$b_{k^{\prime}l^{\prime}mn}$ in terms of new objects $B_{(kl)(mn)}$. These new
objects are motivated as follows: note that Eq. (142) implies

$\displaystyle\dfrac{1}{2}\bigg{[}\mu_{k}^{2}+\mu_{l}^{2}-\mu_{m}^{2}-\mu_{n}^{2}\bigg{]}\,a_{klmn}$
$\displaystyle=b_{k^{\prime}l^{\prime}mn}-b_{klm^{\prime}n^{\prime}}$ (144)

Equivalently, we may write this as

$\displaystyle
b_{k^{\prime}l^{\prime}mn}+\dfrac{1}{2}\bigg{[}\mu_{m}^{2}+\mu_{n}^{2}\bigg{]}\,a_{klmn}$
$\displaystyle=b_{klm^{\prime}n^{\prime}}+\dfrac{1}{2}\bigg{[}\mu_{k}^{2}+\mu_{l}^{2}\bigg{]}\,a_{klmn}$
(145)

In other words, the quantity on the LHS possesses a symmetry under the pair
swap $(k,l)\leftrightarrow(m,n)$. Furthermore, this symmetry is maintained
under the addition of any quantity $\tilde{B}_{(kl)(mn)}$ which is also
symmetric under this pair swap. Inspired by Eq. (145), we define

$\displaystyle B_{(kl)(mn)}\equiv
b_{k^{\prime}l^{\prime}mn}+\dfrac{1}{2}\bigg{[}\mu_{m}^{2}+\mu_{n}^{2}\bigg{]}\,a_{klmn}+\tilde{B}_{(kl)(mn)}$
(146)