# Gravitational Equivalence Theorem and Double-Copy
for Kaluza-Klein Graviton Scattering Amplitudes

Yan-Feng Hang<EMAIL_ADDRESS>Tsung-Dao Lee Institute $\&$ School of
Physics and Astronomy,
Key Laboratory for Particle Astrophysics and Cosmology (MOE),
Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai Jiao Tong University, Shanghai, China  Hong-Jian He<EMAIL_ADDRESS>Tsung-Dao Lee Institute $\&$ School of Physics and Astronomy,
Key Laboratory for Particle Astrophysics and Cosmology (MOE),
Shanghai Key Laboratory for Particle Physics and Cosmology,
Shanghai Jiao Tong University, Shanghai, China  Institute of Modern Physics
$\&$ Physics Department, Tsinghua University, Beijing, China;
Center for High Energy Physics, Peking University, Beijing, China

###### Abstract

We analyze the structure of scattering amplitudes of the Kaluza-Klein (KK)
gravitons and of the KK gravitational Goldstone bosons in the compactified 5d
General Relativity (GR). Using a general $R_{\xi}$ gauge-fixing, we study the
geometric Higgs mechanism for the massive spin-2 KK gravitons. We newly
propose and prove a Gravitational Equivalence Theorem (GRET) to connect the
scattering amplitudes of longitudinal KK gravitons to that of the KK
gravitational Goldstone bosons, which formulates the geometric gravitational
Higgs mechanism at the scattering $S$-matrix level. We demonstrate that the
GRET provides a general energy-cancellation mechanism guaranteeing the
$N$-point longitudinal KK graviton scattering amplitudes to have their leading
energy dependence cancelled down by a large power factor of $E^{2N}$
($N\\!\geqslant\\!4$) up to any loop order. We propose an extended double-copy
approach to construct the massive KK graviton (Goldstone) amplitudes from the
KK gauge boson (Goldstone) amplitudes. With these we establish a new
correspondence between the two types of energy cancellations in the four-point
longitudinal KK amplitudes at tree level: $\,E^{4}\\!\\!\rightarrow\\!E^{0}\,$
in the KK gauge theory and $\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$ in the KK GR
theory.
Journal-Ref: Research, vol. 2022 (2022), Article ID 9860945.
https://doi.org/10.34133/2022/9860945

##  I Introduction

Kaluza-Klein (KK) compactification KK  of the extra spatial dimensions leads
to infinite towers of massive KK excitation states in the low energy 4d
effective field theory. This serves as an essential ingredient of all extra
dimensional models Exd  and the string/M theories string . The KK
compactification realizes the geometric “Higgs” mechanisms for mass
generations of KK gravitons GHiggs  and of KK gauge bosons 5DYM2002  without
invoking any extra Higgs boson of the conventional Higgs mechanism higgsM .

In this work, we formulate the geometric gravitational “Higgs” mechanism for
the compactified 5d General Relativity (GR5) by quantizing the KK GR5 under a
general $R_{\xi}$ gauge-fixing at both the Lagrangian level and the $S$-matrix
level. We prove that the KK graviton propagator is free from the longstanding
problem of van Dam-Veltman and Zakharov (vDVZ) discontinuity vDVZ  in the
conventional Fierz-Pauli massive gravity PF Hinterbichler:2012  and the KK GR5
theory can consistently realize the mass-generation for spin-2 KK gravitons.
Then, we propose and prove a new Gravitational Equivalence Theorem (GRET)
which quantitatively connects each scattering amplitude of the (helicity-zero)
longitudinally-polarized KK gravitons to that of the corresponding KK
Goldstone bosons. The GRET takes a highly nontrivial form and differs
substantially from the KK Gauge Equivalence Theorem (GAET) of the 5d KK gauge
theories 5DYM2002 5DYM2002-2 KK-ET-He2004 , because each massive KK graviton
$h_{n}^{\mu\nu}$ has 5 helicity states ($\lambda\\!=\\!0,\pm 1,\pm 2$) where
the $\lambda\\!=\\!0,\pm 1\,$ components arise from absorbing a scalar
Goldstone boson $h_{n}^{55}$ ($\lambda\\!\\!=\\!\\!0$) and a vector Goldstone
boson $h_{n}^{\mu 5}$ ($\lambda\\!\\!=\\!\\!\pm 1$) in the 5d graviton field.
We demonstrate that the GRET provides a general energy-cancellation mechanism
guaranteeing that the leading energy dependence of $N$-particle longitudinal
KK graviton amplitudes ($\propto\\!E^{2(N+1+L)}$) must cancel down to a much
lower energy power ($\propto\\!E^{2(1+L)}$) by an energy factor of $E^{2N}$,
as enforced by matching the energy dependence of the corresponding leading
gravitational KK Goldstone amplitudes, where $L$ denotes the loop number of
the relevant Feynman diagram. For the four-point longitudinal KK graviton
scattering amplitudes at tree level, this proves the energy cancellations
$\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$, which explains the result of the recent
explicit calculations of 4-longitudinal KK graviton amplitudes Chivukula:2019S
Chivukula:2019L Kurt .
The double-copy approach has profound importance for understanding the quantum
gravity because it uncovers the deep gauge-gravity connection at the
scattering $S$-matrix level, $\mathrm{GR}\\!=\\!(\mathrm{Gauge}$
$\mathrm{Theory})^{2}$ Elvang:2013 . The conventional double-copy method with
color-kinematics (CK) duality of Bern-Carrasco-Johansson (BCJ) BCJ:2008
BCJ:2019  was proposed to connect scattering amplitudes between the massless
Yang-Mills (YM) gauge theories and the massless GR theories. It was inspired
by the Kawai-Lewellen-Tye (KLT) relation KLT  which connects the product of
two scattering amplitudes of open strings to that of the closed string at tree
level Tye-2010 .
Extending the conventional double-copy approach, we construct the massive KK
graviton (Goldstone) amplitudes from the massive KK YM gauge (Goldstone)
amplitudes under high energy expansion at the leading order (LO) and at the
next-to-leading order (NLO). This provides an extremely efficient way to
derive the complicated massive KK graviton amplitudes from the massive KK
gauge boson amplitudes, and gives a deep understanding on the structure of the
KK graviton amplitudes.
Because the LO amplitudes of the longitudinal KK gauge bosons and of their KK
Goldstone bosons have $\mathcal{O}(E^{0}M_{n}^{0})$ and are equal (leading to
the KK GAET) 5DYM2002 , our double-copy approach demonstrates that the
reconstructed LO amplitudes of the longitudinal KK gravitons and of the KK
Goldstone bosons have $\mathcal{O}(E^{2}M_{n}^{0})$, and must be equal to each
other (leading to the KK GRET), where $M_{n}$ denotes the relevant KK mass.
Our double-copy construction further proves that the residual term of the GRET
belongs to the NLO, which has $\mathcal{O}(E^{0}M_{n}^{2})$ and is suppressed
relative to the LO KK Goldstone boson amplitude of
$\mathcal{O}(E^{2}M_{n}^{0})$. Finally, we further construct an exact double-
copy of the KK graviton scattering amplitudes at the NLO.

##  II $R_{\xi}$ Gauge-Fixing and Geometric Higgs Mechanism

We consider the compactified GR5 under the orbifold $S^{1}\\!/\mathbb{Z}_{2}$
where the fifth dimension is a line segment $0\\!\leqslant x^{5}$
$\leqslant\\!L\,(=\\!\pi r_{c})$, with $r_{c}$ being the compactification
radius. Extension to the case of warped 5d space RS  does not cause conceptual
change regarding our current study. Thus, the 5d Einstein-Hilbert (EH) action
takes the following form:

$S_{\mathrm{EH}}\,=\int\\!\\!\text{d}^{5}x\,\hat{\mathcal{L}}_{\mathrm{EH}}\,=\int\\!\\!\text{d}^{5}x\frac{2}{\,\hat{\kappa}^{2}\,}\sqrt{-\hat{g}\,}\hat{R}\,,$
(1)

where the coupling constant $\,\hat{\kappa}=\\!\sqrt{32\pi\hat{G}\,}\,$.

Then, we expand the 5d EH action (1) under the metric perturbation
$\,\hat{g}_{AB}\\!=\\!\hat{\eta}_{AB}\\!+\\!\hat{\kappa}\hskip
0.85358pt\hat{h}_{AB}$ , where $\,\hat{\eta}_{AB}\\!=\text{diag}$
$(-1,1,1,1,1)$ is the 5d Minkowski metric. Thus, we can express the 5d
graviton field $\,\hat{h}_{AB}$ as follows:

$\hat{h}_{AB}\,=\,\begin{pmatrix}\hat{h}_{\mu\nu}\\!-\\!\frac{1}{2}\eta_{\mu\nu}\hat{\phi}~{}&~{}\
\hat{h}_{\mu 5}\\\\[4.2679pt] \hat{h}_{5\nu}&\ \hat{\phi}\end{pmatrix}\\!.$
(2)

Under the compactification of $S^{1}\\!/\mathbb{Z}_{2}$ , the spin-2 field
$\hat{h}_{\mu\nu}$ and scalar field $\hat{\phi}\,(\equiv\hat{h}^{55})$ are
$\mathbb{Z}_{2}$ even, while the vector field
$\hat{\mathcal{A}}_{\mu}\,(\equiv\hat{h}_{\mu 5})$ is $\mathbb{Z}_{2}$ odd.
After compactification, we derive the 4d effective Lagrangian for both the
zero-modes and KK-modes ($h_{n}^{\mu\nu},\,\mathcal{A}_{n}^{\mu}\hskip
0.85358pt,\,\phi_{n}$) supp .

We further construct a general $R_{\xi}$-type gauge-fixing term as follows:

$\mathcal{L}_{\mathrm{GF}}\,=\,-\sum_{n=0}^{\infty}\frac{1}{\xi_{n}}\\!\left[\,\\!(F_{n}^{\mu})^{2}+(F_{n}^{5})^{2}\\!\,\right],$
(3)

where the gauge-fixing functions $\hskip
0.85358pt(F_{n}^{\mu},\,F_{n}^{5})\hskip 0.85358pt$ take the following form
Hang:2021fmp ,

$\displaystyle F_{n}^{\mu}$
$\displaystyle\,=\,\partial_{\nu}h^{\mu\nu}_{n}\\!-\\!\left(\\!1\\!-\\!\mbox{$\frac{\,{1}\,}{\,2\xi_{n}}$}\right)\\!\partial^{\mu}h_{n}+\xi_{n}M_{n}\mathcal{A}_{n}^{\mu}\,,$
(4a) $\displaystyle F_{n}^{5}$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}\\!\\!\left(M_{n}h_{n}\\!-\\!3\xi_{n}M_{n}\phi_{n}\\!+2\partial_{\mu}\mathcal{A}_{n}^{\mu}\right)\\!.$
(4b)

The above $R_{\xi}$ gauge-fixing can ensure the kinetic terms and propagators
of the KK fields $(h_{n}^{\mu\nu}\\!,\,\mathcal{A}_{n}^{\mu},\,\phi_{n})$ to
be diagonal. In the limit of $\,\xi_{n}\\!\\!\rightarrow\\!\infty\hskip
0.85358pt$, we recover the unitary gauge where the KK Goldstone bosons
$(\mathcal{A}_{n}^{\mu},\,\phi_{n})$ are fully absorbed (eaten) by the
corresponding KK gravitons $h_{n}^{\mu\nu}$ at each KK level-$n$ . This
realizes a Geometric Gravitational “Higgs” Mechanism for KK graviton mass-
generations.

Then, we derive the propagators of KK gravitons and KK Goldstone bosons under
the $R_{\xi}$ gauge-fixing (3) supp . For Feynman-’t Hooft gauge
($\xi_{n}\\!=\\!1$), the KK propagators take the following simple forms:

$\displaystyle\mathcal{D}_{nm}^{\mu\nu\alpha\beta}(p)$
$\displaystyle=-\frac{\,\text{i}\delta_{nm}\,}{2}\frac{\,\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta}\,}{\,p^{2}+M_{n}^{2}\,}\,,$
(5a) $\displaystyle\mathcal{D}^{\mu\nu}_{nm}(p)$
$\displaystyle=-\frac{\,\text{i}\eta^{\mu\nu}\delta_{nm}\,}{\,p^{2}+M_{n}^{2}\,}\,,\quad\mathcal{D}_{nm}(p)=-\frac{\,\text{i}\delta_{nm}\,}{\,p^{2}+M_{n}^{2}\,},$
(5b)

which all share the same mass-pole $p^{2}\\!=\\!-M_{n}^{2}$ .

Strikingly, we observe that our massive KK graviton propagator (5a) has a
smooth limit for $\\!M_{n}\\!\\!\rightarrow\\!0$ , under which Eq.(5a) reduces
to the conventional massless graviton propagator of Einstein gravity. Hence,
we have proven that the KK graviton propagator is free from the vDVZ
discontinuity vDVZ  which is a longstanding problem plaguing the conventional
Fierz-Pauli massive gravity theory and alike PF Hinterbichler:2012 . This is
because the GHM under KK compactification guarantees that the physical degrees
of freedom of each KK graviton are conserved before and after taking the
massless limit $\\!M_{n}\\!\\!\rightarrow\\!0$, i.e., $5=2+2+1\hskip
0.85358pt$. This demonstrates that the compactified KK GR theory can
consistently realize the mass-generation for spin-2 KK gravitons. We can also
derive the unitary-gauge propagator of KK gravitons by taking the limit
$\,\xi_{n}\\!\\!\rightarrow\infty$ Hang:2021fmp .

##  III GRET Formulation for the GHM

In the previous section, we analyzed the geometric Higgs mechanism at the
Lagrangian level. In this section, we further formulate the GRET, which
realizes the geometric gravitational Higgs mechanism at the $S$-matrix level.
Using the gauge-fixing terms (3)-(4) and following the method of Ref. ET94 ,
we derive a Slavnov-Taylor-type identity in the momentum space:

$\langle 0|F_{n_{1}}^{\mu_{1}}(k_{1})F_{n_{2}}^{\mu_{2}}(k_{2})\cdots
F_{m_{1}}^{5}(p_{1})F_{m_{2}}^{5}(p_{2})\cdots\Phi|0\rangle=0\,,$ (6)

where $\Phi$ denotes any other on-shell physical fields after the Lehmann-
Symanzik-Zimmermann (LSZ) amputation and each external momentum obeys the on-
shell condition $\,k_{j}^{2}\\!=\\!-M_{n_{j}}^{2}$ or
$\,p_{j}^{2}\\!=\\!-M_{m_{j}}^{2}$. The identity (6) is a direct consequence
of the diffeomorphism (gauge) invariance of the compactified KK theory
Hang:2021fmp ET94 .

Under the Feynman-’t Hooft gauge ($\xi_{n}\\!=\\!1$) and at the tree level, we
can directly amputate each external state by multiplying the propagator-
inverse $(k^{2}\\!+\\!M_{n}^{2})\\!\rightarrow\\!0$ for Eq.(6). Thus, we
derive Hang:2021fmp  the following GRET identity which connects the
longitudinal KK graviton amplitude to the corresponding KK Goldstone amplitude
plus a residual term:

$\displaystyle\mathcal{M}[h^{L}_{n_{1}}\\!,\cdots\\!,h^{L}_{n_{N}},\Phi]\,=\,\mathcal{M}[\phi_{n_{1}}\\!,\cdots\\!,\phi_{n_{N}}\hskip
0.85358pt,\Phi]+\mathcal{M}_{\\!\Delta}\hskip 0.85358pt,$ (7a)
$\displaystyle\mathcal{M}_{\\!\Delta}=\\!\sum_{1\leqslant k\leqslant
N}\\!\\!\\!\mathcal{M}[\\{\widetilde{\Delta}_{n_{k}},\phi_{n}\\},\Phi]\,,$
(7b)

where $\widetilde{\Delta}_{n}\\!=\tilde{v}_{n}\\!-\tilde{h}_{n}$,
$\tilde{v}_{n}\\!=\\!\tilde{v}_{\mu\nu}h_{n}^{\mu\nu}$, and
$\tilde{h}_{n}\\!\\!=\\!\\!\sqrt{2/3}\,\eta_{\mu\nu}h_{n}^{\mu\nu}$. The
tensor
$\,\tilde{v}^{\mu\nu}\\!\\!=\\!\varepsilon_{L}^{\mu\nu}\\!\\!-\\!\sqrt{2/3\,}\,\varepsilon_{S}^{\mu\nu}\\!\\!=\mathcal{O}(E^{0})$,
and $\,(\varepsilon_{L}^{\mu\nu},\,\varepsilon_{S}^{\mu\nu})$ are the
(longitudinal, scalar) polarization tensors of the KK graviton
$h_{n}^{\mu\nu}$. We can extend the GRET (7) up to loop levels and valid for
all $R_{\xi}$ gauges by using the gravitational BRST identities GET-2 ,
similar to the ET formulation in the 5d KK YM theories KK-ET-He2004  and in
the 4d standard model (SM) ET94 ; ET96 ; ET-Rev .

Inspecting the scattering amplitudes in the GRET identity (7a), we can make
direct power counting on the leading $E$-dependence of individual Feynman
diagrams for each amplitude. For the four-particle scattering, the
longitudinal KK graviton amplitude on the left-hand-side of Eq.(7a) contains
individual contributions via quartic interactions or via exchanging KK-mode
(zero-mode) gravitons. Since each external longitudinal KK graviton has
polarization tensor
$\,\varepsilon_{L}^{\mu\nu}\\!\\!\supset\\!k^{\mu}k^{\nu}\\!/M_{n}^{2}\hskip
0.85358pt$, the leading individual contributions behave as
$\mathcal{O}(E^{10})\hskip 0.85358pt$. But we observe that on the right-hand-
side (RHS) of Eq.(7a), the external states in all amplitudes have no
superficial enhancement or suppression factor. Thus, by power counting on the
KK amplitudes, we find that the RHS of Eq.(7a) (including the residual term
$\mathcal{M}_{\\!\Delta}$) scales as $\mathcal{O}(E^{2})\hskip 0.85358pt$.
Hence, the GRET identity (7) provides a general mechanism for the large energy
cancellations of $E^{10}\\!\\!\\!\rightarrow\\!E^{2}$ in the four-longitudinal
KK graviton amplitudes.

We have further developed a generalized energy-power counting method supp  for
the massive KK gauge and gravity theories, by extending the conventional 4d
power counting rule of Steven Weinberg for the nonlinear sigma model of low
energy QCD weinberg steve-foot . With this and the GRET (7), we can prove a
general energy cancellation $E^{2(N+1+L)}\\!\\!\rightarrow\\!\\!E^{2(1+L)}$ in
the $N$-point longitudinal KK graviton amplitudes, which cancels the leading
energy-dependence by $E^{2N}\\!$ powers supp . For $N$-point longitudinal KK
gauge boson amplitudes, we also prove Hang:2021fmp  a general energy
cancellation of $\,E^{4}\\!\\!\rightarrow\\!E^{4-N-\delta},\hskip 0.85358pt$
which cancels the leading energy powers by $\,E^{N+\delta}$, with
$\,\delta\\!=\\![1-(-1)^{N}]/2\,$. We will establish a new correspondence
between the two types of energy cancellations in the $N$-point KK gauge boson
scattering amplitudes and KK graviton scattering amplitudes in Sec.V.

##  IV KK Graviton Scattering Amplitudes from GRET

In the following, we demonstrate explicitly how the GRET holds. For this
purpose, we compute the gravitational KK Goldstone boson scattering amplitude
$\widetilde{\mathcal{M}}[\phi_{n_{1}}\phi_{n_{2}}\\!\\!\\!\rightarrow\\!\phi_{n_{3}}\phi_{n_{4}}]$
($n_{j}\\!\\!\geqslant\\!\\!1$). The relevant Feynman diagrams having leading
energy contributions are shown in Fig. 1.

For the elastic scattering, we set the KK numbers of all external states as
$\,n_{i}\\!\\!=\\!n\,$ and of internal states as $\,N_{j}\\!=\\!0,\hskip
0.85358pt2n\,$. Then, summing up the contributions of Fig. 1 and making high
energy expansion, we derive the following LO scattering amplitude of the
gravitational KK Goldstone bosons:

$\vspace*{-1.5mm}\widetilde{\mathcal{M}}_{0}\,=\,\frac{\,3\kappa^{2}\,}{\,128\,}\frac{~{}(7\\!+\cos
2\theta)^{2}\,}{\sin^{2}\\!\theta}\,s\,,\vspace*{-1.mm}$ (8)

where
$\,\widetilde{\mathcal{M}}_{0}=\widetilde{\mathcal{M}}_{0}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]\hskip
0.85358pt$. To compare our Eq.(8) with the corresponding longitudinal KK
graviton amplitude of Refs. Chivukula:2019S Chivukula:2019L , we rescale our
coupling $\,\kappa\\!\\!\rightarrow\\!\kappa/\\!\sqrt{2\,}$ to match their
normalization and find that the two amplitudes are equal at the LO :

$\mathcal{M}_{0}[h^{n}_{L}h^{n}_{L}\\!\rightarrow\\!h^{n}_{L}h^{n}_{L}]\,=\,\widetilde{\mathcal{M}}_{0}[\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{n}\phi_{n}]\hskip
0.85358pt.$ (9)

Namely, $\mathcal{M}_{0}\\!=\\!\widetilde{\mathcal{M}}_{0}\hskip 0.85358pt$,
where we denote
$\,\mathcal{M}_{0}\\!=\\!\mathcal{M}_{0}[h^{n}_{L}h^{n}_{L}\\!\\!\rightarrow\\!\\!h^{n}_{L}h^{n}_{L}]\,$
and
$\,\widetilde{\mathcal{M}}_{0}\\!=\\!\widetilde{\mathcal{M}}_{0}[\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{n}\phi_{n}]\,$.
From the GRET identity (7a) [and Eq.(18)], this means that the residual term
(7b) belongs to the NLO:

$\mathcal{M}_{\\!\Delta}\\!=\mathcal{M}-\widetilde{\mathcal{M}}=\delta\mathcal{M}\\!-\delta\widetilde{\mathcal{M}}=\mathcal{O}(E^{0}M_{n}^{2})\hskip
0.85358pt.$ (10)

and thus is much smaller. We have further computed the exact tree-level
Goldstone boson amplitude $\widetilde{\mathcal{M}}\hskip 0.85358pt$ by
including all the subleading diagrams supp .

Figure 1: Feynman diagrams for the scattering of gravitational KK Goldstone
bosons,
$\phi_{n_{1}}\phi_{n_{2}}\\!\\!\\!\rightarrow\\!\phi_{n_{3}}\phi_{n_{4}}$, by
exchanging the KK gravitons of level-$N_{j}$ ($\,j=s,t,u)$ at the tree level,
which contribute the leading energy-dependence of $\mathcal{O}(E^{2})$.

For inelastic scattering of gravitational KK Goldstone bosons, we compute the
four-point amplitudes and find that the LO inelastic amplitude is connected to
the LO elastic amplitude (8) by the following relation:

$\displaystyle\widetilde{\mathcal{M}}[\phi_{n_{1}}\phi_{n_{2}}\\!\\!\\!\rightarrow\\!\phi_{n_{3}}\phi_{n_{4}}]\,=\,\zeta\widetilde{\mathcal{M}}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\\!\phi_{n}\phi_{n}]\hskip
0.85358pt,$ (11)

where $\,\zeta\\!=\\!2/3\,$ for
$\,n_{1}\\!=\\!n_{2}\\!\neq\\!n_{3}\\!=\\!n_{4}\,$, and $\,\zeta\\!=\\!1/3\,$
for the cases with KK numbers $(n_{1},n_{2},$ $n_{3},n_{4})$ having no more
than one equality.

Table 1: Kinematic numerators of the LO and NLO scattering amplitudes for KK longitudinal gauge bosons and KK Goldstones as defined in Eq.(12), where $(\mathcal{N}_{j},\,\widetilde{\mathcal{N}}_{j})=(\mathcal{N}_{j}^{0},\,\widetilde{\mathcal{N}}_{j}^{0})\\!+\\!(\delta\mathcal{N}_{j},\,\delta\widetilde{\mathcal{N}}_{j})=\mathcal{O}(E^{2}M_{n}^{0})+\mathcal{O}(E^{0}M_{n}^{2})$, and $(s_{\theta},\,c_{\theta})=(\sin\theta,\,\cos\theta)$. Numerators |  $\mathcal{N}_{s}$ | $\mathcal{N}_{t}$ | $\mathcal{N}_{u}$ | $\quad\widetilde{\mathcal{N}}_{s}\quad$ | $\widetilde{\mathcal{N}}_{t}$ | $\widetilde{\mathcal{N}}_{u}$ | $\,\mathcal{N}_{s}\\!-\\!\widetilde{\mathcal{N}}_{s}\,$ | $\,\mathcal{N}_{t}\\!-\\!\widetilde{\mathcal{N}}_{t}\,$ | $\,\mathcal{N}_{u}\\!-\\!\widetilde{\mathcal{N}}_{u}\,$
---|---|---|---|---|---|---|---|---|---
$\mathcal{N}^{0}_{j}/s_{j}$ |  $\frac{\,5c_{\theta}\,}{2}$ | $\frac{\,13+5c_{\theta}+4c_{2\theta}\,}{2(1+c_{\theta})\,}$ | $-\frac{\,13-5c_{\theta}+4c_{2\theta}\,}{2(1-c_{\theta})\,}$ | $-\frac{\,3c_{\theta}\,}{2}$ | $\frac{\,3(3-c_{\theta})\,}{2(1+c_{\theta})\,}$ | $-\frac{\,3(3+c_{\theta})\,}{2(1-c_{\theta})\,}$ | $4c_{\theta}$ | $4c_{\theta}$ | $4c_{\theta}$
$\delta\mathcal{N}_{j}/M_{n}^{2}$ |  $4c_{\theta}$ | $\frac{\,2(2-3c_{\theta}-2c_{2\theta}-c_{3\theta})\,}{1+c_{\theta}\,}$ | $-\frac{\,2(2+3c_{\theta}-2c_{2\theta}+c_{3\theta})\,}{1-c_{\theta}\,}$ | $\,4c_{\theta}\,$ | $-\frac{\,8c_{\theta}\,}{\,1+c_{\theta}\,}$ | $-\frac{\,8c_{\theta}\,}{\,1-c_{\theta}\,}$ | $0$ | $8s_{\theta}^{2}$ | $-8s_{\theta}^{2}$

##  V Double-Copy Construction of Massive
KK Scattering Amplitudes

The double-copy construction for the massive KK gauge/gravity scattering
amplitudes are highly nontrivial. We make the first serious attempt for an
explicit double-copy construction of KK amplitudes under high energy
expansion. We present the four-point elastic scattering amplitudes of
longitudinal KK gauge bosons (Goldstones) at the LO and NLO:

$\displaystyle\mathcal{T}$
$\displaystyle=\sum_{j}\\!\frac{\,g^{2}\mathcal{C}_{j}\mathcal{N}_{j}}{s_{j}}=\sum_{j}\\!\frac{\,g^{2}\mathcal{C}_{j}(\mathcal{N}^{0}_{j}\\!+\delta\mathcal{N}_{j})}{s_{j}}=\mathcal{T}_{0}\\!+\delta\mathcal{T},$
(12a) $\displaystyle\widetilde{\mathcal{T}}$
$\displaystyle=\sum_{j}\\!\frac{\,g^{2}\mathcal{C}_{j}\widetilde{\mathcal{N}}_{j}}{s_{j}}=\sum_{j}\\!\frac{\,g^{2}\mathcal{C}_{j}(\widetilde{\mathcal{N}}^{0}_{j}\\!+\delta\widetilde{\mathcal{N}}_{j})}{s_{j}}=\widetilde{\mathcal{T}}_{0}\\!+\delta\widetilde{\mathcal{T}},$
(12b)

where we have denoted
$\,\mathcal{T}\\!\equiv\\!\mathcal{T}[A^{an}_{L}A^{bn}_{L}\\!\\!\rightarrow\\!\\!A^{cn}_{L}A^{dn}_{L}]$
and
$\widetilde{\mathcal{T}}\\!\equiv\widetilde{\mathcal{T}}[A^{an}_{5}A^{bn}_{5}\\!\\!\rightarrow\\!\\!A^{cn}_{5}A^{dn}_{5}]$.
We also define the SU($N$) color factors as
$(\mathcal{C}_{s},\,\mathcal{C}_{t},\,\mathcal{C}_{u})\\!\equiv\\!\left(C^{abe}C^{cde}\\!,\,C^{ade}C^{bce}\\!,\,C^{ace}C^{dbe}\right)$,
which obey the Jacobi identity
$\,\mathcal{C}_{s}\\!+\mathcal{C}_{t}\\!+\mathcal{C}_{u}\\!=0\hskip
0.85358pt$.

We present in Table 1 the numerator factors
($\mathcal{N}_{j},\,\widetilde{\mathcal{N}}_{j}$) of Eqs.(12a)-(12b). Table 1
shows that
$(\mathcal{T}_{0},\,\widetilde{\mathcal{T}}_{0})\\!\\!=\mathcal{O}(E^{0}M_{n}^{0})$
and
$(\delta\mathcal{T},\,\delta\widetilde{\mathcal{T}})=\mathcal{O}(M_{n}^{2}/E^{2})\hskip
0.85358pt$. We find that the sum of each set of the LO, NLO, and NNLO
numerators of the KK gauge (Goldstone) scattering amplitudes in Eq.(12)
violate the kinematic Jacobi identity by terms of $\mathcal{O}(M_{n}^{2})$ and
$\mathcal{O}(M_{n}^{4}/E^{2})$, respectively:

$\displaystyle\sum_{j}\\!\mathcal{N}_{j}^{0}=10\hskip
0.85358ptc_{\theta}M_{n}^{2}\,,~{}~{}~{}~{}\sum_{j}\\!\widetilde{\mathcal{N}}_{j}^{0}=-6\hskip
0.85358ptc_{\theta}M_{n}^{2}\,,$ (13a)
$\displaystyle\sum_{j}\\!\delta_{1}\mathcal{N}_{j}=\sum_{j}\\!\delta_{1}\widetilde{\mathcal{N}}_{j}=-2\hskip
0.85358pt(7+c_{2\theta})c_{\theta}\csc^{2}\\!\theta\hskip
0.85358ptM_{n}^{2}\,,$ (13b)
$\displaystyle\sum_{j}\\!\delta_{2}\mathcal{N}_{j}=8\hskip
0.85358pt(31+c_{4\theta})\hskip 0.85358ptc_{\theta}\csc^{4}\\!\theta\hskip
0.85358ptM_{n}^{4}/s\,,$ (13c)
$\displaystyle\sum_{j}\\!\delta_{2}\widetilde{\mathcal{N}}_{j}=32\hskip
0.85358pt(7+c_{2\theta})\hskip 0.85358ptc_{\theta}\csc^{4}\\!\theta\hskip
0.85358ptM_{n}^{4}/s\,,$ (13d)

where $\,c_{n\theta}\\!=\cos(n\theta)\hskip 0.85358pt$,
$\delta\mathcal{N}_{j}\\!=\\!\delta_{1}\mathcal{N}_{j}\\!+\delta_{2}\mathcal{N}_{j}$,
and $\,\delta\widetilde{\mathcal{N}}_{j}\\!=$
$\delta_{1}\widetilde{\mathcal{N}}_{j}\\!+\delta_{2}\widetilde{\mathcal{N}}_{j}\hskip
0.85358pt$. Hence, we cannot naively apply color-kinematics duality for BCJ-
type double-copy construction without making further modifications on these
numerators.

Inspecting the scattering amplitudes in Eq.(12), we first observe that they
are invariant under the following generalized gauge transformations of their
numerators:

$\displaystyle\mathcal{N}_{j}^{\prime}=\mathcal{N}_{j}+s_{j}\hskip
0.85358pt\Delta\,,~{}~{}~{}~{}\widetilde{\mathcal{N}}_{j}^{\prime}=\widetilde{\mathcal{N}}_{j}+s_{j}\hskip
0.85358pt\widetilde{\Delta}\,.$ (14)

We can determine the gauge-parameters $(\Delta,\,\widetilde{\Delta})$ by
requiring the gauge-transformed numerators to obey the Jacobi identities
$\sum_{j}\mathcal{N}_{j}^{\prime}\\!=0\hskip 0.85358pt$ and
$\sum_{j}\widetilde{\mathcal{N}}_{j}^{\prime}\\!=0\,$. Thus, we derive the
following general solutions:

$\displaystyle\Delta=-\frac{1}{\,4M_{n}^{2}\,}\\!\sum_{j}\mathcal{N}_{j}\,,~{}~{}~{}~{}\widetilde{\Delta}=-\frac{1}{\,4M_{n}^{2}\,}\\!\sum_{j}\widetilde{\mathcal{N}}_{j}\,,~{}~{}~{}~{}~{}$
(15)

which realize the BCJ-respecting numerators
$(\mathcal{N}_{j}^{\prime},\,\widetilde{\mathcal{N}}_{j}^{\prime})$. Making
high energy expansions on both sides of Eq. (15), we derive the expressions of
the gauge-parameters
$(\Delta,\,\widetilde{\Delta})=(\Delta_{0}+\Delta_{1},\,\widetilde{\Delta}_{0}+\widetilde{\Delta}_{1})\,$
at the LO and NLO :

$\displaystyle\Delta_{0}$ $\displaystyle=\mbox{$\frac{\,{1}\,}{4}$}(9+7\hskip
0.85358ptc_{2\theta})\hskip 0.85358ptc_{\theta}\csc^{2}\\!\theta\,,~{}~{}~{}$
(16) $\displaystyle\widetilde{\Delta}_{0}$
$\displaystyle=\mbox{$\frac{\,{1}\,}{4}$}(17\\!-c_{2\theta})\hskip
0.85358ptc_{\theta}\csc^{2}\\!\theta\,,$ $\displaystyle\Delta_{1}$
$\displaystyle=-2(31+c_{4\theta})c_{\theta}\csc^{4}\\!\theta\,(M_{n}^{2}/s)\,,\qquad$
$\displaystyle\widetilde{\Delta}_{1}$
$\displaystyle=-8(7+c_{2\theta})c_{\theta}\csc^{4}\\!\theta\,(M_{n}^{2}/s)\,.$

With these, we further compute the new numerators
$(\mathcal{N}_{j}^{\prime},\,\widetilde{\mathcal{N}}_{j}^{\prime})$, and
derive explicitly the LO results in Eq.(19) and the NLO results in the
Supplemental Material supp .

For the 5d KK YM (YM5) and 5d KK GR (GR5) theories, we expect the double-copy
correspondence between the KK gauge fields and KK graviton fields:

$\displaystyle A_{n}^{a\mu}\\!\otimes\\!A_{n}^{a\nu}$
$\displaystyle\longrightarrow$ $\displaystyle~{}h^{\mu\nu}_{n}\,,$
$\displaystyle A_{n}^{a5}\\!\otimes\\!A_{n}^{a5}$
$\displaystyle\longrightarrow$ $\displaystyle~{}h^{55}_{n}\,,$ (17)
$\displaystyle A_{n}^{a\mu}\\!\otimes\\!A_{n}^{a5}$
$\displaystyle\longrightarrow$ $\displaystyle~{}h^{\mu 5}_{n}\,.$

The physical spin-2 KK graviton field $h^{\mu\nu}_{n}$ arises from two copies
of spin-1 KK gauge fields. The KK Goldstone boson $A_{n}^{a5}$ of the YM5 has
its double-copy counterparts $h^{55}_{n}\,(=\\!\\!\phi_{n})$ and $h^{\mu
5}_{n}\,(=\\!\\!\mathcal{A}^{\mu}_{n})$ which correspond to the scalar and
vector KK Goldstone bosons in the compactified GR5 theory. The double-copy
correspondence between the longitudinal KK modes,
$A_{L}^{an}\\!\\!\otimes\\!A_{L}^{an}\\!\\!\rightarrow\\!\\!h_{L}^{n}$, is
highly nontrivial even at the LO of high energy expansion, because
$(A_{L}^{an},\hskip 0.85358pth_{L}^{n})$ do not exist in
$M_{n}\\!\\!\rightarrow\\!0$ limit and the KK Goldstone bosons
$(A_{5}^{an},\hskip 0.85358pt\phi_{n})$ become physical states in massless
limit. Hence, this double-copy is consistently realized only because we can
use the KK GRET (GAET) to connect $h_{L}^{n}\,(A_{L}^{an}$) amplitudes to the
$\phi_{n}\,(A_{5}^{an}$) amplitudes under the $M_{n}/E\\!\rightarrow\\!\hskip
0.85358pt0$ limit where we can hold the KK mass $M_{n}$ fixed and take the
energy $\hskip 0.85358ptE\\!\rightarrow\\!\infty\hskip 0.85358pt$.

Then, we extend the conventional double-copy method BCJ:2008 BCJ:2019  to the
massive KK YM theory under high energy expansion. We apply the correspondence
of color-kinematics duality
$\,\mathcal{C}_{j}\\!\\!\rightarrow\\!\mathcal{N}^{\prime}_{j}\,$ to Eq.(12a)
and
$\,\mathcal{C}_{j}\\!\\!\rightarrow\\!\widetilde{\mathcal{N}}^{\prime}_{j}$ to
Eq.(12b). Thus, we can construct the following four-particle KK graviton
(Goldstone) amplitudes:

$\displaystyle\mathcal{M}$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358pt\sum_{j}\\!\frac{~{}c_{0}\hskip 0.85358ptg^{2}(\mathcal{N}^{0\hskip
0.85358pt\prime}_{j}\\!+\delta\mathcal{N}_{j}^{\prime})^{2}\,}{s_{j}}=\,\mathcal{M}_{0}\\!+\delta\mathcal{M}\,,$
(18a) $\displaystyle\widetilde{\mathcal{M}}$ $\displaystyle\hskip
0.85358pt=\hskip 0.85358pt\sum_{j}\\!\frac{~{}c_{0}\hskip
0.85358ptg^{2}(\widetilde{\mathcal{N}}^{0\hskip
0.85358pt\prime}_{j}\\!+\delta\widetilde{\mathcal{N}}_{j}^{\prime})^{2}\,}{s_{j}}=\hskip
0.85358pt\widetilde{\mathcal{M}}_{0}+\delta\widetilde{\mathcal{M}}\,,$ (18b)

where we have denoted the scattering amplitudes
$\mathcal{M}\\!\equiv\\!\mathcal{M}[h_{L}^{n}h_{L}^{n}\\!\\!\rightarrow\\!\\!h_{L}^{n}h_{L}^{n}]$
and
$\widetilde{\mathcal{M}}\\!\equiv\\!\widetilde{\mathcal{M}}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]$,
and $c_{0}$ is a conversion constant.

From Table 1 and using Eqs.(14)(16), we find that the LO numerators
$(\mathcal{N}_{j}^{0\hskip
0.85358pt\prime},\,\widetilde{\mathcal{N}}_{j}^{0\hskip 0.85358pt\prime})$ are
mass-independent and equal to each other :

$\displaystyle\mathcal{N}_{s}^{0\hskip 0.85358pt\prime}$
$\displaystyle\\!=\widetilde{\mathcal{N}}_{s}^{0\hskip
0.85358pt\prime}\\!=\frac{~{}s\hskip
0.85358pt(7\\!+\\!c_{2\theta})c_{\theta}~{}}{2\sin^{2}\\!\theta}\hskip
0.85358pt,$ (19a) $\displaystyle\mathcal{N}_{t}^{0\hskip 0.85358pt\prime}$
$\displaystyle\\!=\widetilde{\mathcal{N}}_{t}^{0\hskip
0.85358pt\prime}\\!=-\frac{~{}s\hskip
0.85358pt(42\\!-\\!15c_{\theta}\\!+\\!6c_{2\theta}\\!-\\!c_{3\theta})~{}}{16\hskip
0.85358pt(1\\!-\\!c_{\theta})}\hskip 0.85358pt,\hskip 14.22636pt$ (19b)
$\displaystyle\mathcal{N}_{u}^{0\hskip 0.85358pt\prime}$
$\displaystyle\\!=\widetilde{\mathcal{N}}_{u}^{0\hskip
0.85358pt\prime}\\!=\frac{~{}s\hskip
0.85358pt(42\\!+\\!15c_{\theta}\\!+\\!6c_{2\theta}\\!+\\!c_{3\theta})~{}}{16\hskip
0.85358pt(1\\!+\\!c_{\theta})}\hskip 0.85358pt.$ (19c)

This demonstrates the equivalence between the two leading-order KK amplitudes
at $\mathcal{O}(E^{0}M_{n}^{0})$,
$\,\mathcal{T}_{0}=\widetilde{\mathcal{T}}_{0}\,$, which explicitly realizes
the KK GAET. With these and using our LO double-copy formulas in Eq.(18), we
can reconstruct the KK GRET:

$\displaystyle\mathcal{M}_{0}[\mathrm{DC}]\,=\,\widetilde{\mathcal{M}}_{0}[\mathrm{DC}]\,,$
(20)

which is of $\,\mathcal{O}(E^{2}M_{n}^{0})\hskip 0.85358pt$. We stress that as
expected, these LO amplitudes are mass-independent and thus the LO double-copy
can hold universally. We further find that after setting the overall
conversion constant of Eq.(18) as $\,c_{0}\\!=\\!-\kappa^{2}/(24g^{2})\hskip
0.85358pt$, the reconstructed LO KK amplitude
$\,\mathcal{M}_{0}\,(\widetilde{\mathcal{M}}_{0})\,$ just equals the LO KK
Goldstone amplitude (8) and the corresponding LO longitudinal KK graviton
amplitude supp . Hence, our double-copy prediction (20) can prove
(reconstruct) the KK GRET
$\,\mathcal{M}_{0}\\!=\\!\widetilde{\mathcal{M}}_{0}$ from the KK GAET
$\,\mathcal{T}_{0}=\widetilde{\mathcal{T}}_{0}\,$. We derived this GRET
relation in Eq.(9) by direct Feynman-diagram calculations. Note that the KK
GAET relation $\,\mathcal{T}_{0}=\widetilde{\mathcal{T}}_{0}\,$ can hold for
general $N$-point longitudinal KK gauge (Goldstone) amplitudes 5DYM2002 KK-ET-
He2004 . Hence, making double-copy on both sides of
$\,\mathcal{T}_{0}\\!=\\!\widetilde{\mathcal{T}}_{0}\,$ can establish the GRET
(20) to hold for $N$-point longitudinal KK graviton (Goldstone) amplitudes.
From this, we can further establish a new correspondence between the two types
of energy cancellations in the $N$-longitudinal KK gauge boson amplitudes and
in the corresponding $N$-longitudinal KK graviton amplitudes (cf. the
discussion around the end of Sec. III).

Next, we use the double-copy formulas (18a)-(18b) to reconstruct the four-
point longitudinal KK graviton amplitude and the corresponding KK Goldstone
boson amplitude at the NLO:

$\displaystyle\frac{\delta\mathcal{M}(\mathrm{DC})\,}{\kappa^{2}M_{n}^{2}}$
$\displaystyle=-\frac{\,5(1642+297\hskip 0.85358ptc_{2\theta}+102\hskip
0.85358ptc_{4\theta}+7\hskip 0.85358ptc_{6\theta})\,}{768\sin^{4}\theta}\hskip
0.85358pt,$ (21a)
$\displaystyle\frac{\delta\widetilde{\mathcal{M}}(\mathrm{DC})}{\kappa^{2}M_{n}^{2}}$
$\displaystyle=-\frac{\,6386+3837\hskip 0.85358ptc_{2\theta}+30\hskip
0.85358ptc_{4\theta}-13\hskip 0.85358ptc_{6\theta}\,}{768\sin^{4}\theta}\hskip
0.85358pt.$ (21b)

They have the same size of $\mathcal{O}(\kappa^{2}M_{n}^{2})$ and the same
angular structure of $\hskip
0.85358pt(1,\,c_{2\theta},\,c_{4\theta},\,c_{6\theta})\\!\times\csc^{4}\theta\,$
as the original NLO amplitudes
$(\delta\mathcal{M},\,\delta\widetilde{\mathcal{M}})$ derived from Feynman
diagram calculations supp , though their numerical coefficients still differ.
Then, using Eq.(21) we compute the difference between the two double-copied
NLO amplitudes
$\,\Delta\mathcal{M}(\mathrm{DC})\\!\\!=\\!\delta\mathcal{M}\\!-\\!\delta\widetilde{\mathcal{M}}\,$
and compare it with the NLO amplitude-difference
$\Delta\mathcal{M}(\mathrm{GR5})$ by Feynman diagram calculations in the KK
GR5 theory:

$\displaystyle\Delta\mathcal{M}(\mathrm{GR5})$
$\displaystyle=-\mbox{$\frac{\,{3}\,}{2}$}\kappa^{2}\\!M_{n}^{2}\left(19.5+c_{2\theta}\right),$
(22a) $\displaystyle\Delta\mathcal{M}(\mathrm{DC})$
$\displaystyle=-\kappa^{2}M_{n}^{2}\,(7\\!+c_{2\theta})\,.$ (22b)

We find that they also have the same size of
$\mathcal{O}(\kappa^{2}M_{n}^{2})$ and the same angular structure of $\hskip
0.85358pt(1,\,c_{2\theta})\hskip 0.85358pt$. Eq.(22a) shows that the
difference $\Delta\mathcal{M}(\text{GR5})$ between the original NLO amplitudes
exhibits a striking precise cancellations of the angular structure $\hskip
0.85358pt(1,\,c_{2\theta},\,c_{4\theta},\,c_{6\theta})\\!\times\csc^{4}\theta\,$
to $\hskip 0.85358pt(1,\,c_{2\theta})\hskip 0.85358pt$. Impressively, our
double-copied NLO amplitude-difference $\Delta\mathcal{M}(\text{DC})$ in
Eq.(22b) can also realize the same type of the precise angular cancellations.

The above extended NLO double-copy results (21) and (22b) are truly
encouraging, because they already give the correct structure of the NLO KK
amplitudes including the precise cancellations of the angular dependence in
Eqs.(21)-(22). These strongly suggest that our massive KK double-copy approach
is on the right track. Its importance is twofold: (i). In practice, for our
proposed KK double-copy method under high energy expansion, the LO double-copy
construction is the most important part because it newly establishes the GRET
relation $\mathcal{M}_{0}\\!=\\!\widetilde{\mathcal{M}}_{0}$ [Eq. (20)] from
the GAET relation $\,\mathcal{T}_{0}\\!=\\!\widetilde{\mathcal{T}}_{0}\,$
[Eq.(19) and below], as will be shown in Eq.(34). The NLO KK graviton
amplitudes are relevant only when we estimate the size of the residual term
$\mathcal{M}_{\\!\Delta}$ of our GRET (7) and here we do not need the precise
form of $\mathcal{M}_{\\!\Delta}$ except to justify its size
$\mathcal{M}_{\\!\Delta}\\!\\!=\mathcal{O}(E^{0}M_{n}^{2})$ by the double-copy
construction [cf. Eq.(33)]. This proves that the residual term
$\mathcal{M}_{\\!\Delta}\\!$ does belong to the NLO amplitudes and is
neligible for our GRET formulation in the high energy limit. Hence, we do not
need any precise NLO double-copy here. (ii). In general, our current KK
double-copy approach as the first serious attempt to construct the massive KK
graviton amplitudes has given strong motivation and important guideline for a
full resolution of the exact double-copy beyond the LO. Our further study has
found out the reasons for the minor mismatch between the numerical
coefficients of the double-copied NLO amplitudes (21) and that of the direct
Feynman-diagram calculations. One reason is due to the double-pole structure
in the KK amplitudes (including exchanges of both the zero-mode and KK-modes)
beyond the conventional massless theories, so the additional KK mass-poles
contribute to our mass-dependent NLO amplitudes and cause a mismatch. Another
reason is because the exact polarization tensor of the (helicity-zero)
longitudinal KK graviton is given by
$\,\varepsilon_{L}^{\mu\nu}\\!=\\!\left(\epsilon_{+}^{\mu}\epsilon_{-}^{\nu}\\!+\epsilon_{-}^{\mu}\epsilon_{+}^{\nu}\\!+2\epsilon_{L}^{\mu}\epsilon_{L}^{\nu}\right)/\sqrt{6\,}$
supp , which constains not only the longitudinal product
$\epsilon_{L}^{\mu}\epsilon_{L}^{\nu}$, but also the transverse products
$\,\epsilon_{+}^{\mu}\epsilon_{-}^{\nu}\\!+\epsilon_{-}^{\mu}\epsilon_{+}^{\nu}\hskip
0.85358pt$. Thus, the other scattering amplitudes containing possible
transversely polarized external KK gauge boson states should be included for a
full double-copy besides the four-longitudinal KK gauge boson amplitude in
Eq.(12).

With these in minds, we have further used a first principle approach of the KK
string theory in our recent work Li:2021yfk  to derive the extended massive
KLT-like relations between the product of the KK open string amplitudes and
the KK closed string amplitude. In the field theory limit, we can derive the
exact double-copy relations between the product of the KK gauge boson
amplitudes and the KK graviton amplitude at tree level Li:2021yfk . In such
exact double-copy relations all the relevant helicity indices of the external
KK gauge boson states are summed over to match the corresponding polarization
tensors of the external KK graviton states. The double-pole structure is also
avoided by first making the 5d compactification under $S^{1}$ (without
orbifold) where the KK numbers ($\pm n\\!=\\!\pm 1,\pm 2,\pm 3,\cdots$) are
strictly conserved and the amplitudes are ensured to have single-pole
structure. Then, we can define the $\mathbb{Z}_{2}$-even (odd) KK states as
$\,|n_{\pm}\rangle=(|+n\rangle\pm|-n\rangle)/\\!\sqrt{2\,}$, and derive the
amplitudes under $S^{1}\\!/\mathbb{Z}_{2}$ compactification from the
combinations of those amplitudes under the $S^{1}$ compactification Li:2021yfk
. Using this improved massive double-copy approach, we can exactly reconstruct
all the massive KK graviton amplitudes at tree level. For the four
longitudinal KK graviton amplitudes under $S^{1}\\!/\mathbb{Z}_{2}$, we derive
the following (BCJ-type) exact massive double-copy formula:

$\mathcal{M}=-\frac{\kappa^{2}}{64}\sum_{\mathsf{P}}\\!\sum_{j}\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\prod_{k}C_{\lambda_{k}\lambda^{\prime}_{k}}\frac{\,N_{j}^{\mathsf{P}}(\lambda_{k})N_{j}^{\mathsf{P}}(\lambda_{k}^{\prime})\,}{D_{j}}\,,$
(23)

where $\\{N_{j}\\}$ denote the kinematic numerators under the $S^{1}$
compactification and each external KK gauge boson has 3 helicity states (with
$\lambda_{k}\\!=\\!\pm 1,L$ and $k\\!=\\!1,2,3,4$). We use
$\mathsf{P}\\!=\\!\\{n_{1},n_{2},n_{3},n_{4}\\}$ to label each possible
combination of the KK numbers for external gauge bosons, which obey the
condition of KK number conservation $\sum_{k=1}^{4}\\!n_{k}\\!=\\!0$ . For the
elastic KK scattering, we have

$\displaystyle\mathsf{P}=$ $\displaystyle\ \\{\pm n,\pm n,\mp n,\mp
n\\},\hskip 5.69054pt\\{\pm n,\mp n,\pm n,\mp n\\},~{}~{}~{}$ (24)
$\displaystyle\ \\{\pm n,\mp n,\mp n,\pm n\\}.$

In Eq.(23), $C_{\lambda_{k}\lambda^{\prime}_{k}}$ denotes the coefficients in
the longitudinal polarization tensor of $k$-th external KK graviton supp ,
$\varepsilon_{L,k}^{\mu\nu}\\!=\\!\sum
C_{\lambda_{k}\lambda^{\prime}_{k}}\epsilon_{\lambda_{k}}^{\mu}\epsilon_{\lambda^{\prime}_{k}}^{\nu}$,
where $\lambda_{k},\lambda^{\prime}_{k}\\!=\\!\pm 1,L$ are the helicity
indices for the $k$-th external gauge boson. The denominator of Eq.(23) is
defined as $D_{j}\\!=\\!s_{j}\\!-\\!M_{j}^{2}$, where
$\,s_{j}\\!\in\\!\\{s,t,u\\}\,$ and
$M_{j}^{2}\\!\in\\!\\{M_{n_{1}+n_{2}}^{2},M_{n_{1}+n_{4}}^{2},M_{n_{1}+n_{3}}^{2}\\}$.

Then, we make high energy expansion for the corresponding elastic amplitude of
KK gauge bosons (under $S^{1}$ compactification) at the LO and NLO:

$\displaystyle\mathcal{T}$
$\displaystyle\,=\,g^{2}\sum_{j}\frac{\,\mathcal{C}_{j}N_{j}^{\mathsf{P}}}{D_{j}}=g^{2}\sum_{j}\\!\frac{\,\mathcal{C}_{j}(N^{0,\mathsf{P}}_{j}\\!+\delta
N^{\mathsf{P}}_{j})}{s_{j}}\hskip 19.91692pt$
$\displaystyle\,=~{}\mathcal{T}_{0}+\delta\mathcal{T}\,,$ (25a) $\displaystyle
N_{j}^{\mathsf{P}}$
$\displaystyle\,=~{}\frac{\,D_{j}\,}{s_{j}}\\!\left(\\!{N_{j}^{0,\mathsf{P}}}+\delta
N^{\mathsf{P}}_{j}\right)\\!.$ (25b)

With this, we expand the exact double-copy formula of the longitudinal KK
graviton amplitude (23) under the high energy expansion of $1/s$ :

$\displaystyle\mathcal{M}$
$\displaystyle=-\frac{~{}\kappa^{2}\,}{64}\,\sum_{\mathsf{P}}\sum_{j}\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\prod_{k}C_{\lambda_{k}\lambda^{\prime}_{k}}\frac{\,D_{j}\,}{s_{j}^{2}}$
$\displaystyle\hskip
14.22636pt\times\\!\left[\,\\!N_{j}^{0,\mathsf{P}}(\lambda_{k})\\!+\delta
N_{j}^{\mathsf{P}}(\lambda_{k})\\!\,\right]\\!\left[\,\\!N_{j}^{0,\mathsf{P}}(\lambda^{\prime}_{k})\\!+\delta
N_{j}^{\mathsf{P}}(\lambda^{\prime}_{k})\\!\,\right]\hskip 11.38109pt$
$\displaystyle=\,\mathcal{M}_{0}+\delta\mathcal{M}\,.$ (26)

It can be proven that the above double-copied LO amplitude $\mathcal{M}_{0}$
is equivalent to the LO amplitude given in Eq.(18a) Li:2021yfk . We explicitly
compute the above LO amplitude $\mathcal{M}_{0}$ and find that
$\mathcal{M}_{0}$ just equals that of Eq.(8) as well as Eq.(S22a) supp . Then,
we further compute the above double-copied NLO amplitude $\delta\mathcal{M}$
as follows:

$\delta\mathcal{M}=-\frac{\kappa^{2}M_{n}^{2}}{256}(1810+93\hskip
0.85358ptc_{2\theta}+126\hskip 0.85358ptc_{4\theta}+19\hskip
0.85358ptc_{6\theta})\csc^{4}\theta\hskip 0.85358pt.$ (27)

We find that this fully agrees with the exact NLO elastic KK graviton
amplitude derived from the direct Feynman diagram calculation in Eq.(S22b) of
the supplemental material supp . The above analysis is an explicit
demonstration that we can realize the exact (BCJ-type) massive double-copy
construction of the four-point KK graviton amplitudes in Eq.(23), as well as
the precise double-copy of the KK graviton amplitudes (V)-(27) at both the LO
and NLO of the high energy expansion. We will systematically pursue this new
direction in our future work.

Finally, it is very impressive that our improved massive double-copy
construction of the longitudinal KK graviton (KK Goldstone) amplitude in
Eq.(18) is based on the pure longitudinal KK gauge (KK Goldstone) amplitude
(12) alone, which can already give not only the precise LO KK graviton (KK
Goldstone) amplitude, but also the correct structure of the NLO KK graviton
(KK Goldstone) amplitude (21). In the following, we will propose another
improved double-copy method to further reproduce the exact longitudinal KK
graviton (KK Goldstone) amplitudes at the NLO and beyond. It only uses the
amplitudes of pure longitudinal KK gauge bosons (KK Goldstone bosons) alone,
hence it is practically simple and valuable. For this, we construct the
following improved NLO numerators:

$\displaystyle(\delta\mathcal{N}_{s}^{\prime\prime},\,\delta\mathcal{N}_{t}^{\prime\prime},\,\delta\mathcal{N}_{u}^{\prime\prime})$
$\displaystyle=(\delta\mathcal{N}_{s}^{\prime},\,\delta\mathcal{N}_{t}^{\prime}\\!-\\!z,\,\delta\mathcal{N}_{u}^{\prime}\\!+\\!z\,)\hskip
0.85358pt,$ (28a)
$\displaystyle(\delta\widetilde{\mathcal{N}}_{s}^{\prime\prime},\,\delta\widetilde{\mathcal{N}}_{t}^{\prime\prime},\,\delta\widetilde{\mathcal{N}}_{u}^{\prime\prime})$
$\displaystyle=(\delta\widetilde{\mathcal{N}}_{s}^{\prime},\,\delta\widetilde{\mathcal{N}}_{t}^{\prime}\\!-\\!\tilde{z},\,\delta\widetilde{\mathcal{N}}_{u}^{\prime}\\!+\\!\tilde{z}\,)\hskip
0.85358pt,$ (28b)

where $(z,\,\tilde{z})$ are functions of $\,\theta\,$ and can be determined by
matching our improved NLO KK amplitudes of double-copy with the original NLO
KK graviton (Goldstone) amplitudes of the GR5. Then, we solve
$(z,\,\tilde{z})$ as

$\displaystyle z$
$\displaystyle\,=\,\frac{\,M_{n}^{2}(1390\\!+\\!603c_{2\theta}\\!+\\!66c_{4\theta}\\!-11c_{6\theta})\,}{12\hskip
0.85358pt(13\\!-\\!12c_{2\theta}\\!-\\!c_{4\theta})}\,,$ (29a)
$\displaystyle\tilde{z}$
$\displaystyle\,=\,\frac{\,M_{n}^{2}(4546\\!-\\!3585c_{2\theta}\\!+\\!1086c_{4\theta}\\!+c_{6\theta})\,}{12\left(13\\!-\\!12c_{2\theta}\\!-\\!c_{4\theta}\right)}\,.$
(29b)

Note that the modified kinematic numerators (28) continue to hold the Jacobi
identity. Because the corresponding NLO gauge (Goldstone) amplitudes
$(\delta\mathcal{T}^{\prime\prime},\,\delta\widetilde{\mathcal{T}}^{\prime\prime})$
are modified only by terms of NLO, so we can still hold the general GAET
identity $\hskip
0.85358pt\mathcal{T}^{\prime\prime}\\!=\widetilde{\mathcal{T}}^{\prime\prime}\\!+\mathcal{T}_{v}^{\prime\prime}\hskip
0.85358pt$ by redefining the residual term as
$\,\mathcal{T}_{v}^{\prime\prime}\\!=\mathcal{T}_{v}\\!-g^{2}(\mathcal{C}_{t}/t\\!-\mathcal{C}_{u}/u)(z\\!-\\!\tilde{z})\hskip
0.85358pt$. Using Eqs.(28)-(29), we can reproduce the exact NLO KK
gravitational scattering amplitudes [shown in Eqs.(S22a)-(S22b) of the
Supplemental Material supp ]. This double-copy procedure can be further
applied to higher orders beyond the NLO when needed.

##  VI GRET Residual Terms and Energy Cancellation

According to Table 1 and the generalized gauge transformation (14), we can
explicitly deduce the equivalence between the KK gauge boson amplitude and the
corresponding KK Goldstone boson amplitude:

$\mathcal{T}_{0}\,=\,\widetilde{\mathcal{T}}_{0}\,,$ (30)

which belongs to the LO of $\mathcal{O}(E^{0}M_{n}^{0})$. Using our double-
copy method, we further derived the GRET relation
$\mathcal{M}_{0}\\!=\\!\widetilde{\mathcal{M}}_{0}$ at the $\hskip
0.85358pt\mathcal{O}(E^{2}M_{n}^{0})\hskip 0.85358pt$ as shown in Eq. (20).
Thus, the residual terms of the GAET and the GRET (7) are given by the
differences between the KK longitudinal amplitude and KK Goldstone amplitude
at the NLO :

$\displaystyle\mathcal{T}_{v}$
$\displaystyle\equiv\sum\\!\mathcal{T}[A_{5}^{n},v_{n}]=\delta\mathcal{T}\\!-\\!\delta\widetilde{\mathcal{T}}\,=\,\mathcal{O}(M_{n}^{2}/E^{2})\,,$
(31a) $\displaystyle\mathcal{M}_{\Delta}$
$\displaystyle\equiv\sum\\!\mathcal{M}[\widetilde{\Delta}_{n},\phi_{n}]=\delta\mathcal{M}\\!-\\!\delta\widetilde{\mathcal{M}}=\mathcal{O}(E^{0}M_{n}^{2})\,.$
(31b)

The size of $\,\mathcal{T}_{v}=\mathcal{O}(M_{n}^{2}/E^{2})\,$ can be easily
understood by using our generalized power counting rule supp . But, making the
direct power counting gives $\,\mathcal{M}_{\Delta}\\!=\mathcal{O}(E^{2})\,$
for its individual amplitudes, which has the same energy dependence as the LO
KK Goldstone amplitude (8).

We can further determine the size of the residual term
$\,\mathcal{M}_{\Delta}$ by the double-copy construction (18) based upon the
KK gauge (Goldstone) boson scattering amplitudes of the YM5 theory alone
(which are well understood 5DYM2002 5DYM2002-2 KK-ET-He2004 5dSM ). From
Eq.(18) and Table 1, we can estimate the residual term by power counting:

$\displaystyle\mathcal{M}_{\Delta}$
$\displaystyle=\,\mathcal{O}(\delta\mathcal{M},\hskip
0.85358pt\delta\widetilde{\mathcal{M}})=\mathcal{O}\\!\left(\\!\frac{\mathcal{N}_{j}^{0\hskip
0.85358pt\prime}\delta\mathcal{N}_{j}^{\prime}}{s_{j}},\,\frac{\widetilde{\mathcal{N}}_{j}^{0\hskip
0.85358pt\prime}\delta\widetilde{\mathcal{N}}_{j}^{\prime}}{s_{j}}\\!\right)$
$\displaystyle=\,\mathcal{O}(E^{0}M_{n}^{2})\hskip 0.85358pt.$ (32)

Thus, we deduce the double-copy correspondence between the residual term
$\hskip 0.85358pt\mathcal{T}_{v}\hskip 0.85358pt$ of the GAET and the residual
term $\mathcal{M}_{\Delta}$ of the GRET:

$\mathcal{T}_{v}~{}\longrightarrow~{}\mathcal{M}_{\Delta}(\mathrm{DC})=\mathcal{O}(E^{0}M_{n}^{2})\hskip
0.85358pt.$ (33)

Hence, our double-copy construction proves that the GRET residual term
$\,\mathcal{M}_{\Delta}$ should have an energy cancellation
$\,\mathcal{O}(E^{2})\\!\rightarrow\\!\mathcal{O}(E^{0})\,$ among its
individual amplitudes in Eq.(7b). This proves that $\mathcal{M}_{\Delta}$ is
much smaller than the leading KK Goldstone amplitude
$\,\widetilde{\mathcal{M}}_{0}=\mathcal{O}(E^{2}M_{n}^{0})$ under the high
energy expansion.

From the above double-copy construction, we can establish a new correspondence
from the GAET of the KK YM5 theory to the GRET of the 5d KK GR (GR5):

$\mathrm{GAET\,(YM5)}\ \Longrightarrow\ \mathrm{GRET\,(GR5)}\,.$ (34)

We will give a systematically expanded analysis in the companison long paper
Hang:2021fmp , which includes our elaborations of the current key points and
our extension of KLT relations KLT  (along with CHY CHY ) to the double-copy
construction of massive KK graviton amplitudes.

##  VII Conclusions

In this work, we newly formulated the geometric “Higgs” mechanism for the mass
generation of Kaluza-Klein (KK) gravitons of the compactified 5d GR (GR5)
theory at both the Lagrangian level and the scattering $S$-matrix level. Using
a general $R_{\xi}$ gauge-fixing of quantization, we proved that the KK
graviton propagator is free from the longstanding problem of the vDVZ
discontinuity vDVZ  in the conventional Fierz-Pauli massive gravity PF
Hinterbichler:2012  and demonstrated that the KK gravity theory can
consistently realize the mass-generation for spin-2 KK gravitons.

We newly proposed and proved a Gravitational Equivalence Theorem (GRET) which
connects the $N$-point scattering amplitudes of the longitudinal KK gravitons
to that of the gravitational KK Goldstone bosons. We computed the four-point
scattering amplitudes of KK Goldstone bosons in comparison with the
longitudinal KK graviton amplitudes, and explicitly proved the equivalence
between the leading amplitudes of the longitudinal KK graviton scattering and
the corresponding KK Goldstone boson scattering at
$\mathcal{O}(E^{2}M_{n}^{0})$.

We developed a generalized power counting method for massive KK gauge and
gravity theories. Using the GRET and the new power counting rules, we
established a general energy-cancellation mechanism under which the leading
energy dependence of $N$-particle longitudinal KK graviton amplitudes
($\propto\\!E^{2(N+1+L)}$) must cancel down to a much lower energy power
($\propto\\!E^{2(1+L)}$) by an energy factor of $E^{2N}$, where $L$ denotes
the loop number of the relevant Feynman diagram. For the case of longitudinal
KK graviton scattering amplitudes with $N\\!=\\!4$ and $L\\!=\\!0$ , this
proves the energy cancellations of $\,E^{10}\\!\\!\rightarrow\\!E^{2}\hskip
0.85358pt$.

Extending the conventional massless double-copy method BCJ:2008 BCJ:2019  to
the compactified massive KK YM and KK GR theories, we derived the Jacobi-
respecting numerators and constructed the scattering amplitudes of
longitudinal KK gravitons (KK Goldstone bosons) under high energy expansion.
Using our extended massive double-copy approach, we constructed exact double-
copy of the KK graviton scattering amplitudes at both the leading order (LO)
and the next-to-leading order (NLO). Applying this massive double-copy method,
we established a new correspondence between the two energy cancellations in
the four-point longitudinal KK amplitudes: $E^{4}\\!\\!\rightarrow\\!E^{0}\,$
in the 5d KK YM gauge theory and $\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$ in the
5d KK GR theory, which is connected to the double-copy correspondence between
the GAET and GRET as we derived in Eq.(34). Furthermore, we analyzed the
structure of the residual term $\,\mathcal{M}_{\\!\Delta}$ in the GRET (7) and
further uncovered a new energy-cancellation mechanism of
$\,E^{2}\\!\rightarrow\\!E^{0}\,$ for the residual term of the GRET.

Finally, we stress that the geometric Higgs mechanism is a general consequence
of the KK compactification of extra spatial dimensions and should be realized
for other KK gravity theories with more than one extra dimensions or with
nonflat extra dimensions. We note that our identity (6) results from the
underlying gravitational diffeomorphism invariance and thus should generally
hold for any compactified 5d KK GR theory with proper gauge-fixing functions.
Thus, we expect that the GRET should generally hold for other 5d KK GR
theories and take similar form as the present Eq.(7) GET-2 . For instance, we
find that the geometric Higgs mechanism and the large energy-cancellations of
the longitudinal KK graviton amplitudes are also realized in the compactified
warped 5d space of the Randall-Sundrum model RS  and our GRET will work in the
similar way. Following the current work, it is encouraging to further study
these interesting issues in our future work GET-2 . In passing, we recently
proposed Hang:2021oso  a brand-new topological equivalence theorem (TET) to
formulate the topological mass-generation in the 3d topologically massive
Yang-Mills theory (TMYM), with which we uncover the nontrivial energy
cancellations in the $N$-point Chern-Simons scattering amplitudes of the
massive physical gauge bosons, $E^{4}\\!\rightarrow\\!E^{4-N}$. We then made
an extended double-copy construction of the four-point massive graviton
scattering amplitude in the 3d topologically massive gravity (TMG) TMG  and
further proved Hang:2021oso  the striking energy cancellations of $\hskip
0.85358ptE^{12}\\!\rightarrow\\!E^{1}\hskip 0.85358pt$ in such massive
graviton scattering amplitude of the TMG theory.

Acknowledgements
This research was supported in part by the National Natural Science Foundation
of China (under grants Nos. 11835005 and 11675086), and by the National Key R
& D Program of China (under grant No. 2017YFA 0402204).

Supplementary Materials
In the following Supplementary Materials, we provide the relevant technical
details for the analyses presented in the main text of this paper. I.
Kinematics of KK scattering; II. Feynman rules for 5d KK GR theory; III. Power
counting and energy cancellations for KK graviton amplitudes; IV. KK graviton
and Goldstone scattering amplitudes.

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Gravitational Equivalence Theorem and Double-Copy
for Kaluza-Klein Graviton Scattering Amplitudes
— Supplemental Material —

Yan-Feng Hang 1 and Hong-Jian He 1,2,3

1 Tsung-Dao Lee Institute $\&$ School of Physics and Astronomy,

Key Laboratory for Particle Astrophysics and Cosmology (MOE),

Shanghai Key Laboratory for Particle Physics and Cosmology,

Shanghai Jiao Tong University, Shanghai, China

2 Institute of Modern Physics $\&$ Physics Department, Tsinghua University,
Beijing, China

3 Center for High Energy Physics, Peking University, Beijing, China

<EMAIL_ADDRESS><EMAIL_ADDRESS>
This Supplemental Material provides in detail the relevant formulas, Feynman
rules, and the KK power counter method for the analyses of the KK scattering
amplitudes in the compactified 5d Yang-Mills (YM5) theory and the compactified
5d General Relativity (GR5) theory.

##  I Kinematics of KK Scattering

We consider $\,2\\!\rightarrow\\!2\,$ KK scattering process, with the four-
momentum of each external state obeying the on-shell condition
$\,p^{2}_{j}\\!=\\!-M_{j}^{2}$, ($j=1,2,3,4$). We number the external lines
clockwise, with their momenta being out-going. Thus, the energy-momentum
conservation gives $\,\sum_{j}p_{j}\\!=0\,$, and the physical momenta of the
two incident particles equal $-p_{1}$ and $-p_{2}$, respectively. For
illustration, we take the elastic scattering
$\,X_{n}X_{n}\\!\\!\rightarrow\\!X_{n}X_{n}$ ($n\\!\geqslant\\!0$) as an
example, where $X_{n}$ denotes any given KK state of level-$n$ and has
$M_{j}\\!=\\!M_{n}$. For the KK theory, the external particle has mass $M_{n}$
for a given KK-state of level-$n$. Thus, in the center-of-mass frame, we
define the momenta as follows:

$\displaystyle p_{1}^{\mu}$ $\displaystyle=-E\left(1,0,0,\beta\right)\,,$
$\displaystyle\hskip 28.45274ptp_{2}^{\mu}=-E\left(1,0,0,-\beta\right)\,,$
(S1) $\displaystyle p_{3}^{\mu}$ $\displaystyle=E\left(1,\beta
s_{\theta},0,\beta c_{\theta}\right)\,,$ $\displaystyle\hskip
28.45274ptp_{4}^{\mu}=E\left(1,-\beta s_{\theta},0,-\beta
c_{\theta}\right)\,,$

where $(s_{\theta},\hskip 0.85358ptc_{\theta})=(\sin\theta,\,\cos\theta)$ and
$\hskip 0.85358pt\beta\\!=\\!(1\\!-\\!M_{n}^{2}/E^{2})^{1/2}\hskip 0.85358pt$.
With the above, we can define the following three Mandelstam variables:

$\displaystyle
s=-\left(p_{1}\\!+p_{2}\right)^{2}\\!=4E^{2},~{}~{}~{}~{}t=-\left(p_{1}\\!+p_{4}\right)^{2}\\!=-\mbox{$\frac{\,{1}\,}{2}$}s\hskip
0.85358pt\beta^{2}(1\\!+\\!c_{\theta})\hskip
0.85358pt,~{}~{}~{}~{}u=-\left(p_{1}\\!+p_{3}\right)^{2}\\!=-\mbox{$\frac{\,{1}\,}{2}$}s\hskip
0.85358pt\beta^{2}(1\\!-\\!c_{\theta})\hskip 0.85358pt.$ (S2)

Then, using the on-shell condition $E^{2}\\!=E^{2}\beta^{2}+M_{n}^{2}\hskip
0.85358pt$, we define a new set of mass-independent Mandelstam variables as
follows:

$\displaystyle s_{0}=4E^{2}\beta^{2},\qquad
t_{0}=-\frac{\,s_{0}\,}{2}(1+c_{\theta})\hskip 0.85358pt,\qquad
u_{0}=-\frac{\,s_{0}\,}{2}(1-c_{\theta})\hskip 0.85358pt,$ (S3)

where $\hskip 0.85358pts_{0}\\!=\\!s-4M_{n}^{2}$ , and thus
$(s_{0},\,t_{0},\,u_{0})=(s\beta^{2},\,t,\,u)$. Summing up the Mandelstam
variables (S2) and (S3) gives the following relations:

$s+t+u=4M_{n}^{2}\,,\qquad s_{0}+t_{0}+u_{0}=0\,.$ (S4)

As we mentioned in the text, a massive KK graviton has 5 helicity states
($\lambda\\!=\pm 2,\pm 1,0$ ). Their polarization tensors take the following
forms:

$\displaystyle\varepsilon_{\pm 2}^{\mu\nu}$
$\displaystyle=\epsilon_{\pm}^{\mu}\epsilon_{\pm}^{\nu}\,,\qquad\varepsilon_{\pm
1}^{\mu\nu}=\frac{1}{\sqrt{2\,}\,}\\!\left(\epsilon_{\pm}^{\mu}\epsilon_{L}^{\nu}\\!+\epsilon_{L}^{\mu}\epsilon_{\pm}^{\nu}\right)\\!,\qquad\varepsilon_{L}^{\mu\nu}\\!=\frac{1}{\sqrt{6\,}\,}\\!\left(\epsilon_{+}^{\mu}\epsilon_{-}^{\nu}\\!+\epsilon_{-}^{\mu}\epsilon_{+}^{\nu}\\!+2\epsilon_{L}^{\mu}\epsilon_{L}^{\nu}\right)\\!,$
(S5)

where $(\epsilon_{\pm}^{\mu},\,\epsilon_{L}^{\mu})$ are the (transverse,
longitudinal) polarization vectors of a vector boson with the same 4-momentum
$p^{\mu}$. These polarization tensors obey the traceless and orthonormal
conditions. They are also orthogonal to the KK graviton’s 4-momentum
$p^{\mu}\hskip 0.85358pt$. Hence, the following conditions are realized:

$\eta_{\mu\nu}\varepsilon^{\mu\nu}=0\,,\qquad\varepsilon_{\lambda}^{\mu\nu}\varepsilon_{\lambda^{\prime}\\!,\,\mu\nu}^{*}=\delta_{\lambda\lambda^{\prime}}\,,\qquad
p_{\mu}\varepsilon^{\mu\nu}=0\,,$ (S6)

where the helicity indices of each KK graviton are
$\,\lambda,\lambda^{\prime}=\pm 2,\hskip 0.85358pt\pm 1,\hskip 0.85358pt0\,$.

##  II Feynman Rules for 5d KK GR Theory

In this section, we summarize the relevant Feynman rules Hang:2021fmp
including propagators and vertices which are used for the amplitude
calculations in the text of this Letter.

We first give the propagators in $R_{\xi}$ gauge for KK graviton
($h^{\mu\nu}_{n}$) and KK Goldstone bosons
$(\mathcal{A}^{\mu}_{n},\,\phi_{n})$ as follows:

$\displaystyle\mathcal{D}_{nm}^{\mu\nu\alpha\beta}(p)\,=\,$
$\displaystyle-\\!\frac{\,\text{i}\delta_{nm}\,}{2}\left\\{\\!\frac{\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta}}{p^{2}\\!+\\!M_{n}^{2}}+\frac{1}{3}\\!\left[\,\\!\frac{1}{\,p^{2}\\!+\\!M_{n}^{2}}-\frac{1}{\,p^{2}\\!+\\!(3\xi_{n}\\!-\\!2)M_{n}^{2}}\\!\,\right]\\!\\!\left(\\!\eta^{\mu\nu}\\!-\\!\frac{\,2p^{\mu}p^{\nu}}{M_{n}^{2}}\\!\right)\\!\\!\left(\\!\eta^{\alpha\beta}\\!-\\!\frac{\,2p^{\alpha}p^{\beta}}{M_{n}^{2}}\\!\right)\right.$
$\displaystyle+\\!\frac{1}{\,M_{n}^{2}\,}\\!\left(\\!\frac{1}{\,p^{2}\\!+\\!M_{n}^{2}}-\frac{1}{\,p^{2}\\!+\\!\xi_{n}M_{n}^{2}\,}\\!\right)\\!(\eta^{\mu\alpha}p^{\nu}p^{\beta}\\!+\\!\eta^{\mu\beta}p^{\nu}p^{\alpha}\\!+\\!\eta^{\nu\alpha}p^{\mu}p^{\beta}\\!+\\!\eta^{\nu\beta}p^{\mu}p^{\alpha})$
$\displaystyle\left.+\frac{\,4p^{\mu}p^{\nu}p^{\alpha}p^{\beta}\,}{\xi_{n}M_{n}^{4}}\\!\left(\\!\frac{1}{\,p^{2}\\!+\\!\xi_{n}^{2}M_{n}^{2}\,}-\frac{1}{\,p^{2}\\!+\\!\xi_{n}M_{n}^{2}\,}\\!\right)\\!\right\\}\\!,$
(S7a) $\displaystyle\mathcal{D}^{\mu\nu}_{nm}(p)\,=\,$
$\displaystyle\frac{-\text{i}\delta_{nm}}{\,\,p^{2}\\!+\\!\xi_{n}M_{n}^{2}\,\,}\\!\left[\eta^{\mu\nu}\\!\\!-\\!\frac{\,p^{\mu}p^{\nu}(1\\!-\\!\xi_{n})\,}{\,p^{2}\\!+\xi_{n}^{2}M_{n}^{2}}\right]\\!,\qquad\mathcal{D}_{nm}(p)=\frac{-\text{i}\delta_{nm}}{~{}p^{2}\\!+\\!(3\xi_{n}\\!\\!-\\!2)M_{n}^{2}~{}}\,.$
(S7b)

For the Feynman-’t Hooft gauge ($\xi_{n}\\!=\\!1$), the above propagators
reduce to the simple forms [cf. Eq.(5) in the main text].

Next, we make the following Fourier expansions for the 5d graviton fields in
terms of their zero modes and KK states:

$\displaystyle\hat{h}^{\mu\nu}(x^{\rho},x^{5})$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\\!\left[\,h^{\mu\nu}_{0}(x^{\rho})+\sqrt{2}\sum_{n=1}^{\infty}h^{\mu\nu}_{n}(x^{\rho})\cos\\!\frac{n\pi
x^{5}}{L}\\!\,\right]\\!,$ (S8a) $\displaystyle\hat{h}^{\mu
5}(x^{\rho},x^{5})$
$\displaystyle\,=\,\sqrt{\frac{2}{L}\,}\sum_{n=1}^{\infty}h^{\mu
5}_{n}(x^{\rho})\sin\\!\frac{n\pi x^{5}}{L}\,,$ (S8b)
$\displaystyle\hat{\phi}(x^{\rho},x^{5})$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\left[\,\phi_{0}(x^{\rho})+\sqrt{2}\sum_{n=1}^{\infty}\phi_{n}(x^{\rho})\cos\\!\frac{n\pi
x^{5}}{L}\\!\,\right]\\!.$ (S8c)

With these, we list the relevant 4d effective Lagrangians including both cubic
and quartic interactions which are used for our analyses:

$\displaystyle\mathcal{L}_{1}[h\phi^{2}]=$
$\displaystyle\,\frac{\kappa}{\sqrt{2\,}\,}\\!\\!\sum_{n,m,\ell=1}^{\infty}\\!\Bigl{\\{}a_{1}\big{[}\sqrt{2}\hskip
0.85358pt(h_{0}^{\mu\nu}\partial_{\mu}\phi_{0}\partial_{\nu}\phi_{0}\\!+\\!h_{0}^{\mu\nu}\partial_{\mu}\phi_{m}\partial_{\nu}\phi_{\ell}\delta_{m\ell}\\!+\\!h_{n}^{\mu\nu}\partial_{\mu}\phi_{m}\partial_{\nu}\phi_{0}\delta_{nm}\\!+\\!h_{n}^{\mu\nu}\partial_{\mu}\phi_{0}\partial_{\nu}\phi_{\ell}\delta_{n\ell})$
$\displaystyle+h_{n}^{\mu\nu}\partial_{\mu}\phi_{m}\partial_{\nu}\phi_{\ell}\hskip
0.85358pt\Delta_{3}(n,m,\ell)\big{]}+a_{2}\big{[}\sqrt{2}\hskip
0.85358pt(h_{0}^{\mu\nu}\phi_{0}\partial_{\mu}\partial_{\nu}\phi_{0}\\!+\\!h_{0}^{\mu\nu}\phi_{m}\partial_{\mu}\partial_{\nu}\phi_{\ell}\delta_{m\ell}\\!+\\!h_{n}^{\mu\nu}\phi_{m}\partial_{\mu}\partial_{\nu}\phi_{0}\delta_{nm}$
$\displaystyle+h_{n}^{\mu\nu}\phi_{0}\partial_{\mu}\partial_{\nu}\phi_{\ell}\delta_{n\ell})\\!+\\!h_{n}^{\mu\nu}\phi_{m}\partial_{\mu}\partial_{\nu}\phi_{\ell}\Delta_{3}(n,m,\ell)\big{]}\\!+a_{3}\big{[}\sqrt{2}(h_{0}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}\\!+\\!h_{0}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{\ell}\delta_{m\ell}$
$\displaystyle+h_{n}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{0}\delta_{nm}\\!+\\!h_{n}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{\ell}\delta_{n\ell})\\!+\\!h_{n}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{\ell}\hskip
0.85358pt\Delta_{3}(n,m,\ell)\big{]}\\!+a_{4}\big{[}\sqrt{2}\hskip
0.85358pt(h_{0}\phi_{0}\partial^{2}_{\mu}\phi_{0}$
$\displaystyle+h_{0}\phi_{m}\partial^{2}_{\mu}\phi_{\ell}\delta_{m\ell}+h_{n}\phi_{m}\partial^{2}_{\mu}\phi_{0}\delta_{nm}\\!+\\!h_{n}\phi_{0}\partial^{2}_{\mu}\phi_{\ell}\delta_{n\ell})\\!+\\!h_{n}\phi_{m}\partial^{2}_{\mu}\phi_{\ell}\hskip
0.85358pt\Delta_{3}(n,m,\ell)\big{]}$ $\displaystyle+a_{5}\hskip
0.85358ptM_{m}M_{\ell}\big{[}\sqrt{2}\hskip
0.85358pth_{0}\phi_{m}\phi_{\ell}\delta_{m\ell}\\!+\\!h_{n}\phi_{m}\phi_{\ell}\hskip
0.85358pt\widetilde{\Delta}_{3}(n,m,\ell)\big{]}\\!-a_{6}M_{\ell}^{2}\big{[}\sqrt{2}\hskip
0.85358pt(h_{0}\phi_{m}\phi_{\ell}\delta_{m\ell}\\!+\\!h_{n}\phi_{0}\phi_{\ell}\delta_{n\ell})$
$\displaystyle+h_{n}\phi_{m}\phi_{\ell}\hskip
0.85358pt\Delta_{3}(n,m,\ell)\big{]}\Bigr{\\}}\,,$ (S9a)
$\displaystyle\mathcal{L}_{1}[\mathcal{A}\hskip 0.85358pt\phi^{2}]=$
$\displaystyle\
-\frac{\kappa}{\sqrt{2}}\sum_{n,m,\ell=1}^{\infty}\\!\Bigl{\\{}b_{1}\hskip
0.85358ptM_{\ell}\big{[}\sqrt{2}\hskip
0.85358pt\mathcal{A}^{\mu}_{n}\partial_{\mu}\phi_{0}\phi_{\ell}\delta_{n\ell}+\mathcal{A}^{\mu}_{n}\partial_{\mu}\phi_{m}\phi_{\ell}\hskip
0.85358pt\widetilde{\Delta}^{\prime}_{3}(n,m,\ell)\big{]}+b_{2}\hskip
0.85358ptM_{\ell}\big{[}\sqrt{2}\hskip
0.85358pt\mathcal{A}^{\mu}_{n}\phi_{0}\partial_{\mu}\phi_{\ell}\delta_{n\ell}$
$\displaystyle+\mathcal{A}^{\mu}_{n}\phi_{m}\partial_{\mu}\phi_{\ell}\hskip
0.85358pt\widetilde{\Delta}^{\prime}_{3}(n,m,\ell)\big{]}\Bigr{\\}}\,,$ (S9b)
$\displaystyle\mathcal{L}_{1}[\phi^{3}]=$
$\displaystyle\,\frac{\kappa}{\sqrt{2}}\sum_{n,m,\ell=1}^{\infty}\\!\Big{\\{}c_{1}\big{[}\sqrt{2}\hskip
0.85358pt(\phi_{0}(\partial_{\mu}\phi_{0})^{2}\\!+\\!\phi_{0}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{\ell}\,\delta_{m\ell}\\!+\\!\phi_{n}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{m}\delta_{nm}\\!+\\!\phi_{n}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{\ell}\,\delta_{n\ell})$
$\displaystyle+\phi_{n}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{\ell}\hskip
0.85358pt\Delta_{3}(n,m,\ell)\Big{]}\\!+c_{2}M_{m}M_{\ell}\big{[}\sqrt{2}\,\phi_{0}\phi_{m}\phi_{\ell}\delta_{m\ell}\\!+\\!\phi_{n}\phi_{m}\phi_{\ell}\widetilde{\Delta}_{3}(n,m,\ell)\big{]}\Big{\\}}\,,$
(S9c) $\displaystyle\mathcal{L}_{2}[\phi^{4}]=$
$\displaystyle\,\frac{\kappa^{2}}{2}\\!\sum_{n,m,\ell,k=1}^{\infty}\\!\Bigl{\\{}d_{1}\\!\left\\{2(\phi_{0}\partial_{\mu}\phi_{0})^{2}\\!+\\!2\big{[}\\!\left(\partial_{\mu}\phi_{0}\right)^{2}\phi_{n}\phi_{m}\delta_{nm}\\!+\\!\phi_{0}\partial_{\mu}\phi_{0}\phi_{n}\partial^{\mu}\phi_{\ell}\delta_{n\ell}\\!+\\!\phi_{0}\partial_{\mu}\phi_{0}\phi_{n}\partial^{\mu}\phi_{k}\delta_{nk}\right.$
$\displaystyle\\!+\phi_{0}\partial_{\mu}\phi_{0}\phi_{m}\partial^{\mu}\phi_{k}\delta_{mk}\\!+\\!\phi_{0}\partial_{\mu}\phi_{0}\phi_{m}\partial^{\mu}\phi_{\ell}\delta_{m\ell}\\!+\\!(\phi_{0})^{2}\partial_{\mu}\phi_{\ell}\partial^{\mu}\phi_{k}\delta_{\ell
k}\big{]}\\!+\\!\sqrt{2}\left[\,\partial_{\mu}\phi_{0}\phi_{n}\phi_{m}\partial^{\mu}\phi_{\ell}\Delta_{3}(n,m,\ell)\right.$
$\displaystyle\left.\\!+\,\partial_{\mu}\phi_{0}\phi_{n}\phi_{m}\partial^{\mu}\phi_{k}\hskip
0.85358pt\Delta_{3}(n,m,k)\\!+\\!\partial_{\mu}\phi_{0}\phi_{n}\phi_{\ell}\partial^{\mu}\phi_{k}\Delta_{3}(n,\ell,k)\\!+\\!\phi_{0}\phi_{m}\partial_{\mu}\phi_{\ell}\partial^{\mu}\phi_{k}\Delta_{3}(m,\ell,k)\,\right]$
$\displaystyle\left.\\!+\,\phi_{n}\phi_{m}\partial_{\mu}\phi_{\ell}\partial^{\mu}\phi_{k}\hskip
0.85358pt\Delta_{4}(n,m,\ell,k)\right\\}\\!+d_{2}M_{\ell}M_{k}\big{[}2(\phi_{0})^{2}\phi_{\ell}\phi_{k}\delta_{\ell
k}\\!+\\!\sqrt{2}\phi_{0}\phi_{m}\phi_{\ell}\phi_{k}\widetilde{\Delta}_{3}(m,\ell,k)$
$\displaystyle\\!+\sqrt{2}\hskip
0.85358pt\phi_{0}\phi_{n}\phi_{\ell}\phi_{k}\hskip
0.85358pt\widetilde{\Delta}_{3}(n,\ell,k)\\!+\\!\phi_{n}\phi_{m}\phi_{\ell}\phi_{k}\hskip
0.85358pt\widetilde{\Delta}_{4}(n,m,\ell,k)\big{]}\Bigr{\\}}\,,$ (S9d)

where the delta functions ($\Delta_{j},\widetilde{\Delta}_{j}$) are defined as
follows:

$\displaystyle\Delta_{3}(n,m,\ell)\,=$ $\displaystyle\
\delta(n\\!+\\!m\\!-\\!\ell)\\!+\\!\delta(n\\!-\\!m\\!-\\!\ell)\\!+\\!\delta(n\\!-\\!m\\!+\\!\ell)\,,$
(S10) $\displaystyle\widetilde{\Delta}_{3}(n,m,\ell)\,=$ $\displaystyle\
\delta(n\\!+\\!m\\!-\\!\ell)\\!-\\!\delta(n\\!-\\!m\\!-\\!\ell)\\!+\\!\delta(n\\!-\\!m\\!+\\!\ell)\,,$
$\displaystyle\widetilde{\Delta}_{3}^{\prime}(n,m,\ell)\,=$ $\displaystyle\
\delta(n\\!+\\!m\\!-\\!\ell)\\!-\\!\delta(n\\!-\\!m\\!+\\!\ell)\\!+\\!\delta(n\\!-\\!m\\!-\\!\ell)\,,$
$\displaystyle\Delta_{4}(n,m,\ell,k)\,=$ $\displaystyle\
\delta(n\\!+\\!m\\!+\\!\ell\\!-\\!k)\\!+\\!\delta(n\\!+\\!m\\!-\\!\ell\\!-\\!k)\\!+\\!\delta(n\\!-\\!m\\!+\\!\ell\\!-\\!k)\\!+\\!\delta(n\\!-\\!m\\!-\\!\ell\\!-\\!k)$
$\displaystyle\
\\!+\\!\delta(n\\!-\\!m\\!-\\!\ell\\!+\\!k)\\!+\\!\delta(n\\!+\\!m\\!-\\!\ell\\!+\\!k)\\!+\\!\delta(n\\!-\\!m\\!+\\!\ell\\!+\\!k)\,,$
$\displaystyle\widetilde{\Delta}_{4}(n,m,\ell,k)\,=$ $\displaystyle\
\delta(n\\!+\\!m\\!+\\!\ell\\!-\\!k)\\!-\\!\delta(n\\!+\\!m\\!-\\!\ell\\!-\\!k)\\!+\\!\delta(n\\!-\\!m\\!+\\!\ell\\!-\\!k)\\!-\\!\delta(n\\!-\\!m\\!-\\!\ell\\!-\\!k)$
$\displaystyle\
\\!+\\!\delta(n\\!-\\!m\\!-\\!\ell\\!+\\!k)\\!-\\!\delta(n\\!+\\!m\\!-\\!\ell\\!+\\!k)\\!+\\!\delta(n\\!-\\!m\\!+\\!\ell\\!+\\!k)\,.$

Then, we derive the Feynman rules based on the interaction Lagrangians in the
above Eq.(S9). We present the relevant 3-point and 4-point vertices as
follows:

$\displaystyle=\frac{-\text{i}\kappa}{\sqrt{1\\!+\\!\delta_{2n,m}\,}\,}\\!\\!\left[\,\begin{array}[]{cl}&a_{1}\\!\left(p_{1}^{\mu}p_{2}^{\nu}\\!+\\!p_{1}^{\nu}p_{2}^{\mu}\right)\\\\[4.2679pt]
+&a_{2}\\!\left(p_{1}^{\mu}p_{1}^{\nu}\\!+\\!p_{2}^{\mu}p_{2}^{\nu}\right)\\\\[4.2679pt]
+&2\hskip 0.85358pta_{3}\eta^{\mu\nu}\\!\left(p_{1}\\!\cdot
p_{2}\right)\\\\[4.2679pt] -&2\hskip
0.85358pt\tilde{a}_{4}\eta^{\mu\nu}M_{n}^{2}\\\\[1.42262pt]
\end{array}\\!\,\right]\\!,$ (S11e) $\displaystyle=\
-\frac{\,\kappa(b_{1}\\!+b_{2})M_{n}\,}{\sqrt{2\,}}(p_{1}^{\mu}\\!+p_{2}^{\mu})\,,$
(S11f) $\displaystyle=\left\\{\begin{aligned} m&=0\\!:\
\text{i}\,2\kappa\\!\left[\,\\!c_{1}(p_{1}^{2}+p_{2}^{2}+p_{1}\\!\cdot
p_{2})\\!+c_{2}M^{2}_{n}\,\right]\\\\[1.42262pt] &\hskip
25.6073pt\xrightarrow{\mathrm{on}\text{-}\mathrm{shell}}\text{i}\,2\kappa\\!\left[\,\\!c_{1}(p_{1}\\!\cdot
p_{2})\\!-(2c_{1}\\!-c_{2})M^{2}_{n}\,\right]\\!,\\\\[2.84526pt]
m&=2n\\!:-\text{i}\sqrt{2}\,\kappa\\!\left[\,\\!c_{1}(p_{1}\\!\cdot
p_{2})+c_{2}M^{2}_{n}\,\right]\\!,\end{aligned}\right.$ (S11g)
$\displaystyle\begin{aligned}
&=\text{i}\,6\kappa^{2}\\!\left[\,\\!d_{1}(p_{1}^{2}+p_{2}^{2}+p_{1}\\!\cdot
p_{2}-p_{3}\\!\cdot p_{4})+2d_{2}M_{n}^{2}\,\right]\\\\[2.84526pt] &\hskip
11.38109pt\xrightarrow{\mathrm{on}\text{-}\mathrm{shell}}-\text{i}\,12\kappa^{2}(d_{1}\\!-d_{2})M_{n}^{2}\,.\end{aligned}$
(S11h)

where $\,\tilde{a}_{4}=a_{4}+(-1)^{\delta_{2n,m}}a_{5}-a_{6}$ with
$m\\!=0,\,2n$ in Eq.(S11e).

##  III Power Counting and Energy Cancellations for KK Graviton Scattering
Amplitudes

We consider a $S$-matrix element $\,\mathbb{S}\,$ having $\,\mathcal{E}\,$
external states and $L$ loops ($L\\!\geqslant\\!0$). Extending the original
power counting rule for the ungauged nonlinear $\sigma$-model of the low
energy QCD by Steven Weinberg weinberg steve-foot , we develop generalized
power counting approach Hang:2021fmp  for the KK gravity theory. The mass
dimension of a given scattering amplitude $\,\mathbb{S}\,$ in 4d is counted as

$D_{\mathbb{S}}\,=\,4-\mathcal{E}\,,$ (S12)

where the number of external states
$\,\mathcal{E}\\!=\mathcal{E}_{B}+\mathcal{E}_{F}\,$ with
$\,\mathcal{E}_{B}\,(\mathcal{E}_{F})\,$ representing the total number of
external bosonic (fermionic) states. In addition, we only consider the SM
fermions whose masses are much smaller than the scattering energy. We denote
the number of vertices of type-$j$ as $\mathcal{V}_{j}$ . Each vertex of
type-$j$ contains $\,d_{j}\,$ derivatives, $\,b_{j}\,$ bosonic lines and
$\,f_{j}\,$ fermionic lines. Then, the energy dependence of coupling constant
in $\,\mathbb{S}\,$ is given by

$D_{C}\,=\,\sum_{j}\mathcal{V}_{j}\\!\left(4-d_{j}\\!-b_{j}\\!-\mbox{$\frac{\,{3}\,}{2}$}f_{j}\right)\\!.$
(S13)

For each Feynman diagram in the amplitude $\mathbb{S}$ , we denote the number
of the internal lines as $\,I=I_{B}+I_{F}\,$ with $\,I_{B}$ ($\,I_{F}\,$)
being the number of the internal bosonic (fermionic) lines. Thus, we have the
following general relations:

$L=1+I-\mathcal{V}\,,\hskip
14.22636pt\mathcal{V}=\sum_{j}\mathcal{V}_{j}\,,\hskip
14.22636pt\sum_{j}\mathcal{V}_{j}b_{j}=2I_{B}+\mathcal{E}_{B}\,,\hskip
14.22636pt\sum_{j}\mathcal{V}_{j}f_{j}=2I_{F}+\mathcal{E}_{F}\,,$ (S14)

where $\,\mathcal{V}\,$ is the total number of vertices in a given Feynman
diagram. The $\,\mathbb{S}\,$ may include $\,\mathcal{E}_{h_{L}}\\!$ external
longitudinal KK graviton states. Thus, taking Eqs.(S12)-(S14), we deduce the
leading energy-power dependence as follows:

$D_{E}\,=\,D_{\mathbb{S}}-D_{C}\,=\,2\hskip
0.85358pt\mathcal{E}_{h_{L}}\\!\\!+(2L\\!+2)+\sum_{j}\mathcal{V}_{j}\\!\left(d_{j}\\!-2+\\!\mbox{$\frac{\,{1}\,}{2}$}f_{j}\right)\\!.$
(S15)

For the pure longitudinal KK graviton scattering amplitude with $\,N\,$
external states, we have $\,\mathcal{E}_{h_{L}}\\!\\!\\!=\\!N\,$ and
$\,f_{j}\\!=\\!0\,$. Each pure KK graviton vertex always contains two partial
derivatives and thus $\,d_{j}\\!=2\,$. For the loop level ($L\geqslant 1$),
the amplitude may contain gravitational ghost loop which involves graviton-
ghost-antighost vertex, but the number of partial derivatives $\,d_{j}$ should
be no more than two. While for the gravitational KK Goldstone boson scattering
amplitude, its leading energy dependence is given by the diagrams containing
the cubic vertices of type $h^{\mu\nu}_{n}$-$\phi_{m}$-$\phi_{\ell}$ and the
pure graviton self-interaction vertices, where each of these vertices includes
two derivatives ($d_{j}\\!=2$). Hence, we can derive the power counting
formula (S15) as:

$D_{E}[Nh_{L}^{n}]\,=\,2(N\\!+\\!1)\\!+2L\,,\qquad
D_{E}[N\phi_{n}]\,=\,2+2L\,,$ (S16)

where the notation $\,[Nh_{L}^{n}]\,$ and $\,[N\phi_{n}]\,$ denote the $N$
external longitudinal KK graviton states and $N$ external KK Goldstone states,
respectively.

Comparing the energy power counting formulas for KK graviton and KK Goldstone
in Eq.(S16), we note that their difference arises from the leading energy-
dependence of the polarization tensors
$\,\varepsilon_{L}^{\mu\nu}\\!\\!\sim\\!k^{\mu}k^{\nu}/M_{n}^{2}\,$ for the
$N$ external longitudinal KK gravitons in the high energy scattering:

$D_{E}[Nh_{L}^{n}]-D_{E}[N\phi_{n}]\,=\,2N\hskip 0.85358pt.$ (S17)

Finally, we examine the leading energy dependence of the individual amplitudes
in the residual term $\,\mathcal{M}_{\Delta}\,$ of the GRET [cf. Eq.(7) in
main text]. A typical leading amplitude can be
$\mathcal{M}[\tilde{v}_{n_{1}},\cdots\\!,\tilde{v}_{n_{N}}]\,$, in which all
the external states are KK gravitons contracted with
$\,\tilde{v}^{\mu\nu}\\!=\varepsilon_{L}^{\mu\nu}\\!\\!-\varepsilon_{S}^{\mu\nu}\\!=\mathcal{O}(E^{0})\,$,
such as $\,\tilde{v}_{n}\\!=\\!\tilde{v}_{\mu\nu}h_{n}^{\mu\nu}\hskip
0.85358pt$. Hence, the leading energy dependence of this amplitude yields:

$D_{E}[N\tilde{v}_{n}]\,=\,2+2L\,,$ (S18)

which gives the same energy power dependence as $\,D_{E}[N\phi_{n}]\hskip
0.85358pt$.

##  IV KK Graviton and Goldstone Scattering Amplitudes

In this section, we first present the four-point scattering amplitudes of KK
gravitons (Goldstone bosons) at the LO and NLO of the high energy expansion,
which are obtained from computing the Feynman diagrams. Then, we present the
four-point scattering amplitudes of the KK gauge bosons (Goldstone bosons) at
the LO and NLO under two kinds of high energy expansions. From these we
provide the detailed formulas for our improved massive double-copy
construction of the KK graviton (Goldstone) amplitudes which are used in the
main text.

###  IV.1 KK Graviton and Goldstone Amplitudes from Feynman Diagrams

In this subsection, we summarize the full elastic amplitudes of the four
longitudinal KK graviton scattering Chivukula:2019L  and of the four
gravitational KK Goldstone boson scattering Hang:2021fmp . For the purpose of
our double-copy analysis, we express these amplitudes in terms of the
dimensionless variable $\,\bar{s}$ :

$\displaystyle\mathcal{M}[4h_{L}^{n}]$
$\displaystyle=-\frac{~{}\kappa^{2}M_{n}^{2}(X_{0}+X_{2}c_{2\theta}+X_{4}c_{4\theta}+X_{6}c_{6\theta})\csc^{2}\\!\theta~{}}{512\hskip
0.85358pt\bar{s}\hskip 0.85358pt(\bar{s}-4)[\hskip
0.85358pt\bar{s}^{2}-(\bar{s}-4)^{2}c_{2\theta}+24\hskip
0.85358pt\bar{s}+16\hskip 0.85358pt]}\,,$ (S19a)
$\displaystyle\widetilde{\mathcal{M}}[4\phi_{n}]$
$\displaystyle=-\frac{~{}\kappa^{2}M_{n}^{2}(\widetilde{X}_{0}+\widetilde{X}_{2}c_{2\theta}+\widetilde{X}_{4}c_{4\theta}+\widetilde{X}_{6}c_{6\theta})\csc^{2}\\!\theta~{}}{512\hskip
0.85358pt\bar{s}\hskip 0.85358pt(\bar{s}-4)[\hskip
0.85358pt\bar{s}^{2}-(\bar{s}-4)^{2}c_{2\theta}+24\bar{s}+16\hskip
0.85358pt]}\,,$ (S19b)

where $\,\bar{s}=s/M_{n}^{2}\,$ and $\,c_{n\theta}=\cos(n\theta)\hskip
0.85358pt$. In the above, the coefficients $(X_{j},\,\widetilde{X}_{j})$ are
defined as follows:

$\displaystyle X_{0}$
$\displaystyle=-2(255\bar{s}^{5}+2824\bar{s}^{4}-19936\bar{s}^{3}+39936\bar{s}^{2}-256\bar{s}+14336)\hskip
0.85358pt,$ (S20) $\displaystyle X_{2}$
$\displaystyle=429\bar{s}^{5}-10152\bar{s}^{4}+30816\bar{s}^{3}-27136\bar{s}^{2}-49920\bar{s}+34816\,,$
$\displaystyle X_{4}$
$\displaystyle=2(39\bar{s}^{5}-312\bar{s}^{4}-2784\bar{s}^{3}-11264\bar{s}^{2}+26368\bar{s}-2048)\hskip
0.85358pt,$ $\displaystyle X_{6}$
$\displaystyle=3\bar{s}^{5}+40\bar{s}^{4}+416\bar{s}^{3}-1536\bar{s}^{2}-3328\bar{s}-2048\,,$
$\displaystyle\widetilde{X}_{0}$
$\displaystyle=-2(255\bar{s}^{5}+8248\bar{s}^{4}\,-4144\bar{s}^{3}+79104\bar{s}^{2}+642560\bar{s}+69632)\hskip
0.85358pt,$ $\displaystyle\widetilde{X}_{2}$
$\displaystyle=429\bar{s}^{5}+4152\bar{s}^{4}\,+21216\bar{s}^{3}-150016\bar{s}^{2}+1142016\bar{s}+182272\,,$
$\displaystyle\widetilde{X}_{4}$
$\displaystyle=2(39\bar{s}^{5}-1992\bar{s}^{4}+17808\bar{s}^{3}-58112\bar{s}^{2}+70144\bar{s}-20480)\,,$
$\displaystyle\widetilde{X}_{6}$
$\displaystyle=3\bar{s}^{5}-56\bar{s}^{4}+416\bar{s}^{3}-1536\bar{s}^{2}+2816\bar{s}-2048\,.$

Then, we expand the KK graviton and KK Goldstone scattering amplitudes
(S19a)-(S19b) under the high energy expansion of $\,1/s\,$:

$\displaystyle\mathcal{M}[4h_{L}^{n}]$
$\displaystyle\,=\,\mathcal{M}_{0}[4h_{L}^{n}]+\delta\mathcal{M}[4h_{L}^{n}]\,,$
(S21a) $\displaystyle\widetilde{\mathcal{M}}[4\phi_{n}]$
$\displaystyle\,=\,\widetilde{\mathcal{M}}_{0}[4\phi_{n}]+\delta\widetilde{\mathcal{M}}[4\phi_{n}]\,,$
(S21b)

where the LO and NLO KK amplitudes take the following forms,

$\displaystyle\mathcal{M}_{0}[4h_{L}^{n}]$
$\displaystyle\,=\,\widetilde{\mathcal{M}}_{0}[4\phi_{n}]=\frac{~{}3\kappa^{2}\,}{~{}128~{}}\hskip
0.85358pts\hskip 0.85358pt(7+c_{2\theta})^{2}\csc^{2}\\!\theta\,,$ (S22a)
$\displaystyle\delta\mathcal{M}[4h_{L}^{n}]$
$\displaystyle\,=\,-\frac{~{}\kappa^{2}M_{n}^{2}~{}}{256}(1810+93\hskip
0.85358ptc_{2\theta}+126\hskip 0.85358ptc_{4\theta}+19\hskip
0.85358ptc_{6\theta})\csc^{4}\theta\,,$ (S22b)
$\displaystyle\delta\widetilde{\mathcal{M}}[4\phi_{n}]$
$\displaystyle\,=\,-\frac{~{}\kappa^{2}M_{n}^{2}~{}}{256}(-\hskip
0.85358pt902+3669c_{2\theta}-714c_{4\theta}-5\hskip
0.85358ptc_{6\theta})\csc^{4}\theta\,.$ (S22c)

If we make instead the high energy expansion in terms of $\hskip
0.85358pt1/s_{0}\hskip 0.85358pt$, we derive the following LO and NLO KK
amplitudes:

$\displaystyle\mathcal{M}_{0}^{\,\prime}[4h_{L}^{n}]$
$\displaystyle\,=\,\widetilde{\mathcal{M}}_{0}^{\,\prime}[4\phi_{n}]\hskip
0.85358pt=\hskip 0.85358pt\frac{3\kappa^{2}}{~{}128~{}}\hskip
0.85358pts_{0}\hskip 0.85358pt(7+c_{2\theta})^{2}\csc^{2}\\!\theta\,,$ (S23a)
$\displaystyle\delta\mathcal{M}^{\,\prime}[4h_{L}^{n}]$
$\displaystyle\,=\,-\frac{\,\kappa^{2}M_{n}^{2}~{}}{128}(650+261c_{2\theta}\\!+102c_{4\theta}\\!+11c_{6\theta})\csc^{4}\\!\theta\,,$
(S23b) $\displaystyle\delta\widetilde{\mathcal{M}}^{\,\prime}[4\phi_{n}]$
$\displaystyle\,=\,-\frac{\,\kappa^{2}M_{n}^{2}~{}}{128}(-706+2049c_{2\theta}-318c_{4\theta}-c_{6\theta})\csc^{4}\\!\theta\,,$
(S23c)

where $\,s_{0}=s-4M_{n}^{2}\hskip 0.85358pt$. We see that the $\hskip
0.85358pt1/s_{0}\hskip 0.85358pt$ expansion has shifted a hidden
$\mathcal{O}(M_{n}^{2})$ subleading term (contained in $\hskip
0.85358pts=s_{0}+4M_{n}^{2}\hskip 0.85358pt)\hskip 0.85358pt$ from the LO
amplitudes (S22a) into the NLO amplitudes (S23b)-(S23c). But this
rearrangement in Eqs.(S23a)-(S23c) does not affect the difference between the
two NLO amplitudes. Thus, we can deduce the contribution of the residual terms
by computing the amplitude-difference from either Eqs.(S22b)-(S22c) or
Eqs.(S23b)-(S23c) as follows:

$\mathcal{M}_{\Delta}=\,\delta\mathcal{M}[4h_{L}^{n}]-\delta\widetilde{\mathcal{M}}[4\phi_{n}]\,=\,-\frac{\,3\hskip
0.85358pt\kappa^{2}M_{n}^{2}\,}{2}\\!\left(\\!\frac{\,39\,}{2}+c_{2\theta}\\!\right)\\!.$
(S24)

This provides Eq.(22a) in the main text.

###  IV.2 KK Graviton and Goldstone Amplitudes from Extended Double-Copy

We expand the scattering amplitudes under the high energy expansion in terms
of $\hskip 0.85358ptM_{n}^{2}/s\hskip 0.85358pt$. Thus, we can express 4-point
elastic KK gauge boson (Goldstone) amplitudes as follows:

$\displaystyle\mathcal{T}[4A^{n}_{L}]$
$\displaystyle\,=\,g^{2}\\!\left(\\!\frac{\,\mathcal{C}_{s}\mathcal{N}_{s}\,}{s}+\frac{\,\mathcal{C}_{t}\mathcal{N}_{t}\,}{t}+\frac{\,\mathcal{C}_{u}\mathcal{N}_{u}\,}{u}\\!\right)\\!,$
(S25a) $\displaystyle\widetilde{\mathcal{T}}[4A^{n}_{5}]$
$\displaystyle\,=\,g^{2}\\!\left(\\!\frac{\,\mathcal{C}_{s}\widetilde{\mathcal{N}}_{s}\,}{s}+\frac{\,\mathcal{C}_{t}\widetilde{\mathcal{N}}_{t}\,}{t}+\frac{\,\mathcal{C}_{u}\widetilde{\mathcal{N}}_{u}\,}{u}\\!\right)\\!,$
(S25b)

which are invariant under the following generalized gauge-transformations:

$\mathcal{N}_{j}^{\prime}=\mathcal{N}_{j}+s_{j}\hskip
0.85358pt\Delta\,,\qquad\widetilde{\mathcal{N}}_{j}^{\prime}=\widetilde{\mathcal{N}}_{j}+s_{j}\hskip
0.85358pt\widetilde{\Delta}\,.$ (S26)

The above Eqs.(S25)-(S26) are given in Eqs.(12)(14) of the main text. This
allows us to find proper solutions of $\\{\Delta\,,\widetilde{\Delta}\\}$
which ensure the gauge-transformed NLO numerators
$(\delta\mathcal{N}_{j}^{\prime},\,\delta\widetilde{\mathcal{N}}_{j}^{\prime})$
to obey the kinematic Jacobi identity, as we demonstrated in Eqs.(15)-(16) of
the main text (cf. Sec.V). Thus, from these we can derive the gauge-
transformed NLO numerators for the elastic KK gauge boson amplitude:

$\displaystyle\delta\mathcal{N}^{\prime}_{s}$
$\displaystyle=-\mbox{$\frac{\,{1}\,}{4}$}M_{n}^{2}\hskip 0.85358pt(246\hskip
0.85358ptc_{\theta}\\!+\\!7\hskip 0.85358ptc_{3\theta}\\!+\\!3\hskip
0.85358ptc_{5\theta})\csc^{4}\\!\theta\,,$ (S27a)
$\displaystyle\delta\mathcal{N}^{\prime}_{t}$
$\displaystyle=\frac{~{}M_{n}^{2}\hskip
0.85358pt(131\\!-\\!8c_{\theta}\\!-\\!4c_{2\theta}\\!+\\!8c_{3\theta}\\!+\\!c_{4\theta})~{}}{8\hskip
0.85358pt(1\\!-\\!c_{\theta})^{2}}\,,$ (S27b)
$\displaystyle\delta\mathcal{N}^{\prime}_{u}$
$\displaystyle=-\frac{~{}M_{n}^{2}\hskip
0.85358pt(131\\!+\\!8c_{\theta}\\!-\\!4c_{2\theta}\\!-\\!8c_{3\theta}\\!+\\!c_{4\theta})~{}}{8\hskip
0.85358pt(1\\!+\\!c_{\theta})^{2}}\,,$ (S27c)

and the gauge-transformed NLO numerators for the corresponding KK Goldstone
boson amplitude:

$\displaystyle\delta\widetilde{\mathcal{N}}^{\prime}_{s}$
$\displaystyle=-\mbox{$\frac{\,{1}\,}{4}$}M_{n}^{2}\hskip 0.85358pt(238\hskip
0.85358ptc_{\theta}\\!+\\!19\hskip
0.85358ptc_{3\theta}\\!-\\!c_{5\theta})\csc^{4}\\!\theta\,,$ (S28a)
$\displaystyle\delta\widetilde{\mathcal{N}}^{\prime}_{t}$
$\displaystyle=\frac{~{}M_{n}^{2}\hskip 0.85358pt(99+8\hskip
0.85358ptc_{\theta}\\!+\\!28c_{2\theta}\\!-8\hskip
0.85358ptc_{3\theta}\\!+c_{4\theta})}{8\hskip
0.85358pt(1\\!-\\!c_{\theta})^{2}}\,,$ (S28b)
$\displaystyle\delta\widetilde{\mathcal{N}}^{\prime}_{u}$
$\displaystyle=-\frac{M_{n}^{2}\hskip 0.85358pt(99\\!-\\!8\hskip
0.85358ptc_{\theta}\\!+\\!28\hskip 0.85358ptc_{2\theta}\\!+\\!8\hskip
0.85358ptc_{3\theta}\\!+c_{4\theta})}{8\hskip
0.85358pt(1\\!+\\!c_{\theta})^{2}}\,.$ (S28c)

Using the double-copy formulas in Eqs.(18a)-(18b) together with the gauge-
transformed numerators
$(\mathcal{N}_{j}^{\prime},\,\widetilde{\mathcal{N}}_{j}^{\prime})$ in Eq.(19)
and Eqs.(S27)-(S28), we construct the following four-point KK gravition
amplitude and gravitational KK Goldstone amplitude at the LO and NLO:

$\displaystyle\mathcal{M}_{0}(\text{DC})$
$\displaystyle=\widetilde{\mathcal{M}}_{0}(\text{DC})=\frac{~{}3\kappa^{2}\,}{~{}128~{}}\hskip
0.85358pts\hskip 0.85358pt(7+c_{2\theta})^{2}\csc^{2}\\!\theta\,,$ (S29a)
$\displaystyle\delta\mathcal{M}(\text{DC})$
$\displaystyle=-\frac{~{}5\kappa^{2}M_{n}^{2}~{}}{768}(1642+297\hskip
0.85358ptc_{2\theta}+102\hskip 0.85358ptc_{4\theta}+7\hskip
0.85358ptc_{6\theta})\csc^{4}\theta\,,$ (S29b)
$\displaystyle\delta\widetilde{\mathcal{M}}(\text{DC})$
$\displaystyle=-\frac{~{}\kappa^{2}M_{n}^{2}~{}}{768}(6386+3837\hskip
0.85358ptc_{2\theta}+30\hskip 0.85358ptc_{4\theta}-13\hskip
0.85358ptc_{6\theta})\csc^{4}\theta\,,$ (S29c)

where we have set the conversion constant
$\,c_{0}\\!=\\!-\kappa^{2}/(24g^{2})\hskip 0.85358pt$. The double-copy
amplitudes of Eq.(S29a) provide the LO gravitational amplitudes (20) and the
NLO gravitational amplitudes (21) in the main text. We can further compute the
gravitational residual term of the GRET from the difference between the two
NLO amplitudes (S29b) and (S29c):

$\Delta\mathcal{M}(\mathrm{DC})\,=\,\delta\mathcal{M}(\mathrm{DC})-\delta\widetilde{\mathcal{M}}(\mathrm{DC})=-\kappa^{2}M_{n}^{2}\,(7+c_{2\theta})\hskip
0.85358pt,$ (S30)

which provides Eq.(22b) in the main text. We see that the above reconstructed
residual term (S30) by the extended double-copy approach does give the same
size of $\mathcal{O}(E^{0}M_{n}^{2})$ and takes the same angular structure of
$(1,\,c_{2\theta})$ as the original residual term (S24) although their
numerical coefficients still differ. As discussed in the main test, it is
impressive to note that Eq.(S30) also demonstrates a very precise cancellation
between the angular structures
$(1,\,c_{2\theta},\,c_{4\theta},\,c_{6\theta})\\!\times\\!\csc^{4}\\!\theta\,$
of the NLO double-copied KK amplitudes (S29b)-(S29c) down to the substantially
simpler angular structure $(1,\,c_{2\theta})\hskip 0.85358pt$. This is the
same kind of angular cancellations as what we found for the original NLO KK
graviton and Goldstone amplitudes (S22b)-(S22c) and their difference (S24).
This demonstrates that the above double-copied NLO KK amplitudes have captured
the essential features of the original KK graviton (Goldstone) amplitudes at
both the LO and NLO. We have presented the further improved NLO numerators
(28)-(29) in the main text, which can realize the double-copied NLO KK
amplitudes in full agreement with the original NLO KK graviton and Goldstone
amplitudes (S22b)-(S22c). A further study based on the first principle
approach of the KK string theory is recently presented in Ref. Li:2021yfk ,
which can realize the exact double-copy construction of the general $N$-point
KK graviton scattering amplitudes at tree level.

Finally, for the sake of comparison, we also give the results of making the
high energy expansion of $M_{n}^{2}/s_{0}\hskip 0.85358pt$ and explain that
within this expansion there is no generalized gauge transformation which could
realize the Jacobi-conserving numerators for KK gauge boson (Goldstone)
scattering amplitudes. For this, we express the elastic scattering amplitude
$\,\mathcal{T}[4A^{n}_{L}]\equiv\mathcal{T}[A^{an}_{L}A^{bn}_{L}\\!\rightarrow\\!A^{cn}_{L}A^{dn}_{L}]$
and
$\widetilde{\mathcal{T}}[4A^{n}_{5}]\equiv\widetilde{\mathcal{T}}[A^{an}_{5}A^{bn}_{5}\\!\rightarrow\\!A^{cn}_{5}A^{dn}_{5}]$
as follows:

$\displaystyle\mathcal{T}[4A^{n}_{L}]$
$\displaystyle\,=\,g^{2}\\!\left(\\!\frac{\,\mathcal{C}_{s}\mathcal{N}_{s}\,}{s_{0}}+\frac{\,\mathcal{C}_{t}\mathcal{N}_{t}\,}{t_{0}}+\frac{\,\mathcal{C}_{u}\mathcal{N}_{u}\,}{u_{0}}\\!\right)\\!,$
(S31a) $\displaystyle\widetilde{\mathcal{T}}[4A^{n}_{5}]$
$\displaystyle\,=\,g^{2}\\!\left(\\!\frac{\,\mathcal{C}_{s}\widetilde{\mathcal{N}}_{s}\,}{s_{0}}+\frac{\,\mathcal{C}_{t}\widetilde{\mathcal{N}}_{t}\,}{t_{0}}+\frac{\,\mathcal{C}_{u}\widetilde{\mathcal{N}}_{u}\,}{u_{0}}\\!\right)\\!.$
(S31b)

We compute their numerators at the LO and NLO,
$(\mathcal{N}_{j},\,\widetilde{\mathcal{N}}_{j})=(\mathcal{N}_{j}^{0},\,\widetilde{\mathcal{N}}_{j}^{0})+(\delta\mathcal{N}_{j},\,\delta\widetilde{\mathcal{N}}_{j})=\mathcal{O}(E^{2}M_{n}^{0})+\mathcal{O}(E^{0}M_{n}^{2})\hskip
0.85358pt$, and present them in the following Tabel S1.

With these, we verify that the LO numerators of KK gauge boson (Goldstone)
scattering amplitude satisfy the Jacobi identity:

$\sum_{j}\mathcal{N}^{0}_{j}=0\,,\qquad\sum_{j}\widetilde{\mathcal{N}}^{0}_{j}=0\,,$
(S32)

where $j\in(s,t,u)$ . But, we find that the Jacobi identity is no longer
obeyed by the NLO numerators:

$\displaystyle\sum_{j}\delta\mathcal{N}_{j}=\sum_{j}\delta\widetilde{\mathcal{N}}_{j}=\chi\neq
0\,,$ (S33a) $\displaystyle~{}\chi\,=-2\hskip
0.85358pt(7\\!+c_{2\theta})\hskip
0.85358ptc_{\theta}\csc^{2}\\!\theta\,M_{n}^{2}\,,$ (S33b)

We further note that the KK amplitudes (S31a)-(S31b) are invariant under the
generalized gauge transformations for the kinematic numerators:

$\mathcal{N}_{j}\rightarrow\mathcal{N}^{\prime}_{j}=\mathcal{N}_{j}+\Delta\\!\times\\!s_{0j}\,,\hskip
17.07164pt\widetilde{\mathcal{N}}_{j}\rightarrow\widetilde{\mathcal{N}}^{\prime}_{j}=\widetilde{\mathcal{N}}_{j}+\widetilde{\Delta}\\!\times\\!s_{0j}\,.$
(S34)

But, because of $\,\sum_{j}\\!s_{0j}\\!=0$ [cf. Eq.(S4)], we deduce
$\,\sum_{j}\\!\delta\mathcal{N}^{\prime}_{j}\\!=\\!\sum_{j}\delta\mathcal{N}_{j}\\!\neq
0\,$ and
$\,\sum_{j}\\!\delta\widetilde{\mathcal{N}}^{\prime}_{j}\\!=\\!\sum_{j}\delta\widetilde{\mathcal{N}}_{j}\\!\neq
0\,$. Hence, under the expansion of $\hskip 0.85358ptM_{n}^{2}/s_{0}\hskip
0.85358pt$, it is impossible to obtain proper solutions of
$\\{\Delta\,,\widetilde{\Delta}\\}$ which are supposed to ensure the gauge-
transformed NLO numerators
$(\delta\mathcal{N}_{j}^{\prime},\,\delta\widetilde{\mathcal{N}}_{j}^{\prime}\hskip
0.85358pt)$ to obey the kinematic Jacobi identity.

Table S1: Kinematic numerators of the LO and NLO scattering amplitudes (S31a)-(S31b) for KK longitudinal gauge bosons and KK Goldstones under the high energy expansion of $M_{n}^{2}/s_{0}\hskip 0.85358pt$, where $(\mathcal{N}_{j},\,\widetilde{\mathcal{N}}_{j})=(\mathcal{N}_{j}^{0},\,\widetilde{\mathcal{N}}_{j}^{0})\\!+\\!(\delta\mathcal{N}_{j},\,\delta\widetilde{\mathcal{N}}_{j})=\mathcal{O}(E^{2}M_{n}^{0})+\mathcal{O}(E^{0}M_{n}^{2})\hskip 0.85358pt$. Numerators | $\quad\mathcal{N}_{s}\quad$ | $\mathcal{N}_{t}$ | $\mathcal{N}_{u}$ | $\quad\widetilde{\mathcal{N}}_{s}\quad$ | $\widetilde{\mathcal{N}}_{t}$ | $\widetilde{\mathcal{N}}_{u}$ | $\,\mathcal{N}_{s}\\!-\\!\widetilde{\mathcal{N}}_{s}\,$ | $\,\mathcal{N}_{t}\\!-\\!\widetilde{\mathcal{N}}_{t}\,$ | $\,\mathcal{N}_{u}\\!-\\!\widetilde{\mathcal{N}}_{u}\,$
---|---|---|---|---|---|---|---|---|---
$\mathcal{N}^{0}_{j}/s_{0}$ | $-\frac{\,11c_{\theta}\,}{2}$ | $\frac{\,-5+11c_{\theta}+4c_{2\theta}\,}{4}$ | $\frac{\,5+11c_{\theta}-4c_{2\theta}\,}{4}$ | $-\frac{\,3c_{\theta}\,}{2}$ | $\frac{\,3(-3+c_{\theta})\,}{4}$ | $\frac{\,3(3+c_{\theta})\,}{4}$ | $-4c_{\theta}$ | $-4c_{\theta}$ | $-4c_{\theta}$
$\delta\mathcal{N}_{j}/M_{n}^{2}$ | $4c_{\theta}$ | $\frac{\,2(2-3c_{\theta}-2c_{2\theta}-c_{3\theta})\,}{1+c_{\theta}\,}$ | $-\frac{\,2(2+3c_{\theta}-2c_{2\theta}+c_{3\theta})\,}{1-c_{\theta}\,}$ | $\,4c_{\theta}\,$ | $-\frac{\,8c_{\theta}\,}{\,1+c_{\theta}\,}$ | $-\frac{\,8c_{\theta}\,}{\,1-c_{\theta}\,}$ | $0$ | $8s_{\theta}^{2}$ | $-8s_{\theta}^{2}$