[ beforeskip=0.3em plus 1pt,pagenumberformat=]toclinesection aainstitutetext:
Department of Physics and Astronomy,
Northwestern University, Evanston, IL, USA bbinstitutetext: T. D. Lee
Institute & School of Physics and Astronomy,
Key Laboratory for Particle Astrophysics and Cosmology,
Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai Jiao Tong University, Shanghai, China ccinstitutetext: Physics
Department & Institute of Modern Physics, Tsinghua University, Beijing, China;
Center for High Energy Physics, Peking University, Beijing, China

# Structure of Massive Gauge/Gravity Scattering Amplitudes, Equivalence
Theorems, and Extended Double-Copy with Compactified Warped Space

Yanfeng Hang<EMAIL_ADDRESS>b Wei-Wei Zhao<EMAIL_ADDRESS>b,c Hong-Jian He<EMAIL_ADDRESS>b Yin-Long Qiu<EMAIL_ADDRESS>
###### Abstract

We study the structure of scattering amplitudes of massive Kaluza-Klein (KK)
states in the compactified 5-dimensional warped gauge and gravity theories. We
present systematic formulations of the gauge theory equivalence theorem (GAET)
and the gravitational equivalence theorem (GRET) for warped KK theories in
$R_{\xi}$ gauge, where the GAET connects the scattering amplitudes of
longitudinal KK gauge bosons to that of the corresponding KK Goldstone bosons
and the GRET connects the scattering amplitudes of KK gravitons of helicity-
zero (helicity-one) to that of the corresponding gravitational KK Goldstone
bosons. We analyze the structure of 3-point and 4-point scattering amplitudes
of massive KK gauge bosons and of massive KK gravitons as well as their
corresponding Goldstone bosons. We first prove the GAET and GRET explicitly
for the fundamental 3-point KK gauge/gravity scattering amplitudes. We then
demonstrate that the validity of the GAET and GRET for 4-point gauge/gravity
scattering amplitudes can be reduced to the validity of GAET and GRET for
3-point gauge/gravity scattering amplitudes at tree level. With these, we
study the double-copy construction of KK scattering amplitudes in the warped
gauge/gravity theories. We newly realize the double-copy for massive 3-point
full gauge/gravity amplitudes at tree level under proper correspondences of
color-kinematics and of gauge/gravity couplings, whereas we can construct the
double-copy for 4-point KK gauge/gravity amplitudes to the leading order (LO)
of high energy expansion. We further demonstrate that this LO double-copy
construction can be extended to $N$-point KK scattering amplitudes with
$N\\!\geqslant\\!4\hskip 0.85358pt$.
arXiv:2406.12713 [hep-th]

##  1 Introduction

Scattering amplitude is an elementary means for studying fundamental forces in
nature, and can bridge theories with experiments. There have been substantial
efforts Amp-Rev  for studying various massless scattering amplitudes over the
past four decades. Studies on massive scattering amplitudes have attracted
increasing interests in recent years, including to consider some models of
massive gauge bosons and gravitons. Among these, the compactified higher
dimensional gauge and gravity theories à la Kaluza-Klein (KK) KK  stand out as
a fundamental approach for consistent mass generation of the excited KK states
of gauge bosons and gravitons.111The KK theories also have important impacts
on the experimental side because the KK compactification predicts an infinite
tower of massive KK states for each known particle of the Standard Model (SM)
and Generality Relativity (GR). The low-lying KK states in such KK theories
have intrigued great phenomenological and experimental efforts over the past
two decades Exd -Exd-Rev , as they may provide the first signatures for the
new physics beyond the SM, ranging from the KK states of the SM particles to
the spin-2 KK gravitons and possible dark matter candidate.

Simple models of massive gauge bosons and gravitons include such as the
massive Yang-Mills (YM) gauge theory mYM  and massive Fierz-Pauli (FP) gravity
FP  and alike FPRev , which add mass terms by hand and spoil the gauge
symmetry or diffeomorphism invariance. The former is plagued with
nonrenormalizability and the latter further suffers the van Dam-Veltman-
Zakharov (vDVZ) discontinuity in the massive graviton propagator vDVZ vDVZ2 .
Moreover, such massive gauge and gravity theories generally exhibit bad high
energy behaviors in their scattering amplitudes involving longitudinally
polarized gauge bosons or gravitons. The reason behind such inconsistencies is
because the nonconservation of physical degrees of freedom prohibits a smooth
massless limit. In contrast, the KK gauge and gravity theories provide a truly
consistent geometric “Higgs” mechanism for mass generations of KK gauge bosons
and KK gravitons whose additional mass-induced physical degrees of freedom
originate from absorbing the extra-dimensional components of the KK gauge
bosons Chivukula:2001esy Chivukula:2002ej  and KK gravitons Duff Hang:2021fmp
Hang:2022rjp . Hence, this mass-generation mechanism naturally conserves
physical degrees of freedom for KK gauge bosons and KK gravitons, and thus has
smooth massless limit. But this geometric mechanism for mass generations of KK
gauge bosons and KK gravitons is realized through the compactification itself
and does not invoke any additional Higgs boson of the conventional Higgs
mechanism higgsM .

The geometric mechanism for mass generations of KK gauge bosons and KK
gravitons can be formulated at the level of scattering $S$-matrix through the
KK Equivalence Theorem (ET), as done previously for the flat 5d KK gauge
theories Chivukula:2001esy Chivukula:2002ej He:2004zr  and the flat 5d KK
gravity theories Hang:2021fmp Hang:2022rjp . It is desirable to extend these
KK ET formulations to both the KK gauge theories and KK gravity theories with
warped 5d compactification RS1 RS-gauge RS-gauge2 , as we briefly discussed in
Refs. He:2004zr Hang:2022rjp  (cf. also a recent related study on the warped
gravity KK ET Chivukula:2023qrt ). The warped 5d KK gravity of Randall and
Sundrum (RS1) RS1  holds the promise to resolve the gauge hierarchy problem of
the SM and to explain the vast discrepancy between the gravity force and other
three gauge forces of the SM. It also predicts distinctive KK mass spectrum
and KK signatures which are important for the on-going experimental searches.

On the other hand, it is conjectured that the scattering amplitudes of
gravitons and gauge bosons are connected by the striking relation of double-
copy, GR$\,=\,$(Gauge Theory)2, which points to the common root of both the
gravity force and gauge forces. It also provides a powerful means for
efficiently computing the highly complex scattering amplitudes of spin-2
gravitons. Such double-copy relation was first found in string theories by
Kawai-Lewellen-Tye (KLT) KLT , which links the scattering amplitudes of
massless closed strings to the products of scattering amplitudes of massless
open strings at tree level, and further leads to the double-copy relation
between the scattering amplitudes of massless gravitons and of massless gauge
bosons in the field theory limit. With the inspiration of KLT, Bern, Carrasco
and Johansson (BCJ) proposed the double-copy method in the quantum field
theory framework via color-kinematics (CK) duality BCJ BCJ-Rev , which
constructs the massless graviton amplitudes from the (squared) scattering
amplitudes of massless gauge bosons. It is highly nontrivial to extend the
massless double-copy formulation to the massive gauge and gravity theories. In
a recent series of works, the extended massive KLT-type and BCJ-type double-
copies were constructed for the compactified KK string theories Li:2021yfk
and for the KK gauge/gravity field theories under toroidal compactification of
flat 5d (without or with orbifold) Hang:2021fmp Hang:2022rjp Li:2022rel . But,
it is more challenging to extend double-copy to the KK gauge/gravity theories
with warped extra dimension such as the RS1. This is because the double-copy
for $N$-point ($N\\!\\!\geqslant\\!4$) massive KK gauge/gravity scattering
amplitudes was proved to directly work out only for the toroidal
compactifications with flat extra dimensions Li:2022rel . In this work, we
will demonstrate that double-copy can be realized for the 3-point full
scatttering amplitudes of warped KK gauge/gravity theories, whereas for the
$4$-point warped KK gauge/gravity scattering amplitudes the double-copy can
hold at the leading order (LO) of high energy expansion (which are just what
we need to formulate the ET in both the warped KK gauge and gravity
theories).222In passing, the extensions of double-copy for the topologically
massive Chern-Simons (CS) gauge/gravity theories were presented in
Hang:2021oso Gonzalez:2021bes . There are other recent works in the literatue
tried to extend the massless double-copy formulation to the massive double-
copies, including the 4d massive YM theory versus Fierz-Pauli-like gravity
dRGT DC-4dx1 , the spontaneously broken YM-Einstein supergravity models with
adjoint Higgs fields SUGRAhiggs , the KK-inspired effective gauge theory with
extra global U(1) Momeni:2020hmc , the 3d CS gauge/gravity theories with or
without supersymmetry 3dCS-susy -3dCS1 , and the massless QCD/gravity coupled
to massive scalars/fermions/vectors Ochirov -1911.06785 .

In this work, we study the structure of scattering amplitudes of massive KK
states in the compactified 5-dimensional warped gauge and gravity theories. We
present systematic formulations of the gauge theory equivalence theorem (GAET)
and the gravitational equivalence theorem (GRET) for warped KK gauge/gravity
theories up to loop level and in the $R_{\xi}$ gauge. The GAET connects the
scattering amplitudes of longitudinal KK gauge bosons to that of the
corresponding KK Goldstone bosons, whereas the GRET connects the scattering
amplitudes of KK gravitons of helicity-0 (helicity-1) to that of the
corresponding gravitational KK Goldstone bosons. As a key idea of this work,
we take the GAET of KK Yang-Mills gauge theories as the truly fundamental ET
formulation, from which we can reconstruct the GRET of the corresponding KK
gravity theories by using the double-copy at the leading order (LO) of high
energy expansion. Then, we analyze the structure of 3-point and 4-point
scattering amplitudes of massive KK gauge bosons and of massive KK gravitons,
as well as their corresponding KK Goldstone bosons. We first prove explicitly
the GAET and GRET for the fundamental 3-point KK gauge/gravity scattering
amplitudes. We then demonstrate that the validity of the GAET and GRET for
4-point gauge/gravity scattering amplitudes can be reduced to the validity of
GAET and GRET for 3-point gauge/gravity scattering amplitudes. With these, we
further study the double-copy construction of the KK scattering amplitudes in
the warped gauge/gravity theories. We newly realize the double-copy for the
3-point full gauge/gravity amplitudes at tree level under proper color-
kinematics correspondence and gauge-gravity coupling correspondence without
high energy expansion. Then, we construct double-copy for 4-point KK
gauge/gravity scattering amplitudes to the leading order (LO) of high energy
expansion. We can derive the GRET from GAET by using the LO double-copy
construction for 3-point and 4-point KK scattering amplitudes. Finally, we
demonstrate that this LO double-copy construction can be extended to the
$N$-point KK gauge/gravity scattering amplitudes with $N\\!\geqslant\\!4\hskip
0.85358pt$.

Figure 1: Equivalence Theorem and Double-Copy Correspondences from 3-point
(3pt) scattering amplitudes to 4-point (4pt) scattering amplitudes and from
massive KK gauge scattering amplitudes to massive KK gravitational scattering
amplitudes in the warped gauge and gravity theories.

In summary, we present the main logic of this study in Fig. 1. This figure
shows the correspondence of the Equivalence Theorems from the GAET to the
GRET, and the correspondence of Double-Copies from the 3-point KK scattering
amplitudes to the 4-point KK scattering amplitudes and from the massive KK
gauge scattering amplitudes to the massive KK gravitational scattering
amplitudes. In the lower-left corner, we start from analyzing the most
fundamental 3-point KK gauge boson scattering amplitudes and establish the
GAET between 3-point scattering amplitudes of longitudinal KK gauge bosons and
of the corresponding KK Goldstone bosons to the LO of high energy expansion.
Then, along the bottom horizontal arrow we reach the lower-right corner where
we compute the 4-point scattering amplitudes of longitudinal KK gauge bosons
and of the KK Goldstone bosons, as well as establishing the GAET for these
4-point amplitudes at the LO of high energy exoansion. Here we use the word
“Reconstruction” or “Reduction” to indicate that the validity of the GAET for
4-point KK gauge/Goldstone boson amplitudes can be reduced to (proved from)
the validity of the GAET for 3-point KK gauge/Goldstone boson amplitudes.
Next, we can go from the lower-left corner to the upper-left corner by the
double-copy construction along the left vertical arrow and from the 3-point KK
gauge boson amplitudes to the 3-point KK graviton amplitudes, as well as from
the LO 3-point KK Goldstone boson amplitudes to the LO 3-point gravitational
KK Goldstone amplitudes. Then, along the top horizontal arrow we go from the
upper-left corner to reach the upper-right corner where the 4-point KK
graviton amplitudes and gravitational KK Goldstone amplitudes are computed;
and thus we establish the GRET for these 4-point gravitational KK amplitudes.
Here we use the word “Reconstruction” or “Reduction” to indicate that along
the top horizontal arrow the validity of the GRET for 4-point KK
graviton/Goldstone boson amplitudes can be reduced to (proved from) the
validity of the GRET for 3-point KK graviton/Goldstone boson amplitudes.
Finally, we can go from the lower-right corner to the upper-right corner by
the double-copy construction along the right vertical arrow and from the LO
4-point KK gauge (Goldstone) boson amplitudes to the LO 4-point KK graviton
(Goldstone) amplitudes.

The rest of this paper is organized as follows. In Section 2, after
introducing the warped 5d compactification with $S^{1}\\!/\mathbb{Z}_{2}$
orbifold (Section 2.1), we present the formulation of GAET in the warped KK
gauge theory (Section 2.2), and then the formulations of the GRET of type-I,II
in the warped KK gravity theory (Section 2.3). In Section 3, we study the
structure of massive scattering amplitudes from the warped GAET and GRET.
These include to first compute the 3-point scattering amplitudes of KK gauge
bosons (KK Goldstone bosons) and of KK gravitons (KK gravitational Goldstone
bosons), and to establish the GAET and GRET for 3-point KK amplitudes at the
leading order of high energy expansion (Section 3.1). Then, we present the
4-point scattering amplitudes of KK gauge bosons (KK Goldstone bosons) and of
KK gravitons (KK gravitational Goldstone bosons), and to establish the GAET
and GRET for 4-point KK amplitudes at the leading order of high energy
expansion (Section 3.2). Our key point is to demonstrate that the validity of
the GRET for 4-point KK amplitude can be reduced to (proved from) the validity
of the GAET for basic 3-point KK amplitudes. In Section 4, we study the
double-copy construction of the KK scattering amplitudes in the warped
gauge/gravity theories. We realize the double-copy for the 3-point full
gauge/gravity amplitudes at tree level under the proper correspondence of
color-kinematics and the gauge-gravity coupling correspondence without high
energy expansion, whereas the double-copy can be constructed for the 4-point
KK gauge/gravity amplitudes to the LO of high energy expansion. We further
study the $N$-point ($N\\!\\!\geqslant\\!4\hskip 0.85358pt$) KK gauge/gravity
scattering amplitudes at the leading order and their LO double-copy. Finally,
we conclude in Section 5. The four Appendices present supporting materials for
our analyses in the main text. Among these, Appendix A provides the kinematics
of three and four KK-particle scattering; Appendix B presents the Becchi-
Rouet-Stora-Tyutin (BRST) quantization for warped 5d KK gravity theory and
further details for deriving the GRET of type-I and type-II; Appendix C
derives the Feynman Rules for warped KK gauge and gravity theories as needed
for the analyses of Section 3; Appendix D proves the relevant 3-point and
4-point warped identities.

##  2 Formulation of Gauge and Gravitational Equivalence Theorems
with Compactified Warped Space

In this section, we formulate the gauge and gravitational equivalence theorems
for the compactified 5d warped gauge and gravity theories. In section 2.1, we
present the warped 5d compactification for RS1. In section 2.2, we formulate
the gauge theory ET (GAET) for the warped KK gauge theory. In section 2.3, we
first present the BRST quantization (Sec. 2.3.1), and then formulate the
gravitational ET (GRET) of type-I (Sec. 2.3.3) and the GRET of type-II (Sec.
2.3.2) for the warped KK gravity theory.

###  2.1 Warped 5d Compactification

We consider the warped 5d compactification à la Randall and Sundrum (RS1) RS1
, where a warped background metric generates weak scale from Planck scale
through an exponential suppression. The background metric involves a five-
dimensional non-factorizable warped geometry, derived from a slice of
$\mathrm{AdS}_{5}$ spacetime. This framework is realized with two 3-branes
situated at fixed points of the $S^{1}\\!/\mathbb{Z}_{2}$ orbifold
compactification, where one 3-brane is the UV brane (Planck brane) at the
fixed point $\hskip 0.85358pty\\!=\\!0\hskip 0.85358pt$ and another 3-brane is
IR brane (TeV brane) at the fixed point $\hskip 0.85358pty\\!=\\!\bar{L}\hskip
0.85358pt$. Here the physical coordinate of the fifth dimension is denoted as
$y$, so we have $\,y\\!\in\\![0,\bar{L}]$ with $\bar{L}\\!=\pi
r_{\mathrm{c}}\hskip 0.85358pt$ denoting the size of fifth dimension and
$r_{\mathrm{c}}$ denoting the compactification radius.

The 5-dimensional warped background metric in the physical coordinate takes
the following nonfactorizable form:

$\text{d}s^{2}=e^{2A(y)}\eta_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}+\text{d}y^{2},$
(1)

where $e^{2A(y)}$ is the warp factor. Then, we can transform the physical
coordinates $(x^{\mu},y)$ into the conformal coordinate $\hskip
0.85358pt(x^{\mu},z)$ by imposing the relation $\hskip
0.85358pt\text{d}y=e^{A(z)}\text{d}z\hskip 0.85358pt$. Thus, we have the
conformally flat 5-dimensional spacetime metric:

$\text{d}s^{2}\hskip
0.85358pt=\,e^{2A(z)}\big{(}\eta_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}\\!+\text{d}z^{2}\hskip
0.85358pt\big{)}\hskip 0.85358pt.$ (2)

Setting $A(z)\\!=-ky\hskip 0.85358pt$, one can resolve the function $A(z)$ as
follows Exd-Rev :

$A(z)=-\ln(1+kz)\,,$ (3)

where $\hskip 0.85358ptk\\!=\\!\\!\sqrt{-\Lambda/6\,}\,$ is the AdS curvature
scale and has the order of Planck mass. (The parameter $\Lambda\,$ denotes a
negative bulk cosmological constant, as will be defined in Sec. 2.3.1.) Thus,
we deduce $z\\!=\\!\\!(e^{\hskip 0.85358ptky}\\!-\\!1)/k\,$. Hence, in the
conformal coordinates, we have $z\\!\in\\![0,{L}]$ with ${L}\equiv[e^{\hskip
0.85358ptk\bar{L}}\\!-\\!1]/k\,$, which corresponds to the physical coordinate
$\,y\\!\in\\![0,\bar{L}]$ with $\bar{L}\\!=\pi r_{\mathrm{c}}\hskip
0.85358pt$.

The Eq.(2) shows that the 5-dimensional warped metric tensor $\hskip
0.85358pt\hat{g}_{MN}$ is conformally flat and can be expressed as

$\hat{g}_{MN}=\hskip 0.85358pte^{2A(z)}\hat{\eta}_{MN}\,,$ (4)

where the flat 5d Minkowski metric $\hskip 0.85358pt\hat{\eta}_{MN}$ takes the
form:

$\hat{\eta}_{MN}=\hat{\eta}^{MN}=\text{diag}(-1,1,1,1,1)\hskip 0.85358pt.$ (5)

For clarity of notations, we denote the 5d Lorentz indices by uppercase Latin
letters (such as $M,N\\!=\\!\mu,5$) and the 4d Lorentz indices by the
lowercase Greek letters (such as $\mu,\nu=0,1,2,3$).

In the next subsections, we will consider the RS1 under
$S^{1}\\!/\mathbb{Z}_{2}$ compactification for both warped 5d Yang-Mills gauge
theory and the warped 5d GR theory. With these, we formulate the KK
equivalence theorems for the warped 5d Yang-Mills gauge theory and the warped
5d GR theory.

###  2.2 Formulating Gauge ET for Warped KK Gauge Theory

The warped bulk gauge theories in RS1 were discussed in the literature RS-
gauge RS-gauge2 Randall:2001gb . We consider the Yang-Mills Lagrangian in
warped 5d spacetime:

$\hat{\mathcal{L}}_{\mathrm{YM}}\,=\,\sqrt{-\hat{g}\,}\hskip
0.85358pt\Big{(}\\!-\frac{1}{4}\hat{g}^{MP}\hat{g}^{NQ}\hat{F}_{MN}^{a}\hat{F}_{PQ}^{a}\Big{)},$
(6)

where the square root of the determinant of the warped 5d background metric
tensor $\hat{g}_{MN}$ is given by $\sqrt{-\hat{g}\,}=\exp[5A(z)]\hskip
0.85358pt$. The 5d field strength tensor $\hat{F}_{MN}^{a}$ takes the form:

$\hat{F}_{MN}^{a}=\partial_{M}\hat{A}_{N}^{a}-\partial_{N}\hat{A}_{M}^{a}+g_{5}f^{abc}\hat{A}_{M}^{b}\hat{A}_{N}^{c}\,,$
(7)

where $(a,b,c)$ denote the gauge-group indices, $f^{abc}$ denotes the gauge-
group structure constant, and $g_{5}$ is the 5d gauge coupling constant having
mass-dimension $-1/2$.

Expanding the Lagrangian (6) to the quadratic order, we derive the following
terms:

$\hat{\mathcal{L}}_{\mathrm{YM}}^{(2)}=-\frac{\,e^{A(z)}}{4}\left\\{\\!\Big{(}\partial_{\mu}\hat{A}_{\nu}^{a}\\!-\partial_{\nu}\hat{A}_{\mu}^{a}\Big{)}^{\\!2}\\!+2\\!\left[\\!\Big{(}\partial_{z}\hat{A}_{\mu}^{a}\Big{)}^{\\!2}\\!+\\!\Big{(}\partial_{\mu}\hat{A}_{5}^{a}\Big{)}^{\\!2}\right]\\!-4\hskip
0.85358pt\partial_{z}\hat{A}_{\mu}^{a}\partial^{\mu}\\!\hat{A}_{5}^{a}\right\\}.$
(8)

The mixing term between the 5d components $\hat{A}_{\mu}^{a}$ and
$\hat{A}_{5}^{a}$ in Eq.(8) can be eliminated by including the following
$R_{\xi}$ gauge-fixing term:

$\hat{\mathcal{L}}_{\mathrm{GF}}=-\frac{~{}e^{A(z)}\hskip
0.85358pt}{2\xi}\big{(}\hat{\mathcal{F}}^{a}\big{)}^{\\!2},~{}~{}~{}~{}~{}\hat{\mathcal{F}}^{a}=\partial^{\mu}\hat{A}_{\mu}^{a}\\!+\xi(A^{\prime}\\!+\partial_{z})\hat{A}_{5}^{a}\,,$
(9)

where $\hskip 0.85358pt\xi\hskip 0.85358pt$ is the gauge-fixing parameter and
$A^{\prime}\\!=\partial_{z}A(z)\hskip 0.85358pt$. We note that the gauge-
fixing term (9) contains an $\hat{A}_{\mu}^{a}$-$\hat{A}_{5}^{a}$ mixing term
which precisely cancels the corresponding mixing term in Eq.(8) upon
integration by part. The fifth component $\hat{A}_{5}^{a}$ will be identified
as the would-be Goldstone boson via a geometric “Higgs” mechanism of the KK
compactification Chivukula:2001esy He:2004zr . Using Eqs.(8)-(9), we derive
the 5d equation of motion (EOM) for the free fields
$(\hat{A}_{\mu}^{a},\hat{A}_{5}^{a}$):

$\displaystyle(\partial_{\nu}^{2}+e^{-A}\partial_{z}e^{A}\partial_{z})\hat{A}_{\mu}^{a}-(1\\!-\xi^{-1})\partial_{\mu}\partial_{\nu}\hat{A}^{a\nu}$
$\displaystyle=\,0\hskip 0.85358pt,$ (10a)
$\displaystyle\partial_{\mu}^{2}\hat{A}_{5}^{a}+\xi\partial_{z}(A^{\prime}\\!+\partial_{z})\hat{A}_{5}^{a}$
$\displaystyle=\,0\hskip 0.85358pt.$ (10b)

Next, we compactify the warped 5d space under $S^{1}\\!/\mathbb{Z}_{2}$
orbifold, which corresponds to compactifying the fifth-dimension to a line
segment $\hskip 0.85358pty\\!\in\\![0,\bar{L}]\hskip 0.85358pt$ or $\hskip
0.85358ptz\\!\in\\![0,L]\hskip 0.85358pt$, with $\bar{L}\\!=\\!\pi\hskip
0.85358ptr_{\mathrm{c}}$ and ${L}\equiv(e^{\hskip
0.85358ptk\bar{L}}\\!-\\!1)/k\,$, where $r_{\mathrm{c}}$ denotes the
compactification radius. We impose the gauge-covariant boundary conditions on
the 5d gauge field strength, $\hat{F}_{M5}^{a}\\!=\\!\hat{F}_{5M}^{a}\\!=0\,$
at $z=0,L\hskip 0.85358pt$. This leads to the Neumann and Dirichlet boundary
conditions on the 5d gauge field components $\hat{A}_{\mu}^{a}$ and
$\hat{A}_{5}^{a}$ as follows:

$\partial_{z}\hat{A}^{a}_{\mu}({x},z)\Big{|}_{z=0,L}\\!=\hskip
0.85358pt0\,,\hskip 28.45274pt\hat{A}^{a}_{5}({x},z)\Big{|}_{z=0,L}\\!=\hskip
0.85358pt0\,.$ (11)

Then, we can make KK expansions for the 5d gauge fields $\hat{A}_{\mu}^{a}$
and $\hat{A}_{5}^{a}$ in terms of the eigenfunctions of warped 5d space:

$\displaystyle\hat{A}^{a\mu}({x}\\!,z)$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\sum_{n=0}^{\infty}A^{a\mu}_{n}({x})\,\mathsf{f}_{n}(z)\,,$
(12a) $\displaystyle\hat{A}^{a5}({x}\\!,z)$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\sum_{n=1}^{\infty}A^{a5}_{n}({x})\,\tilde{\mathsf{f}}_{n}(z)\,.$
(12b)

Substituting the KK expansions (12) into the 5d EOM (10) and using the 4d EOM
for free KK gauge fields,

$\displaystyle(\partial_{\nu}^{2}-\\!M_{n}^{2})A_{n}^{a\mu}-(1\\!-{\xi^{-1}_{n}})\partial_{\mu}\partial_{\nu}A_{n}^{a\nu}$
$\displaystyle=\,0\hskip 0.85358pt,$ (13a)
$\displaystyle(\partial_{\nu}^{2}-\\!\tilde{M}_{n}^{2})A_{5}^{a\mu}$
$\displaystyle=\,0\hskip 0.85358pt,$ (13b)

we deduce the following EOMs for the 5d eigenfunctions $\mathsf{f}_{n}(z)$ and
$\tilde{\mathsf{f}}_{n}(z)$:

$\displaystyle(A^{\prime}+\partial_{z})\partial_{z}\mathsf{f}_{n}(z)$
$\displaystyle=-M_{n}^{2}\,\mathsf{f}_{n}(z)\hskip 0.85358pt,$ (14a)
$\displaystyle\partial_{z}(A^{\prime}+\partial_{z})\tilde{\mathsf{f}}_{n}(z)$
$\displaystyle=-{\xi^{-1}_{n}}\\!\tilde{M}_{n}^{2}\,\tilde{\mathsf{f}}_{n}(z)\hskip
0.85358pt,$ (14b)

where $M_{n}$ and $\tilde{M}_{n}$ are the mass eigenvalue of KK level-$n\hskip
0.85358pt$. Applying the partial derivative $\partial_{z}$ on both sides of
Eq.(14a) and comparing it with Eq.(14b), we find that $\hskip
0.85358pt\partial_{z}\mathsf{f}_{n}\\!\propto\tilde{\mathsf{f}}_{n}$ and
$\tilde{M}_{n}^{2}\\!=\\!\xi_{n}M_{n}^{2}$. Thus, with proper normalization we
infer the following equations from the above Eq.(14a)-(14b):

$\displaystyle\partial_{z}\mathsf{f}_{n}(z)$ $\displaystyle=-M_{n}\hskip
0.85358pt\tilde{\mathsf{f}}_{n}(z)\hskip 0.85358pt,$ (15a)
$\displaystyle(A^{\prime}\\!+\partial_{z})\hskip
0.85358pt\tilde{\mathsf{f}}_{n}(z)$
$\displaystyle=M_{n}\mathsf{f}_{n}(z)\hskip 0.85358pt,$ (15b)

which connect 5d wavefunctions $\mathsf{f}_{n}(z)$ and
$\tilde{\mathsf{f}}_{n}(z)$. Substituting the KK expansions (12) into Eq.(11),
we obtain the boundary conditions imposed on
$(\mathsf{f}_{n},\tilde{\mathsf{f}}_{n})$:

$\partial_{z}\mathsf{f}_{n}(z)\big{|}_{z=0,L}=0\,,\qquad\tilde{\mathsf{f}}_{n}(z)\big{|}_{z=0,L}=0\,.$
(16)

They also obey the orthonormal conditions:

$\frac{1}{L\,}\\!\int_{0}^{L}\\!\\!\\!\text{d}z\,e^{A(z)}\hskip
0.85358pt\mathsf{f}_{n}(z)\hskip 0.85358pt\mathsf{f}_{m}(z)\hskip
0.85358pt=\hskip 0.85358pt\delta_{nm}\hskip 0.85358pt,\hskip
17.07164pt\frac{1}{L\,}\\!\int_{0}^{L}\\!\\!\\!\text{d}z\,e^{A(z)}\hskip
0.85358pt\tilde{\mathsf{f}}_{n}(z)\hskip
0.85358pt\tilde{\mathsf{f}}_{m}(z)\hskip 0.85358pt=\hskip
0.85358pt\delta_{nm}\,.$ (17)

The 5d wavefunctions $\mathsf{f}_{n}(z)$ and $\tilde{\mathsf{f}}_{n}(z)$ can
be solved from the equation of motion (14) in terms of Bessel functions of
order $\,\nu=\\!1,0$:

$\displaystyle\mathsf{f}_{n}(z)$
$\displaystyle=\frac{\,e^{-A(z)}\,}{N_{n}}\\!\left[J_{1}\\!\left(\\!\frac{M_{n}}{k}e^{-A(z)}\\!\right)\\!+b_{n0}\hskip
0.85358ptY_{1}\\!\left(\\!\frac{M_{n}}{k}e^{-A(z)}\\!\right)\\!\right]\\!,$
(18a) $\displaystyle\tilde{\mathsf{f}}_{n}(z)$
$\displaystyle=\frac{\,e^{-A(z)}\,}{{N}_{n}}\\!\left[J_{0}\\!\left(\\!\frac{M_{n}}{k}e^{-A(z)}\\!\right)\\!+b_{n0}Y_{0}\\!\left(\\!\frac{M_{n}}{k}e^{-A(z)}\\!\right)\\!\right]\\!,$
(18b)

where the solution $\mathsf{f}_{n}(z)$ agrees with Refs. RS-gauge RS-gauge2 .
In the above, the normalization factors $N_{n}$ can be fixed by the
orthonormal conditions (17), whereas the coefficient $b_{n0}$ is derived as
follows:

$b_{n0}=-\frac{~{}J_{0}\\!\left(\\!\frac{M_{n}}{k}\\!\right)~{}}{~{}Y_{0}\\!\left(\\!\frac{M_{n}}{k}\\!\right)~{}}\,.$
(19)

The KK mass $M_{n}$ is determined by roots of the eigenvalue equation:

$J_{0}\\!\left(\\!\\!\frac{\,M_{n}}{k}e^{-A(L)}\\!\right)+b_{n0}\hskip
0.85358ptY_{0}\\!\left(\\!\\!\frac{\,M_{n}}{k}e^{-A(L)}\\!\right)=0\,.$ (20)

For the massless zero-mode wavefunction $\mathsf{f}_{0}\hskip 0.85358pt$, we
can solve it from the eigenvalue equation (14a) and obtain:

$\mathsf{f}_{0}\,=\hskip
0.85358pt\sqrt{\\!\frac{~{}1\\!-e^{-A(L)}~{}}{A(L)}}\,.$ (21)

Under the KK expansions for the 5d gauge fields (12), the 4d effective KK
Lagrangian is derived by integrating over the fifth coordinate $z$ on the
interval $[\hskip 0.85358pt0,L\hskip 0.85358pt]$. The quadratic KK Yang-Mills
Lagrangian is given by

$\mathcal{L}_{\mathrm{YM}}^{(2)}=\\!\int_{0}^{L}\\!\\!\text{d}z\,\hat{\mathcal{L}}_{\mathrm{YM}}^{(2)}=-\frac{1}{4}\\!\sum_{n=0}^{\infty}\\!\left(\\!\partial_{\mu}A_{\nu}^{an}\\!-\\!\partial_{\nu}A_{\mu}^{an}\\!\right)^{\\!2}\\!-\frac{1}{2}\\!\sum_{n=1}^{\infty}\\!\left(\\!\partial_{\mu}A_{5}^{an}\\!+\\!M_{n}A_{\mu}^{an}\\!\right)^{\\!2}.$
(22)

Then, using Eq.(9) and making the KK expansions (12), we can derive the
following $R_{\xi}$ gauge-fixing term:

$\displaystyle\mathcal{L}_{\mathrm{GF}}$
$\displaystyle\,=\,\int_{0}^{L}\\!\\!\text{d}z\,\hat{\mathcal{L}}_{\mathrm{GF}}\,=\sum_{n=0}^{\infty}\frac{-1}{\,2\xi_{n}\,}(\mathcal{F}^{a}_{n})^{2}\hskip
0.85358pt,$ (23a) $\displaystyle\mathcal{F}^{a}_{n}$
$\displaystyle\,=\,\partial_{\mu}A^{a\mu}_{n}\\!+\xi_{n}M_{n}A_{n}^{a5}=\mathbf{K}_{n}^{T}\mathbf{A}_{n}^{a}\hskip
0.85358pt,$ (23b)

in agreement with Chivukula:2001esy He:2004zr , where
$\,\mathbf{K}_{n}\\!=\\!(\partial_{\mu},\hskip 0.85358pt\xi_{n}M_{n})^{T}$ and
$\mathbf{A}_{n}^{a}\\!=\\!(A_{n}^{a\mu},\hskip 0.85358ptA_{n}^{a5})^{T}$. In
the above gauge-fixing term, we have introduced a gauge-fixing parameter
$\xi_{n}$ for each KK level-$n\hskip 0.85358pt$.

Corresponding to the gauge-fixing (9), we have the 5d Faddeev-Popov ghost
term,

$\hat{\mathcal{L}}_{\mathrm{FP}}=e^{A(z)}\hat{\bar{c}}^{a}\widehat{\tt
s}\hat{\mathcal{F}}^{a}\,,$ (24)

where $\widehat{\tt s}$ is the Becchi-Rouet-Stora-Tyutin (BRST) operator and
the 5d BRST transformations take the form:

$\widehat{\tt
s}\hat{A}^{aN}\\!=D_{b}^{aN}\\!(\hat{A})\hat{c}^{b},~{}~{}~{}~{}~{}\widehat{\tt
s}\hat{c}^{a}\\!=\mbox{$\frac{\,{1}\,}{2}$}\hat{g}_{5}C^{abc}\hat{c}^{b}\hat{c}^{c},~{}~{}~{}~{}~{}\widehat{\tt
s}\hat{\bar{c}}^{a}\\!=-\xi^{-1}\hat{\mathcal{F}}^{a}.$ (25)

We make KK expansions for ghost fields,

$\displaystyle\hat{c}^{a}(x,z)$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\sum_{n=0}^{\infty}c^{a}_{n}(x)\hskip
0.85358pt\mathsf{f}_{n}(z)\,,$ (26a) $\displaystyle\hat{\bar{c}}^{a}(x,z)$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\sum_{n=1}^{\infty}\bar{c}^{a}_{n}(x)\hskip
0.85358pt\tilde{\mathsf{f}}_{n}(z)\,,$ (26b)

and derive the BRST transformations for the KK fields,

$\displaystyle\begin{aligned} \widehat{\tt
s}{A}^{a\mu}_{n}&=D^{ab,\mu}_{nm}{c}_{m}^{b}\,,~{}~{}~{}~{}~{}&\widehat{\tt
s}{A}^{a5}_{n}&=D^{ab,5}_{nm}{c}_{m}^{b}\,,\\\\[2.84526pt] \widehat{\tt
s}{c}_{n}^{a}&=\mbox{$\frac{\,{1}\,}{2}$}{g}\hskip
0.85358ptC^{abc}\mathfrak{D}^{ab,m\ell}_{n}{c}_{m}^{b}{c}_{\ell}^{c}\,,~{}~{}~{}~{}~{}&\widehat{\tt
s}{\bar{c}}_{n}^{a}&=-\xi_{n}^{-1}\mathcal{F}_{n}^{a}\,,\end{aligned}$ (27)

where the following notations are defined,

$\displaystyle D^{ab,\mu}_{nm}$
$\displaystyle=-\delta^{ab}\delta_{nm}\partial^{\mu}+\left(\\!\\!\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{2A}\hskip
0.85358pt\mathsf{f}_{n}\mathsf{f}_{m}\mathsf{f}_{\ell}\\!\right)\\!g\hskip
0.85358ptC^{abc}A^{c\mu}_{\ell}\,,$ $\displaystyle D^{ab,5}_{nm}$
$\displaystyle=-\delta^{ab}\delta_{nm}M_{m}+\left(\\!\\!\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{2A}\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\mathsf{f}_{m}\tilde{\mathsf{f}}_{\ell}\\!\right)\\!g\hskip
0.85358ptC^{abc}A^{c5}_{\ell}\,,$ (28)
$\displaystyle\mathfrak{D}^{ab,m\ell}_{n}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{2A}\hskip
0.85358pt\mathsf{f}_{n}\mathsf{f}_{m}\mathsf{f}_{\ell}\,.$

Using these, we can further derive a BRST identity for the propagator of the
gauge fields $\mathbf{A}_{n}^{a}\\!=\\!(A_{n}^{a\mu},\hskip
0.85358ptA_{n}^{a5})^{T}$ as follows:

$\mathbf{K}_{n}^{T}\bm{\mathcal{D}}_{nm}^{ab}(k)=-{\mathbf{X}}_{nm}^{ab}(k)^{T}\,,$
(29)

where we introduce the notations,

$\displaystyle\bm{\mathcal{D}}_{nm}^{ab}(k)$
$\displaystyle=\big{\langle}0|T\mathbf{A}_{n}^{a}{\mathbf{A}_{m}^{b}}^{T}|0\big{\rangle}\hskip
0.85358pt,~{}~{}~{}~{}~{}\SS_{nm}(k)\delta^{ab}=\big{\langle}0|Tc_{n}^{a}{\bar{c}_{m}^{b}}|0\big{\rangle}\hskip
0.85358pt,$ $\displaystyle{\mathbf{X}}_{nm}^{ab}(k)$
$\displaystyle=\underline{\mathbf{X}}_{mj}^{ab}(k)\SS_{jn}(k)\,,$
$\displaystyle\underline{\mathbf{X}}_{mj}^{ab}(k)$
$\displaystyle=\big{\langle}0|T\widehat{\tt
s}\mathbf{A}_{m}^{b}|\bar{c}_{j}^{a}\big{\rangle}=\\!\left(\\!\begin{matrix}\big{\langle}0|T\widehat{\tt
s}A_{m}^{b\mu}|\bar{c}_{j}^{a}\big{\rangle}\\\\[2.84526pt]
\big{\langle}0|T\widehat{\tt
s}A_{m}^{b5}|\bar{c}_{j}^{a}\big{\rangle}\end{matrix}\\!\right)\equiv\\!\left(\\!\begin{matrix}-\text{i}k^{\mu}\delta^{ab}[\delta_{jm}\\!+\\!\Delta^{a}_{jm}(k^{2})]\\\\[2.84526pt]
-M_{m}\delta^{ab}[\delta_{jm}\\!+\\!\tilde{\Delta}^{a}_{jm}(k^{2})]\end{matrix}\right)\\!,$
(30)

with $\Delta^{a}_{jm}(k^{2})$ and $\tilde{\Delta}^{a}_{jm}(k^{2})$ being loop-
level quantities.

In the following, we derive the KK gauge boson equivalence theorem (KK GAET)
for the compactified warped 5d Yang-Mills gauge theory. The KK GAET was
discussed He:2004zr  for the compactified 5d Yang-Mills gauge theories with a
general metric in connection to its deconstruction formulation, as an
extension of the formulations in the flat 5d KK gauge theories
Chivukula:2001esy Chivukula:2002ej . Our current derivation differs from
He:2004zr  since we focus on the warped RS1 and choose the conformal
coordinate frame of $(x^{\mu},z)$ instead of the physical coordinate frame of
$(x^{\mu},y)$. Besides, we will further study explicitly the three-point and
four-point KK gauge (Goldstone) boson scattering amplitudes in Section 3 and
their double-copy in Section 4.

Due to the 5d gauge invariance, we can derive a Slavnov-Taylor (ST) type of
identity:

$\big{\langle}0|T\hat{\mathcal{F}}^{\hskip
0.85358pta_{1}^{~{}}}(\hat{x}_{1})\hat{\mathcal{F}}^{\hskip
0.85358pta_{2}}(\hat{x}_{2})\cdots\hat{\mathcal{F}}^{\hskip
0.85358pta_{\\!N}}(\hat{x}_{\\!N}){\hat{\tilde{\Phi}}}|0\big{\rangle}\,=\,0\,,$
(31)

where $\hat{\tilde{\Phi}}$ denotes any possible external physical states in 5d
spacetime. This is analogous to the ST identity for external gauge-fixing
functions in the 4d SM with spontaneous electroweak gauge symmetry breaking
He:1993yd . Then, after making the KK expansion, we obtain the following ST
identities for the external KK gauge-fixing functions:

$\big{\langle}0|T\mathcal{F}^{\hskip
0.85358pta_{1}}_{n_{1}}(x_{1})\mathcal{F}^{\hskip
0.85358pta_{2}}_{n_{2}}(x_{2})\cdots\mathcal{F}^{\hskip
0.85358pta_{\\!N}}_{n_{\\!N}}(x_{\\!N}){\tilde{\Phi}}|0\big{\rangle}\,=\,0\,.$
(32)

Using the BRST identity (29), we can amputate each external $\mathcal{F}^{an}$
function in the momentum space and obtain a corresponding identity for the
$S$-matrix amplitude:

$\mathcal{T}[\underline{\mathcal{F}}^{\hskip
0.85358pta_{1}}_{n_{1}}(k_{1}),\hskip 0.85358pt\underline{\mathcal{F}}^{\hskip
0.85358pta_{2}}_{n_{2}}(k_{2}),\hskip 0.85358pt\cdots\\!,\hskip
0.85358pt\underline{\mathcal{F}}^{\hskip
0.85358pta_{\\!N}}_{n_{\\!N}}(k_{N}),{\Phi}]\,=\,0\,,$ (33)

where each external momentum $k_{j}$ ($j\\!=\\!1,2,\cdots$) is set on-shell
with $k_{j}^{2}\\!=\\!\\!-M_{n_{j}}^{2}$, each
$\underline{\mathcal{F}}^{\hskip 0.85358pta_{j}}_{n_{j}}\\!(k_{j})$ represents
the amputated external gauge-fixing function, and ${\Phi}$ denotes any
possible physical states after the LSZ (Lehmann-Symanzik-Zimmermann)
amputation. We can express the amputated gauge-fixing function as follows:

$\displaystyle\underline{\mathcal{F}}^{\hskip 0.85358pta}_{n}$
$\displaystyle=k_{\mu}A^{a\mu}_{n}\\!-M_{n}\hat{C}^{a}_{nm}A_{m}^{a5}=M_{n}(A^{an}_{S}\\!-\hat{C}^{a}_{nm}A_{5}^{am})=M_{n}(A_{L}^{an}\\!-{\mathbb{Q}}^{a}_{n})\hskip
0.85358pt,$ (34a) $\displaystyle{\mathbb{Q}}^{\hskip 0.85358pta}_{n}$
$\displaystyle=\hat{C}^{a}_{nm}A_{5}^{am}\\!+v_{n}^{a}\hskip
0.85358pt,~{}~{}~{}A_{S}^{an}=\epsilon_{S}^{\mu}A_{\mu}^{an},~{}~{}~{}A_{L}^{an}=\epsilon_{L}^{\mu}A_{\mu}^{an},~{}~{}~{}v^{a}_{n}=v_{n}^{\mu}A_{\mu}^{an},$
(34b)

with notations defined as

$\displaystyle\hat{C}^{a}_{nm}$
$\displaystyle=\text{i}(M_{m}/M_{n})\big{[}(\mathbf{1}\\!+\\!\bm{\tilde{\Delta}}^{\\!a}(k^{2}))(\mathbf{1}\\!+\\!\bm{\Delta}^{\\!a}(k^{2}))^{-1}\big{]}_{mn}=\text{i}\hskip
0.85358pt\delta_{nm}\\!+O(\mathrm{loop})\hskip 0.85358pt,$ (35a)
$\displaystyle\epsilon_{S}^{\mu}$
$\displaystyle=k^{\mu}\\!/M_{n}\\!=\epsilon_{L}^{\mu}\\!-v_{n}^{\mu}\hskip
0.85358pt,~{}~{}~{}v_{n}^{\mu}\\!=O(M_{n}/E_{n})\hskip 0.85358pt,$ (35b)

where the matrix form is used such that
$\big{(}\bm{\tilde{\Delta}}^{\\!a}\big{)}_{\\!jj^{\prime}}\\!\equiv\tilde{\Delta}^{\\!a}_{jj^{\prime}}$
and
$\big{(}\bm{\Delta}^{\\!a}\big{)}_{\\!jj^{\prime}}\\!\equiv\Delta^{\\!a}_{jj^{\prime}}\hskip
0.85358pt$. In the above, $\epsilon_{L}^{\mu}$ and $\epsilon_{S}^{\mu}$ denote
the KK gauge boson’s longitudinal polarization vector and scalar polarization
vector respectively. Thus, from the identity (33) we deduce the KK gauge
equivalence theorem (GAET) identity:

$\displaystyle\mathcal{T}[\hskip 0.85358ptA_{L\hskip
0.85358ptn_{1}}^{a_{1}},\hskip 0.85358pt\cdots\\!,\hskip 0.85358ptA_{L\hskip
0.85358ptn_{\\!N}}^{a_{N}}\\!,{\Phi}\hskip 0.85358pt]=\mathcal{T}[\hskip
0.85358pt{\mathbb{Q}}^{\hskip 0.85358pta_{1}}_{n_{1}},\hskip
0.85358pt\cdots\\!,\hskip 0.85358pt{\mathbb{Q}}_{n_{\\!N}}^{\hskip
0.85358pta_{N}}\\!,{\Phi}\hskip 0.85358pt]$ (36a) $\displaystyle\hskip
102.43008pt=\hat{C}_{\mathrm{mod}}^{n_{j}m_{j}}\mathcal{T}[\,A_{5}^{a_{1}m_{1}}\\!,\hskip
0.85358pt\cdots\\!,\hskip
0.85358ptA_{5}^{a_{N}m_{N}}\\!,{\Phi}\,]+\mathcal{T}_{v}\,,$ (36b)
$\displaystyle\mathcal{T}_{v}=\sum_{\ell=1}^{N}\tilde{C}_{\\!\mathrm{mod},\ell}^{n_{j}m_{j}}\hskip
0.85358pt\mathcal{T}[\,v^{a_{1}n_{1}},\cdots\\!,v^{a_{\ell}n_{\ell}},A_{5}^{a_{\ell+1}m_{\ell+1}},\cdots\\!,A_{5}^{a_{N}m_{N}},{\Phi}\,]\,,$
(36c)

where the multiplicative modification factors are given by

$\displaystyle\hat{C}_{\mathrm{mod}}^{n_{j}m_{j}}$
$\displaystyle=\hat{C}^{a_{1}}_{n_{1}m_{1}}\\!\cdots\hat{C}^{a_{N}}_{n_{\\!N}m_{\\!N}}=\,\text{i}^{N}\delta_{n_{1}m_{1}}\\!\\!\cdots\hskip
0.85358pt\delta_{n_{\\!N}m_{\\!N}}\\!+O(\mathrm{loop})\hskip 0.85358pt,$ (37a)
$\displaystyle\tilde{C}_{\\!\mathrm{mod},\ell}^{n_{j}m_{j}}$
$\displaystyle=\hat{C}^{a_{\ell+1}}_{n_{\ell+1}m_{\ell+1}}\\!\cdots\hat{C}^{a_{N}}_{n_{\\!N}m_{\\!N}}=\,\text{i}^{N-\ell}\delta_{n_{\ell+1}m_{\ell+1}}\\!\\!\cdots\hskip
0.85358pt\delta_{n_{\\!N}m_{\\!N}}\\!+O(\mathrm{loop})\hskip 0.85358pt.$ (37b)

Such radiative modification factors also exist in the ET formulation for the
4d electroweak gauge theory of the SM He:1993yd -He:1997zm . Note that the
residual term $\mathcal{T}_{v}$ in the KK-ET identity is suppressed under high
energy expansion due to $v_{n}^{\mu}\\!=O(M_{n}/E_{n})$. With these, we derive
the KK Equivalence Theorem for warped gauge theory (KK GAET) for the high
energy scattering:

$\mathcal{T}[\,A_{L}^{a_{1}n_{1}},\hskip 0.85358pt\cdots\\!,\hskip
0.85358ptA_{L}^{a_{N}n_{N}}\\!,{\Phi}\,]\hskip
0.85358pt=\,\hat{C}_{\mathrm{mod}}^{n_{j}m_{j}}\mathcal{T}[\,A_{5}^{a_{1}m_{1}}\\!,\hskip
0.85358pt\cdots\\!,\hskip
0.85358ptA_{5}^{a_{N}m_{N}}\\!,{\Phi}\,]+O(M_{n}/E_{n})\hskip 0.85358pt,$ (38)

where $\Phi$ denotes any other external physical state(s) after LSZ
amputation, and at tree-level the multiplicative factor
$\hat{C}_{\mathrm{mod}}^{n_{j}m_{j}}$ just reduces to a simple factor
$(\text{i}^{N}\delta_{n_{1}m_{1}}\\!\\!\cdots\hskip
0.85358pt\delta_{n_{\\!N}m_{\\!N}})\hskip 0.85358pt$.

###  2.3 Formulating Gravitational ET for Warped KK Gravity Theory

In this subsection, we first present the BRST quantization for the warped KK
gravity theory in Section 2.3.1. Then, we formulate the warped Gravitational
Equivalence Theorem (GRET) for the KK gravitons with helicity $\pm 1$ (called
GRET of type-II) in Section 2.3.2 and for the longitudinal KK gravitons with
helicity-0 (called GRET of type-I) in Section 2.3.3, respectively. Using the
GRET and our generalized energy-power counting method Hang:2021fmp , we
further prove the different large energy cancellations for the scattering
amplitudes of KK gravitons with helicity-0 and with helicity $\pm 1$ around
the end of this subsection.

####  2.3.1 Warped KK Gravity Theory with BRST Quantization

In this subsection, we consider the 5d RS1 with the fifth spatial dimension
compactified on an interval $[0,\hskip 0.85358pt{L}]$. The setup of RS1
contains warped 5d Einstein-Hilbert action and is compactified on a 5d
interval $[0,\hskip 0.85358pt{L}]$ with the two 3-branes located at the
boundaries. The relevant part of the warped 5d Lagrangian can be written as

$\hat{\mathcal{L}}_{\mathrm{RS}}=\frac{1}{2}\sqrt{-\hat{g}\,}\hat{M}_{\mathrm{P}}^{3}\big{(}\hat{R}-2\Lambda\big{)}\hskip
0.85358pt,$ (39)

where $\hat{R}$ and $\Lambda\,(<0)$ are the 5d Ricci scalar curvature and
cosmological constant respectively. The 5d Planck mass $\hat{M}_{\mathrm{P}}$
is given by
$\hat{M}_{\mathrm{P}}\\!=\\!M_{\mathrm{P}}^{\frac{2}{3}}\bar{L}^{-\frac{1}{3}}$,
where $M_{\mathrm{P}}$ denotes the 4d reduced Planck mass and is connected to
the 4d Newton constant $G$ via $M_{\mathrm{P}}\\!=\\!(8\hskip
0.85358pt\pi\hskip 0.85358ptG)^{-1/2}$. The possible brane terms RS1  are
irrelevant for the current analysis.

To study the linearized theory, we apply the following weak field expansion
for the 5d Lagrangian (39):

$\hat{g}_{MN}\,=\,e^{2A(z)}\big{(}\hat{\eta}_{MN}+\hat{\kappa}\hskip
0.85358pt\hat{h}_{MN}\big{)}\,,$ (40)

where the 5d gravitational coupling $\hat{\kappa}$ has mass-dimension $-3/2$
and is connected to the 4d gravitational coupling ${\kappa}$ via
$\,\hat{\kappa}=\kappa\sqrt{L\,}\\!=2M_{\mathrm{P}}^{-1}\sqrt{L\,}\,$. (Here
$M_{\mathrm{P}}$ is the 4d reduced Planck mass.) The 5d graviton field
$\hat{h}_{MN}$ can be further parametrized as follows Hang:2021fmp ;
Hang:2022rjp :

$\hat{h}_{MN}=\begin{pmatrix}\hat{h}_{\mu\nu}\\!-\\!\frac{1}{2}\eta_{\mu\nu}\hat{\phi}&~{}~{}\hat{\mathcal{V}}_{\mu}\\\\[5.69054pt]
\hat{\mathcal{V}}_{\nu}&~{}~{}\hat{\phi}\end{pmatrix}\\!,$ (41)

where we denote the (off-diagonal) vector component as
$\,\hat{\mathcal{V}}_{\mu}\equiv\hat{h}_{\mu 5}\hskip 0.85358pt$ and the
(diagonal) scalar component as $\hskip
0.85358pt\hat{\phi}\equiv\hat{h}_{55}\hskip 0.85358pt$. Then, we derive the
Lagrangian at quadratic order, which takes the following form:

$\displaystyle\hat{\mathcal{L}}_{\mathrm{RS}}^{(2)}=$
$\displaystyle~{}e^{3A(z)}\bigg{\\{}\\!-\\!\frac{1}{\,4\,}\hat{h}^{\mu\nu}\Big{\\{}\\!\big{(}\eta_{\mu\alpha}\partial_{\nu}\partial_{\beta}+\eta_{\mu\beta}\partial_{\nu}\partial_{\alpha}+\eta_{\nu\alpha}\partial_{\mu}\partial_{\beta}+\eta_{\nu\beta}\partial_{\mu}\partial_{\alpha}\big{)}\\!-2\big{(}\eta_{\mu\nu}\partial_{\alpha}\partial_{\beta}$
$\displaystyle\,+\eta_{\alpha\beta}\partial_{\mu}\partial_{\nu}\big{)}\\!-(\eta_{\mu\alpha}\eta_{\nu\beta}+\eta_{\mu\beta}\eta_{\nu\alpha}-2\eta_{\mu\nu}\eta_{\alpha\beta})\Big{[}\partial^{2}\\!+\\!(3A^{\prime}\\!+\partial_{z})\partial_{z}\Big{]}\Big{\\}}\hat{h}^{\alpha\beta}$
$\displaystyle\,+\hat{\mathcal{V}}^{\mu}\big{(}\eta_{\mu\nu}\partial^{2}\\!-\partial_{\mu}\partial_{\nu}\big{)}\hat{\mathcal{V}}^{\nu}\\!+\\!\frac{3}{\,4\,}\hat{\phi}\Big{[}\partial^{2}\\!-2(A^{\prime}\\!+\partial_{z})(2A^{\prime}\\!+\partial_{z})\Big{]}\hat{\phi}$
$\displaystyle-\\!\frac{1}{\,2\,}\Big{[}\hat{h}^{\mu\nu}(\eta_{\mu\alpha}\partial_{\nu}+\eta_{\nu\alpha}\partial_{\mu}\\!-2\eta_{\mu\nu}\partial_{\alpha})(3A^{\prime}\\!+\partial_{z})\hat{\mathcal{V}}^{\alpha}\\!+\hat{\mathcal{V}}^{\mu}(\eta_{\mu\alpha}\partial_{\beta}\\!+\eta_{\mu\beta}\partial_{\alpha}$
$\displaystyle-2\hskip
0.85358pt\eta_{\alpha\beta}\partial_{\mu})\partial_{z}\hat{h}^{\alpha\beta}\Big{]}+\frac{3}{\,4\,}\Big{[}\hat{h}(3A^{\prime}\\!+\partial_{z})(2A^{\prime}\\!+\partial_{z})\hat{\phi}+\hat{\phi}(A^{\prime}\\!+\partial_{z})\partial_{z}\hat{h}\Big{]}$
$\displaystyle-\\!\frac{3}{\,2\,}\Big{[}\hat{\mathcal{V}}^{\mu}(2A^{\prime}\\!+\partial_{z})\partial_{\mu}\hat{\phi}+\hat{\phi}(A^{\prime}\\!+\partial_{z})\partial_{\mu}\hat{\mathcal{V}}^{\mu}\Big{]}\bigg{\\}}\hskip
0.85358pt.$ (42)

This agrees with the literature Chacko:2003yp ; Callin:2004zm ;
Gherghetta:2005se ; Lim:2008 . To eliminate the mixing terms appearing in
Eq.(2.3.1), we choose the gauge-fixing terms as follows:

$\hat{\mathcal{L}}_{\mathrm{GF}}=-\frac{\,e^{3A(z)}\,}{\xi}\big{(}\hat{\mathcal{F}}_{\mu}^{2}+\hat{\mathcal{F}}_{5}^{2}\,\big{)}\hskip
0.85358pt,$ (43)

where $\,\xi\,$ is a gauge-fixing parameter and the gauge-fixing functions
$\hat{\mathcal{F}}_{\mu}$ and $\hat{\mathcal{F}}_{5}$ take the following form:

$\displaystyle\hat{\mathcal{F}}_{\mu}$
$\displaystyle=\partial^{\nu}\hat{h}_{\mu\nu}\\!-\\!\left(\\!1\\!-\\!\frac{1}{\,2\xi\,}\right)\\!\partial_{\mu}\hat{h}+\xi(3A^{\prime}\\!+\partial_{z})\hat{\mathcal{V}}_{\mu}\,,$
(44a) $\displaystyle\hat{\mathcal{F}}_{5}$
$\displaystyle=\frac{1}{\,2\,}\\!\left[\partial_{z}\hat{h}-3\hskip
0.85358pt\xi(2A^{\prime}\\!+\partial_{z})\hat{\phi}-2\partial_{\mu}\hat{\mathcal{V}}^{\mu}\right]\\!.$
(44b)

In the above, we denote the gravitational gauge-fixing parameter by $\,\xi\,$
for notational convenience, but it is independent of the gauge-fixing
parameter $\,\xi\,$ for gauge theories as defined in Eq.(9).

For the BRST quantization, the 5d gravitational ghost fields
$(\hat{c}^{M}\\!,\hskip 0.85358pt\hat{\bar{c}}^{M})$ are introduced to
compensate the gravitational gauge-fixing term (9) in the path integral
formulation. The 5d Faddeev-Popov Lagrangian takes the following form:

$\hat{\mathcal{L}}_{\mathrm{FP}}\,=\,e^{3A(z)}\,\hat{\bar{c}}^{M}\widehat{\tt
s}\hat{\mathcal{F}}_{M}\,,$ (45)

where the gauge-fixing functions,
$\hat{\mathcal{F}}_{M}\\!=\\!(\hat{\mathcal{F}}_{\mu},\,\hat{\mathcal{F}}_{5})$,
are defined as in Eq.(44), with the 5d Lorentz index $M\\!=\\!\mu,5\hskip
0.85358pt$.

Under the $S^{1}\\!/\mathbb{Z}_{2}$ orbifold compactification, the tensor
component of graviton field $\hat{h}_{\mu\nu}$ and the scalar component
$\hat{\phi}$ are assigned to be even under the $\mathbb{Z}_{2}$ parity,
whereas the vector component $\hat{\mathcal{V}}_{\mu}$ is $\mathbb{Z}_{2}$
odd:

$\displaystyle\hat{h}_{\mu\nu}(x,z)=\hat{h}_{\mu\nu}(x,-z)\hskip
0.85358pt,~{}~{}~{}~{}\hat{\mathcal{V}}_{\mu}(x,z)=-\hat{\mathcal{V}}_{\mu}(x,-z)\hskip
0.85358pt,~{}~{}~{}~{}\hat{\phi}(x,z)=\hat{\phi}(x,-z)\hskip 0.85358pt.$ (46)

We find that the ghost fields $(\hat{c}^{\hskip 0.85358pt\mu},\hat{c}^{\hskip
0.85358pt5})$ obey the same equations of motion as that of the gravitational
fields $(\hat{h}^{\mu\nu},\hskip 0.85358pt\hat{\mathcal{V}}^{\mu})$
respectively. Thus, we assign the following $\mathbb{Z}_{2}$ parities for the
ghost fields:

$\hat{c}^{\mu}(x,z)=\hat{c}^{\mu}(x,-z)\hskip
0.85358pt,~{}~{}~{}~{}\hat{c}^{5}(x,z)=-\hat{c}^{5}(x,-z)\hskip 0.85358pt,$
(47)

and their corresponding anti-ghost fields share the same $\mathbb{Z}_{2}$
parities.

With these, we make the following KK expansions for the 5d graviton fields in
terms of Fourier series containing the zero-modes and KK-modes:

$\displaystyle\hat{h}^{\mu\nu}(x,z)$
$\displaystyle=\frac{1}{\sqrt{L\,}\,}\sum_{n=0}^{\infty}h^{\mu\nu}_{n}(x)\,\mathsf{u}_{n}(z)\,,$
(48a) $\displaystyle\hat{\mathcal{V}}^{\mu}(x,z)$
$\displaystyle=\frac{1}{\sqrt{L\,}\,}\sum_{n=1}^{\infty}\mathcal{V}^{\mu}_{n}(x)\,\mathsf{v}_{n}(z)\,,$
(48b) $\displaystyle\hat{\phi}(x,z)$
$\displaystyle=\frac{1}{\sqrt{L\,}\,}\sum_{n=0}^{\infty}\phi_{n}(x)\,\mathsf{w}_{n}(z)\,.$
(48c)

Using the above KK decompositions, we derive the following equations of motion
(EOM) for the 5d gravitational KK wavefunctions:333Note that our analyses
below and elsewhere in this work are self-contained and well understood. Thus
it does not invoke any additional symmetries such as SUSY and alike.

$\displaystyle(3A^{\prime}\\!+\partial_{z})\partial_{z}\mathsf{u}_{n}(z)$
$\displaystyle=-\mathbb{M}_{n}^{2}\,\mathsf{u}_{n}(z)\hskip 0.85358pt,$ (49a)
$\displaystyle\partial_{z}(3A^{\prime}\\!+\partial_{z})\mathsf{v}_{n}(z)$
$\displaystyle=-\mathbb{M}_{n}^{2}\,\mathsf{v}_{n}(z)\hskip 0.85358pt,$ (49b)
$\displaystyle(2A^{\prime}\\!+\partial_{z})(A^{\prime}+\partial_{z})\mathsf{v}_{n}(z)$
$\displaystyle=-\mathbb{M}_{n}^{2}\,\mathsf{v}_{n}(z)\hskip 0.85358pt,$ (49c)
$\displaystyle(A^{\prime}\\!+\partial_{z})(2A^{\prime}\\!+\partial_{z})\mathsf{w}_{n}(z)$
$\displaystyle=-\mathbb{M}_{n}^{2}\,\mathsf{w}_{n}(z)\hskip 0.85358pt.$ (49d)

In the above and hereafter, we use the notation $\mathbb{M}_{n}$ to denote the
mass eigenvalue of KK gravitons at level-$n\hskip 0.85358pt$, which differs
from the KK gauge boson mass eigenvalue $M_{n}$ of level-$n\hskip 0.85358pt$
and is in contrast to the case of compactified flat 5d extra dimension. In
addition, Eqs.(49b) and (49c) are actually equivalent and we give both
formulas for convenience of the later analysis. The 5d wavefunctions
$\big{\\{}\mathsf{u}_{n}(z),\hskip 0.85358pt\mathsf{v}_{n}(z),\hskip
0.85358pt\mathsf{w}_{n}(z)\big{\\}}$ obey the following boundary conditions,

$\left.\partial_{z}\mathsf{u}_{n}(z)\right|_{z=0,L}=0\hskip
0.85358pt,~{}~{}~{}~{}\left.\mathsf{v}_{n}(z)\right|_{z=0,L}=0\hskip
0.85358pt,~{}~{}~{}~{}\left.(2A^{\prime}\\!+\partial_{z})\mathsf{w}_{n}(z)\right|_{z=0,L}=0\hskip
0.85358pt,$ (50)

and the orthonormal conditions,

$\displaystyle\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\\!\text{d}z\,e^{3A(z)}\hskip
0.85358pt\chi_{n}(z)\,\chi_{m}(z)=\delta_{nm}\,,$ (51)

where we denote $\chi_{n}\\!=\\{\mathsf{u}_{n},\hskip
0.85358pt\mathsf{v}_{n},\hskip 0.85358pt\mathsf{w}_{n}\\}\hskip 0.85358pt$.
The 5d wavefunctions $\mathsf{u}_{n}(z)$ and $\mathsf{v}_{n}(z)$ are connected
to each other by the equations:

$\displaystyle\partial_{z}\mathsf{u}_{n}(z)$
$\displaystyle=-\mathbb{M}_{n}\mathsf{v}_{n}(z)\,,$ (52a)
$\displaystyle(3A^{\prime}\\!+\partial_{z})\mathsf{v}_{n}(z)$
$\displaystyle=\mathbb{M}_{n}\mathsf{u}_{n}(z)\,,$ (52b)

whereas the 5d wavefunctions $\mathsf{w}_{n}(z)$ and $\mathsf{v}_{n}(z)$ are
connected by the equations,

$\displaystyle(2A^{\prime}\\!+\partial_{z})\mathsf{w}_{n}(z)$
$\displaystyle=-\mathbb{M}_{n}\mathsf{v}_{n}(z)\hskip 0.85358pt,$ (53a)
$\displaystyle(A^{\prime}\\!+\partial_{z})\mathsf{v}_{n}(z)$
$\displaystyle=\mathbb{M}_{n}\mathsf{w}_{n}(z)\hskip 0.85358pt.$ (53b)

Moreover, by using Eqs.(52b) and (53b), we obtain the following identity
connecting the three types of wavefunctions:

$\mathsf{u}_{n}(z)=\mathsf{w}_{n}(z)+\frac{\,2A^{\prime}\,}{\mathbb{M}_{n}}\mathsf{v}_{n}(z)\,.$
(54)

In passing, the relation of Eq.(54) to the quantum-mechanical SUSY algebra was
discussed in the literature SUSY .

By solving the equation of motion (49), the 5d wavefunctions
$\mathsf{u}_{n}(z),\hskip 0.85358pt\mathsf{v}_{n}(z)$ and $\mathsf{w}_{n}(z)$
can be expressed in terms of the Bessel functions of order $\nu=2,1,0\hskip
0.85358pt$ respectively:

$\displaystyle\mathsf{u}_{n}(z)$
$\displaystyle\,=\,\frac{\,e^{-2A(z)\,}}{N_{n}^{\prime}}\\!\left[J_{2}\\!\left(\\!\frac{\,\mathbb{M}_{n}}{k}e^{-A(z)}\\!\right)\\!+b_{n1}Y_{2}\\!\left(\frac{\mathbb{M}_{n}}{k}e^{-A(z)}\\!\right)\right]\\!,$
(55a) $\displaystyle\mathsf{v}_{n}(z)$
$\displaystyle\,=\,\frac{e^{-2A(z)}}{N_{n}^{\prime}}\left[J_{1}\\!\left(\\!\frac{\,\mathbb{M}_{n}}{k}e^{-A(z)}\\!\right)\\!+b_{n1}Y_{1}\\!\left(\frac{\,\mathbb{M}_{n}}{k}e^{-A(z)}\\!\right)\right]\\!,$
(55b) $\displaystyle\mathsf{w}_{n}(z)$
$\displaystyle\,=\,\frac{e^{-2A(z)}}{N_{n}^{\prime}}\left[J_{0}\\!\left(\,\frac{\mathbb{M}_{n}}{k}e^{-A(z)}\\!\right)\\!+b_{n1}Y_{0}\\!\left(\frac{\,\mathbb{M}_{n}}{k}e^{-A(z)}\\!\right)\right]\\!,$
(55c)

where $N_{n}^{\prime}$ is the normalization factor and $b_{n1}$ is given by

$b_{n1}\,=\hskip
0.85358pt-\frac{\,J_{1}\\!\left(\\!\frac{\,\mathbb{M}_{n}\,}{k}\\!\right)}{~{}Y_{1}\\!\left(\\!\frac{\,\mathbb{M}_{n}\,}{k}\\!\right)\,}\,.$
(56)

The KK mass eigenvalue $\mathbb{M}_{n}$ is determined by solving the following
equation:

$J_{1}\\!\left(\\!\frac{\,\mathbb{M}_{n}}{k}e^{-A(L)}\\!\right)\\!+b_{n1}Y_{1}\\!\left(\\!\frac{\,\mathbb{M}_{n}}{k}e^{-A(L)}\\!\right)=0\,.$
(57)

We note that the KK graviton mass $\mathbb{M}_{n}$ solved from the above
eigenvalue equation (57) differs from the KK gauge boson mass $M_{n}$ solved
from Eq.(20). Then, the zero-mode wavefunctions $\mathsf{u}_{0}(z)$ and
$\mathsf{w}_{0}(z)$ can be solved as follows:

$\displaystyle\mathsf{u}_{0}$
$\displaystyle\,=\sqrt{2\,}\big{[}e^{A(L)}\\!+e^{2A(L)}\big{]}^{-\frac{1}{2}}\,,$
(58a) $\displaystyle\mathsf{w}_{0}(z)$ $\displaystyle=\,\sqrt{2\,}\hskip
0.85358pte^{-2A(z)}\big{[}1\\!+e^{-A(L)}\big{]}^{-\frac{1}{2}},$ (58b)

where $\mathsf{u}_{0}$ has no dependence on $z\hskip 0.85358pt$.

As shown in Eqs.(46) and (47), the 5d ghost fields $(\hat{c}^{\hskip
0.85358pt\mu},\hat{c}^{\hskip 0.85358pt5})$ share the same $\mathbb{Z}_{2}$
parities as that of the gravitational fields $(\hat{h}^{\mu\nu},\hskip
0.85358pt\hat{\mathcal{V}}^{\mu})$ respectively. So we make the following KK
expansions for the gravitational ghost and anti-ghost fields,

$\displaystyle\hat{c}^{\mu}(x,z)$
$\displaystyle=\frac{1}{\sqrt{L\,}\,}\\!\sum_{n=0}^{\infty}\\!c^{\mu}_{n}(x)\mathsf{u}_{n}(z)\hskip
0.85358pt,\qquad\hat{\bar{c}}^{\mu}(x,z)=\frac{1}{\sqrt{L\,}\,}\\!\sum_{n=0}^{\infty}\\!\bar{c}_{n}^{\mu}(x)\mathsf{u}_{n}(z)\hskip
0.85358pt,$ (59a) $\displaystyle\hat{c}^{5}(x,z)$
$\displaystyle=\frac{1}{\sqrt{L\,}\,}\\!\sum_{n=1}^{\infty}\\!c_{n}^{5}(x)\mathsf{v}_{n}(z)\hskip
0.85358pt,\qquad\hat{\bar{c}}^{5}(x,z)=\frac{1}{\sqrt{L\,}\,}\\!\sum_{n=1}^{\infty}\\!\bar{c}_{n}^{5}(x)\mathsf{v}_{n}(z)\hskip
0.85358pt.$ (59b)

We can further derive the BRST transformations for the KK fields, which are
given in Appendix B.

Making KK expansions for the 5d Lagrangian (2.3.1) and integrating $z$ over
the interval $[\hskip 0.85358pt0,\hskip 0.85358ptL\hskip 0.85358pt]$, we
derive the 4d KK Lagrangian at quadratic order:

$\displaystyle\mathcal{L}^{(2)}_{\mathrm{RS}}\,=\,$
$\displaystyle\sum_{n=0}^{\infty}\bigg{[}\,\frac{{1}}{\,2\,}(\partial_{\mu}h_{n})^{2}\\!+\\!\frac{{1}}{\,2\,}\mathbb{M}_{n}^{2}(h_{n})^{2}\\!-\\!\frac{{1}}{\,2\,}(\partial^{\rho}h^{\mu\nu}_{n})^{2}\\!-\\!\frac{{1}}{\,2\,}\mathbb{M}_{n}^{2}(h^{\mu\nu}_{n})^{2}\\!-\\!(\partial^{\mu}\mathcal{V}^{\nu}_{n})^{2}\\!$
$\displaystyle\hskip
24.46938pt+\\!(\partial_{\mu}\mathcal{V}_{n}^{\mu})^{2}\\!-\\!\frac{3}{\,4\,}(\partial_{\mu}\phi_{n})^{2}+\frac{3}{2}\mathbb{M}_{n}^{2}\phi_{n}^{2}-\partial_{\mu}h^{\mu\nu}_{n}\partial_{\nu}h_{n}\\!+\partial_{\mu}h^{\mu\rho}_{n}\partial^{\nu}h_{\nu\rho,n}$
$\displaystyle\hskip
24.46938pt+2\mathbb{M}_{n}h_{n}\partial_{\mu}\mathcal{V}^{\mu}_{n}\\!-2\mathbb{M}_{n}h^{\mu\nu}_{n}\partial_{\mu}\mathcal{V}_{\nu,n}\\!-\\!\frac{3}{\,2\,}\mathbb{M}_{n}^{2}h_{n}\phi_{n}\\!-3\mathbb{M}_{n}\partial_{\mu}\mathcal{V}^{\mu}_{n}\phi_{n}\bigg{]}.$
(60)

We further derive the 4d $R_{\xi}$-type gauge-fixing terms under the KK
expansions:

$\mathcal{L}_{\mathrm{GF}}=\int_{0}^{L}\\!\\!\text{d}z\,\hat{\mathcal{L}}_{\mathrm{GF}}\hskip
0.85358pt=\hskip 0.85358pt-\sum_{n=0}^{\infty}\\!\frac{1}{\,\xi_{n}\hskip
0.85358pt}\\!\left[(\mathcal{F}_{n}^{\mu})^{2}\\!+\\!(\mathcal{F}_{n}^{5})^{2}\right]\\!,$
(61)

where $\hskip 0.85358pt\xi_{n}$ is the gauge-fixing parameter for KK
level-$n\hskip 0.85358pt$, and the gauge-fixing functions
$\mathcal{F}_{n}^{\mu}$ and $\mathcal{F}_{n}^{5}$ take the following forms:

$\displaystyle\mathcal{F}_{n}^{\mu}$
$\displaystyle\,=\,\partial_{\nu}h_{n}^{\mu\nu}\\!-\\!\left(\\!1\\!-\\!{\frac{1}{\,2\hskip
0.85358pt\xi_{n}}\,}\right)\\!\partial^{\mu}h_{n}\\!+\xi_{n}\mathbb{M}_{n}\mathcal{V}^{\mu}_{n}\,,$
(62a) $\displaystyle\mathcal{F}_{n}^{5}$ $\displaystyle\,=\hskip
0.85358pt{\frac{1}{2}}\\!\left(\mathbb{M}_{n}h_{n}\\!-\\!3\xi_{n}\mathbb{M}_{n}\phi_{n}\\!+\\!2\partial_{\mu}\mathcal{V}^{\mu}_{n}\right)\\!.$
(62b)

They take the same forms as what we obtained for the flat 5d compactification
Hang:2021fmp Hang:2022rjp . The above gauge-fixing terms are designed to
cancel the mixing terms among the KK graviton fields $h_{n}^{\mu\nu}$ and the
KK gravitational Goldstone fields $\mathcal{V}_{n}^{\mu}$ and $\phi_{n}\hskip
0.85358pt$. To ensure proper normalization of the kinetic terms of
$\mathcal{V}_{n}^{\mu}$ and $\phi_{n}$, we make the following rescaling,

$\displaystyle\mathcal{V}^{\mu}_{n}\rightarrow\frac{1}{\sqrt{2\,}\,}\mathcal{V}^{\mu}_{n}\,,\quad~{}~{}~{}\phi_{n}\rightarrow\sqrt{\frac{2}{3}\,}\phi_{n}\,,$
(63)

under which the gauge-fixing functions (62a)-(62b) become:

$\displaystyle\mathcal{F}_{n}^{\mu}$
$\displaystyle\,=\,\partial_{\nu}h_{n}^{\mu\nu}\\!-\\!\left(\\!1\\!-\\!{\frac{1}{\,2\hskip
0.85358pt\xi_{n}}\,}\right)\\!\partial^{\mu}h_{n}\\!+\frac{1}{\sqrt{2\,}\,}\xi_{n}\mathbb{M}_{n}\mathcal{V}^{\mu}_{n}\,,$
(64a) $\displaystyle\mathcal{F}_{n}^{5}$ $\displaystyle\,=\hskip
0.85358pt{\frac{1}{\hskip 0.85358pt2\hskip
0.85358pt}}\,\\!\mathbb{M}_{n}h_{n}\\!-\\!{\sqrt{\frac{3}{2}}}\,\xi_{n}\mathbb{M}_{n}\phi_{n}\\!+\\!{\frac{1}{\sqrt{2\,}\,}}\,\partial_{\mu}\mathcal{V}^{\mu}_{n}\,.$
(64b)

Thus, we derive the propagators of KK fields $(h_{n}^{\mu\nu},\hskip
0.85358pt\mathcal{V}_{n}^{\mu},\hskip 0.85358pt\phi_{n})$ at tree level (as in
Hang:2021fmp ) which take the simple diagonal forms in the Feynman-’t Hooft
gauge ($\xi_{n}\\!=1$):

$\displaystyle\mathcal{D}_{h,nm}^{\mu\nu\alpha\beta}(k)$
$\displaystyle=-\frac{\,\text{i}\delta_{nm}\,}{2}\frac{\,\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta}\,}{\,k^{2}\\!+\\!\mathbb{M}_{n}^{2}\,}\,,$
(65a) $\displaystyle\mathcal{D}^{\mu\nu}_{\mathcal{V},nm}(k)$
$\displaystyle=-\frac{~{}\text{i}\eta^{\mu\nu}\delta_{nm}~{}}{\,k^{2}\\!+\\!\mathbb{M}_{n}^{2}\,}\,,$
(65b) $\displaystyle\mathcal{D}_{\\!\phi,nm}(k)$
$\displaystyle=-\frac{\,\text{i}\delta_{nm}\,}{~{}k^{2}\\!+\\!\mathbb{M}_{n}^{2}~{}}\,.$
(65c)

We will present the BRST transformations in Appendix B.

Then, we derive a general (Slavnov-Taylor-type) identity in the momentum space
for the gravitational gauge-fixing functions (64a)-(64b) as external lines of
the Green function Hang:2021fmp Hang:2022rjp :

$\big{\langle}0\hskip 0.85358pt|\hskip 0.85358pt\mathcal{F}_{n_{1}}^{\hskip
0.85358pt\mu_{1}}(k_{1})\hskip 0.85358pt\mathcal{F}_{n_{2}}^{\hskip
0.85358pt\mu_{2}}(k_{2})\cdots\mathcal{F}_{m_{1}}^{\hskip
0.85358pt5}\\!(p_{1})\hskip 0.85358pt\mathcal{F}_{m_{2}}^{\hskip
0.85358pt5}\\!(p_{2})\cdots\Phi\hskip 0.85358pt|\hskip
0.85358pt0\big{\rangle}=\,0\,,$ (66)

where each external momentum ($k_{j}$ or $p_{j}$) is chosen as incoming and
$\Phi$ denotes any possible external physical states after the LSZ reduction.

####  2.3.2 Formulating GRET for Warped KK Gravity Theory (I)

In this subection, we first formulate a KK Gravitational Equivalence Theorem
(GRET of type-II)444For convenience, we call this GRET (for KK gravitons with
helicity $\pm 1$) the GRET of type-II, and call the other GRET (for KK
gravitons with helicity-0) the GRET of type-I as will be given in Section
2.3.3. to quantitatively connect the scattering amplitude of KK gravitons
$h^{\mu\nu}_{n}$ with helicity $\pm 1$ to that of the corresponding
gravitational KK vector Goldstone bosons $\mathcal{V}_{n}^{\mu}$. We derived
the general identity (66) for the gravitational gauge-fixing functions
$\mathcal{F}_{n}^{\hskip 0.85358pt\mu}$ and $\mathcal{F}_{n}^{\hskip
0.85358pt5}$ of Eq.(64).

We reexpress the gauge-fixing function (64a) in matrix form:

$\displaystyle\mathcal{F}_{n}^{\mu}$
$\displaystyle=-\frac{\,\text{i}\mathbb{M}_{n}\,}{\sqrt{2\,}\,}\mathbb{F}_{n}^{\mu}\,,\quad\mathbb{F}_{n}^{\mu}=\mathbf{K}_{n}^{T}\mathbf{H}_{n}\,,$
(67a) $\displaystyle\mathbf{K}_{n}$
$\displaystyle=\Big{(}\\!\frac{1}{\,\sqrt{2\,}\mathbb{M}_{n}}\\!\big{[}2k_{\beta}\tensor{\eta}{{}_{\alpha}^{\mu}}\\!-k^{\mu}(2\\!-\xi_{n}^{-1})\eta_{\alpha\beta}\big{]},\,\text{i}\xi_{n}\Big{)}^{\\!T},\quad\mathbf{H}_{n}\\!=\big{(}h_{n}^{\alpha\beta},\,\mathcal{V}_{n}^{\mu}\big{)}^{\\!T}$

Then, we derive the following BRST identity including the gauge-fixing
function $\mathbb{F}_{n}^{\mu}\hskip 0.85358pt$,

$\big{\langle}0\big{|}T\,\mathbb{F}_{n}^{\mu}\mathbf{H}_{m}^{T}\big{|}0\big{\rangle}=-\frac{\text{i}\xi_{n}}{\sqrt{2}\hskip
0.85358pt\mathbb{M}_{n}}\big{\langle}0\big{|}T\,\widehat{\tt
s}\mathbf{H}_{m}^{T}\bar{c}^{\mu}_{n}\big{|}0\big{\rangle}\,.$ (67bp)

Using Eq.(67), we can rewrite Eq.(67bp) as follows:

$\mathbf{K}_{n}^{T}\bm{\mathcal{D}}_{nm}(k)=-\mathbf{X}^{T}_{nm}(k)\,,$ (67bq)

where we have defined the notations,

$\displaystyle\bm{\mathcal{D}}_{nm}(k)$
$\displaystyle=\big{\langle}0\big{|}T\hskip
0.85358pt\mathbf{H}_{n}\mathbf{H}_{m}^{T}\big{|}0\big{\rangle}(k)\hskip
0.85358pt,~{}~{}~{}$ $\displaystyle\mathbf{X}_{nm}(k)$
$\displaystyle=\underline{\mathbf{X}}_{mj}(k)\SS_{jn}(k)\hskip
0.85358pt,~{}~{}~{}{\SS_{jn}(k)}\hskip
0.85358pt{\eta}^{\lambda\mu}\\!=\big{\langle}0\big{|}T{c_{j}^{\lambda}\bar{c}_{n}^{\hskip
0.85358pt\mu}}\big{|}0\big{\rangle}(k)\hskip 0.85358pt,$ (67br)
$\displaystyle\underline{\mathbf{X}}_{mj}(k)$
$\displaystyle=\frac{\,\text{i}\hskip
0.85358pt{\eta}^{\lambda\mu}}{\,\sqrt{2\,}\hskip
0.85358pt{\mathbb{M}_{j}}\,}\\!\\!\begin{pmatrix}\\!\big{\langle}0\big{|}\hskip
0.85358ptT\,\widehat{\tt s}h^{\sigma\rho}_{m}\big{|}\bar{c}_{j,\lambda}\hskip
0.85358pt\big{\rangle}(k)\\\\[2.84526pt] \big{\langle}0\big{|}\hskip
0.85358ptT\,\widehat{\tt
s}\mathcal{V}_{m}^{\nu}\big{|}\bar{c}_{j,\lambda}\hskip
0.85358pt\big{\rangle}(k)\end{pmatrix}\\!=\\!\frac{1}{\sqrt{2\,}\hskip
0.85358pt{\mathbb{M}_{j}}}\\!\\!\begin{pmatrix}\\!(k^{\sigma}{\eta}^{\rho\mu}\\!+k^{\rho}{\eta}^{\sigma\mu})[\hskip
0.85358pt\delta_{mj}\\!+\\!\Delta^{\\!(1)}_{mj}(k^{2})\hskip
0.85358pt]\\\\[2.84526pt] \text{i}\sqrt{2\,}\hskip
0.85358pt{\mathbb{M}_{m}}{\eta}^{\mu\nu}[\hskip
0.85358pt\delta_{mj}\\!+\\!\widetilde{\Delta}^{\\!(2)}_{mj}(k^{2})\hskip
0.85358pt]\end{pmatrix}\\!,\hskip 11.38109pt$

where $\Delta^{(1)}_{mj}(k^{2})$ and $\widetilde{\Delta}^{(2)}_{mj}(k^{2})$
are the loop-level quantities, and are further defined in Eq.(67kz) of
Appendix B. In the above, we have chosen the Feynman-’t Hooft gauge
$\xi_{n}\\!=\\!1\hskip 0.85358pt$ for simplicity of the
$\underline{\mathbf{X}}_{mj}$ formula. The case of general $R_{\xi}$ gauge
will be discussed in Appendix B.

Then, we analyze the identity (66) and construct an external state as a
combination of gauge-fixing functions ($\mathcal{F}_{n}^{\mu}$ or
$\mathbb{F}_{n}^{\mu}$) as given in Eq.(67). Thus, we amputate this external
state by using the BRST identity (67bq) and obtain:

$\displaystyle 0\,=\,\big{\langle}0|\hskip 0.85358pt\mathbb{F}_{n}^{\hskip
0.85358pt\mu}(k)\cdots{{\Phi}}\hskip
0.85358pt|0\big{\rangle}=-\mathbf{X}_{nm}^{T}(k)\hskip
0.85358pt\mathcal{M}\big{[}\underline{\mathbf{H}}_{m}(k),\cdots\\!,{{\Phi}}\hskip
0.85358pt\big{]}.$ (67bs)

From the above, we can derive the following identity:

$\displaystyle\mathcal{M}\big{[}\underline{\mathbb{F}}_{n}^{\hskip
0.85358pt\mu}(k),\cdots\\!,{\Phi}\hskip 0.85358pt\big{]}=0\,,$ (67bt)

where $\mathcal{M}[\cdots]$ denotes the amputated scattering amplitude. In
Eq.(67bt), the amputated external state $\underline{\mathbb{F}}_{n}^{\hskip
0.85358pt\mu}$ takes the following form (after removing an irrelevant overall
coefficient):

$\underline{\mathbb{F}}_{n}^{\hskip
0.85358pt\mu}(k)=\frac{\sqrt{2\,}\,}{\,\mathbb{M}_{n}}\hskip
0.85358ptk_{\nu}h_{n}^{\mu\nu}\\!-\hat{C}^{\hskip 0.85358ptnm}\hskip
0.85358pt{\eta}^{\hskip 0.85358pt\mu\nu}\mathcal{V}_{m,\nu}\,,$ (67bu)

where the loop-induced modification factor is derived as follows:

$\hat{C}_{\hskip
0.85358ptnm}(k^{2})=-\frac{\,\text{i}\mathbb{M}_{m}\,}{\mathbb{M}_{n}}\\!\\!\left[\\!\frac{\,{\mathbf{1}}\\!+\\!\tilde{\bm{\Delta}}^{\\!(2)}(k^{2})\,}{\,{\mathbf{1}}\\!+\\!\bm{\Delta}^{\\!(1)}(k^{2})\,}\\!\right]_{mn}\\!=-\text{i}\hskip
0.85358pt\delta_{nm}+O(\mathrm{loop})\hskip 0.85358pt.$ (67bv)

In the above, the matrix form is used such that the loop-level quantities
$(\bm{\Delta}^{\\!(1)})_{jj^{\prime}}\\!={\Delta}^{\\!(1)}_{jj^{\prime}}$ and
$(\bm{\tilde{\Delta}}^{\\!(2)})_{jj^{\prime}}\\!=\\!\tilde{\Delta}^{\\!(2)}_{jj^{\prime}}\hskip
0.85358pt$. The derivations of Eqs.(67bs)-(67bv) are given in more detail
around Eqs.(B.3)-(67lb) of Appendix B.

Then, we utilize a transverse polarization vector $\epsilon^{\mu}_{\pm}$ to
contract with $\underline{\mathbb{F}}_{n}^{\hskip 0.85358pt\mu}(k)$ in
Eq.(67bu) and obtain the formula:

$\underline{\mathbb{F}}_{n}(k)\equiv\epsilon_{\mu}^{\pm}\hskip
0.85358pt\underline{\mathbb{F}}_{n}^{\hskip 0.85358pt\mu}(k)=h_{n}^{S,\pm
1}-\hat{C}_{\hskip 0.85358ptnm}\hskip 0.85358pt\mathcal{V}^{\pm 1}_{m}\,,$
(67bw)

where we have used the fact $\hskip 0.85358pt\epsilon^{\mu}_{\pm}\hskip
0.85358ptk_{\mu}\\!=0\hskip 0.85358pt$, and defined the external states
$h^{S,\pm 1}_{n}$ and $\mathcal{V}^{\pm 1}_{m}$ as follows:

$h_{n}^{S,\pm 1}\\!=\varepsilon^{S,\pm 1}_{\mu\nu}h^{\mu\nu}_{n}\hskip
0.85358pt,~{}~{}~{}~{}\mathcal{V}^{\pm 1}_{m}\\!=\epsilon_{\mu}^{\pm}\hskip
0.85358pt\mathcal{V}_{m}^{\mu}\hskip 0.85358pt,~{}~{}~{}~{}\varepsilon^{S,\pm
1}_{\mu\nu}\\!=\\!{\frac{1}{\sqrt{2\,}\,}}\big{(}\epsilon^{S}_{\mu}\epsilon_{\nu}^{\pm}\\!+\epsilon^{S}_{\nu}\epsilon_{\mu}^{\pm}\big{)},$
(67bx)

with $\epsilon^{S}_{\mu}\\!\\!=\\!\\!k_{\mu}/\mathbb{M}_{n}$ being the scalar-
polarization vector. A massive spin-2 KK graviton has 5 physical helicity
states $(\lambda\\!=\\!\pm 2,\pm 1,0)$. The polarization tensors of KK
graviton with helicities $\pm 1$ are given by

$\varepsilon_{\pm
1}^{\mu\nu}={\frac{1}{\sqrt{2\,}\,}}\\!\left(\epsilon_{\pm}^{\mu}\epsilon_{L}^{\nu}\\!+\epsilon_{L}^{\mu}\epsilon_{\pm}^{\nu}\right)\\!,$
(67by)

where $(\epsilon_{\pm}^{\mu},\,\epsilon_{L}^{\mu})$ denote the 3 polarization
vectors of a massive spin-1 KK gauge boson. Under high energy expansion, we
have $\epsilon_{L}^{\mu}\\!=\epsilon_{S}^{\mu}+v^{\mu}$ with
$\epsilon_{S}^{\mu}\\!=\\!k^{\mu}\\!/\mathbb{M}_{n}$ and
$v^{\mu}\\!=O(\mathbb{M}_{n}/E_{n})\hskip 0.85358pt$. Thus, we can decompose
the polarization tensor $\varepsilon_{\pm 1}^{\mu\nu}$ as follows:

$\displaystyle\varepsilon_{\pm 1}^{\mu\nu}$ $\displaystyle=\varepsilon_{S,\pm
1}^{\mu\nu}+v_{\pm 1}^{\mu\nu}\,,~{}~{}~{}~{}\varepsilon_{S,\pm
1}^{\mu\nu}=\\!\mbox{$\frac{\,{1}\,}{\sqrt{2\,}\,}$}\big{(}\epsilon_{S}^{\mu}\epsilon^{\nu}_{\pm}\\!+\epsilon_{S}^{\nu}\epsilon^{\mu}_{\pm}\big{)},$
(67bza) $\displaystyle v_{\pm 1}^{\mu\nu}$
$\displaystyle=\\!\mbox{$\frac{\,{1}\,}{\sqrt{2\,}\,}$}\big{(}v^{\mu}\epsilon^{\nu}_{\pm}\\!+v^{\nu}\epsilon^{\mu}_{\pm}\big{)}=O(\mathbb{M}_{n}/E)\hskip
0.85358pt.$ (67bzb)

With these, we can reexpress the function $\underline{\mathbb{F}}_{n}(k)$ of
Eq.(67bw) in the form:

$\displaystyle\underline{\mathbb{F}}_{n}(k)=h_{n}^{\pm 1}\\!-\Theta_{n}\hskip
0.85358pt,~{}~{}~{}~{}\Theta_{n}\\!=\hat{C}^{\hskip
0.85358ptnm}\mathcal{V}_{m}^{\pm 1}\\!+v_{n}^{\pm 1}\hskip 0.85358pt,$ (67ca)

where we have used the notations,

$h_{n}^{\pm 1}\\!=\varepsilon^{\pm
1}_{\mu\nu}h^{\mu\nu}_{n},\quad~{}\mathcal{V}_{m}^{\pm
1}\\!=\epsilon^{\pm}_{\mu}\mathcal{V}^{\mu}_{m}\hskip
0.85358pt,\quad~{}v_{n}^{\pm 1}\\!=v_{\mu\nu}^{\pm 1}h^{\mu\nu}_{n}\hskip
0.85358pt,$ (67cb)

with the tensor $v_{\pm 1}^{\mu\nu}\\!=O(\mathbb{M}_{n}/E_{n})$ under high
energy expansion and thus $v_{n}^{\pm 1}$ is suppressed by an energy factor of
$O(\mathbb{M}_{n}/E_{n})\hskip 0.85358pt$. For later analysis, we can also
multiply $\,v^{\mu}\\!=\\!\epsilon_{L}^{\mu}\\!-\epsilon_{S}^{\mu}\,$ on both
sides of Eq.(67bt) and use the expression (67bu) to obtain the following
identity:

$\epsilon_{\mu}^{S}v_{\nu}\hskip
0.85358pt\mathcal{M}\big{[}h_{n}^{\mu\nu}(k),\cdots\\!,{\Phi}\hskip
0.85358pt\big{]}=\hat{C}_{nm}\hskip 0.85358ptv_{\mu}\hskip
0.85358pt\mathcal{M}\big{[}\mathcal{V}_{m}^{\mu}(k),\cdots\\!,{\Phi}\hskip
0.85358pt\big{]},$ (67cc)

where we see that the right-hand side is suppressed by the external factor
$v_{\mu}\\!=O(\mathbb{M}_{n}/E_{n})\hskip 0.85358pt$. Hence the above identity
(67cc) ensures that its left-hand side must receive a suppression such that
its overall energy-power dependence is the same as the right-hand side
[despite that the external factor of the left-hand side is
$\epsilon_{\mu}^{S}v_{\nu}\\!=O(E^{0}_{n})$].

Then, we reformulate the identity (67bt) and extend it to the general case of
$N$ external amputated gauge-fixing functions:

$\mathcal{M}\big{[}\underline{\mathbb{F}}_{n_{1}}\\!(k_{1}),\cdots\\!,\underline{\mathbb{F}}_{n_{\\!N}}\\!(k_{\\!N}),{\Phi}\hskip
0.85358pt\big{]}=0\hskip 0.85358pt.$ (67cd)

Using this identity, we can further derive the gravitational equivalence
theorem (GRET) identity which connects the $h_{n}^{\pm 1}$ amplitude to the
amplitude of $\Theta_{n}$:

$\displaystyle\mathcal{M}\big{[}h_{n_{1}}^{\pm
1}(k_{1}),\cdots\\!,h_{n_{\\!N}}^{\pm 1}(k_{N}),{\Phi}\hskip
0.85358pt\big{]}\hskip 0.85358pt=\hskip
0.85358pt\mathcal{M}\big{[}\Theta_{n_{1}}\\!(k_{1}),\cdots\\!,\Theta_{n_{\\!N}}\\!(k_{N}),{\Phi}\hskip
0.85358pt\big{]}\hskip 0.85358pt.$ (67ce)

The above identity can be proven by computing the amplitude on its right-hand
side and using the relation $\Theta_{n}\\!\\!=\\!h_{n}^{\pm
1}\\!-\\!\underline{\mathbb{F}}_{n}$ in Eq.(67ca). Expanding
$\Theta_{n}(=\\!\hat{C}_{\hskip 0.85358ptnm}\mathcal{V}_{m}^{\pm
1}\\!+\\!v_{n}^{\pm 1})$ for each external state on the right-hand side of the
identity (67ce), we derive the following gravitational equivalence theorem
(GRET) identity:

$\displaystyle\mathcal{M}[\,h^{\pm 1}_{n_{1}},\cdots\\!,h^{\pm
1}_{n_{N}},\Phi\,]\,=\,\hat{C}_{\mathrm{mod}}^{n_{j}m_{j}}\mathcal{M}[\,{\mathcal{V}^{\pm
1}_{m_{1}},\cdots\\!,\mathcal{V}_{m_{N}}^{\pm
1}},\Phi\,]+\mathcal{M}_{v}\hskip 0.85358pt,$ (67cfa)
$\displaystyle\mathcal{M}_{v}=\sum_{\ell=1}^{N}\tilde{C}_{\mathrm{mod},\ell}^{n_{j}m_{j}}\,\mathcal{M}[\,v_{n_{1}}^{\pm
1},\cdots\\!,v_{n_{\ell}}^{\pm 1},\mathcal{V}_{m_{\ell+1}}^{\pm
1},\cdots\\!,\mathcal{V}_{m_{N}}^{\pm 1},\Phi\,]\hskip 0.85358pt,$ (67cfb)

where the multiplicative modification factors are given by

$\displaystyle\hat{C}_{\mathrm{mod}}^{n_{j}m_{j}}$
$\displaystyle=\,\hat{C}_{n_{1}m_{1}}\\!\cdots\hat{C}_{n_{\\!N}m_{\\!N}}=\,(-\text{i})^{N}\delta_{n_{1}m_{1}}\\!\\!\cdots\hskip
0.85358pt\delta_{n_{\\!N}m_{\\!N}}\\!+O(\mathrm{loop})\hskip 0.85358pt,$
(67cga) $\displaystyle\tilde{C}_{\mathrm{mod},\ell}^{n_{j}m_{j}}$
$\displaystyle=\,\hat{C}_{n_{\ell+1}m_{\ell+1}}\\!\cdots\hat{C}_{n_{\\!N}m_{\\!N}}=\,(-\text{i})^{N-\ell}\delta_{n_{\ell+1}m_{\ell+1}}\\!\\!\cdots\hskip
0.85358pt\delta_{n_{\\!N}m_{\\!N}}\\!+O(\mathrm{loop})\hskip 0.85358pt.$
(67cgb)

Under the high energy expansion, the residual term is suppressed by the energy
factor of $O(\mathbb{M}_{n}/E)\hskip 0.85358pt$. Hence, from the identity
(67cfa) we derive the following warped GRET of type-II:

$\mathcal{M}[\,h^{\pm 1}_{n_{1}},\cdots\\!,h^{\pm
1}_{n_{N}},\Phi\,]\,=\,\hat{C}_{\mathrm{mod}}^{n_{j}m_{j}}\mathcal{M}[\,{\mathcal{V}^{\pm
1}_{m_{1}},\cdots\\!,\mathcal{V}_{m_{N}}^{\pm 1}},\Phi\,]\hskip
0.85358pt+\hskip
0.85358ptO({\mathbb{M}_{n}}/{E_{n}}\mathrm{-suppressed})\hskip 0.85358pt.$
(67ch)

This connects the scattering amplitude of the KK gravitons ($h_{n}^{\pm 1}$)
with helicity $\pm 1$ to the corresponding gravitational KK vector-Goldstone
bosons ($\mathcal{V}_{m}^{\pm 1}$).

####  2.3.3 Formulating GRET for Warped KK Gravity Theory (II)

In this subsection, we further derive the KK GRET to connect the scattering
amplitude of helicity-0 longitudinal KK gravitons ($h_{L}^{n}$) to the
corresponding gravitational KK scalar Goldstone bosons ($\phi_{n}$) under the
high energy expansion. This GRET will be called GRET of type-I.

We use the combination of gauge-fixing functions
$\partial_{\mu}\mathcal{F}_{n}^{\mu}\\!-\xi_{n}\mathbb{M}_{n}\mathcal{F}_{n}^{5}\,$
to eliminate the KK vector-Goldstone field $\,\mathcal{V}_{n}^{\mu}\,$ and
derive an expression in momentum space:

$-\text{i}k_{\mu}\mathcal{F}_{n}^{\mu}\\!-\xi_{n}\mathbb{M}_{n}\mathcal{F}_{n}^{5}\,=\,-k_{\mu}k_{\nu}h_{n}^{\mu\nu}\\!+\mbox{$\frac{\,{1}\,}{2}$}\\!\left[\\!(2\\!-{\xi_{n}^{-1}})k^{2}\\!-\xi_{n}\mathbb{M}_{n}^{2}\right]\\!h_{n}\\!+\\!\sqrt{\\!\mbox{$\frac{\,{3}\,}{2}$}\,}{\xi_{n}^{2}}\mathbb{M}_{n}^{2}\phi_{n}\,,$
(67ci)

where we have made the rescaling of $\hskip 0.85358pt\phi_{n}$ as in Eq.(63).
Imposing the on-shell condition $k^{2}\\!=\\!-\mathbb{M}_{n}^{2}\hskip
0.85358pt$, we can rewrite the above formula as follows:

$\displaystyle\text{i}k_{\mu}\mathcal{F}_{n}^{\mu}\\!+\xi_{n}\mathbb{M}_{n}\mathcal{F}_{n}^{5}\,=\sqrt{\\!\mbox{$\frac{\,{3}\,}{2}$}\,}\mathbb{M}_{n}^{2}\hskip
0.85358pt\widetilde{\mathbb{F}}_{n}\,,$ (67cja)
$\displaystyle\widetilde{\mathbb{F}}_{n}=\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}h_{n}^{S}+\mbox{$\frac{\,{1}\,}{\sqrt{6\,}\,}$}\big{(}2+\xi_{n}\\!-\xi_{n}^{-1}\big{)}h_{n}-\xi_{n}^{2}\phi_{n}=\mathbf{K}^{T}_{n}\mathbf{H}_{n}$
(67cjb) $\displaystyle\hskip
11.66563pt=\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}\big{(}h_{n}^{S}+h_{n}\big{)}\\!-\phi_{n}\hskip
0.85358pt,\hskip 19.91692pt(\text{for}~{}\xi_{n}\\!=\\!1)\hskip 0.85358pt,$
(67cjc)
$\displaystyle\mathbf{K}_{n}\\!=\Big{(}\\!\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}\varepsilon^{S}_{\mu\nu}\\!+\\!\mbox{$\frac{\,{1}\,}{\sqrt{6\,}\,}$}\big{(}2+\xi_{n}\\!-\xi_{n}^{-1}\big{)}\eta_{\mu\nu},\,-\xi_{n}^{2}\Big{)}^{\\!\\!T}\\!,~{}~{}~{}~{}\mathbf{H}_{n}=(h_{n}^{\mu\nu},\,\phi_{n})^{T}\hskip
0.85358pt,$ (67cjd)

where we have defined the scalar KK graviton
$\,h_{n}^{S}\\!=\\!\varepsilon_{\mu\nu}^{S}{h_{n}^{\mu\nu}}\,$ with its
scalar-polarization tensor
$\varepsilon_{\mu\nu}^{S}\\!=\\!k_{\mu}k_{\nu}/\mathbb{M}_{n}^{2}\hskip
1.42262pt$. Then, we inspect the KK graviton’s longitudinal polarization
tensor $\varepsilon_{L}^{\mu\nu}$ and make high energy expansion of it as
follows:

$\displaystyle\varepsilon_{L}^{\mu\nu}=\frac{1}{\sqrt{6\,}\,}\\!\left(\epsilon_{+}^{\mu}\epsilon_{-}^{\nu}\\!+\epsilon_{+}^{\nu}\epsilon_{-}^{\mu}\\!+2\epsilon_{L}^{\mu}\epsilon_{L}^{\nu}\right)=\sqrt{\frac{2}{3}\,}\varepsilon_{S}^{\mu\nu}\\!+\tilde{v}^{\mu\nu},$
(67ck)

where the longitudinal polarization vector
$\epsilon_{L}^{\mu}\\!=\\!(k^{0}\\!/\mathbb{M}_{n})(|\vec{k}|/k^{0},\,\vec{k}/|\vec{k}|)\\!=\\!\epsilon_{S}^{\mu}\\!+\\!v^{\mu}$
with $\,\epsilon_{S}^{\mu}=k^{\mu}\\!/\mathbb{M}_{n}\,$ and
$\,v^{\mu}\\!=\\!\mathcal{O}(\mathbb{M}_{n}/E_{n})\,$. In the above, the
residual term

$\tilde{v}^{\mu\nu}\equiv\,\varepsilon_{L}^{\mu\nu}\\!\\!-\\!\sqrt{\frac{2}{3}\,}\varepsilon_{S}^{\mu\nu}=\frac{1}{\sqrt{6\,}\,}\\!\left[\big{(}\epsilon_{+}^{\mu}\epsilon_{-}^{\nu}\\!+\epsilon_{+}^{\nu}\epsilon_{-}^{\mu}\big{)}\\!+2\big{(}\epsilon_{S}^{\mu}v^{\nu}\\!+\\!\epsilon_{S}^{\nu}v^{\mu}\\!+\\!v^{\mu}v^{\nu}\big{)}\right]\\!,$
(67cl)

where $\epsilon_{S}^{\mu}\\!=\\!k^{\mu}/\mathbb{M}_{n}$ and
$\,\tilde{v}^{\mu\nu}\\!\\!=\\!\mathcal{O}(E_{n}^{0})$ under high energy
expansion.

According to Appendix B, we derive a BRST identity on the full propagator of
the matrix field $\mathbf{H}_{n}\\!=\\!(h_{n}^{\mu\nu},\,\phi_{n})^{T}$,

$\mathbf{K}_{n}^{T}\bm{\mathcal{D}}_{nm}(k)=-\mathbf{X}_{nm}^{T}(k)\,,$ (67cm)

where we have defined the following notations:

$\displaystyle\bm{\mathcal{D}}_{nm}(k)$
$\displaystyle=\big{\langle}0\big{|}T\mathbf{H}_{n}\mathbf{H}_{m}^{T}\big{|}0\big{\rangle}(k)\,,\quad\mathbf{X}_{nm}(k)=\underline{\mathbf{X}}_{mj}(k)\SS_{jn}(k)\hskip
0.85358pt,$ $\displaystyle\underline{\mathbf{X}}_{mj}(k)$
$\displaystyle=\frac{1}{\,2\sqrt{6\hskip 0.85358pt}\hskip
0.85358pt\mathbb{M}_{j}^{2}\,}\\!\begin{pmatrix}\big{\langle}0\big{|}T\,\widehat{\tt
s}h^{\mu\nu}_{m}\big{|}\bar{\mathsf{c}}_{j}\big{\rangle}(k)\\\\[2.84526pt]
\big{\langle}0\big{|}T\,\widehat{\tt
s}\phi_{m}\big{|}\bar{\mathsf{c}}_{j}\big{\rangle}(k)\end{pmatrix}\\!=\\!{\frac{1}{\,2\sqrt{6\hskip
0.85358pt}\,}}\\!\begin{pmatrix}\\!(2k^{\mu}k^{\nu}\\!/\mathbb{M}_{j}^{2}\\!-\\!\eta^{\mu\nu})[\delta_{mj}\\!+\\!\Delta^{\\!(3)}_{mj}(k^{2})]\\\\[2.84526pt]
-\sqrt{6\,}[\delta_{mj}\\!+\\!\tilde{\Delta}_{mj}^{\\!(4)}(k^{2})]\end{pmatrix}\\!,$
$\displaystyle\SS_{jn}(k)$
$\displaystyle=\big{\langle}0\big{|}T{\mathsf{c}_{j}\bar{\mathsf{c}}_{n}}\big{|}0\big{\rangle}(k)\hskip
0.85358pt,$ (67cn)

with
$\,{\mathsf{c}}_{n}\\!\equiv\text{i}\epsilon_{\mu}^{S}{c}^{\mu}_{n}+{c}_{n}^{5}\hskip
0.85358pt$, $\hskip
0.85358pt\bar{\mathsf{c}}_{n}\\!\equiv\text{i}\epsilon_{\mu}^{S}\bar{c}^{\mu}_{n}+\bar{c}_{n}^{5}\hskip
0.85358pt$, and $\epsilon_{\mu}^{S}\\!=\\!k_{\mu}/\mathbb{M}_{n}\hskip
0.85358pt$. We also note that the content of Eq.(67cm) actually differs from
that of Eq.(67bq) although they are expressed in the same notations. In the
above, we choose Feynman-’t Hooft gauge $\xi_{n}\\!=\\!1$ for simplicity, and
the quantities $\Delta^{\\!(3)}_{mj}(k^{2})$ and
$\tilde{\Delta}^{\\!(4)}_{mj}(k^{2})$ belong to loop level and are generated
by the non-linear terms of the BRST transformations (67ke) and (67ke) of
$h_{n}^{\mu\nu}$ and $h_{n}^{55}(\hskip 0.85358pt\equiv\\!\phi_{n})\hskip
0.85358pt$.

Using the identity (66) and Eq.(67cj) together with Eq.(67cm), we can obtain a
new identity with external state $\widetilde{\mathbb{F}}_{n}(k)\hskip
0.85358pt$,

$\displaystyle 0\,=\,\big{\langle}0\hskip
0.85358pt|\,\widetilde{\mathbb{F}}_{n}(k)\cdots\Phi\,|\hskip
0.85358pt0\big{\rangle}=-\mathbf{X}_{nm}^{T}(k)\hskip
0.85358pt\mathcal{M}\big{[}\underline{\mathbf{H}}_{m}(k),\cdots\\!,\Phi\hskip
0.85358pt\big{]},$ (67co)

from which we derive the following identity:

$\displaystyle\mathcal{M}\big{[}\underline{\widetilde{\mathbb{F}}}_{n}(k),\cdots\\!,\Phi\hskip
0.85358pt\big{]}=0\,,$ (67cp)

where the amputated external state $\underline{\widetilde{\mathbb{F}}}_{n}(k)$
is given as follows:

$\displaystyle\underline{\widetilde{\mathbb{F}}}_{n}$
$\displaystyle=\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}(h_{n}^{S}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}h_{n})-C_{nm}\phi_{m}=h_{n}^{L}-\Omega_{n}\hskip
0.85358pt,$ (67cqa) $\displaystyle\Omega_{n}$ $\displaystyle=C_{nm}\hskip
0.85358pt\phi_{m}\\!+\tilde{\Delta}_{n}\hskip
0.85358pt,~{}~{}~{}~{}\tilde{\Delta}_{n}=\mbox{$\frac{\,{1}\,}{\sqrt{6\,}\,}$}h_{n}\\!+\tilde{v}_{n}\hskip
0.85358pt,~{}~{}~{}~{}\tilde{v}_{n}\\!=v_{\mu\nu}h_{n}^{\mu\nu}\hskip
0.85358pt.$ (67cqb)

In the above, $C_{nm}$ is a multiplicative modification factor induced at loop
level:

$C_{nm}(k^{2})=\left[\\!\frac{~{}\mathbf{1}\\!+\\!\bm{\tilde{\Delta}}^{\\!(4)}(k^{2})~{}}{\,\mathbf{1}\\!+\\!\bm{\Delta}^{\\!(3)}(k^{2})}\\!\right]_{mn}\\!\equiv\,\delta_{nm}+O(\mathrm{loop})\hskip
0.85358pt,$ (67cr)

where the matrix form is used such that
$(\bm{\Delta}^{\\!(3)})_{jj^{\prime}}\\!\\!=\\!{\Delta}^{\\!(3)}_{jj^{\prime}}$
and
$(\bm{\tilde{\Delta}}^{\\!(4)})_{jj^{\prime}}\\!\\!=\\!\\!\tilde{\Delta}^{\\!(4)}_{jj^{\prime}}$
with the matrix elements $({\Delta}^{\\!(3)}_{jj^{\prime}},\hskip
0.85358pt\widetilde{\Delta}^{\\!(4)}_{jj^{\prime}})$ from Eq.(67cn). More
details of the derivations are given in Appendix B.2.

Then, using Eq.(67cp) we can derive the following general identity for the
gravitational equivalence theorem (GRET):

$\mathcal{M}\big{[}\underline{\widetilde{\mathbb{F}}}_{n_{1}}\\!(k_{1}),\cdots\\!,\underline{\widetilde{\mathbb{F}}}_{n_{\\!N}}\\!(k_{N}),{\Phi}\hskip
0.85358pt\big{]}=0\,.$ (67cs)

From this, we further derive the following GRET identity:

$\displaystyle\mathcal{M}\big{[}h_{n_{1}}^{L}\\!(k_{1}),\cdots\\!,h_{n_{\\!N}}^{L}\\!(k_{N}),{\Phi}\hskip
0.85358pt\big{]}\hskip 0.85358pt=\hskip
0.85358pt\mathcal{M}\big{[}\Omega_{n_{1}}\\!(k_{1}),\cdots\\!,\Omega_{n_{\\!N}}\\!(k_{N}),{\Phi}\hskip
0.85358pt\big{]}\hskip 0.85358pt,$ (67ct)

which can be proven by computing the amplitude on its right-hand side and
using the relation
$\Omega_{n}\\!=\\!h_{n}^{L}\\!-\underline{\widetilde{\mathbb{F}}}_{n}$ in
Eq.(67cqa). Expanding the right-hand side of the GRET identity (67ct), we
connect the scattering amplitude of helicity-zero longitudinal KK gravitons
($h_{n}^{L}$) to that of the corresponding KK Goldstone bosons
$(\phi_{m})\hskip 0.85358pt$:

$\displaystyle\mathcal{M}[\hskip
0.85358pth^{L}_{n_{1}},\cdots\\!,h^{L}_{n_{\\!N}},\Phi\hskip
0.85358pt]\,=\,{C}_{\mathrm{mod}}^{n_{j}m_{j}}\mathcal{M}[\hskip
0.85358pt\phi_{m_{1}},\cdots\\!,\phi_{m_{\\!N}},\Phi\hskip
0.85358pt]+\mathcal{M}_{\\!\Delta}\,,$ (67cua)
$\displaystyle\mathcal{M}_{\\!\Delta}=\,\sum_{\ell=1}^{N}\\!\widetilde{C}_{\mathrm{mod},\ell}^{n_{j}m_{j}}\,\mathcal{M}\big{[}\tilde{\Delta}_{n_{1}},\cdots\\!,\tilde{\Delta}_{n_{\ell}},\phi_{m_{\ell+1}},\cdots\\!,\phi_{m_{N}},\Phi\hskip
0.85358pt\big{]},$ (67cub)

where $\tilde{\Delta}_{n}\\!=\\!\frac{1}{\sqrt{6\,}\,}h_{n}+\tilde{v}_{n}$
with $\tilde{v}_{n}\\!=\\!v_{\mu\nu}h_{n}^{\mu\nu}$, and $\Phi$ denotes any
other external on-shell physical state(s) after the LSZ reduction. In the
above, the multiplicative modification factors are defined as follows:

$\displaystyle C_{\mathrm{mod}}^{n_{\\!j}m_{\\!j}}$
$\displaystyle=\,C_{n_{1}m_{1}}\\!\\!\cdots
C_{n_{\\!N}m_{\\!N}}=\,\delta_{n_{1}m_{1}}\\!\\!\cdots\hskip
0.85358pt\delta_{n_{\\!N}m_{\\!N}}\\!+O(\mathrm{loop})\hskip 0.85358pt,$
(67cva) $\displaystyle\widetilde{C}_{\mathrm{mod},\ell}^{n_{\\!j}m_{\\!j}}$
$\displaystyle=\,C_{n_{\ell+1}m_{\ell+1}}\\!\\!\cdots
C_{n_{\\!N}m_{\\!N}}=\,\delta_{n_{\ell+1}m_{\ell+1}}\\!\\!\cdots\hskip
0.85358pt\delta_{n_{\\!N}m_{\\!N}}\\!+O(\mathrm{loop})\hskip 0.85358pt,$
(67cvb)

which are renormalization-scheme-dependent and can be normalized to one under
proper setup of the renormalization scheme. They reduce to one at tree level.
We can further justify that in the residual term (67cub) each external state
$\tilde{\Delta}_{n}\hskip
0.85358pt(=\\!\frac{1}{\sqrt{6\,}\,}h_{n}+\tilde{v}_{n})$ is suppressed by an
energy factor $\mathbb{M}_{n}/E_{n}$ under high energy expansion. For this,
using the expression (67cl) and the normalization condition
$\sum_{\sigma}\epsilon_{\sigma}^{\mu}(\epsilon_{\sigma}^{\nu})^{*}\\!=\eta^{\mu\nu}\\!+k^{\mu}k^{\nu}/\mathbb{M}_{n}^{2}\hskip
0.85358pt$ (with helicity $\sigma\\!=\\!\pm,L$), we derive the following:

$\tilde{\Delta}_{n}=\mbox{$\frac{\,{1}\,}{\sqrt{6\,}\,}$}h_{n}\\!+\tilde{v}_{n}=\sqrt{\frac{3}{2}\,}\big{(}2\epsilon^{S}_{\mu}v_{\nu}\\!+\\!v_{\mu}v_{\nu}\big{)}h_{n}^{\mu\nu}\,,$
(67cw)

where $v_{\mu}v_{\nu}\\!=\\!O(\mathbb{M}_{n}^{2}/E_{n}^{2})$ and
$\epsilon^{S}_{\mu}v_{\nu}\\!\\!=\\!\\!O(E_{n}^{0})$. The external state of
$(v_{\mu}v_{\nu}h_{n}^{\mu\nu})$ is clearly suppressed by the extra energy
factor $v_{\mu}v_{\nu}\\!\sim\\!\mathbb{M}_{n}^{2}/E_{n}^{2}$ in the high
energy limit. As proven by Eq.(67cc), despite the external factor
$\epsilon_{\mu}^{S}v_{\nu}\\!=O(E^{0}_{n})$, the amplitude with external state
$\epsilon^{S}_{\mu}v_{\nu}h_{n}^{\mu\nu}$ is actually suppressed by an overall
energy factor $v^{\mu}\\!=\\!O(\mathbb{M}_{n}/E_{n})\hskip 0.85358pt$. Thus,
as the external state, we can reexpress $\tilde{\Delta}_{n}$ as follows:

$\tilde{\Delta}_{n}=\,\sqrt{\frac{3}{2}\,}\big{(}\hat{C}_{nm}2\hskip
0.85358ptv_{\\!\mu}\mathcal{V}^{\mu}_{m}\\!+\\!v_{\\!\mu}v_{\nu}h_{n}^{\mu\nu}\big{)}\hskip
0.85358pt,$ (67cx)

where the two terms on right-hand side is suppressed by external factors of
$v_{\\!\mu}\\!\\!=\\!O(E^{-1})$ and $v_{\\!\mu}v_{\nu}\\!\\!=\\!O(E^{-2})$,
respectively. Hence, the residual term $\mathcal{M}_{\\!\Delta}$ in Eq.(67cub)
must be suppressed (relative to the leading KK Goldstone amplitude) at least
by an overall energy factor $v^{\mu}\\!=\\!O(\mathbb{M}_{n}/E_{n})\hskip
0.85358pt$ under the high energy expansion. Thus, we can further formulate the
following warped GRET of type-I:

$\mathcal{M}[\hskip
0.85358pth^{L}_{n_{1}},\cdots\\!,h^{L}_{n_{\\!N}},\Phi\hskip
0.85358pt]\,=\,{C}_{\mathrm{mod}}^{n_{j}m_{j}}\mathcal{M}[\hskip
0.85358pt\phi_{m_{1}},\cdots\\!,\phi_{m_{\\!N}},\Phi\hskip
0.85358pt]+O({\mathbb{M}_{n}}/{E_{n}}\mathrm{-suppressed})\hskip 0.85358pt.$
(67cy)

This ensures that the GRET identity (67cua) can establish the equivalence
between the helicity-zero longitudinal KK graviton amplitude
$\mathcal{M}[h_{n_{1}}^{L},\cdots]$ and the corresponding gravitational KK
Goldstone amplitude
${C}_{\mathrm{mod}}^{n_{j}m_{j}}\mathcal{M}[\phi_{m_{1}},\cdots]$ in the high
energy limit. Thus, the identity (67cu) gives in Eq.(67cy) the KK gravitional
equivalence theorem (GRET) for the warped 5d GR theory under high energy
expansion, which extends our previous formulation of the KK GRET for the
compactified flat 5d GR Hang:2021fmp ; Hang:2022rjp .

We note that the generalized power-counting method for the compactified KK
gauge theories and KK GR theories as established in Ref. Hang:2021fmp  does
hold for both flat and warped extra dimensions. Thus, we can deduce the
leading energy-power dependence of $N$-point helicity-zero longitudinal KK
graviton amplitudes and of the corresponding $N$-point KK Goldstone
amplitudes:

$\displaystyle D_{E}(Nh^{L}_{n})=2(N\\!+\\!L\\!+\\!1),\hskip
19.91692ptD_{E}(N\phi_{n})=2(L\\!+\\!1),$ (67cza)
$\displaystyle\Longrightarrow~{}~{}D_{E}(Nh^{L}_{n})-D_{E}(N\phi_{n})=2N,$
(67czb)

where $L\\!\geqslant\\!0\hskip 0.85358pt$ denotes the number of loops
involved. For the scattering amplitudes of $N$ longitudinal KK gravitons (with
helicity-zero) at tree level ($L\\!=\\!0$), we use the right-hand side of the
GRET identity (67cua) and prove the large energy cancellations of
$E^{2N+2}\\!\\!\rightarrow\\!E^{2}$ by an energy factor $E^{2N}$ in the $N$
longitudinal KK graviton amplitude.

The case of 3-point KK graviton amplitudes needs a further discussion.
According to the above power counting rule (67cza), the tree-level 3-point
longitudinal KK graviton amplitude ($3h_{n}^{L}$) have a possible leading
energy dependence of $E^{2(N+1)}\\!\\!=\\!E^{8}$ for $N\\!\\!=\\!3\,$. The
tree-level amplitude of 3 gravitational KK Goldstone bosons ($3\phi_{n}$)
arises from trilinear KK Goldstone vertex whose coupling factor contains
products of two external momenta $\,p_{i}\cdot p_{j}$. So the naive power
counting shows that this KK Goldstone amplitude would have leading energy
dependence of $E^{2}$ in agreement with Eq.(67cza). Thus, according to the
GRET (67cu), the 3-point tree-level longitudinal KK graviton amplitude should
have a total energy-power deduction of $E^{8-2}\\!=\\!E^{6}$, which agrees to
Eq.(67czb). But, as we will further explain at the end of the following
Section 3.1.2, the $N\\!=\\!3$ is a special case where the momentum
conservation and on-shell condition of each external state plays additional
roles to reduce the leading energy dependence. Namely, after imposing the
momentum conservation at the trilinear vertex and the on-shell condition for
each momentum, the individual leading-energy dependence of 3-point
longitudinal KK graviton amplitude reduces to $E^{4}$ and that of the
corresponding KK Goldstone amplitude becomes $E^{0}$. Hence the GRET (67cu)
further ensures a nontrivial energy cancellation of
$E^{4}\\!\rightarrow\\!E^{0}$.

Then, we can further consider the scattering amplitudes of $N$ KK gravitons of
$h_{n}^{\pm 1}$ (with helicity $\pm 1$) and of the corresponding gravitational
KK vector-Goldstone bosons ($\mathcal{V}_{n}^{\pm 1}$). Using the generalized
power-counting method Hang:2021fmp , we derive their leading energy-power
dependence as follows:

$\displaystyle D_{E}(Nh_{n}^{\pm 1})=N\\!+\\!2(L\\!+\\!1),\hskip
19.91692ptD_{E}(N\mathcal{V}_{n}^{\pm 1})=2(L\\!+\\!1),$ (67daa)
$\displaystyle\Longrightarrow~{}~{}D_{E}(Nh_{n}^{\pm
1})-D_{E}(N\mathcal{V}_{n}^{\pm 1})=N.$ (67dab)

For the scattering amplitudes of $N$ KK gravitons ($h_{n}^{\pm 1}$) at tree
level, we use the right-hand side of the GRET identity (67cf) [or (67ch)] and
prove the large energy cancellations of $E^{N+2}\\!\\!\rightarrow\\!E^{2}$ by
an energy factor $E^{N}$ in the scattering amplitude of $N$ KK gravitons (with
helicity $\pm 1$).

For the warped KK gauge theories as studied in Section 2.2, using the
generalized power-counting method Hang:2021fmp  and the GAET identity (36), we
derive their leading energy-power dependence for $N$ longitudinal as follows:

$\displaystyle D_{E}(NA_{L}^{a\hskip 0.85358ptn})=4\hskip 0.85358pt,\hskip
19.91692ptD_{E}(NA_{5}^{a\hskip
0.85358ptn})=4-N-N_{v}-\bar{V}_{3}^{\min}\hskip 0.85358pt,$ (67dba)
$\displaystyle\Longrightarrow~{}~{}D_{E}(NA_{L}^{a\hskip
0.85358ptn})-D_{E}(NA_{5}^{a\hskip
0.85358ptn})=N\\!+N_{v}\\!+\bar{V}_{3}^{\min}\hskip 0.85358pt,$ (67dbb)

where $\bar{V}_{3}^{\min}$ denotes the involved minimal number of non-
derivative cubic vertices in the KK Goldstone boson amplitude, with
$\bar{V}_{3}^{\min}=0\,(1)$ for $N\\!=\hskip 0.85358pt$even (odd). Hence, this
proves that the GAET identity (36) guarantees nontrivial energy cancellations
by an energy power of $E^{N}$ (or, $E^{N+1}$) for $N\\!\\!=\hskip
0.85358pt$even (or, $N\\!\\!=\hskip 0.85358pt$odd) in the scattering amplitude
of $N$ longitudinal KK gauge bosons. In the above, $N_{v}$ denotes the number
of possible external $v_{\mu}A_{n}^{a\mu}$ lines in the KK Goldstone boson
amplitude, where $v_{\mu}\\!\\!=\\!O(M_{n}/E)$ in the high energy limit. We
find that the above large energy cancellations [as shown in Eqs.(67czb),
(67dab) and (67dbb)] hold for both the flat and warped 5d KK gravity (gauge)
theories. Before ending this discussion, we further comment on the $N\\!=\\!3$
case for KK gauge boson (KK Goldstone boson) scattering amplitudes. The
3-point longitudinal KK gauge boson amplitude ($3A_{L}^{an}$) has a leading
energy dependence of $E^{4}$ by the power counting rule (67dba), which arises
from the trilinear KK gauge boson vertex (containing one external momentum)
and the three external longitudinal polarization vectors (each of which scales
like $\epsilon_{L}^{\mu}\\!\sim\\!p_{j}^{\mu}/M_{n_{j}}$ in the high energy
limit). On the other hand, the corresponding 3-point KK Goldstone amplitude
vanishes at tree level due to the absence of cubic vertex $3A_{5}^{an}$.
According to the GAET identity (36), the leading nonzero contribution on its
right-hand side arises from the residual terms with the trilinear vertex
having two KK Goldstone bosons and one KK gauge boson (contracting with the
external vector $v^{\mu}$). So we have $(N,N_{v})\\!=\\!(3,1)$ and
$\bar{V}_{3}^{\min}\\!=\\!0$, and thus this residual term has energy power
$D_{E}\\!=\\!4\\!-\\!3\\!-\\!1\\!-\\!0\\!=\\!0$, showing that the right-hand
side of the GAET identity (36) has leading energy dependence of $E^{0}$.
Hence, the total energy reduction of the 3-point longitudinal KK gauge boson
amplitude is $E^{4}\\!\rightarrow\\!E^{0}$.

We also note that for the Goldberger-Wise (GW) model GW , it further
introduces a bulk scalar field $\hat{\mathcal{S}}$ to stabilize the size of
the 5d extra dimension of the RS1. Its zero-mode $\mathcal{S}_{0}$ can mix
with the radion field ${h}_{0}^{55}$, so that the physical radion after
stabilization becomes a mixture of $\SS_{0}$ and ${h}_{0}^{55}$, whereas the
scalar KK mode $\SS_{n}$ mixes with the Goldstone KK mode $h_{n}^{55}$
(denoted as $\phi_{n}$ in our notation) at each KK level, leading to a mass-
eigenstate $\SS_{n}^{\prime}$ and the true Goldstone KK state
$h_{n}^{55\prime}$. Thus, the gauge-fixing term (43)-(44) should be extended
to contain the true KK Goldstone boson field ($h_{n}^{55\prime}$).
Consequently, the GRET (67cy) can be extended to hold the equivalence between
the scattering amplitudes of the longitudinal KK graviton states ($h_{n}^{L}$)
and of the true KK Goldstone boson states ($h_{n}^{55\prime}$) in the high
energy limit.555In passing, Ref. chivukula-GW  studied spin-2 KK scattering
amplitudes with massive radion in the GW model.

In passing, during the finalization of this paper, we received a recent paper
Chivukula:2023qrt  which studied the tree-level formulation of the
gravitational ET for warped 5d RS1 model in the ’t Hooft-Feynman gauge and the
$O(s)$ high-energy behavior of the 4-point amplitudes therein.666We thank the
colleagues Sekhar Chivukula and Elizabeth Simmons for discussions on their
paper Chivukula:2023qrt  and related issues. Our general formulations in this
section significantly differ from Chivukula:2023qrt  in several aspects. We
first derived the new GAET for the warped KK gauge theory in Sec. 2.2, in
addition to deriving the general GRET for warped KK gravity theory in Sec.
2.3. The GAET of the warped KK gauge theory is the focus of our present work
and serves as the most fundamental KK ET formulation, because we will prove in
Sec. 4 that the GRET of the warped KK gravity can be derived (reconstructed)
from the warped GAET by our leading-order double-copy method. This is
important for us to derive the warped GRET from warped GAET via double-copy
construction in Sec. 4.2. We derived in Sec. 2.3 two new BRST identities
(67bq) and (67cm) to generally amputate the mixed external KK graviton states
($h_{n}^{\mu\nu}$) and KK Goldstone states ($\phi_{n}$ or
$\mathcal{V}_{n}^{\mu}$), which are needed because such mixings generally
reappear at loop levels. We presented GRET type-II in general $R_{\xi}$ gauge
up to loop level (Sec. 2.3.3) and GRET type-I up to loop level in ’t Hooft-
Feynman gauge for simplicity (Sec. 2.3.2), whereas in Appendix B.1 we also
formulated the GRET type-I (including the LSZ reduction for external states)
in the general $R_{\xi}$ gauge at tree level. Using the generalized power-
counting method Hang:2021fmp  for the compactified KK GR theories [with the
GRET (67cu) and (67cf)] and the KK gauge theories [with the GAET (36)], we
proved the different energy cancellations for the scattering amplitudes of
longitudinal KK gravitions (with helicity-0) in Eq.(67cz), of the KK gravitons
(with helicity $\pm 1$) in Eq.(67da), and of the KK gauge bosons in Eq.(67db).

##  3 Structure of Massive Scattering Amplitudes from Warped Gauge and
Gravitational Equivalence Theorems

In this section, we systematically compute the three-point and four-point
scattering amplitudes of KK gauge bosons (KK gravitons) and of their
corresponding KK Goldstone bosons. Then, we explicitly demonstrate the
validity of KK GAET (38) and GRET (67ch)(67cy) by comparing the scattering
amplitudes of KK gauge bosons (KK gravitons) with their corresponding KK
Goldstone boson amplitudes at the leading order of high energy expansion.

###  3.1 Warped Gauge and Gravitational ETs for 3-Point KK Amplitudes

In this subsection, we systematically compute the tree-level 3-point
scattering amplitudes of massive KK gauge bosons in Section 3.1.1, and the
3-point scattering amplitudes of KK gravitons in Section 3.1.2.

####  3.1.1 Warped GAET for 3-Point KK Gauge Boson Amplitudes

To compute the on-shell 3-point KK gauge and Goldstone scattering amplitudes,
we derive the Feynman rules for cubic KK gauge/Goldstone vertices in Appendix
C.1. Then, we derive the following general 3-point on-shell scattering
amplitude of KK gauge bosons:

$\mathcal{T}[A^{an_{1}}_{\lambda_{1}}\\!A^{bn_{2}}_{\lambda_{2}}\\!A^{cn_{3}}_{\lambda_{3}}]=-\text{i}2\hskip
0.85358ptg\hskip
0.85358pta_{n_{1}n_{2}n_{3}}f^{abc}\left[(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot
p_{1})+(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{1}\\!\cdot
p_{2})+(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{2}\\!\cdot
p_{3})\right]\\!,$ (67dc)

where the subscript $\lambda_{j}\\!\\!=\\!\pm,0$ ($j=1,2,3$) denotes the
helicity of each external state and
$\epsilon_{j}^{\mu}(=\\!\epsilon_{\lambda_{j}}^{\mu})$ is the shorthand
notation for the polarization vector of each external KK auge boson. All the
three external momenta $\\{p_{j}\\}$ are outgoing and obey the momentum
conservation $p_{1}\\!+p_{2}\\!+p_{3}\\!=0\hskip 0.85358pt$. In the above
tree-level scattering amplitude, $g$ is the Yang-Mills gauge coupling and
$a_{n_{1}n_{2}n_{3}}$ denotes coupling coefficient induced by integrating over
the compactified 5th-dimension coordinate as defined in Eq.(67lj).

With these, we compute the 3-point scattering amplitude of KK gauge bosons of
$LLT$ type at tree level, which contains two longitudinal KK gauge bosons
[chosen as particles 1 and 2 and denoted by $A_{L}^{a\hskip
0.85358ptn_{1}}(p_{1})$ and $A_{L}^{b\hskip 0.85358ptn_{2}}(p_{2})$], and one
transverse KK gauge boson [chosen as the particle 3 and denoted by
$A_{\pm}^{c\hskip 0.85358ptn_{3}}(p_{3})$]. Thus, we derive the following
3-point KK gauge boson amplitude:

$\displaystyle\mathcal{T}[A_{L}^{an_{1}}\\!A_{L}^{bn_{2}}\\!A_{\pm}^{cn_{3}}]\,$
$\displaystyle=\,\mp\text{i}\hskip
0.85358ptgf^{abc}\frac{\,[E_{1}^{2}\\!+\\!E_{2}^{2}\\!+\\!E_{3}^{2}\\!-\\!2E_{1}E_{2}(1\\!+\\!c_{2\theta})\\!-\\!2k^{2}]k\hskip
0.85358pts_{\theta}\,}{\sqrt{2\,}M_{n_{1}}\\!M_{n_{2}}}\hskip
0.85358pta_{n_{1}n_{2}n_{3}},$ $\displaystyle=\,\mp\text{i}\hskip
0.85358ptg\hskip
0.85358ptf^{abc}\frac{~{}(M_{n_{3}}^{2}\\!\\!-\\!M_{n_{1}}^{2}\\!\\!-\\!M_{n_{2}}^{2})k\hskip
0.85358pts_{\theta}~{}}{\sqrt{2\,}M_{n_{1}}\\!M_{n_{2}}}\hskip
0.85358pt{a_{n_{1}n_{2}n_{3}}}+O(E^{0})\hskip 0.85358pt,$
$\displaystyle=\,-\text{i}\hskip 0.85358ptg\hskip
0.85358ptf^{abc}(\epsilon_{3}\\!\cdot{p_{1}})\frac{~{}(M_{n_{3}}^{2}\\!\\!-\\!M_{n_{1}}^{2}\\!\\!-\\!M_{n_{2}}^{2})~{}}{M_{n_{1}}M_{n_{2}}}\hskip
0.85358pt{a_{n_{1}n_{2}n_{3}}}+O(E^{0})\hskip 0.85358pt,$ (67dd)

where each external momentum is on-shell ($\hskip 0.85358ptp_{j\mu}\hskip
0.85358ptp_{j}^{\mu}\\!=-M_{n_{j}}^{2}$) and $s_{\theta}\\!=\\!\sin\theta$
with $\theta$ being the angle between the moving directions of particles 1 and
3 . The cubic coupling coefficient of KK gauge bosons $a_{n_{1}n_{2}n_{3}}\\!$
is given by Eq.(67lj). We further compute the corresponding 3-point scattering
amplitude with particles 1 and 2 replaced by the KK Goldstone bosons,
$A_{5}^{a\hskip 0.85358ptn_{1}}(p_{1})A_{5}^{a\hskip 0.85358ptn_{2}}(p_{2})$.
Thus, we derive the following 3-point scattering amplitudes of KK Goldstone
and gauge bosons:777From this section and hereafter, we denote scattering
amplitudes containing external Goldstone states with a tilde symbol, such as
$\widetilde{\mathcal{T}}$ and $\widetilde{\mathcal{M}}$, to distinguish them
from the amplitudes with all external states being physical particles.

$\widetilde{\mathcal{T}}[A_{5}^{an_{1}}\\!A_{5}^{bn_{2}}\\!A_{\pm}^{cn_{3}}]\,=\,-\text{i}\hskip
0.85358pt2\hskip 0.85358ptgf^{abc}(\epsilon_{3}\\!\cdot p_{1})\hskip
0.85358pt\tilde{a}_{n_{1}n_{2}n_{3}}=\mp\text{i}\hskip 0.85358ptg\hskip
0.85358ptf^{abc}\sqrt{2\,}k\hskip 0.85358pts_{\theta}\hskip
0.85358pt\tilde{a}_{n_{1}n_{2}n_{3}}\hskip 0.85358pt,$ (67de)

where the cubic KK Goldstone-gauge coupling coefficient $\hskip
0.85358pt\tilde{a}_{n_{1}n_{2}n_{3}}\\!$ is given by Eq.(67lj).

According to the KK GAET (38), the equivalence between the two KK scattering
amplitudes (3.1.1) and (67de) in the high energy limit,

$\mathcal{T}_{0}[A_{L}^{an_{1}}\\!A_{L}^{bn_{2}}\\!A_{\pm}^{cn_{3}}]\hskip
0.85358pt=\hskip
0.85358pt-\widetilde{\mathcal{T}}_{0}[A_{5}^{an_{1}}\\!A_{5}^{bn_{2}}\\!A_{\pm}^{cn_{3}}]\hskip
0.85358pt,$ (67df)

where $\mathcal{T}_{0}$ and $\widetilde{\mathcal{T}}_{0}$ denote the leading-
order KK amplitudes under high energy expansion. The 3-point GAET (67df) is
equivalent to the following sum rule condition:

$\left(M_{n_{1}}^{2}\\!\\!+\\!M_{n_{2}}^{2}\\!-\\!M_{n_{3}}^{2}\right)a_{n_{1}n_{2}n_{3}}=\,2{M_{n_{1}}\\!M_{n_{2}}}\tilde{a}_{n_{1}n_{2}n_{3}}\hskip
0.85358pt,$ (67dg)

which connects the trilinear KK gauge boson coupling $\,a_{n_{1}n_{2}n_{3}}$
to the corresponding trilinear KK Goldstone-gauge boson coupling
$\,\tilde{a}_{n_{1}n_{2}n_{3}}$. The condition (67dg) does hold for the warped
5d KK Yang-Mills gauge theory as we directly prove it in Eq.(67mi) of Appendix
D. For the special case of $(n_{1},\hskip 0.85358ptn_{2},\hskip
0.85358ptn_{3})=(n,\hskip 0.85358ptn,\hskip 0.85358ptj)$, the condition (67dg)
takes the following simple form:

$\big{(}1\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}\hskip
0.85358ptr_{j}^{2}\big{)}a_{nnj}=\,\tilde{a}_{nnj}\hskip 0.85358pt,$ (67dh)

where we denote $\,r_{j}\\!=\\!M_{j}/M_{n}\hskip 0.85358pt$.

Next, we compute the 3-point KK gauge boson scattering amplitude including one
longitudinal and two transverse external states and the corresponding KK
Goldstone-gauge boson scattering amplitude:

$\displaystyle\mathcal{T}_{0}[A^{an_{1}}_{\pm}\\!A^{bn_{2}}_{\pm}\\!A^{cn_{3}}_{L}]$
$\displaystyle=-\text{i}\hskip
0.85358ptgf^{abc}{a_{n_{1}n_{2}n_{3}}}\Big{\\{}\\!(\epsilon_{1}\\!\cdot\epsilon_{2})\\!\\!\left[\\!(M_{n_{1}}^{2}\\!\\!-\\!M_{n_{2}}^{2})M_{n_{3}}^{-1}\\!+\\!(p_{1}\\!-p_{2})\\!\cdot\\!v_{3}\right]$
$\displaystyle\hskip 11.38109pt+\\!2(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{2}\\!\cdot v_{3})-\\!2(\epsilon_{1}\\!\cdot
v_{3})(\epsilon_{2}\\!\cdot p_{1})\\!\Big{\\}}$ $\displaystyle=0+O(E^{0})\,,$
(67dia)
$\displaystyle\widetilde{\mathcal{T}}_{0}[A^{an_{1}}_{\pm}\\!A^{bn_{2}}_{\pm}\\!A^{cn_{3}}_{5}]$
$\displaystyle=gf^{abc}\big{(}\epsilon_{1}\\!\cdot\epsilon_{2}\big{)}(\tilde{a}_{n_{1}n_{3}n_{2}}M_{n_{1}}\\!-\tilde{a}_{n_{2}n_{3}n_{1}}M_{n_{2}})$
$\displaystyle=gf^{abc}{a}_{n_{1}n_{2}n_{3}}(\epsilon_{1}\\!\cdot\epsilon_{2})\big{(}M_{n_{1}}^{2}\\!\\!-\\!M_{n_{2}}^{2}\big{)}M_{n_{3}}^{-1}=0+O(E^{0})\hskip
0.85358pt,$ (67dib)

where
$v_{3}^{\mu}\\!=\epsilon_{3,L}^{\mu}\\!-p_{3}^{\mu}/{M_{n_{3}}}\\!\\!=\\!O(M_{n_{3}}/E_{3})$,
and we have used the sum rules (67mh) to simplify the second KK amplitude
(67dib). From the above, we see that the leading contributions of $O(E^{1})$
vanishes, and the remaining contributions are of the same $O(E^{0})$ as the
residual term $\hskip
0.85358pt\mathcal{T}_{v}\\!=\\!\mathcal{T}[A^{an_{1}}_{\pm}\\!A^{bn_{2}}_{\pm}\\!A^{cn_{3}}_{\mu}]v^{\mu}$.

We have also analyzed the 3-point scattering amplitudes of the pure
longitudinal KK gauge bosons and of their corresponding KK Goldstone bosons.
The 3-point longitudinal KK gauge boson amplitude ($3A_{L}^{an}$) has the
leading energy dependence of $E^{4}$ by the power counting rule (67dba), which
arises from the trilinear KK gauge boson vertex (containing one external
momentum) and the three external longitudinal polarization vectors, each of
which has the high energy behavior

$\epsilon_{L}^{\mu}=\epsilon_{S}^{\mu}+v^{\mu}=O(E^{1})+O(E^{-1})\hskip
0.85358pt,$ (67dj)

according to Eq.(35b), where $\hskip
0.85358pt\epsilon_{S}^{\mu}\\!=\\!p_{j}^{\mu}/M_{n_{j}}$ for the $j$-th
external state ($j\\!=\\!1,2,3$). The four external momenta (from either the
trilinear vertex or the $\epsilon_{S}^{\mu}$ part of the external longitudinal
polarization vector $\epsilon_{L}^{\mu}$) will form two bilinear contractions
like $\,p_{i}\\!\cdot p_{j}$ or $p_{j}^{2}$, which can be converted into
squared KK-masses due to the momentum conservation $\hskip
0.85358ptp_{1}\\!+\\!p_{2}\\!+\\!p_{3}\\!=\\!0\hskip 0.85358pt$ and the on-
shell condition of each external momentum $p_{j}^{2}\\!=\\!-M_{n_{j}}^{2}$. So
all the leading energy dependence of $E^{4}$ will be reduced to $E^{0}$. The
remaining contributions arise from replacing the external longitudinal
polarization vectors by one, two, and three vectors of
$v^{\mu}\\!\\!=\\!O(E^{-1})$, which can contribute to the 3-point KK gauge
boson amplitude with leading energy dependence of $E^{0}$, $E^{0}$, and
$E^{-2}$ respectively. Hence, the 3-point longitudinal KK gauge boson
amplitude should actually have a deduction on its leading energy dependence,
$E^{4}\\!\rightarrow\\!E^{0}$. We can show that this is consistent with the
requirement of the GAET identity (36). We note that the corresponding 3-point
KK Goldstone amplitude vanishes at tree level due to the absence of cubic
vertex $A_{5}^{an}$-$A_{5}^{am}$-$A_{5}^{a\ell}$. Thus, according to the GAET
identity (36), the leading nonzero contribution on its right-hand side arises
from the residual terms with the trilinear vertex having two KK Goldstone
bosons and one KK gauge boson (contracting with the external vector $v^{\mu}$)
which is apparently of $O(E^{0})$. Note that the second power counting rule in
Eq.(67dba) gives: $D_{E}\\!=\\!4\\!-\\!3\\!-\\!1\\!=\\!0\hskip 0.85358pt$,
where we have inputs $(N,\hskip 0.85358ptN_{v})\\!=\\!(3,\hskip 0.85358pt1)$
and $\bar{V}_{3}^{\min}\\!=\\!0\hskip 0.85358pt$. Hence, the GAET identity
(36) also enforces the energy deduction of $E^{4}\\!\\!\rightarrow\\!E^{0}$
for the 3-point longitudinal KK gauge boson amplitude on its left-hand side.

Then, we explicily compute the 3-point scattering amplitude of pure
longitudinal KK gauge bosons, in which each external longituding polarization
vector can be split into two parts,
$\epsilon_{L}^{\mu}=\epsilon_{S}^{\mu}+v^{\mu}$ as shown in Eq.(67dj). Thus,
we derive the expanded scattering amplitude of 3 longitudinal KK gauge bosons:

$\displaystyle\mathcal{T}[A^{an_{1}}_{L}\\!A^{bn_{2}}_{L}\\!A^{cn_{3}}_{L}]=-\text{i}gf^{abc}({M_{n_{1}}M_{n_{2}}M_{n_{3}}})^{-1}a_{n_{1}n_{2}n_{3}}\times$
$\displaystyle\hskip 17.07164pt\Big{[}\,(p_{1}\\!\cdot p_{2})(p_{3}\\!\cdot
v_{2})-(p_{1}\\!\cdot p_{2})(p_{3}\\!\cdot v_{1})-(p_{2}\\!\cdot
p_{3})(p_{1}\\!\cdot v_{2})+(p_{1}\\!\cdot p_{3})(p_{2}\\!\cdot v_{1})$
$\displaystyle\hskip 11.38109pt+(p_{2}\\!\cdot p_{3})(p_{1}\\!\cdot
v_{3})-(p_{1}\\!\cdot p_{3})(p_{2}\\!\cdot v_{3})+(p_{1}\\!\cdot
v_{3})(p_{2}\\!\cdot v_{1})-(p_{1}\\!\cdot p_{2})(v_{1}\\!\cdot
v_{3})-(p_{1}\\!\cdot v_{2})(p_{2}\\!\cdot v_{3})$ $\displaystyle\hskip
11.38109pt+(p_{1}\\!\cdot p_{2})(v_{2}\\!\cdot v_{3})-(p_{1}\\!\cdot
v_{2})(p_{3}\\!\cdot v_{1})+(p_{1}\\!\cdot p_{3})(v_{1}\\!\cdot
v_{2})+(p_{1}\\!\cdot v_{3})(p_{3}\\!\cdot v_{2})-(p_{1}\\!\cdot
p_{3})(v_{2}\\!\cdot v_{3})$ $\displaystyle\hskip 11.38109pt+(p_{1}\\!\cdot
v_{3})(v_{1}\\!\cdot v_{2})-(p_{1}\\!\cdot v_{2})(v_{1}\\!\cdot
v_{3})-(p_{2}\\!\cdot p_{3})(v_{1}\\!\cdot v_{2})+(p_{2}\\!\cdot
v_{1})(p_{3}\\!\cdot v_{2})+(p_{2}\\!\cdot p_{3})(v_{1}\\!\cdot v_{3})$
$\displaystyle\hskip 11.38109pt-(p_{2}\\!\cdot v_{3})(p_{3}\\!\cdot
v_{1})-(p_{2}\\!\cdot v_{3})(v_{1}\\!\cdot v_{2})+(p_{2}\\!\cdot
v_{1})(v_{2}\\!\cdot v_{3})+(p_{3}\\!\cdot v_{2})(v_{1}\\!\cdot
v_{3})-(p_{3}\\!\cdot v_{1})(v_{2}\\!\cdot v_{3})\Big{]}.$ (67dk)

In the above amplitude, we see that all the terms having leading energy
dependence of $E^{4}$ disappear as expected, and all the remaining terms can
be classfied into the following 4 types according to the numbers of $v_{j}$’s
contained therein:

$(p_{i}\cdot p_{j})(p_{k}\cdot v_{\ell}),\quad(p_{i}\cdot v_{j})(p_{k}\cdot
v_{\ell}),\quad(p_{i}\cdot p_{j})(v_{k}\cdot v_{\ell}),\quad(p_{i}\cdot
v_{j})(v_{k}\cdot v_{\ell}).$ (67dl)

Due to the momentum conservation $\hskip
0.85358ptp_{1}+p_{2}+p_{3}\\!=\\!0\hskip 0.85358pt$ and the on-shell condition
of each external momentum, the bilinear momentum products can all be converted
into squared KK-masses, $\hskip 0.85358ptp_{i}\cdot
p_{j}\\!=\\!\frac{1}{2}(M_{n_{i}}^{2}\\!+\\!M_{n_{j}}^{2}\\!-\\!M_{n_{k}}^{2})$
(with $i\\!\neq\\!j\\!\neq\\!k$) and $p_{j}^{2}\\!=\\!-M_{n_{j}}^{2}$. Since
$v_{j}\\!=\\!O(M_{n_{j}}/E_{j})$, the 3 longitudinal KK gauge boson amplitude
(67dk) could only have leading energy dependence of $E^{0}$ and are given by
the terms of the first and second types in Eq.(67dl). Thus, the GAET identity
(36) for the 3-point tree-level scattering amplitude of longitudinal KK gauge
bosons take the following form:

$\mathcal{T}[A^{an_{1}}_{L}\\!A^{bn_{2}}_{L}\\!A^{cn_{3}}_{L}]\hskip
0.85358pt=\hskip 0.85358pt\text{i}^{3}\hskip
0.85358pt\widetilde{\mathcal{T}}[A^{an_{1}}_{5}\\!A^{bn_{2}}_{5}\\!A^{cn_{3}}_{5}]+\mathcal{T}_{v}\hskip
0.85358pt=\hskip 0.85358pt0+O(E^{0})\hskip 0.85358pt,$ (67dm)

where on the right-hand side the 3-point KK Goldstone amplitude vanishes and
the residual term $\mathcal{T}_{v}$ gives the leading contributions to the
corresponding longitudinal KK gauge boson amplitude on the left-hand side.
Thus, we further deduce the $O(E^{0})$ contributions to the 3-point scattering
amplitude of pure longitudinal KK gauge bosons as follows:

$\displaystyle\mathcal{T}_{0}[A^{an_{1}}_{L}\\!A^{bn_{2}}_{L}\\!A^{cn_{3}}_{L}]=-\text{i}\hskip
0.85358ptgf^{abc}(M_{n_{1}}M_{n_{2}}M_{n_{3}})^{-1}a_{n_{1}n_{2}n_{3}}\times$
$\displaystyle\Big{[}M_{13}(p_{2}\\!\cdot v_{1})\\!-\\!M_{12}(p_{3}\\!\cdot
v_{1})\\!-\\!M_{23}(p_{1}\\!\cdot v_{2})\\!+\\!M_{23}(p_{1}\\!\cdot
v_{3})\\!+\\!M_{12}(p_{3}\\!\cdot v_{2})\\!-\\!M_{13}(p_{2}\\!\cdot v_{3})$
$\displaystyle+(p_{1}\\!\cdot v_{3})(p_{2}\\!\cdot v_{1})\\!-\\!(p_{1}\\!\cdot
v_{2})(p_{2}\\!\cdot v_{3})-(p_{1}\\!\cdot v_{2})(p_{3}\\!\cdot
v_{1})\\!+\\!(p_{1}\\!\cdot v_{3})(p_{3}\\!\cdot v_{2})$
$\displaystyle+(p_{2}\\!\cdot v_{1})(p_{3}\\!\cdot v_{2})-(p_{2}\\!\cdot
v_{3})(p_{3}\\!\cdot v_{1})\Big{]},$ (67dn)

where we have defined ${M_{ij}}\equiv p_{i}\cdot
p_{j}\\!=\\!\frac{1}{2}(M_{n_{i}}^{2}\\!+\\!M_{n_{j}}^{2}\\!-\\!M_{n_{k}}^{2})$
with $i\\!\neq\\!j\\!\neq\\!k\hskip 0.85358pt$.

####  3.1.2 Warped GRET for 3-Point KK Graviton Amplitudes

In this subsection, we compute the 3-point massive scattering amplitudes of KK
gravitons and of the corresponding gravitational KK Goldstone bosons. With
these we explicitly demonstrate that the KK GRETs (67cy) and (67ch) hold at
tree level.

Using the Feynman rules of Appendix C.2, we explicitly compute the most
general 3-point on-shell scattering amplitudes of KK gravitons at tree level
as given in Eq.(C.2). Using this, we derive the 3-point KK graviton scattering
amplitude with the external states (1, 2, 3) having helicities $(\pm 1,\,\pm
1,\,\pm 2)$. We also compute the corresponding on-shell 3-point scattering
amplitudes with external states (1, 2) replaced by the gravitational KK vector
Goldstone bosons having helicities $\pm 1\hskip 0.85358pt$. Thus, we derive
the following 3-point gravitational KK scattering amplitudes:

$\displaystyle\mathcal{M}[h^{\pm 1}_{n_{1}}h^{\pm 1}_{n_{2}}h^{\pm
2}_{n_{3}}\hskip 0.85358pt]={-\kappa}\Big{[}(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{1}^{L}\\!\cdot\epsilon_{2}^{L})(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{3}\\!\cdot
p_{1})+(\epsilon_{1}\\!\cdot\epsilon_{2}^{L})(\epsilon_{1}^{L}\\!\cdot
p_{2})(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{3}\\!\cdot p_{1})$
$\displaystyle\hskip 14.22636pt+(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{1}^{L}\\!\cdot\epsilon_{2})(\epsilon_{2}^{L}\\!\cdot\epsilon_{3})(\epsilon_{3}\\!\cdot
p_{1})+(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{1}^{L}\\!\cdot
p_{2})(\epsilon_{2}^{L}\\!\cdot\epsilon_{3})(\epsilon_{3}\\!\cdot p_{1})$
$\displaystyle\hskip
14.22636pt+(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{1}^{L}\\!\cdot\epsilon_{2}^{L})(\epsilon_{2}\\!\cdot
p_{3})(\epsilon_{3}\\!\cdot
p_{1})+(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{1}^{L}\\!\cdot\epsilon_{2})(\epsilon_{2}^{L}\\!\cdot
p_{3})(\epsilon_{3}\\!\cdot p_{1})$ $\displaystyle\hskip
14.22636pt+(\epsilon_{1}\\!\cdot\epsilon_{2}^{L})(\epsilon_{3}\\!\cdot\epsilon_{1}^{L})(\epsilon_{2}\\!\cdot
p_{3})(\epsilon_{3}\\!\cdot
p_{1})+(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot\epsilon_{1}^{L})(\epsilon_{2}^{L}\\!\cdot
p_{3})(\epsilon_{3}\\!\cdot p_{1})$ $\displaystyle\hskip
14.22636pt+(\epsilon_{1}\\!\cdot\epsilon_{2}^{L})(\epsilon_{1}^{L}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot
p_{1})^{2}+(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{1}^{L}\\!\cdot\epsilon_{2}^{L})(\epsilon_{3}\\!\cdot
p_{1})^{2}$ $\displaystyle\hskip
14.22636pt+(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{1}^{L}\\!\cdot
p_{2})(\epsilon_{2}\\!\cdot
p_{3})(\epsilon_{2}^{L}\\!\cdot\epsilon_{3})+(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{1}^{L}\\!\cdot
p_{2})(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{2}^{L}\\!\cdot p_{3})$
$\displaystyle\hskip 14.22636pt+(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{3}\\!\cdot\epsilon_{1}^{L})(\epsilon_{2}\\!\cdot
p_{3})(\epsilon_{2}^{L}\\!\cdot\epsilon_{3})+(\epsilon_{1}\cdot
p_{2})(\epsilon_{3}\\!\cdot\epsilon_{1}^{L})(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{2}^{L}\\!\cdot
p_{3})$ $\displaystyle\hskip 14.22636pt+2(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{1}^{L}\\!\cdot
p_{2})(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{2}^{L}\\!\cdot\epsilon_{3})+2(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{3}\\!\cdot\epsilon_{1}^{L})(\epsilon_{2}\\!\cdot
p_{3})(\epsilon_{2}^{L}\\!\cdot p_{3})\Big{]},$ (67doa)
$\displaystyle\widetilde{\mathcal{M}}[\mathcal{V}^{\pm
1}_{n_{1}}\mathcal{V}^{\pm 1}_{n_{2}}h^{\pm 2}_{n_{3}}\hskip
0.85358pt]={\kappa}\Big{[}(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot
p_{1})^{2}\\!+(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{3}\\!\cdot
p_{1})+(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{2}\\!\cdot
p_{3})(\epsilon_{3}\\!\cdot p_{1})$ $\displaystyle\hskip
14.22636pt-(\mathbb{M}_{n_{1}}^{2}\\!\\!+\\!\mathbb{M}_{n_{2}}\mathbb{M}_{n_{3}}\\!\\!+\\!\mathbb{M}_{n_{3}}^{2})(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{3}\\!\cdot\epsilon_{2})\Big{]}\tilde{\alpha}_{n_{1}n_{2}n_{3}}\hskip
0.85358pt,$ (67dob)

where $\epsilon_{j}\\!=\\!\epsilon_{j}^{\pm}$ denotes transverse polarization
vector associated with the external KK graviton $j\hskip
0.85358pt(=\\!1,2,3)$, and $\epsilon_{j^{\prime}}^{L}$
($j^{\prime}\\!=\\!1,2$) denotes longitudinal polarization vector associated
with the external KK graviton $j^{\prime}$ . In the above, the three-point
gravitational KK coupling coefficients
$(\alpha_{n_{1}n_{2}n_{3}},\,\tilde{\alpha}_{n_{1}n_{2}n_{3}})$ are defined in
Eq.(67lj) of Appendix C.1. Then, we make high energy expansion and derive the
3-point KK scattering amplitudes at the leading order:

$\displaystyle\mathcal{M}_{0}[h^{\pm 1}_{n_{1}}h^{\pm 1}_{n_{2}}h^{\pm
2}_{n_{3}}\hskip 0.85358pt]$
$\displaystyle=\frac{\,\kappa\big{(}\mathbb{M}_{n_{3}}^{2}\\!\\!-\\!\mathbb{M}_{n_{1}}^{2}\\!\\!-\\!\mathbb{M}_{n_{2}}^{2}\big{)}\,}{2\hskip
0.85358pt\mathbb{M}_{n_{1}}\\!\mathbb{M}_{n_{2}}}\Big{[}\\!(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot
p_{1})^{2}\\!+(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{3}\\!\cdot p_{1})$ $\displaystyle\hskip
14.22636pt+(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{2}\\!\cdot
p_{3})(\epsilon_{3}\\!\cdot p_{1})\Big{]}\alpha_{n_{1}n_{2}n_{3}},$ (67dpa)
$\displaystyle\widetilde{\mathcal{M}}_{0}[\mathcal{V}^{\pm
1}_{n_{1}}\mathcal{V}^{\pm 1}_{n_{2}}h^{\pm 2}_{n_{3}}\hskip 0.85358pt]$
$\displaystyle={\kappa}\Big{[}(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot
p_{1})^{2}\\!+(\epsilon_{2}\\!\cdot\epsilon_{3})(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{3}\\!\cdot p_{1})$ $\displaystyle\hskip
14.22636pt+(\epsilon_{3}\\!\cdot\epsilon_{1})(\epsilon_{2}\\!\cdot
p_{3})(\epsilon_{3}\\!\cdot p_{1})\Big{]}\tilde{\alpha}_{n_{1}n_{2}n_{3}}.$
(67dpb)

The equivalence between the two leading-order KK scattering amplitudes
$\mathcal{M}_{0}$ and $\widetilde{\mathcal{M}}_{0}$ in Eqs.(67dp)-(67dp) is
realized by the identity,

$\mathcal{M}_{0}[h^{\pm 1}_{n_{1}}h^{\pm 1}_{n_{2}}h^{\pm 2}_{n_{3}}\hskip
0.85358pt]\hskip 0.85358pt=\hskip
0.85358pt-\widetilde{\mathcal{M}}_{0}[\mathcal{V}^{\pm
1}_{n_{1}}\mathcal{V}^{\pm 1}_{n_{2}}h^{\pm 2}_{n_{3}}\hskip 0.85358pt]\hskip
0.85358pt,$ (67dq)

where the minus sign on the right-hand side comes from the modification factor
$\hat{C}_{\mathrm{mod}}\\!=(-\text{i})^{2}$, and Eq.(67dq) enforces the
following condition:

$\left(\mathbb{M}_{n_{1}}^{2}\\!\\!+\\!\mathbb{M}_{n_{2}}^{2}\\!-\\!\mathbb{M}_{n_{3}}^{2}\right)\\!\alpha_{n_{1}n_{2}n_{3}}\\!=\,2\hskip
0.85358pt\mathbb{M}_{n_{1}}\\!\mathbb{M}_{n_{2}}\tilde{\alpha}_{n_{1}n_{2}n_{3}}\hskip
0.85358pt.$ (67dr)

We have directly proved that the above condition (67dr) does hold for the
warped 5d KK gravity theory as shown in Eq.(67mj) of Appdenix D. We compare
the above Eq.(67dr) with Eq.(67dg) which holds the GAET (67dg) for 3-point KK
gauge boson (KK Goldstone boson) amplitude. We observe that the condition
(67dr) has identical structure to the condition (67dg) and can be inferred
from Eq.(67dg) under the following mass and coupling correspondences
(replacements):

$M_{n_{j}}\\!\\!\rightarrow\mathbb{M}_{n_{j}},~{}~{}~{}~{}a_{n_{1}n_{2}n_{3}}\\!\\!\rightarrow\alpha_{n_{1}n_{2}n_{3}},~{}~{}~{}~{}\tilde{a}_{n_{1}n_{2}n_{3}}\\!\\!\rightarrow\tilde{\alpha}_{n_{1}n_{2}n_{3}}.$
(67ds)

As we will see in Section 4.1, the above correspondences (replacements) are
consistent with the double-copy construction of the GRET from the GAET for the
3-point KK scattering amplitudes.

Besides, for the special case of $(n_{1},\hskip 0.85358ptn_{2},\hskip
0.85358ptn_{3})\\!=\\!(n,\hskip 0.85358ptn,\hskip 0.85358ptj)$, the condition
(67dr) can be simplified as follows:

$\left(1\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}\hskip
0.85358pt\hat{r}_{j}^{2}\right)\\!\alpha_{nnj}=\,\tilde{\alpha}_{nnj}\hskip
0.85358pt,$ (67dt)

with the defined mass ratio
$\,\hat{r}_{j}\\!=\\!\mathbb{M}_{j}/\mathbb{M}_{n}\hskip 0.85358pt$.

Next, we derive the 3-point KK graviton scattering amplitude with external
states (1, 2) being longitudinal KK gravitons and the external state 3 having
helicities $\pm 2\hskip 0.85358pt$. We also compute the 3-point KK Goldstone
scattering amplitude with the external states (1, 2) replaced by the
corresponding gravitational KK Goldstone bosons. Thus, using the general
3-point KK amplitudes (C.2) of Appendix C.2, we infer the following 3-point
gravitational KK scattering amplitudes at the leading order of high energy
expansion:

$\displaystyle\mathcal{M}_{0}[h^{L}_{n_{1}}h^{L}_{n_{2}}h^{\pm
2}_{n_{3}}\hskip 0.85358pt]$ $\displaystyle=\frac{\,\kappa\hskip
0.85358pt(\epsilon_{3}\\!\cdot p_{1})^{2}\,}{\,6\hskip
0.85358pt\mathbb{M}_{n_{1}}^{2}\\!\mathbb{M}_{n_{2}}^{2}\,}\\!\left[2\mathbb{M}_{n_{1}}^{2}\\!\mathbb{M}_{n_{2}}^{2}\\!+\\!\left(\mathbb{M}_{n_{1}}^{2}\\!\\!+\\!\mathbb{M}_{n_{2}}^{2}\\!-\\!\mathbb{M}_{n_{3}}^{2}\right)^{\\!2}\,\right]\\!\alpha_{n_{1}n_{2}n_{3}}\hskip
0.85358pt,$ (67dua)
$\displaystyle\widetilde{\mathcal{M}}_{0}[\phi_{n_{1}}\\!\phi_{n_{2}}\\!h_{n_{3}}^{\pm
2}\hskip 0.85358pt]$ $\displaystyle=\kappa(\epsilon_{3}\\!\cdot
p_{1})^{2}\tilde{\beta}_{n_{1}n_{2}n_{3}}\hskip 0.85358pt,$ (67dub)

where the trilinear KK coupling coefficient
$\tilde{\beta}_{n_{1}n_{2}n_{3}}\\!$ is defined in Eq.(67lpc) of Appendix C.1
and the KK Goldstone boson amplitude (67dub) actually equals its full
amplitude without expansion, i.e.,
$\widetilde{\mathcal{M}}[\phi_{n_{1}}\\!\phi_{n_{2}}\\!h_{n_{3}}^{\pm 2}\hskip
0.85358pt]\\!=\kappa(\epsilon_{3}\\!\cdot
p_{1})^{2}\tilde{\beta}_{n_{1}n_{2}n_{3}}$. Then, to hold the KK GRET (67cu)
for the 3-point KK scattering amplitudes,

$\mathcal{M}_{0}[h^{L}_{n_{1}}h^{L}_{n_{2}}h^{\pm 2}_{n_{3}}\hskip
0.85358pt]\hskip 0.85358pt=\hskip
0.85358pt\widetilde{\mathcal{M}}_{0}[\phi_{n_{1}}\phi_{n_{2}}h_{n_{3}}^{\pm
2}\hskip 0.85358pt]\hskip 0.85358pt,$ (67dv)

we derive the following condition to ensure the equivalence between the two
gravitational KK scattering amplitudes (67dua) and (67dub):

$\left[\\!\left(\mathbb{M}_{n_{1}}^{2}\\!+\\!\mathbb{M}_{n_{2}}^{2}\\!-\\!\mathbb{M}_{n_{3}}^{2}\right)^{2}\\!+2\hskip
0.85358pt\mathbb{M}_{n_{1}}^{2}\\!\mathbb{M}_{n_{2}}^{2}\right]\\!\alpha_{n_{1}n_{2}n_{3}}=\,6\hskip
0.85358pt\mathbb{M}_{n_{1}}^{2}\mathbb{M}_{n_{2}}^{2}\tilde{\beta}_{n_{1}n_{2}n_{3}}\hskip
0.85358pt.$ (67dw)

The proof of the condition (67dw) is highly nontrivial and we present a direct
proof of it around Eq.(67mw) of Appdenix D. For the case of $(n_{1},\hskip
0.85358ptn_{2},\hskip 0.85358ptn_{3})\\!=\\!(n,\hskip 0.85358ptn,\hskip
0.85358ptj)$, the condition (67dw) can be simplified as follows:

$(\hat{r}_{j}^{4}\\!-\\!4\hskip
0.85358pt\hat{r}_{j}^{2}\\!+\\!6)\alpha_{nnj}=\,6\hskip
0.85358pt\tilde{\beta}_{nnj}\hskip 0.85358pt,$ (67dx)

with the mass ratio $\,\hat{r}_{j}\\!=\\!\mathbb{M}_{j}/\mathbb{M}_{n}\hskip
0.85358pt$.

Then, we consider a mixed 3-point scattering amplitudes including two external
KK gravitons with helicities $\pm 1$ and one external KK graviton with
helicity zero. Thus, we derive the following 3-point scattering amplitudes of
KK gravitons and of the corresponding KK Goldstone bosons at the leading order
of high energy expansion:

$\displaystyle\mathcal{M}_{0}\big{[}h^{\pm 1}_{n_{1}}h^{\pm
1}_{n_{2}}h^{L}_{n_{3}}\big{]}$
$\displaystyle=\frac{\,\kappa\big{[}2\\!-\\!(\hat{r}_{13}^{2}\\!+\\!\hat{r}_{23}^{2})\\!-\\!(\hat{r}_{13}^{2}\\!-\\!\hat{r}_{23}^{2})^{2}\big{]}\,}{\,2\sqrt{6\,}\hskip
0.85358pt\hat{r}_{13}\hat{r}_{23}\,}(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{2}\\!\cdot p_{1})\alpha_{n_{1}n_{2}n_{3}}\hskip 0.85358pt,$
(67dya) $\displaystyle{\widetilde{\mathcal{M}}}\big{[}\mathcal{V}^{\pm
1}_{n_{1}}\mathcal{V}^{\pm 1}_{n_{2}}\phi_{n_{3}}\big{]}$
$\displaystyle={-\kappa}\Bigg{[}\\!\sqrt{\frac{3}{2}\,}(\epsilon_{1}\\!\cdot
p_{2})(\epsilon_{2}\\!\cdot
p_{1})\\!-\\!\frac{\,{\mathbb{M}_{n_{3}}^{2}}\hskip
0.85358pt}{\sqrt{6\,}\hskip
0.85358pt}(\epsilon_{1}\\!\cdot\epsilon_{2})(\hat{r}_{13}\hat{r}_{23}\\!+\\!2\hskip
0.85358pt\hat{r}_{13}\\!+\\!2\hskip
0.85358pt\hat{r}_{23})\\!\Bigg{]}\tilde{\rho}_{n_{1}n_{2}n_{3}}$
$\displaystyle=\mathcal{M}_{0}\big{[}\mathcal{V}^{\pm
1}_{n_{1}}\mathcal{V}^{\pm 1}_{n_{2}}\phi_{n_{3}}\big{]}+O(E^{0})\hskip
0.85358pt,$ (67dyb)
$\displaystyle{\widetilde{\mathcal{M}}}_{0}\big{[}\mathcal{V}^{\pm
1}_{n_{1}}\mathcal{V}^{\pm 1}_{n_{2}}\phi_{n_{3}}\big{]}$
$\displaystyle={-\kappa}\sqrt{\frac{3}{2}\,}\hskip
0.85358pt(\epsilon_{1}\\!\cdot p_{2})(\epsilon_{2}\\!\cdot
p_{1})\tilde{\rho}_{n_{1}n_{2}n_{3}}\hskip 0.85358pt,$ (67dyc)

where we have defined the KK mass ratios $\hskip
0.85358pt\hat{r}_{13}\\!=\\!\mathbb{M}_{n_{1}}\\!/\mathbb{M}_{n_{3}}$ and
$\hskip 0.85358pt\hat{r}_{23}\\!=\\!\mathbb{M}_{n_{2}}/\mathbb{M}_{n_{3}}$,
and the $\mathcal{V}_{n_{1}}^{\pm 1}$-$\mathcal{V}_{n_{2}}^{\pm 1}$-$\phi_{n}$
trilinear coupling coefficient $\tilde{\rho}_{n_{1}n_{2}n_{3}}$ is defined in
Eq.(67lp) of Appendix C. There is another helicity combination for such mixed
3-point KK scattering amplitudes, but they vanish at the leading order of high
energy expansion:

$\mathcal{M}_{0}\big{[}h^{\pm 1}_{n_{1}}h^{\mp
1}_{n_{2}}h^{L}_{n_{3}}\big{]}=0\hskip
0.85358pt,~{}~{}~{}~{}~{}{\widetilde{\mathcal{M}}}_{0}\big{[}\mathcal{V}^{\pm
1}_{n_{1}}\mathcal{V}^{\mp 1}_{n_{2}}{\phi_{n_{3}}}\big{]}=0\hskip 0.85358pt.$
(67dz)

Using the 3-point gravitational KK scattering amplitudes (67dya)-(67dyc), we
find that holding the GRET,

$\mathcal{M}_{0}\big{[}h^{\pm 1}_{n_{1}}h^{\pm
1}_{n_{2}}h^{L}_{n_{3}}\big{]}\,=\,-\widetilde{\mathcal{M}}_{0}\big{[}\mathcal{V}^{\pm
1}_{n_{1}}\mathcal{V}^{\pm 1}_{n_{2}}\phi_{n_{3}}\big{]},$ (67ea)

will impose the following condition:

$\Big{[}2\\!-\\!(\hat{r}_{13}^{2}\\!+\\!\hat{r}_{23}^{2})\\!-\\!(\hat{r}_{13}^{2}\\!-\\!\hat{r}_{23}^{2})^{2}\Big{]}{\alpha_{n_{1}n_{2}n_{3}}}\\!=\hskip
0.85358pt6\hskip 0.85358pt\hat{r}_{13}\hat{r}_{23}\hskip
0.85358pt\tilde{\rho}_{n_{1}n_{2}n_{3}}\hskip 0.85358pt,$ (67eb)

with the KK mass ratios $(\hat{r}_{13},\hskip 0.85358pt\hat{r}_{23})$ and the
trilinear KK coupling coefficient $\tilde{\rho}_{n_{1}n_{2}n_{3}}$ are defined
below Eq.(67dy). The above identity (67eb) can be directly proven as in (67my)
of Appendix D. Hence, we have explicitly established the GRET in its mixed
form (67ea) for the 3-point gravitational KK scattering amplitudes.

Besides, we can consider a specific case with $(n_{1},\hskip
0.85358ptn_{2},\hskip 0.85358ptn_{3})\\!=\\!(n,\hskip 0.85358ptn,\hskip
0.85358ptj)$. So we have $\hskip
0.85358pt\hat{r}_{13}\\!=\\!\hat{r}_{23}\hskip 0.85358pt$ and thus the above
GRET condition (67eb) is simplified as follows:

$(\hat{r}_{j}^{2}\\!-\\!1)\hskip 0.85358pt\alpha_{nnj}=3\hskip
0.85358pt\tilde{\rho}_{nnj}\hskip 0.85358pt,$ (67ec)

where the mass ratio $\hskip
0.85358pt\hat{r}_{j}\\!=\\!\mathbb{M}_{j}/\mathbb{M}_{n}\hskip 0.85358pt$. For
the case of compactified flat 5d under $S^{1}\\!/\mathbb{Z}_{2}$, the KK index
$j$ could only take values of $j=0,2n$ in Eq.(67ec).

Finally, we discuss the 3-point scattering amplitudes of pure longitudinal KK
gravitons ($3h_{n}^{L}$) and of the corresponding gravitational KK Goldstone
bosons ($3\phi_{n}$). We gave a brief comment on this in a separate paragraph
below Eq.(67cz). The tree-level 3-point longitudinal KK graviton amplitude
($3h_{n}^{L}$) have a possible leading energy dependence of $E^{8}$ by using
the power counting rule (67cza) for $N\\!\\!=\\!3\,$. But, we note that the
3-point KK graviton amplitude arises from the trilinear KK graviton vertex
(whose coupling factor contains products of two external momenta) and is
expected to have individual leading energy dependence of $E^{8}$ after
including the 3 external longitudinal polarization tenors of KK gravitons,
each of which has the high energy behavior,

$\varepsilon^{\mu\nu}_{L}=\sqrt{\frac{2}{3}\,}\hskip
0.85358pt\varepsilon_{S}^{\mu\nu}\\!+\tilde{v}^{\mu\nu}=O(E^{2})+O(E^{0})\hskip
0.85358pt,$ (67ed)

based on Eqs.(67ck)-(67cl), where for the $j$-th external KK graviton the
scalar polarization tensor
$\varepsilon_{S}^{\mu\nu}(p_{j})\\!=\\!p_{j}^{\mu}p_{j}^{\nu}/\mathbb{M}_{n_{j}}^{2}$
(with $j\\!=\\!1,2,3$). Thus, each leading energy term of $O(E^{8})$ is given
by picking up the leading energy term $\varepsilon_{S}^{\mu\nu}$ from all 3
external longitudinal polarization tenors ($\varepsilon^{\mu\nu}_{L}\hskip
0.85358pt$) which have contractions with the two-momentum-factors in of the
trilinear KK graviton vertex. These momenta will form up 4 pairs of bilinear
contraction ($\,p_{i}\cdot p_{j}$) which can be all converted into certain
combinations of squared KK-masses by using the momentum conservation $\hskip
0.85358ptp_{1}\\!+\\!p_{2}\\!+\\!p_{3}\\!=\\!0\hskip 0.85358pt$ and the on-
shell condition of each external momentum $p_{j}^{2}\\!=\\!-M_{n_{j}}^{2}$.
Hence, all the leading energy terms of $O(E^{8})$ are reduced to $O(E^{0})$ in
this way. The remaining contributions include to replace one, two, and three
external longitudinal polarization tensors ($\varepsilon^{\mu\nu}_{L}\hskip
0.85358pt$) by the tensor $\tilde{v}^{\mu\nu}\\!\\!=\\!O(E^{0})$ respectively.
For those external momentum factors contracting among themselves, they will be
converted into squared KK-masses due to the momentum conservation and the on-
shell conditions on all the external momenta. Hence, a given external momentum
$p_{j}^{\mu}$ could contribute a nontrivial energy factor $E^{1}$ to the
3-point KK graviton amplitude only if it contracts with a
$\tilde{v}^{\mu\nu}\\!=\\!O(E^{0})$ tensor from Eq.(67ed). The terms having
maximal number of such contractions arise from picking up two
$\tilde{v}^{\mu\nu}$ tensors from two of the external KK graviton states via
Eq.(67ed); so the third external longitudinal polarization tensor contributes
a leading momentum term of $\varepsilon_{S}^{\mu\nu}$ via Eq.(67ed) and the
trilinear KK graviton vertex contributes the leading coupling factors with two
external momenta; thus all the 4 external momenta may contract with the two
external $\tilde{v}^{\mu\nu}$ tensors [via Eq.(67ed)] and thus makes the
leading energy contributions of $O(E^{4})$. On the other hand, the tree-level
amplitude of 3 gravitational KK Goldstone bosons ($3\phi_{n}$) arises from
trilinear KK Goldstone vertex whose coupling factor contains products of two
external momenta $\,p_{i}\cdot p_{j}$. So the naive power counting shows that
this KK Goldstone amplitude would have leading energy dependence of $E^{2}$ in
agreement with Eq.(67cza). But the momentum conservation and the on-shell
condition of each external state convert all the bilinear momentum products
($\,p_{i}\cdot p_{j}$) in the trilinear Goldstone vertex coupling into squared
KK-masses, so the 3-point KK Goldstone amplitude actually has energy
dependence $E^{0}$. Then, we can apply the GRET (67cu) and note that on its
right-hand side the corresponding 3-point KK Goldstone amplitude has the
actual leading energy dependence $E^{0}$. Hence, the GRET ensures a large
energy cancellation of $E^{4}\\!\rightarrow\\!E^{0}\hskip 0.85358pt$ in the
3-point longitudinal KK graviton amplitude.

Then, we explicitly compute the 3-point scattering amplitudes of pure
longitudinal KK gravitons ($3h_{n}^{L}$) and of the corresponding
gravitational KK Goldstone bosons ($3\phi_{n}$). Under high energy expansion,