aainstitutetext: 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, Chinabbinstitutetext: Institute of
Modern Physics & Physics Department, Tsinghua University, Beijing, China;
Center for High Energy Physics, Peking University, Beijing, China

# Massive Color-Kinematics Duality and Double-Copy
for Kaluza-Klein Scattering Amplitudes

Yao Li<EMAIL_ADDRESS>a Yan-Feng Hang<EMAIL_ADDRESS>a,b Hong-Jian
He<EMAIL_ADDRESS>
###### Abstract

In this work, we study the scattering amplitudes of massive Kaluza-Klein (KK)
states under toroidal compactification. We present a shifting method to
quantitatively derive the scattering amplitudes of massive KK gauge bosons and
KK gravitons from the corresponding massless amplitudes in the noncompactified
higher dimensional theories. With these we construct the massive KK scattering
amplitudes by extending the double-copy relations of massless scattering
amplitudes within the field theory formulations, including both the BCJ method
and CHY method. We present the massive BCJ-type double-copy construction of
the $N$-point KK gauge boson/graviton scattering amplitudes, and as the
applications we derive explicitly the four-point KK scattering amplitudes as
well as the five-point KK scattering amplitudes. We further study the
nonrelativistic limit of these massive scattering amplitudes with the heavy
external KK states and discuss the impact of the compactified extra dimensions
on the low energy gravitational potential. Finally, we analyze the four-point
and $N$-point mass spectral conditions and newly propose a group theory
approach to prove that only the KK theories under toroidal compactification
can satisfy these conditions for realizing the massive double-copy.

##  1 Introduction

Scattering amplitude is an important means for studying the fundamental forces
in nature,
and can bridge theories with experiments. One of the greatest challenges in
modern physics
is the unification of gravitational force with the gauge forces of
electromagnetic, weak and strong interactions of all elementary particles.
This leads to the explorations of a higher dimensional spacetime structure,
with extra spatial dimensions compactified on the boundaries and being smaller
than the existing observational limits. Such attempts started a century ago
from the Kaluza-Klein (KK) theory for unifying the gravitational and
electromagnetic forces with a compactified fifth dimension (5d) KK . This
opened up a truly fundamental direction, and has been seriously pursued and
extensively developed in various contexts, including the string/M theories
string  and the extra dimensional field theories with large or small extra
dimensions Exd0 Exd ExdRS . The KK compactification predicts an infinite tower
of massive KK excitations for each known particle of the Standard Model (SM).
These have intrigued substantial phenomenological and experimental efforts
over the past two decades to search for the low-lying KK states of the extra
dimensional KK theories exdpheno exd-rev , which could produce the first
signatures of new physics beyond the Standard Model (SM), including the KK
states of gravitons and of the SM particles as well as the possible dark
matter candidate.

The longstanding difficulty of unifying the gravitational force with the gauge
forces arises from the intricate nonlinearity and perturbative
nonrenormalizability of the Einstein theory of General Relativity (GR), in
contrast to the modern renormalizable gauge theories of the electroweak and
strong interactions in the SM of particle physics. However, the scattering
amplitudes of gravitons and gauge bosons are connected through the conjectured
deep relation of double-copy, $\hskip
0.85358pt\text{GR}\\!=\\!(\text{Gauge~{}Theory})^{2}\hskip 0.85358pt$, even
though it does not manifest at the Lagrangian (Hamiltonian) level. The double-
copy relation points to the common fundamental root of both the gravity force
and gauge forces. It has also become a powerful tool for efficiently computing
the highly intricate scattering amplitudes of spin-2 gravitons. The first
double-copy was constructed by Kawai-Lewellen-Tye (KLT) KLT  to link the
scattering amplitudes of closed strings to the products of scattering
amplitudes of open strings at tree level. The KLT relation leads to the
connection between the scattering amplitudes of massless gravitons and the
products of the color-ordered amplitudes of massless gauge bosons in the low
energy field-theory limit. Then, Bern-Carrasco-Johansson (BCJ) constructed the
double-copy within the field theory formalism through the color-kinematics
duality BCJ:2008 BCJ:2019 , which connects the (squared) scattering amplitudes
of massless gauge bosons to the corresponding massless graviton amplitudes.
The massless double-copy relations of BCJ may be proven and refined by
analyzing the tree-level scattering amplitudes of the heterotic strings and
open strings Tye-2010 . They may also be proven BCJ:2019  by using the on-
shell recursion relation of Britto-Cachazo-Feng-Witten (BCFW) BCFW  and be
extended to loop levels in the field theory framework. Afterwards, the
worldsheet Cachazo-He-Yuan (CHY) method CHY  was motivated by Witten’s twistor
string theory Witten:2003nn , it further shows that the KLT kernel can be
interpreted as the inverse amplitudes of bi-adjoint scalars; it can also
extend the double-copy relations to other field theories than the
gauge/gravity theories CHY14 . But, all these works were to formulate and test
double-copy for the scattering amplitudes of massless gauge bosons/gravitons
BCJ:2019 . Some recent works attempted to generalize the double-copy method to
the 4d massive Yang-Mills (YM) theory versus Fierz-Pauli-like massive gravity
dRGT DC-4dx1 , to spontaneously broken YM-Einstein supergravity models with
adjoint Higgs fields SUGRAhiggs , to the KK-inspired effective gauge theory
with extra global U(1) Momeni:2020hmc , to the 5d KK gauge/gravity theories
with orbifold compactification Hang:2021fmp , to the 3d Chern-Simons (CS)
gauge/gravity theories with or without supersymmetry 3dCS-susy 3dCS1
Hang:2021oso , and to certain massive scalar theories DC-scalar .

We note that extensions of the conventional massless double-copy formalism to
the massive gauge/gravity theories are generally difficult, because many
theories of this kind violate gauge symmetry and diffeomorphism invariance
(which include the massive YM theory and the massive Fierz-Pauli gravity FP ).
By adding extra non-linear polynomial interaction terms in the literature dRGT
DC-4dx1 , one could realize the double-copy between the massive YM and Fierz-
Pauli gravity theories, but the high energy behavior of the four-point massive
graviton amplitudes could be improved to no better than $E^{6}$ Cheung2016
Kurt-E6  and is still much worse than the final energy-dependence of
$O(E^{2})$ in the tree-level massive KK graviton scattering amplitudes
Hang:2021fmp Chivukula:2020 Kurt-2019 . The KK compactification can realize a
geometric “Higgs” mechanism for mass generation of KK gravitons at both the
Lagrangian level GHiggs Hang:2021fmp  and scattering $S$-matrix level
Hang:2021fmp . This is shown Hang:2021fmp  to make the massive KK GR theory
free from the vDVZ discontinuity vDVZ  and exhibit much better high energy
behaviors (as good as the massless Einstein gravity). Hence the KK GR theory
provides a truly consistent realization of the massive gravity in the
effective field theory formulation. Another important realization of
consistent massive double-copy is given by the 3d topologically massive Chern-
Simons gauge and gravity theories TMG  because they have topological mass-
generations for the gauge bosons and gravitons in a gauge-invariant manner and
guarantee the good high energy behaviors of the massive gauge boson/graviton
scattering amplitudes Hang:2021oso .

Our recent works studied the massive double-copy constructions for the KK
gauge/gravity field theories under the 5d compactification with the
$S^{1}\\!/\mathbb{Z}_{2}$ orbifold Hang:2021fmp , and for the 26d KK bosonic
string theory under the compactification of $\mathbb{R}^{1,24}\otimes S^{1}$
KKString .111In passing, a recent paper wKLT  studied the general KLT
factorization of winding string amplitudes in the bosonic string theory and
computed explicitly the four-point tachyon amplitudes, without considering the
amplitudes in the low energy field-theory limit. Hence, Ref. wKLT  does not
overlap with our present study. We found Hang:2021fmp  that the massive
double-copy construction for the 5d KK theories with $S^{1}\\!/\mathbb{Z}_{2}$
orbifold compactification is highly involved because the KK gauge boson
scattering amplitudes exhibit double-pole structures and the naive extension
of the conventional massless BCJ method does not work even at tree level. By
making the high energy expansion, we proved Hang:2021fmp  that the leading
order (LO) KK gauge boson amplitudes are mass-independent and obey the color-
kinematics duality. So the massive double-copy works at the LO [which is
enough for our double-copy construction of the KK Gravitational Equivalence
Theorem (GRET) Hang:2021fmp ], but it does not exactly work at the next-to-
leading-order (NLO) and beyond, unless certain special treatment is made.
Then, by using the first principle approach of KK string theory, we realized
KKString  that the massive extension of the KLT-type double-copy construction
could exactly work for the 5d $S^{1}$ compactification (without orbifold)
under which the KK amplitudes have single-pole structure, exhibit the massive
color-kinematics duality, and obey the mass spectral condition. As the
resolution for the nontrivial case of the orbifold compactification, we found
KKString  that the correct double-copied KK graviton amplitudes can be
constructed in terms of proper combinations of the KK graviton amplitudes
(derived under the compactification without orbifold). We obtained these
insights and the exact double-copy construction of KK gauge boson/graviton
amplitudes by first deriving the massive KLT relations (connecting the KK
closed string amplitudes to the products of the KK open string amplitudes) for
the bosonic string theory under the compactification of
$\mathbb{R}^{1,24}\otimes S^{1}$. With these, we took the low energy field-
theory limit (with the string tension $\alpha^{\prime}\\!\\!\rightarrow\\!0$)
and derived the exact double-copied KK graviton amplitudes. Then, we found
that the KK scattering amplitudes under the $S^{1}\\!/\mathbb{Z}_{2}\hskip
0.85358pt$ orbifold compactification can be constructed in terms of proper
combinations of the corresponding KK amplitudes (computed without the
$\mathbb{Z}_{2}$ orbifold). We note that the massive KK KLT-like relations
derived in Ref. KKString  rely on the color-ordered amplitudes of KK gauge
bosons. It is thus desirable to further construct the BCJ-type massive double-
copy with extended color-kinematics duality for the KK gauge boson/graviton
scattering amplitudes and build its quantitative connection to the massive
KLT-like relations in the present work. Especially, the BCJ double-copy
approach has its own advantages BCJ:2019 , hence studying its massive
extension to the compactified KK gauge/gravity theories is valuable. It is
also desirable to extend the conventional massless CHY method to the massive
KK double-copy construction, and build up its connections to the massive KLT-
like relations and the massive BCJ-like construction.

In this work, we study the scattering amplitudes of massive KK states under
toroidal compactification, and analyze the realization of massive color-
kinematics duality and KK gauge/gravity double-copy construction. We will
present a shifting method to quantitatively derive the scattering amplitudes
of massive KK gauge bosons and KK gravitons from the corresponding massless
amplitudes in the noncompactified higher dimensional theories. The massive KK
amplitudes derived in this way correspond to the toroidal compactification
without orbifold and can serve as the basis KK amplitudes to further construct
other types of KK amplitudes under the orbifold toroidal compactification.
With these we construct the massive KK scattering amplitudes by extending the
double-copy relations of massless scattering amplitudes within the field
theory formulations, including both the BCJ method and CHY method. We present
the massive BCJ-type double-copy construction of the $N$-point KK gauge
boson/graviton scattering amplitudes. As the applications, we derive
explicitly the four-point KK scattering amplitudes and the five-point KK
scattering amplitudes. Under the KK compactification without orbifold and
using the generalized gauge invariance, we will derive a mass spectral
condition for the four-point massive KK graviton amplitudes. We also use an
extended fundamental BCJ relation for the massive KK theories and prove that
the four-point KK scattering amplitudes have to obey the same mass spectral
condition. We further study the nonrelativistic limit of these massive
scattering amplitudes with the heavy external KK states and discuss the impact
of the compactified extra dimensions on the low energy gravitational
potential. We show that the elastic scattering amplitudes of the heavy KK
states can induce a leading-order behavior of the classical potential scaling
as $1/r$ at low energies. Finally, we study the possible solutions to the
four-point KK mass spectral condition which is a necessary and sufficient
condition for realizing the massive KK double-copy. We newly propose a group
theory approach to prove that it gives a unique consistent solution of the
mass spectrum which could be realized only by the KK theories under the
toroidal compactifications without orbifold. We further extend this four-point
spectral condition to the general $N$-point spectral condition. We prove that
the KK mass spectrum is also the solution to the $N$-point spectral condition,
so it is thus a truly consistent solution.

This paper is organized as follows. In Section 2, we establish an extended
double-copy approach for scattering amplitudes of massive KK gauge bosons and
KK gravitons under the toroidal compactification within the quantum field
theory (QFT) formulation. We construct the massive KK amplitudes from the
massless amplitudes of zero-modes by using the “shifting” method, with which
we build up a correspondence from the conventional massless BCJ double-copy
construction to the extended massive KK double-copy construction. In Section
3, we present the extended four-point massive BCJ-type double-copy
construction of scattering amplitudes of the KK gauge bosons and KK gravitons
under the 5d toroidal compactification with or without orbifold. Under the
toroidal compactification without orbifold and by using either the generalized
gauge invariance or the extended massive fundamental BCJ relation, we will
derive a mass spectral condition for a consistent double-copy construction of
the four-point KK graviton amplitude. We also derive the nonrelativistic KK
scattering amplitudes for the heavy KK gauge bosons, heavy KK gravitons, and
heavy KK scalars respectively. We further compute the five-point KK graviton
scattering amplitudes from double-copy construction. In Section 4, we
generalize the conventional massless CHY method and give an extended massive
CHY formulation of the KK gauge boson/graviton scattering amplitudes. We will
derive a massive scattering equation for the KK scattering amplitudes and
construct the KK bi-adjoint scalar amplitudes. We use this extended massive
CHY approach to further construct the scattering amplitudes of KK gauge bosons
and of KK gravitons, and derive their relations to the extended BCJ-type KK
amplitudes (Section 3) and to the extended KLT-type KK amplitudes (as given by
Ref. KKString ). In Section 5, we study the possible solutions to the mass
spectral conditions for the four-point KK scattering amplitudes and for the
general $N$-point KK scattering amplitudes. For this we propose a group theory
approach to prove that the four-point mass spectral condition can uniquely
determine the allowed mass spectrum to be that of the KK theories under
toroidal compactification. Finally, we summarize and conclude in Section 6.
For the analyses shown in the main text, we also define the kinematics of the
four-point scattering for both the massless 5d theories and the compactified
massive 4d KK theories in Appendix A. The kinematic numerators for KK gauge
boson scattering amplitudes are given in Appendix B, and the double-copied
full scattering amplitudes of KK gravitons are presented in Appendix C.

##  2 KK Scattering Amplitudes under Toroidal Compactification

In the recent work KKString , we studied the scattering amplitudes of massive
Kaluza-Klein (KK) states of open and closed bosonic strings under toroidal
compactification. We demonstrated that the $N$-point scattering amplitudes of
the massive KK gauge bosons and KK gravitons can be derived by taking the
field-theory limit $\hskip 0.85358pt\alpha^{\prime}\\!\\!\rightarrow\\!0\,$
for the corresponding open and closed amplitudes of the massive KK bosonic
strings. We demonstrated that the extended massive KLT-like relations can
realize the exact double-copy construction under the toroidal compactification
(without orbifold) which conserves the KK numbers and ensures the massive KK
amplitudes to have single-pole structure in each kinematic channel. For the
toroidal compactification with $\mathbb{Z}_{2}$ orbifold, we showed that any
$N$-point KK amplitude (with external states being $\mathbb{Z}_{2}$ even or
odd) can be decomposed into a sum of sub-amplitudes which belong to the
toroidal compactification without orbifold.

In Section, we will show that the eigenfunctions of Laplace operator on a
compact manifold can be chosen as the exponential functions with which the
massive KK scattering amplitudes of a higher dimensional theory under toroidal
compactification can be obtained by replacing the extra-dimensional momentum-
components in the corresponding massless amplitudes of the noncompactified
theory by their discretized values (given by the KK compactification). The
physical scattering amplitudes are independent of which basis of
eigenfunctions is chosen. Thus, the amplitudes defined under other
eigenfunction bases (such as the trigonometric functions) can be obtained by
proper transformations of the external states. Using such a “shifting” method,
we can directly construct the massive KK scattering amplitudes from the
corresponding massless amplitudes of the noncompactified higher dimensional
theory. Thus, we can establish an extended BCJ-type double-copy approach for
scattering amplitudes of the massive KK gauge bosons and KK gravitons in the
QFT formulation. We also stress the importance of using the toroidal
compactification without orbifold as the base construction of the massive KK
double-copy, with which the double-copy constructions in other KK theories
under the orbifold compactification (such as $S^{1}\\!/\mathbb{Z}_{2}$) can be
formulated by proper transformations.

###  2.1 Toroidal Compactification with Different Eigenbases

Consider a generic extra-dimensional model defined on the manifold
$\,\mathbb{M}^{1,3}\otimes\mathbb{N}^{\delta}$, where $\,\mathbb{M}^{1,3}$
denotes the (1+3)-dimensional Minkowski spacetime and $\mathbb{N}^{\delta}\\!$
($\hskip 0.85358pt\delta\\!\geqslant\\!1$) denotes the extra
$\delta$-dimensional space under toroidal compactification. The extra-
dimensional mass operator $\Delta$ is a Laplacian defined on
$\mathbb{N}^{\delta}\hskip 0.85358pt$ and has the following KK eigenvalue
equation:

$\Delta\mathcal{O}_{I}\,=\,M_{I}^{2}\hskip 0.85358pt\mathcal{O}_{I}\,.$ (1)

where $\\{\mathcal{O}_{I}\\}$ denote the eigenfunctions with KK index $I$ and
$\\{M_{I}\\}$ are the corresponding mass-eigenvalues Kurt-2019 . According to
Ref. Nakahara , we can express the Laplacian $\Delta$ as follows:

$\Delta=(\mathbf{d}+\mathbf{d}^{\dagger})^{2}=\hskip
0.85358pt\mathbf{d}\mathbf{d}^{\dagger}+\mathbf{d}^{\dagger}\mathbf{d}\,,$ (2)

where $\hskip 0.85358pt\mathbf{d}\hskip 0.85358pt$ denotes the exterior
derivative operator for the extra dimensional space and $\hskip
0.85358pt\mathbf{d}^{\dagger}$ is the adjoint exterior derivative operator. It
can be shown Nakahara  that the nilpotency holds,
$\,\mathbf{d}\mathbf{d}=\mathbf{d}^{\dagger}\mathbf{d}^{\dagger}=0\,$. Thus we
can prove $M_{I}^{2}$ to be positive definite:

$\displaystyle
M_{I}^{2}\,=\int_{\mathbb{N}}\\!\mathcal{O}_{I}\Delta\mathcal{O}_{I}\hskip
0.85358pt=\hskip 0.85358pt(\mathcal{O}_{I},\Delta\mathcal{O}_{I})\hskip
0.85358pt=\hskip
0.85358pt(\mathbf{d}\mathcal{O}_{I},\mathbf{d}\mathcal{O}_{I})+(\mathbf{d}^{\dagger}\mathcal{O}_{I},\mathbf{d}^{\dagger}\mathcal{O}_{I})\geqslant
0\,,$ (3)

for $\mathbb{N}$ being a Riemann manifold, where the integration is performed
over the extra dimensional coordinates.

For the case without degenerate KK states, the KK eigenfunctions
$\\{\mathcal{O}_{I}\\}$ with different eigenvalues should be orthonormal to
each other:

$\int_{\mathbb{N}}\\!\mathcal{O}_{I}\hskip
0.85358pt\mathcal{O}_{J}\,=\,\delta_{IJ}\,.$ (4)

For the case with degenerate KK states, the same condition can be retained by
using the Schmidt orthogonalization. We can further define the general
$N$-point coupling constants among the KK states as follows:

$\displaystyle\int_{\mathbb{N}}\mathcal{O}_{I_{1}}\mathcal{O}_{I_{2}}\cdots\mathcal{O}_{I_{N}}$
$\displaystyle=\,C_{I_{1}\cdots I_{N}}\,,$ (5a)
$\displaystyle\int_{\mathbb{N}}\partial\hskip
0.85358pt\mathcal{O}_{J_{1}}\mathcal{O}_{I_{2}}\cdots\mathcal{O}_{I_{N}}$
$\displaystyle=\,D_{J_{1}I_{2}\cdots I_{N}}\,,$ (5b)

and so on. In Eq.(5b), the symbol $\hskip 0.85358pt\partial\hskip
0.85358pt\mathcal{O}_{J_{1}}$ denotes the possible extra-dimensional
derivative(s) in a given interaction vertex.

Under the toroidal compactification, there are two common choices of the
eigenfunctions $\,\mathcal{O}_{I}\,$, one is based on the Fourier expansion
with trigonometric functions, and another is the choice of exponential
functions:

$\displaystyle\\{\mathcal{O}_{I}\\}\,=\,$ $\displaystyle\left\\{\hskip
1.42262pt1,\,\sqrt{2\hskip 0.85358pt}\cos(n_{i}\hskip
0.85358pty_{i}/\\!R_{i}),\,\sqrt{2\hskip 0.85358pt}\sin(n_{i}\hskip
0.85358pty_{i}/\\!R_{i})\right\\},~{}~{}$ $n_{i}\in\mathbb{Z}^{+}$, (6a)
$\displaystyle\\{\mathcal{O}_{I}\\}\,=\,$
$\displaystyle\left\\{\exp(\text{i}\hskip
0.85358ptn_{i}y_{i}/\\!R_{i})\right\\},~{}~{}$ $n_{i}\in\mathbb{Z}$ , (6b)

where $\,y_{i}\\!\in\\![0,2\pi R_{i}]\,$ represents the $i$-th coordinate of
the extra dimensional space $\mathbb{N}^{\delta}$ and $R_{i}$ denotes the
radius of the $i$-th dimension of $\mathbb{N}^{\delta}$. In addition, we note
that the eigenfunctions (6b) will lead to Feynman rules which are analogous to
that of the non-compactified flat space, but withe the continuous momenta
replaced by the corresponding discrete values, namely,
$\,q_{i}\\!\rightarrow\\!n_{i}/\\!R_{i}$ . In fact, as we will show, it is
really advantageous to first analyze the KK scattering amplitudes under the
toroidal compactification without orbifold and using the exponential
eigenfunctions (6b); and then we can use these to derive the corresponding KK
scattering amplitudes in another given basis of eigenfunctions such as
Eq.(6a).

We note that for a massless theory in the $d$-dimensional spacetime
$\,\mathbb{M}^{1,d-1}$, the general $N$-point amplitude,222In the following
text, the variables with an extra “hat” symbol denote the quantities defined
in the higher $d$-dimensional spacetime with $\hskip 0.85358ptd>4\hskip
0.85358pt$. $\,\widehat{\mathcal{A}}_{N}(\\{\hat{p}_{i},\hat{\zeta}_{i}\\})$,
is an analytical function of the momenta and polarizations333If we consider
the scalar fields, then the polarization $\,\hat{\zeta}=1\,$ and has no
effect. living in the full $d$-dimensional spacetime. We can decompose the
momentum $\hat{p}_{i}=(p_{i},\,q_{i})$ into a $(1+3)$-dimensional momenta
$p_{i}$ and an extra $\delta$-dimensional momentum $q_{i}$. Thus, we can
obtain the expression of an $N$-point KK scattering amplitude (in the KK
theory with the compactified extra-dimensional space $\mathbb{N}^{\delta}$)
from the corresponding massless amplitude (in the non-compactified massless
$d$-dimensional theory with $d=4+\delta\hskip 0.85358pt$),

$\mathcal{A}_{N}\big{[}\\{p_{i},\hat{\zeta}_{i},I_{i}\\}\big{]}\,=\,\sum_{q_{j}}\\!\widehat{\mathcal{A}}_{N}\big{[}\\{p_{i},\hat{\zeta}_{i},q_{i}\\}\big{]}\prod_{j}\\!\int_{\mathbb{N}}\\!\\!\text{d}y_{j}\hskip
0.85358pte^{\text{i}\hskip 0.85358ptq_{j}y_{j}}\hskip
0.85358pt\mathcal{O}_{I_{j}}(y_{j})\,,$ (7)

where $\,y_{j}\,$ denotes the coordinates of the extra-dimensional space
$\mathbb{N}^{\delta}$ in the eigenfunction $\hskip
0.85358pt\mathcal{O}_{I_{j}}(y_{j})\hskip 0.85358pt$. Since the final physical
KK amplitude should not depend on choosing which base of extra-dimensional
eigenfunctions, the KK amplitudes defined by using the base $X$ should be
connected to the KK amplitudes defined under another base $Y$ through the
linear transformations of external KK states,

$\Big{|}\mathcal{O}^{X}_{i}\Big{\rangle}\,=\,\int\\!\mathcal{O}^{X}_{i}\hskip
0.85358pt\mathcal{O}^{Y}_{j}\Big{|}\mathcal{O}^{Y}_{j}\Big{\rangle}\,.$ (8)

The KK scattering amplitude may be expressed in the following form,

$\mathcal{A}_{N}\,\sim\,\langle
0|S|\mathcal{O}_{1}\mathcal{O}_{2}\ldots\mathcal{O}_{N}\rangle\,,$ (9)

where we choose all the external particles to be incoming. Thus, the
scattering amplitudes in the compactified KK theories defined under different
bases of KK eigenfunctions can be transformed into each other under the
transformation (8) of their external states. We note that only the amplitudes
under the toroidal compactification can be connected to amplitudes in flat
spacetime and obeys such simple rules, as shown in Eq.(7).

For instance, consider an $N$-point amplitude in $\,\mathbb{M}^{1,3}\otimes
S^{1}$ spacetime under the $S^{1}$ toroidal compactification. Substituting an
eigenfunction $\sqrt{2}\cos(n_{j}y_{j}/\\!R)$ (with
$\,n_{j}\\!\in\mathbb{Z}^{+}$) of Eq.(6a) into Eq.(7) for each external state,
we derive:

$\displaystyle\mathcal{A}_{N}[\\{p_{i},q_{i},\hat{\zeta}_{i}\\}]$
$\displaystyle=\,\sum_{q_{j}}\\!\widehat{\mathcal{A}}_{N}[\\{p_{i},\hat{\zeta}_{i},q_{i}\\}]\prod_{j}\\!\int^{2\pi
R}_{0}\\!\\!\\!\text{d}y_{j}\exp(\text{i}\hskip
0.85358ptq_{j}y_{j})\\!\left(\\!\\!\sqrt{2}\cos\frac{\hskip
0.85358ptn_{j}y_{j}\,}{R}\\!\\!\right)$
$\displaystyle=\,2^{-N/2}\sum_{q_{j}}\\!\widehat{\mathcal{A}}_{N}[\\{p_{i},\hat{\zeta}_{i},q_{j}\\}]\\!\prod_{j}\\!\left[\delta\\!\left(\\!q_{j},\frac{\,n_{j}\,}{R}\\!\right)\\!+\delta\\!\left(\\!q_{j},-\frac{\,n_{j}\,}{R}\\!\right)\right]$
$\displaystyle=\,2^{-N/2}\\!\\!\sum_{\\{\text{sgn}_{i}\\}}\\!\\!\widehat{\mathcal{A}}_{N}[\\{p_{i},\hat{\zeta}_{i},\text{sgn}_{i}\\!\times\\!n_{i}/\\!R\\}]\hskip
0.85358pt,$ (10)

where in the last line we have defined
$\,\text{sgn}_{i}\\!=\text{sign}(n_{i})=\\!\pm 1$ and the sum of
$\\{\text{sgn}_{i}\\}$ runs over all possible independent combinations of the
$N$-external KK states. In the second line of the above formula, the delta
function is defined as $\,\delta(a,\hskip 0.85358ptb)\equiv\delta_{ab}\hskip
0.85358pt$. In Eq.(10), we have used the conservation of the extra-
dimenstional momenta $\,\sum_{i}q_{i}\\!=0\,$ for the amplitude
$\widehat{\mathcal{A}}_{N}(\\{p_{i},\hat{\zeta}_{i},q_{i}\\})$ because the
flat non-compactified extra-dimensional space holds the translation symmetry.
Under the periodic boundary conditions, the toroidal compactification of extra
spatial dimensions also holds the translation symmetry and thus the
conservation of KK numbers. Eq.(10) explicitly demonstrates that under the
toroidal compactification of the extra-dimensional space
$\mathbb{N}^{\delta}$, a given $N$-point massive KK amplitude with external
particles being the $\mathbb{Z}_{2}$-even eigenfunctions of Eq.(6a) can be
obtained from a sum of the corresponding massless amplitudes defined in a
higher dimensional spacetime $\mathbb{M}^{1,3+\delta}$ with each extra-
dimensional momentum $q_{i}$ replaced by the discretized one ($\hskip
0.85358ptq_{i}\\!=\\!+n_{i}/\\!R\,$, or, $\hskip
0.85358ptq_{i}\\!=\\!-n_{i}/\\!R\,$), which we will call the sub-amplitudes
hereafter. For the external states being $\mathbb{Z}_{2}$-odd or certain more
complicated combination, their KK amplitudes can be derived in the similar
fashion by using Eq.(7). We stress that the above presentation gives a fairly
general approach for deriving the massive KK scattering amplitudes (under the
toroidal compactification) from the corresponding massless scattering
amplitudes of the non-compactified higher dimensional theory. This
construction can be applied to any physical KK fields and is not limited to
the KK gauge bosons and KK gravitons. Because the double-copy method has been
well established for the massless field theories, our current extended double-
copy approach for the massive KK scattering amplitudes in the following
section can be formulated within pure field-theory framework without relying
on the KK string theory construction given in our previous work KKString .

As a final remark, we note that the polarization $\hskip
0.85358pt\hat{\zeta}_{i}\hskip 0.85358pt$ of each external state as appeared
on the left-hand-side (LHS) of Eq.(7) or Eq.(10) is still the one defined in
the full $\hskip 0.85358ptd=4+\delta\hskip 0.85358pt$ dimensions. In the next
subsection, we will further derive the exact formulation of the KK scattering
amplitudes with the polarization $\hat{\zeta}_{i}$ reduced to that of the
compactified KK theory in the 4-dimensional spacetime.

###  2.2 Connecting Amplitudes Before and After Toroidal Compactification

In Eqs.(7) and (10), the scattering amplitudes on their LHS still have the
polarization $\hat{\zeta}$ defined in the full $\,d=4+\delta\,$ dimensions. To
obtain the $4$-dimensional KK scattering amplitudes, we first note that the
compactified components and non-compactified components of $\,\hat{\zeta}\,$
transform in different representations of the Lorentz group for the non-
compactified spacetime $\mathbb{M}^{1,3}$. Hence, the polarization vector
$\hat{\zeta}$ can be separated into two parts in terms of the representations
of Lorentz group. One part of the ploarization $\hat{\zeta}$ is restricted to
(1+3)-dimensions, $\hskip
0.85358pt\hat{\zeta}\\!=\\{\zeta,0,0,\cdots\\!,0\\}\\!=\\!\\{\zeta,\hskip
0.85358pt\vec{0}\hskip 0.85358pt\\}\hskip 0.85358pt$, whereas the other part
of the ploarization vector $\hat{\zeta}$ only has compactified components
$\hskip 0.85358pt\hat{\zeta}\\!=\\!\\{0,\cdots\\!,0,\vec{\eta}\hskip
0.85358pt\\}\hskip 0.85358pt$.

In the following we analyze the polarization vector $\hskip
0.85358pt\hat{\zeta}\\!=\\{\zeta,\vec{0}\hskip 0.85358pt\\}\hskip 0.85358pt$.
We consider the compactification from the $(4+\delta)$-dimensional Minkowski
spacetime to the $4$-dimensional spacetime. We denote the momenta and
polarization vectors as $\hat{p}_{i}$ and $\hat{\zeta}_{i}$ in
$(4+\delta)$-dimensions, and those in the $4$-dimensional Minkowski spacetime
by $\hskip 0.85358ptp_{i}\hskip 0.85358pt$ and $\hskip
0.85358pt\zeta_{i}\hskip 0.85358pt$. Under the toroidal compactification, the
momentum and polarization vector in the full $(4+\delta)$-dimensional
spacetime are connected to those in the $4$-dimensional spacetime:

$\hat{p}^{M}\\!=(p^{\mu},\,n_{1}/\\!R_{1},\cdots,n_{\delta}/\\!R_{\delta})\,,\qquad\hat{\zeta}^{M}\\!=(\zeta^{\mu},0,\cdots,0)\,,$
(11)

where the integer $\hskip 0.85358ptn\\!\in\\!\mathbb{Z}$ is the KK number. The
tree-level scattering amplitudes should be rational functions of the Lorentz
invariants, such as $\,\hat{p}_{i}\cdot\hat{p}_{j}\hskip 0.85358pt$,
$\,\hat{p}_{i}\cdot\hat{\zeta}_{j}\hskip 0.85358pt$, and
$\,\hat{\zeta}_{i}\cdot\hat{\zeta}_{j}\,$. Using Eq.(11), we can rewrite these
Lorentz invariants as follows:

$\displaystyle\hat{s}_{ij}=s_{ij}-\sum_{k=1}^{\delta}\\!\frac{\,(n_{k,i}\\!+n_{k,j})^{2}\,}{R_{k}^{2}}\hskip
0.85358pt=\,s_{ij}-\sum_{k=1}^{\delta}\\!\left(\text{sgn}_{i}M_{k,i}\\!+\text{sgn}_{j}M_{k,j}\right)^{\\!2},$
(12a) $\displaystyle\hat{\zeta}_{i}\cdot\hat{p}_{j}=\zeta_{i}\cdot
p_{j}\,,\qquad\hat{\zeta}_{i}\cdot\hat{\zeta}_{j}=\zeta_{i}\cdot\zeta_{j}\,,$
(12b)

where $\hskip 0.85358pt\text{sgn}_{i}\\!\equiv\mathrm{sign}(n_{k,i})\\!=\\!\pm
1\hskip 0.85358pt$ and $M_{k,i}\\!=\\!|n_{k,i}|/\\!R_{k}\hskip 0.85358pt$.
According to the above, we can derive the massive KK scattering amplitude
$\widehat{\mathcal{A}}_{N}(\\{\hat{\zeta}_{i}\\},\\{\hat{p}_{i}\\})$ from the
corresponding massless zero-mode scattering amplitude
$\mathcal{A}_{N}(\\{\zeta_{i}\\},\\{p_{i}\\})\hskip 0.85358pt$ by shifting its
Mandelstam variables: $\,s_{ij}\\!\\!\rightarrow\\!\hat{s}_{ij}\,$, where
$\,\hat{s}_{ij}$ is defined in Eq.(12). We note that since the physical
degrees of freedom for each external state are conserved before and after the
toroidal compactification, the number of physical polarization vectors will
remain the same after the compactification, except that each massive KK gauge
boson acquires a physical longitudinal component with polarization vector
$\hskip 0.85358pt\zeta_{L}^{\mu}\hskip 0.85358pt$ which is ensured by the
geometric Higgs mechanism 5DYM2002  of KK compactification and charaterized by
the KK gauge boson equivalence theorem (GAET) 5DYM2002 5DYM2002-2 KK-ET-He2004
at the $S$-matrix level. This shifting method provides a powerful way to
efficiently compute both the KK gauge boson amplitudes and KK graviton
amplitudes under toroidal compactification.

We first consider an $N$-point scattering amplitude for massless gauge boson
in 5d. It can be generally expressed in the following form:

$\displaystyle\widehat{\mathcal{T}}\Big{[}\widehat{A}^{a_{1}}_{\hat{\lambda}_{1}}\widehat{A}^{a_{2}}_{\hat{\lambda}_{2}}\ldots\widehat{A}^{a_{N}}_{\hat{\lambda}_{N}}\Big{]}\,=\,\hat{g}^{N-2}\sum_{j}\frac{~{}\mathcal{C}_{j}\,\widehat{\mathcal{N}}_{j}(\\{\hat{\lambda}\\})~{}}{\llbracket\widehat{\mathcal{D}}_{j}\rrbracket}\,,$
(13)

where on the left-hand-side (LHS) the subscripts
$\hat{\lambda}_{i}\\!=\\!\\{\pm 1,0\\}\hskip 0.85358pt$ denote the helicity
states for each 5d gauge boson, and on the right-hand-side (RHS) the sum runs
over all $\,(2N\\!-5)!!\,$ distinct diagrams and
$\,\llbracket\widehat{\mathcal{D}}_{j}\rrbracket\,$ denotes the product of
denominators of the Feynman propagators. In each numerator,
$\,\\{\mathcal{C}_{j}\\}\,$ and $\,\\{\widehat{\mathcal{N}}_{j}\\}\,$ are the
color and kinematic factors, obeying the color and kinematic Jacobi identities
respectively:

$\mathcal{C}_{\alpha}+\mathcal{C}_{\beta}+\mathcal{C}_{\gamma}=0\,,\hskip
28.45274pt\widehat{\mathcal{N}}_{\alpha}+\widehat{\mathcal{N}}_{\beta}+\widehat{\mathcal{N}}_{\gamma}=0\,,$
(14)

which contain only $(2N\\!-5)!!-(N\\!-2)!\,$ independent equations for each
group of equations in Eq.(14) BCJ:2008 BCJ:2019 . Thus, according to Eq.(12),
the $N$-point scattering amplitude for the massive KK gauge bosons can be
derived from the corresponding non-compactified higher dimensional massless
scattering amplitude (13):

$\mathcal{T}\\!\left[A_{\lambda_{1}}^{a_{1}\hskip
0.85358pt\mathfrak{n}_{1}}A_{\lambda_{2}}^{a_{2}\hskip
0.85358pt\mathfrak{n}_{2}}\cdots A_{\lambda_{N}}^{a_{N}\hskip
0.85358pt\mathfrak{n}_{N}}\right]=\,g^{N-2}\sum_{j}\\!\frac{\mathcal{C}_{j}\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{P}}(\\{\lambda\\})}{~{}\llbracket\mathcal{D}_{j}\\!-M_{\mathfrak{n}\mathfrak{n}_{j}}^{2}\rrbracket~{}}\,,$
(15)

where on the left side the subscripts $\lambda_{i}\\!=\\!\\{\pm
1,0\\}\\!\equiv\\!\\{\pm 1,L\\}$ denote the helicities of each external KK
gauge boson state, including two transverse polarizations and one longitudinal
polarization. On the right side of Eq.(15), the 4d gauge coupling $\hskip
0.85358ptg\hskip 0.85358pt$ is connected to the 5d gauge coupling $\hskip
0.85358pt\hat{g}\hskip 0.85358pt$ via $\,g\\!=\\!\hat{g}/\sqrt{2\pi R\,}\,$.
The superscript $\mathsf{P}$ labels every possible combination of the signs of
the KK indices $\\{\mathfrak{n}_{i}\\}$ of the external gauge bosons obeying
the $N$-point neutral condition $\sum_{i=1}^{N}\mathfrak{n}_{i}\\!=\\!0$
KKString . In the denominator of Eq.(15), the symbol $\hskip
0.85358pt\llbracket\mathcal{D}_{j}-M_{\mathfrak{n}\mathfrak{n}_{j}}^{2}\rrbracket\hskip
0.85358pt$ denotes the product of the denominators of the propagators with
$M_{\mathfrak{n}\mathfrak{n}_{j}}$ being the relevant KK mass. For instance,
considering the four-point scattering process, we have $\hskip
0.85358pt\mathcal{D}_{j}\\!\in\\!\\{s,t,u\\}$ and $\hskip
0.85358pt\mathfrak{n}\mathfrak{n}_{j}\\!\in\\!\\{\mathfrak{n}_{1}\\!+\mathfrak{n}_{2},\,\mathfrak{n}_{1}+\mathfrak{n}_{4},\,\mathfrak{n}_{1}\\!+\mathfrak{n}_{3}\\}\hskip
0.85358pt$, where $\,j\\!=\\!1,2,3$ . We note that the massive KK gauge boson
amplitude (15) obtained by the shifting method just corresponds to the 5d
toroidal compactification without orbifold.

Then, we consider the 5d compactification under $S^{1}\\!/\mathbb{Z}_{2}\hskip
0.85358pt$ orbifold. The KK gauge boson $A_{\lambda}^{a\hskip
0.85358ptn}\\!=\zeta^{\mu}_{\lambda}A^{an}_{\mu}$ is even under
$\mathbb{Z}_{2}$ and is defined as

$\displaystyle A_{\lambda}^{a\hskip
0.85358ptn}\,=\,\frac{1}{\sqrt{2\,}\,}\left[A_{\lambda}^{a(+n)}\\!+A_{\lambda}^{a(-n)}\right]\\!.$
(16)

Also a $\mathbb{Z}_{2}$-odd state $A_{\lambda(-)}^{a\hskip 0.85358ptn}$ can be
defined by flipping the plus sign in the center of the brackets of Eq.(16)
into minus sign. Thus, we can derive the $N$-point KK gauge boson scattering
amplitude as follows:

$\mathcal{T}\big{[}A_{\lambda_{1}}^{a\hskip
0.85358ptn_{1}}A_{\lambda_{2}}^{b\hskip 0.85358ptn_{2}}\cdots
A_{\lambda_{N}}^{a_{N}\hskip
0.85358ptn_{N}}\big{]}=\frac{~{}g^{N-2}~{}}{2^{{\scriptsize{\hbox
to0.0pt{\hskip
1.66495pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}}}}/2}}\sum_{\mathsf{P}}\sum_{j}\frac{\mathcal{C}_{j}\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{P}}(\lambda)}{~{}\llbracket\mathcal{D}_{j}-M_{\mathfrak{n}\mathfrak{n}_{j}}^{2}\rrbracket~{}}\hskip
0.85358pt,\hskip 17.07164pt$ (17)

where  $\overline{\hbox{}}$$N$ denotes the number of the external KK states
(with $n_{j}\\!\neq\\!0$) and the difference $(N-\hbox to0.0pt{\hskip
4.94992pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}})$
equals the number of possible external zero-mode states.

Next, using the color-kinematics duality of the BCJ double-copy method, we can
construct the $N$-point 5d massless graviton amplitude from the corresponding
$N$-point massless gauge boson amplitude (13):

$\widehat{\mathcal{M}}\big{[}\hat{h}_{\hat{\sigma}_{1}}\hat{h}_{\hat{\sigma}_{2}}\cdots\hat{h}_{\hat{\sigma}_{N}}\big{]}=(-)^{N+1}\\!\left(\frac{\hat{\kappa}}{4}\right)^{\\!N-2}\sum_{j}\\!\\!\sum_{\hat{\lambda}_{k},\hat{\lambda}^{\prime}_{k}}\\!\\!\Big{(}\\!\prod_{k}\\!C_{\hat{\lambda}_{k}\hat{\lambda}^{\prime}_{k}}^{\hskip
0.85358pt\hat{\sigma}_{k}}\Big{)}\frac{~{}\widehat{\mathcal{N}}_{j}(\hat{\lambda}_{k})\hskip
0.85358pt\widehat{\mathcal{N}}_{j}(\hat{\lambda}^{\prime}_{k})~{}}{\llbracket\widehat{\mathcal{D}}_{j}\rrbracket}\,,$
(18)

where the helicity index $\,\hat{\sigma}_{k}\\!=\\!\\{\pm 2,\pm 1,0\\}\,$
labels the five helicity states of each external 5d massless graviton. The 5d
polarization tensor of the $k$-th external graviton state is given by the
following formulas:

$\hat{\zeta}^{\mu\nu}_{\hskip 0.85358pt\hat{\sigma}_{k}}\hskip
0.85358pt=\sum_{\hat{\lambda}_{k},\hat{\lambda}^{\prime}_{k}}\\!\\!C_{\hat{\lambda}_{k}\hat{\lambda}^{\prime}_{k}}^{\hskip
0.85358pt\hat{\sigma}_{k}}\hskip
0.85358pt\hat{\zeta}_{\hat{\lambda}_{k}}^{\mu}\hskip
0.85358pt\hat{\zeta}_{\hat{\lambda}^{\prime}_{k}}^{\nu}\,,$ (19)

where on the right-hand-side summations over the repeated helicity indices
$(\hat{\lambda}_{k},\hat{\lambda}^{\prime}_{k})$ are implied. In Eq.(19), the
normalization coefficients in each polarization tensor of graviton are defined
as follows:

$\displaystyle\hat{h}_{\pm 2}=\hat{\zeta}^{\mu\nu}_{\pm 2}\hskip
0.85358pt\hat{h}_{\mu\nu}\\!:$ $\displaystyle\hskip
11.38109ptC_{\hat{\lambda}_{k}\hat{\lambda}^{\prime}_{k}}^{\hskip
0.85358pt\hat{\sigma}_{k}}\\!=C_{\pm 1,\pm 1}^{\pm 2}=1\hskip 0.85358pt,$
(20a) $\displaystyle\hat{h}_{\pm 1}=\hat{\zeta}^{\mu\nu}_{\pm 1}\hskip
0.85358pt\hat{h}_{\mu\nu}\\!:$ $\displaystyle\hskip
11.38109ptC_{\hat{\lambda}_{k}\hat{\lambda}^{\prime}_{k}}^{\hskip
0.85358pt\hat{\sigma}_{k}}\\!=C_{\pm 1,0}^{\pm 1}=C_{0,\pm 1}^{\pm
1}=\sqrt{\mbox{$\frac{\,{1}\,}{2}$}\hskip 0.85358pt}\hskip 0.85358pt,$ (20b)
$\displaystyle\hat{h}_{0}=\hat{\zeta}^{\mu\nu}_{0}\hskip
0.85358pt\hat{h}_{\mu\nu}\\!:$ $\displaystyle\hskip
11.38109ptC_{\hat{\lambda}_{k}\hat{\lambda}^{\prime}_{k}}^{\hskip
0.85358pt\hat{\sigma}_{k}}\\!=C_{\pm 1,\mp 1}^{\hskip
0.85358pt0}=\\!\sqrt{\mbox{$\frac{\,{1}\,}{6}$}\hskip 0.85358pt}\hskip
0.85358pt,~{}~{}~{}C_{0,0}^{\hskip
0.85358pt0}=\\!\sqrt{\mbox{$\frac{\,{2}\,}{3}$}\hskip 0.85358pt}\hskip
0.85358pt,$ (20c)

where the polarization tensors of the 5d massless graviton are defined by
“doubling up” the three transverse polarization vectors of the 5d massless
gauge boson via

$\displaystyle\hat{\zeta}^{\mu\nu}_{\pm 2}$ $\displaystyle\hskip
0.85358pt=\hskip 0.85358pt\hat{\zeta}_{\pm 1}^{\mu}\hat{\zeta}_{\pm
1}^{\nu}\,,\quad\hat{\zeta}_{\pm 1}^{\mu\nu}\hskip
0.85358pt=\mbox{$\frac{\,{1}\,}{\sqrt{2\,}\,}$}\\!\left(\hat{\zeta}_{\pm
1}^{\mu}\hat{\zeta}_{0}^{\nu}+\hat{\zeta}_{0}^{\mu}\hat{\zeta}_{\pm
1}^{\nu}\right)\\!,$ (21a) $\displaystyle\hat{\zeta}^{\mu\nu}_{0}$
$\displaystyle=\mbox{$\frac{\,{1}\,}{\sqrt{6\,}\,}$}\\!\left(\hat{\zeta}^{\mu}_{+1}\hat{\zeta}^{\nu}_{-1}+\hat{\zeta}^{\mu}_{-1}\hat{\zeta}^{\nu}_{+1}+2\hskip
0.85358pt\hat{\zeta}^{\mu}_{0}\hat{\zeta}^{\nu}_{0}\right)\\!.$ (21b)

In the double-copy formula (18), we have made the following replacement
between the gauge couplings and gravitational couplings KKString :

${\hat{g}^{N-2}}\,\longrightarrow~{}(-)^{N+1}(\hat{\kappa}/4)^{N-2}\,.$ (22)

With the 5d compactification we have relations between the 5d and 4d
couplings, $(\hat{g},\hskip 0.85358pt\hat{\kappa})=(g,\hskip
0.85358pt\kappa)\sqrt{L\hskip 0.85358pt}\hskip 0.85358pt$, where $L$ denotes
the length of 5d.

Then, under the $S^{1}$ compactification, we can use our shifting method in
Eq.(12) to derive the 4d $N$-point massive KK graviton scattering amplitude
from the corresponding double-copied 5d massless graviton amplitude (18) as
follows:

$\mathcal{M}\big{[}h_{\sigma_{1}}^{\mathfrak{n}_{1}}h_{\sigma_{2}}^{\mathfrak{n}_{2}}\cdots
h_{\sigma_{N}}^{\mathfrak{n}_{N}}\big{]}=\,(-)^{N+1}\\!\left(\\!\frac{\hskip
0.85358pt\kappa\,}{4}\\!\right)^{\\!\\!N-2}\\!\\!\sum_{j}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\\!\\!\Big{(}\\!\prod_{k}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\Big{)}\frac{~{}\mathcal{N}_{j}^{\mathsf{P}}(\lambda_{k})\hskip
0.85358pt\mathcal{N}^{\hskip
0.85358pt\mathsf{P}}_{j}(\lambda^{\prime}_{k})~{}}{\llbracket\mathcal{D}_{j}\\!-\\!M_{\mathfrak{n}\mathfrak{n}_{j}}^{2}\rrbracket}\hskip
0.85358pt,$ (23)

where the helicity index $\,\sigma_{k}\\!=\\!\\{\pm 2,\pm
1,0\\}\\!\equiv\\!\\{\pm 2,\pm 1,L\\}\,$ labels the five helicity states of
each external 4d massive KK graviton. Equivalently, we can also construct the
above $N$-point massive KK graviton scattering amplitude by directly applying
the color-kinematics duality to the corresponding $N$-point massive KK gauge
boson amplitude (15). In Eq.(23), the polarization tensors of the external KK
graviton states are defined as

$\displaystyle\zeta^{\mu\nu}_{\hskip
0.85358pt\sigma_{k}}=\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\hskip 0.85358pt\zeta_{\lambda_{k}}^{\mu}\hskip
0.85358pt\zeta_{\lambda^{\prime}_{k}}^{\nu}\,,$ (24a) $\displaystyle C_{\pm
1,\pm 1}^{\pm 2}\\!=1\hskip 0.85358pt,~{}~{}C_{\pm 1,0}^{\pm 1}=C_{0,\pm
1}^{\pm 1}=\sqrt{\mbox{$\frac{\,{1}\,}{2}$}\hskip 0.85358pt}\hskip
0.85358pt,~{}~{}C_{\pm 1,\mp 1}^{\hskip
0.85358pt0}=\\!\sqrt{\mbox{$\frac{\,{1}\,}{6}$}\hskip 0.85358pt}\hskip
0.85358pt,~{}~{}C_{0,0}^{\hskip
0.85358pt0}=\\!\sqrt{\mbox{$\frac{\,{2}\,}{3}$}\hskip 0.85358pt}\hskip
0.85358pt.$ (24b)

Thus, the polarization tensors of a massive KK graviton take the following
forms:

$\displaystyle\zeta^{\mu\nu}_{\pm 2}$ $\displaystyle=\hskip
0.85358pt\zeta_{\pm 1}^{\mu}\zeta_{\pm 1}^{\nu}\hskip
0.85358pt,\quad\zeta_{\pm
1}^{\mu\nu}=\mbox{$\frac{\,{1}\,}{\sqrt{2\,}\,}$}\big{(}\zeta_{\pm
1}^{\mu}\zeta_{L}^{\nu}\\!+\zeta_{L}^{\mu}\zeta_{\pm 1}^{\nu}\big{)}\hskip
0.85358pt,$ (25a) $\displaystyle\zeta^{\mu\nu}_{L}$
$\displaystyle=\mbox{$\frac{\,{1}\,}{\sqrt{6\,}\,}$}\big{(}\zeta^{\mu}_{+1}\zeta^{\nu}_{-1}+\zeta^{\mu}_{-1}\zeta^{\nu}_{+1}+2\hskip
0.85358pt\zeta^{\mu}_{L}\zeta^{\nu}_{L}\big{)}\hskip 0.85358pt,$ (25b)

where we denote $\hskip 0.85358pt\zeta^{\mu}_{L}\equiv\zeta_{\hskip
0.85358pt0}^{\mu}\hskip 0.85358pt$ and $\hskip
0.85358pt\zeta^{\mu\nu}_{L}\equiv\zeta_{\hskip 0.85358pt0}^{\mu\nu}$.

Finally, for the KK compactification under the orbifold
$S^{1}\\!/\mathbb{Z}_{2}$, we can apply the color-kinematics duality to the
corresponding $N$-point massive KK gauge boson amplitude (17) and derive the
following $N$-point massive KK graviton scattering amplitude:

$\mathcal{M}\big{[}h_{\sigma_{1}}^{n_{1}}h_{\sigma_{2}}^{n_{2}}\ldots
h_{\sigma_{N}}^{n_{N}}\big{]}=\frac{\,(-)^{N+1}}{2^{{\scriptsize{\hbox
to0.0pt{\hskip
1.66495pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}}}}/2}}\\!\left(\\!\frac{\hskip
0.85358pt\kappa\,}{4}\\!\right)^{\\!\\!N-2}\\!\\!\sum_{\mathsf{P}}\\!\sum_{j}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\\!\Big{(}\\!\prod_{k}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\,\sigma_{k}}\Big{)}\frac{~{}\mathcal{N}_{j}^{\mathsf{P}}(\lambda_{k})\mathcal{N}^{\hskip
0.85358pt\mathsf{P}}_{j}(\lambda^{\prime}_{k})~{}}{\llbracket\mathcal{D}_{j}\\!-\\!M_{\mathfrak{n}\mathfrak{n}_{j}}^{2}\rrbracket}\,,$
(26)

where  $\overline{\hbox{}}$$N$ denotes the number of $\mathbb{Z}_{2}$-even
external KK states and the $(N\\!-\\!\hbox to0.0pt{\hskip
4.94992pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}})$
equals the number of possible external zero-mode states.

##  3 Extended Color-Kinematics Duality for Massive KK Double-Copy

In the previous section, we have established the general correspondence
between the massless scattering amplitudes in the noncompactified higher
dimensional theory and the massive KK scattering amplitudes in the
compactified 4d KK theory. With these, we present systematically in this
section the explicit massive double-copy construction for the four-point and
five-point scattering amplitudes of the massive KK gauge bosons and KK
gravitons. In Section 3.1, we present the general four-point massive KK gauge
boson amplitudes and the double-copied KK graviton amplitudes from extending
the corresponding massless 5d scattering amplitudes. Imposing the generalized
massive gauge invariance on the double-copied KK graviton amplitudes, we will
derive a mass spectral condition. Then, we will derive the same mass spectral
condition by using the massive fundamental BCJ relation. In Section 3.2, we
present the explicit double-copy constructions of the four-point KK graviton
amplitudes from the corresponding KK gauge boson amplitudes under the 5d
toroidal compactification of $S^{1}$. Then, in Section 3.3 we extend this
analysis to the case of the 5d orbifold compactification of
$S^{1}\\!/\mathbb{Z}_{2}$. In Section 3.4, we further present the five-point
KK graviton scattering amplitudes from double-copy. Finally, in Section 5 we
will derive the nonrelativistic KK scattering amplitudes for the heavy KK
gauge bosons, heavy KK gravitons, and heavy KK scalars respectively.

###  3.1 KK Amplitudes and Double-Copy under Toroidal Compactification

Consider the massless gauge boson amplitude at tree level in the 5d Yang-Mills
theory. The four-point scattering amplitudes can be generally expressed as the
sum of the three kinematic channels with distinct color structures,
corresponding to the three pole-diagrams plus the contributions of the contact
diagram (which are absorbed into the pole terms):

$\widehat{\mathcal{T}}\\!\left[\widehat{A}^{a}_{\hat{\lambda}_{1}}\widehat{A}^{b}_{\hat{\lambda}_{2}}\widehat{A}^{c}_{\hat{\lambda}_{3}}\widehat{A}^{d}_{\hat{\lambda}_{4}}\right]\,=\,\hat{g}^{2}\sum_{j}\frac{~{}\mathcal{C}_{j}\hskip
0.85358pt\widehat{\mathcal{N}}_{j}(\hat{\lambda})~{}}{\hat{s}_{j}}\,.$ (27)

On the left side of Eq.(27), the subscript $\,\hat{\lambda}_{i}\\!=\\!\\{\pm
1,0\\}\,$ denotes the three transverse helicity states for each 5d gauge
boson, whereas on the right side the subscript $\,j\\!\in\\!\\{s,t,u\\}\,$
runs over the three kinematic channels and $\,\hat{g}\,$ is the 5d gauge
coupling constant. In the numerator of Eq.(27), the color factors
$\mathcal{C}_{j}$ are defined as follows:

$(\mathcal{C}_{s},\hskip 0.85358pt\mathcal{C}_{t},\hskip
0.85358pt\mathcal{C}_{u})\hskip 0.85358pt=\hskip
0.85358pt(f^{abe}f^{cde},\hskip 0.85358ptf^{ade}f^{bce},\hskip
0.85358ptf^{ace}f^{dbe})\hskip 0.85358pt,$ (28)

and the kinematic factors $\widehat{\mathcal{N}}_{j}$ are given in Eq.(202) of
Appendix B. They obey the color and kinematic Jacobi identities respectively
BCJ:2008 BCJ:2019 :

$\mathcal{C}_{s}+\mathcal{C}_{t}+\mathcal{C}_{u}=0\,,\hskip
22.76219pt\widehat{\mathcal{N}}_{s}+\widehat{\mathcal{N}}_{t}+\widehat{\mathcal{N}}_{u}=0\,.$
(29)

In the denominator of Eq.(27), $\hat{s}_{j}$ denotes the 5d Mandelstam
variables and has the correspondence with the general denominator of the
$N$-point amplitude (13):
$\llbracket\widehat{\mathcal{D}}_{j}\rrbracket\\!\rightarrow\\!\hat{s}_{j}\,$,
their definitions in the center-of-mass frame are given by Eq.(192) of
Appendix A. We note that because of the color Jacobi identity as given by the
first formula of Eq.(29), the gauge boson amplitude (27) is invariant under
the following generalized gauge transformations BCJ:2008 :

$\widehat{\mathcal{N}}_{j}^{\hskip
0.85358pt\prime}\,=\,\widehat{\mathcal{N}}_{j}+\hat{s}_{j}\\!\times\\!\widehat{\Delta}\,,$
(30)

where the gauge parameter $\widehat{\Delta}\hskip 0.85358pt$ is an arbitrary
local function of kinematic variables.

Then, we compactify the 5d Yang-Mills theory on a flat $S^{1}$ space with the
fifth coordinate $\,0\\!\leqslant y\leqslant\\!2\pi R$ . Under such
compactification, the 5d massless gauge bosons $\,\widehat{A}^{a}_{M}\,$
acquire masses through the geometric Higgs mechanism Hang:2021fmp 5DYM2002
and result in a tower of massive KK gauge boson states, whereas the zero-mode
gauge bosons remain massless. As generally shown in Eq.(15), we can further
derive the 4d massive KK gauge boson scattering amplitude from the
corresponding 5d massless amplitude (27) as follows:

$\mathcal{T}\\!\left[A_{\lambda_{1}}^{a\hskip
0.85358pt\mathfrak{n}_{1}}A_{\lambda_{2}}^{b\hskip
0.85358pt\mathfrak{n}_{2}}A_{\lambda_{3}}^{c\hskip
0.85358pt\mathfrak{n}_{3}}A_{\lambda_{4}}^{d\hskip
0.85358pt\mathfrak{n}_{4}}\right]=\,g^{2}\sum_{j}\frac{~{}\mathcal{C}_{j}\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{P}}(\lambda)~{}}{~{}s_{j}-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}~{}}\,,$
(31)

where on the left side the index $\lambda_{i}\,$ denote the helicities of each
external gauge boson state which corresponds to two transverse polarizations
and one longitudinal polarization, $\lambda_{i}\\!=\\!\\{\pm
1,0\\}\\!\equiv\\!\\{\pm 1,L\\}$. In Eq.(31), the KK number of each external
gauge boson is $\,\mathfrak{n}_{i}\\!=\pm n_{i}\,$ with
$\,n_{i}\\!\in\mathbb{Z}^{+}$, where the sum of the KK numbers of all external
states should satisfy the neutral condition:

$\mathfrak{n}_{1}+\mathfrak{n}_{2}+\mathfrak{n}_{3}+\mathfrak{n}_{4}=0\,.$
(32)

On the right-hand side of Eq.(31), the mass pole is defined as $\hskip
0.85358ptM_{\mathfrak{n}\mathfrak{n}_{j}}\\!\\!=\\!\mathfrak{n}\mathfrak{n}_{j}/R$
, with
$\mathfrak{n}\mathfrak{n}_{j}\\!\in\\!\\{\mathfrak{n}_{1}\\!+\mathfrak{n}_{2},\,\mathfrak{n}_{1}\\!+\mathfrak{n}_{4},\,\mathfrak{n}_{1}\\!+\mathfrak{n}_{3}\\}\,$
and $\,j\\!\\!\in\\!\\!\\{s,t,u\\}$, whereas the symbol $\mathsf{P}$ labels
each allowed combination of the KK numbers of the external gauge bosons
obeying the neutral condition (32). An essential property of the amplitude
(31) is that it contains only the simple poles in the denominator. Thus, by
setting all the internal momenta be on-shell, we can factorize the higher-
point amplitude into the products of lower-point amplitudes at the presence of
simple poles. We further note that the massive KK gauge boson amplitude (31)
is invariant under the following generalized gauge transformation:

$\mathcal{N}_{j}^{\mathsf{P}\hskip
0.85358pt\prime}\,=\,\mathcal{N}_{j}^{\mathsf{P}}+\left(s_{j}-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}\right)\\!\times\\!\Delta\,,$
(33)

where $\Delta$ denotes an arbitrary gauge parameter which is a local function
of kinematic variables.

Finally, using the shifting method of Ref. KKString , we also can directly
derive the massive KK gauge boson scattering amplitude (31) from the
corresponding scattering amplitude of the 4d massless zero-mode gauge bosons:

$\mathcal{T}\big{[}A_{\lambda_{1}}^{a\hskip
0.85358pt\mathfrak{n}_{1}}A_{\lambda_{2}}^{b\hskip
0.85358pt\mathfrak{n}_{2}}A_{\lambda_{3}}^{c\hskip
0.85358pt\mathfrak{n}_{3}}A_{\lambda_{4}}^{d\hskip
0.85358pt\mathfrak{n}_{4}}\big{]}\,=\,\mathcal{T}\big{[}A_{\lambda_{1}}^{a\hskip
0.85358pt0}A_{\lambda_{2}}^{b\hskip 0.85358pt0}A_{\lambda_{3}}^{c\hskip
0.85358pt0}A_{\lambda_{4}}^{d\hskip
0.85358pt0}\big{]}\Big{|}_{\begin{subarray}{c}s_{j}\rightarrow\hskip
0.85358pts_{j}-M_{\mathfrak{n}\mathfrak{n}_{j}}^{2}\end{subarray}},$ (34)

which agress to Eq.(31). We note that the massive KK amplitude (31) is deduced
from the 5d massless gauge boson amplitude (27) by using the shifting method
of Eq.(12), whereas the massive KK amplitude (34) is obtained from the 4d
massless gauge boson amplitude by using of the shifting method of Ref.
KKString . In fact, the two methods of deriving Eq.(31) and Eq.(34) give the
same result and are thus practically equivalent.

Next, we study how to explicitly construct the four-point massive KK graviton
scattering amplitudes. We note that the four-point massless graviton ampliutde
in 5d can be obtained by applying the color-kinematics (CK) duality BCJ:2008
BCJ:2019  to the corresponding 5d massless gauge boson amplitude (27). Because
the color factors and kinematic factors of the gauge boson amplitude (27)
satisfies the Jacobi identities of Eq.(29), this reflects an exchangeability
between those two kinds of factors
$\,\mathcal{C}_{j}\\!\leftrightarrow\\!\widehat{\mathcal{N}}_{j}\hskip
0.85358pt$. Hence, we can replace the color factors by the kinematic factors
and replace gauge coupling $\,\hat{g}\,$ with the gravity coupling
$\,\hat{\kappa}/4\,$ simultaneously, which lead to the following:

$\displaystyle\widehat{\mathcal{M}}\big{[}\hat{h}_{\hat{\sigma}_{1}}\hat{h}_{\hat{\sigma}_{2}}\hat{h}_{\hat{\sigma}_{3}}\hat{h}_{\hat{\sigma}_{4}}\big{]}$
$\displaystyle=\,-\,\widehat{\mathcal{T}}\\!\left[\widehat{A}^{a}_{\hat{\lambda}_{1}}\widehat{A}^{b}_{\hat{\lambda}_{2}}\widehat{A}^{c}_{\hat{\lambda}_{3}}\widehat{A}^{d}_{\hat{\lambda}_{4}}\right]\\!\bigg{|}_{\vspace*{-10mm}\begin{subarray}{c}\mathcal{C}_{j}\,\rightarrow\,\widehat{\mathcal{N}}_{j}\\\
{\hskip 1.42262pt\hat{g}\,\rightarrow\,\hat{\kappa}/4}\\\ {}\\\ {}\\\
{}\end{subarray}}$
$\displaystyle=\,-\frac{~{}\hat{\kappa}^{2}~{}}{16}\sum_{j}\\!\\!\sum_{\hat{\lambda}_{k},\hat{\lambda}^{\prime}_{k}}\\!\\!\\!\Big{(}\\!\prod_{k=1}^{4}\\!C^{\hskip
0.85358pt\hat{\sigma}_{k}}_{\hat{\lambda}_{k}\hat{\lambda}^{\prime}_{k}}\Big{)}\frac{~{}\widehat{\mathcal{N}}_{j}(\hat{\lambda}_{k})\hskip
0.85358pt\widehat{\mathcal{N}}_{j}(\hat{\lambda}^{\prime}_{k})~{}}{\hat{s}_{j}}\,,$
(35)

where helicity index $\,\hat{\sigma}_{k}=\\{\pm 2,\pm 1,0\\}\,$ denotes the
five helicity states for each 5d massless graviton. Here the overall minus
sign on the right-hand side of Eq.(35) comes from the contour integral on the
complex plane for the closed string string KLT Bjerrum-Bohr:2010pnr . In the
above, the polarization tensor of each external graviton state is defined in
Eqs.(19)-(20).

Then, we apply the generalized gauge transformation (30) to the double-copied
5d massless graviton scattering amplitude (35) and require it to be gauge-
invariant. This leads to the conditions $\hskip
0.85358pt\sum_{j}\widehat{\mathcal{N}}_{j}\\!=\\!0\hskip 0.85358pt$ and
$\hskip 0.85358pt\sum_{j}\hat{s}_{j}\\!=0\hskip 0.85358pt$ (with
$j\\!=\\!s,t,u$), where the first condition is just the kinematic Jacobi
identity in Eq.(29), and the second condition always holds for the massless
external states as shown by the first formula of Eq.(200).

In addition, we can substitute the numerators of Eq.(202) into Eq.(35) and re-
express the four-point graviton scattering amplitude (35) in a Lorentz-
invariant form which is a function of all the relevant kinematic variables:

$\widehat{\mathcal{M}}\big{[}\hat{h}_{\hat{\sigma}_{1}}\hat{h}_{\hat{\sigma}_{2}}\hat{h}_{\hat{\sigma}_{3}}\hat{h}_{\hat{\sigma}_{4}}\big{]}=\frac{\,\hat{\kappa}^{2}\,}{\,\hat{s}\hskip
0.85358pt\hat{t}\hskip
0.85358pt\hat{u}~{}}\\!\\!\sum_{\hat{\lambda}_{k},\hat{\lambda}^{\prime}_{k}}\\!\\!\\!\Big{(}\\!\prod_{k=1}^{4}\\!C^{\hskip
0.85358pt\hat{\sigma}_{k}}_{\hat{\lambda}_{k}\hat{\lambda}^{\prime}_{k}}\Big{)}K_{1234}\\!\times\\!K_{1^{\prime}2^{\prime}3^{\prime}4^{\prime}}\hskip
0.85358pt,$ (36)

where the factor $K_{1234}\equiv
K_{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}$ is given by

$\displaystyle K=$ $\displaystyle\ \hat{s}\hskip 0.85358pt\hat{t}\hskip
0.85358pt(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{3})\hskip
0.85358pt(\hat{\zeta}_{2}\\!\cdot\hat{\zeta}_{4})+\hat{s}\hskip
0.85358pt\hat{u}\hskip 0.85358pt(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{4})\hskip
0.85358pt(\hat{\zeta}_{2}\\!\cdot\hat{\zeta}_{3})+\hat{t}\hskip
0.85358pt\hat{u}\hskip 0.85358pt(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{2})\hskip
0.85358pt(\hat{\zeta}_{3}\\!\cdot\hat{\zeta}_{4})$ $\displaystyle\ -2\hskip
0.85358pt\hat{s}\,[(\hat{p}_{1}\\!\cdot\hat{\zeta}_{4})(\hat{p}_{3}\\!\cdot\hat{\zeta}_{2})(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{3})+(\hat{p}_{1}\\!\cdot\hat{\zeta}_{3})(\hat{p}_{4}\\!\cdot\hat{\zeta}_{2})(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{4})+(\hat{p}_{2}\\!\cdot\hat{\zeta}_{3})(\hat{p}_{4}\\!\cdot\hat{\zeta}_{1})(\hat{\zeta}_{2}\\!\cdot\hat{\zeta}_{4})$
$\displaystyle\
+(\hat{p}_{2}\\!\cdot\hat{\zeta}_{4})(\hat{p}_{3}\\!\cdot\hat{\zeta}_{1})(\hat{\zeta}_{2}\\!\cdot\hat{\zeta}_{3})]-2\hskip
0.85358pt\hat{t}\,[(\hat{p}_{1}\\!\cdot\hat{\zeta}_{2})(\hat{p}_{3}\\!\cdot\hat{\zeta}_{4})(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{3})+(\hat{p}_{1}\\!\cdot\hat{\zeta}_{3})(\hat{p}_{2}\\!\cdot\hat{\zeta}_{4})(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{2})$
$\displaystyle\
+(\hat{p}_{2}\\!\cdot\hat{\zeta}_{1})(\hat{p}_{4}\\!\cdot\hat{\zeta}_{3})(\hat{\zeta}_{2}\\!\cdot\hat{\zeta}_{4})+(\hat{p}_{3}\\!\cdot\hat{\zeta}_{1})(\hat{p}_{4}\\!\cdot\hat{\zeta}_{2})(\hat{\zeta}_{3}\\!\cdot\hat{\zeta}_{4})]-2\hskip
0.85358pt\hat{u}\big{[}(\hat{p}_{1}\\!\cdot\hat{\zeta}_{2})(\hat{p}_{4}\\!\cdot\hat{\zeta}_{3})(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{4})$
$\displaystyle\
+(\hat{p}_{1}\\!\cdot\hat{\zeta}_{4})(\hat{p}_{2}\\!\cdot\hat{\zeta}_{3})(\hat{\zeta}_{1}\\!\cdot\hat{\zeta}_{2})+(\hat{p}_{3}\\!\cdot\hat{\zeta}_{2})(\hat{p}_{4}\\!\cdot\hat{\zeta}_{1})(\hat{\zeta}_{3}\\!\cdot\hat{\zeta}_{4})+(\hat{p}_{3}\\!\cdot\hat{\zeta}_{4})(\hat{p}_{2}\\!\cdot\hat{\zeta}_{1})(\hat{\zeta}_{2}\\!\cdot\hat{\zeta}_{3})\big{]}\hskip
0.85358pt,$ (37)

and the factor $K_{1^{\prime}2^{\prime}3^{\prime}4^{\prime}}\equiv
K_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}\lambda_{3}^{\prime}\lambda_{4}^{\prime}}$
takes the same form as Eq.(3.1). As can be checked, the four-point 5d massless
graviton scattering amplitude given by Eq.(35) or Eqs.(36)-(3.1) agrees with
that of the direct Feynman diagram calculation. It also agrees with the field
theory limit of the four-point genus-zero closed string amplitude Schwarz:1982
.

Then, under the toroidal compactification of $S^{1}$ and from the $N$-point
massive double-copy formula (23), we can deduce the four-point massive KK
graviton scattering amplitude as follows:

$\mathcal{M}\\!\left[h_{\sigma_{1}}^{\mathfrak{n}_{1}}h_{\sigma_{2}}^{\mathfrak{n}_{2}}h_{\sigma_{3}}^{\mathfrak{n}_{3}}h_{\sigma_{4}}^{\mathfrak{n}_{4}}\right]=\,-\frac{~{}\kappa^{2}~{}}{16}\sum_{j}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\\!\\!\Big{(}\\!\prod_{k=1}^{4}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\Big{)}\frac{~{}\mathcal{N}_{j}^{\mathsf{P}}(\lambda_{k})\hskip
0.85358pt\mathcal{N}^{\mathsf{P}}_{j}(\lambda^{\prime}_{k})~{}}{s_{j}-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}}\,,$
(38)

where $\\{C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip 0.85358pt\sigma_{k}}\\}$
denote the coefficients of the polarization tensor of the $k$-th external KK
graviton state as given by Eqs.(24)-(25). We can apply the generalized gauge
transformation (33) to the double-copied KK graviton scattering amplitude (38)
and require it to be gauge-invariant. From this we deduce the following two
conditions:

$\displaystyle\sum_{j}\mathcal{N}_{j}^{\mathsf{P}}=0\hskip
0.85358pt,~{}~{}~{}~{}~{}\sum_{j}\\!\left(s_{j}-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}\right)=0\hskip
0.85358pt.$ (39)

where $j\\!=\\!s,t,u\hskip 0.85358pt$. The first condition in Eq.(39) is the
massive kinematic Jacobi identity for the numerators of the four-point KK
gauge boson amplitude (31) and can be inferred from the massless kinematic
Jacobi identity in the second formula of Eq.(29) by using the shifting method
of Sec. 2.2. The second condition in Eq.(39) is also important and can be
derived from the 5d massless kinematic condition $\hskip
0.85358pt\sum_{j}\hat{s}_{j}\\!=0\hskip 0.85358pt$ in Eq.(200) by using the
shifting formula (12a). Thus, according to Eq.(200), we have the sum of
Mandelstam variables for the four KK gauge boson scattering, $\hskip
0.85358pt\sum_{j}s_{j}\\!=\\!\sum_{i=1}^{4}\\!M_{\mathfrak{n}_{i}}^{2}\hskip
0.85358pt$. Hence, from the second condition in Eq.(39), we derive the
following four-point mass spectral condition:

$\sum_{i=1}^{4}\\!M_{\mathfrak{n}_{i}}^{2}=\hskip
0.85358ptM^{2}_{\mathfrak{n}\mathfrak{n}_{s}}\\!+M^{2}_{\mathfrak{n}\mathfrak{n}_{t}}\\!+M^{2}_{\mathfrak{n}\mathfrak{n}_{u}}\hskip
0.85358pt,$ (40)

where the left-hand side sums up the squared-mass $M_{\mathfrak{n}_{i}}^{2}$
of each external KK state and the right-hand side is the sum of the internal
pole-mass-squared of all three kinematic channels. This condition requires
that the sum of the squared-masses of all the external KK states equals the
sum of the internal squared-mass-poles in the $(s,t,u)$ channels. It should
hold for any four-point scattering amplitudes of the KK gauge bosons and of
the KK graviton in the KK gauge/gravity theories under the 5d toroidal $S^{1}$
compactification (without orbifold).

For the 5d toroidal $S^{1}$ compactification, the KK mass $\hskip
0.85358ptM_{\mathfrak{n}_{i}}\\!\\!=\mathfrak{n}_{i}/R\,$ and the KK numbers
are conserved in each interaction vertex. Thus, the pole mass in each of the
$(s,t,u)$ channels is given by

$M_{\mathfrak{n}\mathfrak{n}_{s}}\\!=M_{\mathfrak{n}_{1}\\!+\mathfrak{n}_{2}}\hskip
0.85358pt,~{}~{}~{}~{}M_{\mathfrak{n}\mathfrak{n}_{t}}\\!=M_{\mathfrak{n}_{1}\\!+\mathfrak{n}_{4}}\hskip
0.85358pt,~{}~{}~{}~{}M_{\mathfrak{n}\mathfrak{n}_{u}}\\!=M_{\mathfrak{n}_{1}\\!+\mathfrak{n}_{3}}\hskip
0.85358pt.$ (41)

With these we can re-express the mass spectral condition (40) as follows:

$\mathfrak{n}_{1}^{2}+\mathfrak{n}_{2}^{2}+\mathfrak{n}_{3}^{2}+\mathfrak{n}_{4}^{2}\,=\,(\mathfrak{n}_{1}\\!+\mathfrak{n}_{2})^{2}\\!+(\mathfrak{n}_{1}\\!+\mathfrak{n}_{4})^{2}\\!+(\mathfrak{n}_{1}\\!+\mathfrak{n}_{3})^{2}\hskip
0.85358pt.$ (42)

This relation can be further reduced to
$\mathfrak{n}_{1}\\!+\mathfrak{n}_{2}\\!+\mathfrak{n}_{3}\\!+\mathfrak{n}_{4}\\!=\\!0\hskip
0.85358pt$, which is just the same as the neutral condition (32) and is thus
guaranteed by the KK number conservation under the toroidal compactification
of $S^{1}$. The neutral condition (32) is again a direct outcome of the
conservation of KK indices under the 5d toroidal $S^{1}$ compactification.
Hence, the mass spectral condition (40) does hold for the 5d toroidal $S^{1}$
compactification.

Regarding the mass spectral condition (40) we have some comments in order.
First, we stress that Eq.(40) is both a necessary and sufficient condition for
directly realizing the massive double-copy construction in any KK
gauge/gravity theories under the 5d toroidal compactification. As we will
explicitly demonstrate in Sec. 3.3, under the 5d compactification with
$S^{1}\\!/\mathbb{Z}_{2}$ orbifold, the condition (40) is violated for the
four-point KK scattering amplitudes and the direct double-copy is impossible.
Instead we could realize the exact double-copy by decomposing each KK graviton
amplitude into a sum of the partial KK graviton amplitudes which are
constructed from the double-copy of the KK gauge boson amplitudes under the
$S^{1}$ compactification without $\hskip 0.85358pt\mathbb{Z}_{2}\\!$ orbifold.
Second, for the gauge/gravity theories other than the KK gauge/gravity
theories under the toroidal compactification, the BCJ-type double-copy could
be realized even without obeying the mass spectral condition (40). An
important example is the double-copy construction for the 3d topological
Chern-Simons gauge/gravity theories TMG , where we find that the BCJ-type
color-kinematics duality and massive double-copy can be realized Hang:2021oso
, but the spectral condition (40) is not obeyed because all physical gauge
bosons (gravitons) in the pure non-Abelian Chern-Simons gauge (gravity)
theories have the same mass $M$ and thus the condition (40) becomes $\hskip
0.85358pt4M^{2}\\!\neq\\!3M^{2}$. Finally, we will further analyze the mass
spectral condition (40) in Section 5 and demonstrate that without assuming an
underlying theory a priori, solving this spectral condition (40) can uniquely
determine the mass spectrum to be that of the KK theories under the toroidal
compactification.

In passing, from the KK gauge boson amplitude (31) and the KK graviton
amplitude (38), we can further derive the double-copy formulation under the
high energy expansion of
$\,s_{j}\\!\gg\\!M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}$. We expand the KK gauge
boson amplitude (31) as follows:

$\displaystyle\mathcal{T}\big{[}A_{\lambda_{1}}^{a\hskip
0.85358pt\mathfrak{n}_{1}}A_{\lambda_{2}}^{b\hskip
0.85358pt\mathfrak{n}_{2}}A_{\lambda_{3}}^{c\hskip
0.85358pt\mathfrak{n}_{3}}A_{\lambda_{4}}^{d\hskip
0.85358pt\mathfrak{n}_{4}}\big{]}$ $\displaystyle=\hskip
0.85358ptg^{2}\sum_{j}\\!\frac{\,\mathcal{C}_{j}\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{P}}(\lambda)\,}{\,s_{j}\\!-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}\,}=g^{2}\sum_{j}\\!\frac{\,\mathcal{C}_{j}\big{[}\mathcal{N}_{j}^{0,\mathsf{P}}(\lambda)\\!+\\!\delta\mathcal{N}_{j}^{\mathsf{P}}(\lambda)\big{]}\,}{\,s_{j}\,}=\mathcal{T}_{0}+\delta\mathcal{T}\hskip
0.85358pt,$
$\displaystyle\frac{\mathcal{N}_{j}^{\mathsf{P}}(\lambda)}{~{}1\\!-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}/s_{j}\,}$
$\displaystyle=\mathcal{N}_{j}^{0,\mathsf{P}}(\lambda)+\delta\mathcal{N}_{j}^{\mathsf{P}}(\lambda)\hskip
0.85358pt.$ (43b)

Then, we can expand the double-copied four-point KK graviton scattering
amplitude (38) for
$h_{\sigma_{1}}^{\mathfrak{n}_{1}}h_{\sigma_{2}}^{\mathfrak{n}_{2}}\\!\rightarrow\\!h_{\sigma_{3}}^{\mathfrak{n}_{3}}h_{\sigma_{4}}^{\mathfrak{n}_{4}}$
as follows:

$\displaystyle\mathcal{M}$ $\displaystyle=-\frac{\,\kappa^{2}\hskip
0.85358pt}{16}\\!\sum_{j}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\prod_{k=1}^{4}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\\!\frac{\,1\\!-\\!M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}/s_{j}\,}{\,s_{j}\,}\Big{[}\mathcal{N}_{j}^{0,\mathsf{P}}(\lambda_{k})\\!+\delta\mathcal{N}^{\mathsf{P}}_{j}(\lambda_{k})\Big{]}\\!\Big{[}\mathcal{N}_{j}^{0,\mathsf{P}}(\lambda_{k}^{\prime})\\!+\delta\mathcal{N}^{\mathsf{P}}_{j}(\lambda_{k}^{\prime})\Big{]}$
$\displaystyle=\,\mathcal{M}_{0}+\delta\mathcal{M}\,,$ (44)

where $\mathcal{M}_{0}$ and $\delta\mathcal{M}$ denote the expanded KK
graviton amplitudes at the leading order (LO) and next-to-leading order (NLO)
respectively,

$\displaystyle\mathcal{M}_{0}$
$\displaystyle=-\frac{\kappa^{2}}{\,16\,}\sum_{j}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\prod_{k}C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\frac{\,\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda_{k})\hskip
0.85358pt\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda^{\prime}_{k})~{}}{s_{j}}\,,$
(45a) $\displaystyle\delta\mathcal{M}$
$\displaystyle=-\frac{\kappa^{2}}{\,16\,}\\!\sum_{j}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\\!\prod_{k}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\\!\frac{\,\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda_{k})\delta\mathcal{N}_{j}^{\mathsf{P}}\\!(\lambda^{\prime}_{k})\\!+\\!\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda^{\prime}_{k})\delta\mathcal{N}_{j}^{\mathsf{P}}\\!(\lambda_{k})\\!-\\!\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda_{k})\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda^{\prime}_{k})M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}/s_{j}\,}{s_{j}}.$
(45b)

Using the general power counting method for the KK theories Hang:2021fmp , we
can estimate that the above four-point LO KK graviton amplitude $\hskip
0.85358pt\mathcal{M}_{0}\\!=\\!O(E^{2}M_{n}^{0})\hskip 0.85358pt$ and the NLO
KK graviton amplitude $\delta\mathcal{M}\\!=\\!O(E^{0}M_{n}^{2})\hskip
0.85358pt$. The above expanded massive double-copy formulas under the high
energy expansion are important for the formulation of the KK gravitational
equivalence theorem (GRET) Hang:2021fmp  which quantitatively connects the
scattering amplitudes of the longitudinal KK gravitons to the scattering
amplitudes of the corresponding KK Goldstone bosons in the high energy limit.

We derived the extended KLT relations for the massive KK open/closed string
amplitudes in Ref. KKString  by using the compactified bosonic string theory.
We note that this is closely related to the massive KK BCJ construction as
generally discussed in the previous section. For the four KK graviton
scattering, we use the massive KLT relation KKString  to express the tree-
level KK graviton scattering amplitude as the products of the two color-
ordered KK gauge boson amplitudes:

$\mathcal{M}\\!\left[h_{\sigma_{1}}^{\mathfrak{n}_{1}}h_{\sigma_{2}}^{\mathfrak{n}_{2}}h_{\sigma_{3}}^{\mathfrak{n}_{3}}h_{\sigma_{4}}^{\mathfrak{n}_{4}}\right]\hskip
0.85358pt=\hskip
0.85358pt\frac{~{}\kappa^{2}~{}}{16}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\\!\\!\left(\prod_{k=1}^{4}\\!\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\\!\right)\\!(t-M_{\mathfrak{n}\mathfrak{n}_{t}}^{2})\hskip
0.85358pt\mathcal{T}_{\lambda_{k}}^{\mathsf{P}}[1234]\hskip
1.42262pt\mathcal{T}_{\lambda^{\prime}_{k}}^{\mathsf{P}}[1324]\hskip
0.85358pt,$ (46)

where the superscript $\mathsf{P}$ on the right-hand-side denotes each given
choice of the KK indices of the external KK gravitons
$\\{\mathfrak{n}_{1},\hskip 0.85358pt\mathfrak{n}_{2},\hskip
0.85358pt\mathfrak{n}_{3},\hskip 0.85358pt\mathfrak{n}_{4}\\}$. For the case
of elastic scattering, there are three types of independent combinations of
the KK numbers for the external state, $\\{\mathfrak{n}_{1},\hskip
0.85358pt\mathfrak{n}_{2},\hskip 0.85358pt\mathfrak{n}_{3},\hskip
0.85358pt\mathfrak{n}_{4}\\}\\!=\\!\\{\pm n,\pm n,\mp n,\mp n\\},\\{\pm n,\mp
n,\mp n,\pm n\\},\\{\pm n,\mp n,\pm n,\mp n\\}$. In the above Eq.(46), we have
used the shorthand notations for the color-ordered amplitudes $\hskip
0.85358pt\mathcal{T}[1^{\mathfrak{n}_{1}}_{\lambda_{1}}2^{\mathfrak{n}_{2}}_{\lambda_{2}}3^{\mathfrak{n}_{3}}_{\lambda_{3}}4^{\mathfrak{n}_{4}}_{\lambda_{4}}]\equiv\mathcal{T}_{\lambda_{k}}^{\mathsf{P}}[1234]\,$
and $\hskip
0.85358pt\mathcal{T}[1^{\mathfrak{n}_{1}}_{\lambda_{1}}3^{\mathfrak{n}_{3}}_{\lambda_{3}}2^{\mathfrak{n}_{2}}_{\lambda_{2}}4^{\mathfrak{n}_{4}}_{\lambda_{4}}]\equiv\mathcal{T}_{\lambda_{k}}^{\mathsf{P}}[1324]\hskip
0.85358pt$.

Then, we can express the above two color-ordered KK gauge boson scattering
amplitudes as follows:

$\displaystyle\left(\\!\\!\begin{aligned}
\mathcal{T}^{\mathsf{P}}[1234]\\\\[2.84526pt]
-\mathcal{T}^{\mathsf{P}}[1324]\end{aligned}\right)\\!$
$\displaystyle=\begin{pmatrix}\\!-\frac{1}{\,s-M_{\mathfrak{n}\mathfrak{n}_{s}}^{2}\,}-\frac{1}{\,t-M_{\mathfrak{n}\mathfrak{n}_{t}}^{2}\,}~{}~{}~{}&-\frac{1}{t-M_{\mathfrak{n}\mathfrak{n}_{t}}^{2}}\\\\[5.69054pt]
-\frac{1}{\,t-M_{\mathfrak{n}\mathfrak{n}_{t}}^{2}\,}\\!~{}&-\frac{1}{\,t-M_{\mathfrak{n}\mathfrak{n}_{t}}^{2}\,}-\frac{1}{\,u-M_{\mathfrak{n}\mathfrak{n}_{u}}^{2}\,}\\!\end{pmatrix}\\!\\!\\!\left(\\!\begin{aligned}
\mathcal{N}_{s}^{\mathsf{P}}\\\\[2.84526pt]
\mathcal{N}_{u}^{\mathsf{P}}\end{aligned}\right)$
$\displaystyle\\!\equiv\Theta\\!\left(\\!\begin{aligned}
\mathcal{N}_{s}^{\mathsf{P}}\\\\[2.84526pt]
\mathcal{N}_{u}^{\mathsf{P}}\end{aligned}\right)\\!,$ (47)

where we have used the notations $\hskip
0.85358pt\mathcal{T}[1^{\mathfrak{n}_{1}}_{\lambda_{1}}2^{\mathfrak{n}_{2}}_{\lambda_{2}}3^{\mathfrak{n}_{3}}_{\lambda_{3}}4^{\mathfrak{n}_{4}}_{\lambda_{4}}]\\!\equiv\\!\mathcal{T}^{\mathsf{P}}[1234]\,$
and $\hskip
0.85358pt\mathcal{T}[1^{\mathfrak{n}_{1}}_{\lambda_{1}}3^{\mathfrak{n}_{3}}_{\lambda_{3}}2^{\mathfrak{n}_{2}}_{\lambda_{2}}4^{\mathfrak{n}_{4}}_{\lambda_{4}}]\\!\equiv\\!\mathcal{T}^{\mathsf{P}}[1324]\hskip
0.85358pt$ with the helicity indices suppressed for simplicity. On the right-
hand side of Eq.(47), the kinematic numerators have encoded the helicity
indices of the external KK gauge bosons,
$\mathcal{N}^{\mathsf{P}}_{s}\\!=\\!\mathcal{N}^{\mathsf{P}}_{s}(\lambda)$ and
$\mathcal{N}^{\mathsf{P}}_{u}\\!=\\!\mathcal{N}^{\mathsf{P}}_{u}(\lambda)$. In
the derivation of Eq.(47), we have used a massive fundamental BCJ identity:

$(s-\\!M_{\mathfrak{n}\mathfrak{n}_{s}}^{2})\hskip
0.85358pt\mathcal{T}^{\mathsf{P}}[1234]\,=\,(u-\\!M_{\mathfrak{n}\mathfrak{n}_{u}}^{2})\hskip
0.85358pt\mathcal{T}^{\mathsf{P}}[1324]\hskip 0.85358pt,$ (48)

which is deduced by applying the shifting method to the conventional massless
fundamental BCJ relation $\,s\hskip 1.42262pt\mathcal{T}[1234]=u\hskip
1.42262pt\mathcal{T}[1324]\hskip 0.85358pt$ BCJ:2008  or by taking the field
theory limit of the corresponding relation for the KK open string amplitudes
KKString . Then, we can substitute the color-ordered KK gauge boson amplitudes
in Eq.(47) into Eq.(46), and derive the following BCJ-type KK graviton
scattering amplitude:

$\mathcal{M}\\!\left[h_{\sigma_{1}}^{\mathfrak{n}_{1}}h_{\sigma_{2}}^{\mathfrak{n}_{2}}h_{\sigma_{3}}^{\mathfrak{n}_{3}}h_{\sigma_{4}}^{\mathfrak{n}_{4}}\right]=\,-\frac{~{}\kappa^{2}~{}}{16}\\!\sum_{j}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\\!\\!\left(\prod_{k=1}^{4}\\!\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\\!\right)\\!\frac{~{}\mathcal{N}_{j}^{\mathsf{P}}\\!(\lambda_{k})\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{P}}\\!(\lambda^{\prime}_{k})~{}}{~{}s_{j}-M_{\mathfrak{n}\mathfrak{n}_{j}}^{2}~{}}\,.$
(49)

The above fully agrees with the extended massive BCJ-type double-copy formula
(38) [which was derived from Eq.(23) by using the shifting method]. This full
agreement demonstrates that the massive BCJ-type double-copy formula (49) [or
(38)] can be also derived from the massive KK KLT relation (46) (which was
based on our previous derivation of the KK KLT relations from analyzing the KK
open/closed string amplitudes KKString ). Hence, the two types of the massive
double-copy constructions of the four-point KK graviton amplitudes are shown
to be equivalent at tree level. We will systematically present the explicit
results of the four-point full scattering amplitudes of the massive
longitudinal KK gravitons in Appendix C.

Next, we further analyze the structure of the color-ordered KK gauge boson
amplitudes (47) in connection to the massive fundamental BCJ relation (48).
This relation (48) shows that the two color-ordered KK gauge boson amplitudes
$\mathcal{T}^{\mathsf{P}}[1234]$ and $\mathcal{T}^{\mathsf{P}}[1324]$ are not
independent, so the kernel matrix $\hskip 0.85358pt\Theta$ of Eq.(47) has rank
$1\hskip 0.85358pt$ and leads to a vanishing determinant
$\det\Theta\\!=\\!0\hskip 0.85358pt$. Using Eq.(47), we directly compute this
determinant:

$\det\Theta\,=\,-\frac{~{}s+t+u-(M_{\mathfrak{n}\mathfrak{n}_{s}}^{2}\\!+M_{\mathfrak{n}\mathfrak{n}_{t}}^{2}\\!+M_{\mathfrak{n}\mathfrak{n}_{u}}^{2})~{}}{(s-M_{\mathfrak{n}\mathfrak{n}_{s}}^{2})(t-M_{\mathfrak{n}\mathfrak{n}_{t}}^{2})(u-M_{\mathfrak{n}\mathfrak{n}_{u}}^{2})}\,,$
(50)

which should vanish once we impose the kinematic condition
$s+t+u=M_{\mathfrak{n}\mathfrak{n}_{s}}^{2}\\!+M_{\mathfrak{n}\mathfrak{n}_{t}}^{2}\\!+M_{\mathfrak{n}\mathfrak{n}_{u}}^{2}$.
Since Eq.(200) gives the sum of Mandelstam variables $\hskip
0.85358pts+t+u\\!=\\!\sum_{i=1}^{4}\\!M_{\mathfrak{n}_{i}}^{2}$, we can
express this kinematic condition as follows:

$\sum_{i=1}^{4}M_{\mathfrak{n}_{i}}^{2}=M_{\mathfrak{n}\mathfrak{n}_{s}}^{2}\\!+M_{\mathfrak{n}\mathfrak{n}_{t}}^{2}\\!+M_{\mathfrak{n}\mathfrak{n}_{u}}^{2}\hskip
0.85358pt.$ (51)

This just gives the same four-point mass spectral condition as Eq.(40) which
we derived earlier by requiring the double-copied KK graviton scattering
amplitude (38) to be invariant under the generalized KK gauge transformation
(33). The above proof of the condition (51) relies on using the massive KK
fundamental BCJ relation (48) for the rank reduction of the kernel matrix
$\hskip 0.85358pt\Theta\hskip 0.85358pt$, where we deduced Eq.(48) to hold for
the KK gauge theories under the 5d toroidal $S^{1}$ compactification. The
exact agreement between the conditions (40) and (51) means that the
generalized KK gauge transformation (33) and the massive KK fundamental BCJ
relation (48) are fully compatible.

In passing, we note that a recent literature DC-4dx1  studied the constraints
on a massive double-copy between the massive YM theory and the dRGT gravity
model dRGT ; they found that the massive kernel matrix of the four-point bi-
adjoint scalar amplitudes could take the minimal rank if a spectral condition
is met; their spectral condition turns out to be similar to our above KK
spectral condition (51). But, the literature dRGT  essentially differs from
our present study because we focus on the KK gauge/gravity theories under the
5d toroidal compactification in which both the massive KK fundamental BCJ
relation (48) and the massive KK double-copy construction (49) or [(38)] can
be proven without additional assumption. Our mass spectral condition (51) is a
prediction of the KK gauge/gravity theories under the 5d toroidal $S^{1}$
compactification. Moreover, our proof of the same condition (40) only relies
on the fact that the double-copied four-point KK graviton scattering
amplitudes (38) should be gauge-invariant under the generalized KK gauge
transformation (33). This independently gives the simplest direct proof of the
spectral condition (40).

###  3.2 Explicit Double-Copy Construction of KK Graviton Amplitudes

In this subsection, we compute explicitly the four-point elastic and inelastic
scattering amplitudes of the KK gauge bosons. Then, using the extended BCJ-
type massive double-copy construction of Section 3.1, we derive the
corresponding four-point KK graviton scattering amplitudes under the toroidal
compactification of $S^{1}$.

For the convenience and simplicity of explicit demonstration of our extended
massive KK BCJ construction of the four-point KK graviton amplitudes in a
compact analytical form, we will consider the leading scattering amplitudes of
the longitudinal KK gravitons, which are defined by using an effective
leading-order polarization tensor $\hskip
0.85358pt\bar{\zeta}^{\mu\nu}_{L}\\!\\!=\\!\zeta^{\mu}_{L}\hskip
0.85358pt\zeta^{\nu}_{L}\hskip 0.85358pt$ of the KK graviton.444We will
present the double-copied full scattering amplitudes of longitudinal KK
gravitons using the exact longitudinal polarization tensor (25b) in Appendix
C. Thus, we define the corresponding on-shell longitudinal KK graviton field
as follows:

$\bar{h}_{L}=\bar{\zeta}^{\mu\nu}_{L}h_{\mu\nu}\,,\qquad\bar{\zeta}^{\mu\nu}_{L}=\zeta^{\mu}_{L}\hskip
0.85358pt\zeta^{\nu}_{L}\,.$ (52)

As we will demonstrate in this subsection, the effective longitudinal KK
graviton amplitudes defined by this effective leading-order polarization
tensor $\hskip
0.85358pt\bar{\zeta}^{\mu\nu}_{L}\\!\\!=\\!\zeta^{\mu}_{L}\hskip
0.85358pt\zeta^{\nu}_{L}\hskip 0.85358pt$ can always give the correct leading-
order results of the exact longitudinal KK graviton scattering amplitudes
under the high energy expansion, and thus have a physical meaning.

####  $\blacklozenge$ Inelastic KK Scattering Amplitudes of $\,\\{0,0,n,n$}:

Since the structures of the inelastic KK scattering amplitudes are simpler
than those of the elastic KK scattering amplitudes, we will start with
analyzing the inelastic KK amplitudes. Thus, we first consider the simplest
nontrivial example of the inelastic scattering process. This is the “mixed”
four-particle scattering process of gauge bosons with the initial states being
zero-mode gauge bosons (gravitons) and the final states being longitudinal KK
gauge bosons (KK gravitons). The neutral condition (32) allows two possible
choices of the external KK states under the $S^{1}$ toroidal compactification:

$\\{\mathfrak{n}_{1},\hskip 0.85358pt\mathfrak{n}_{2},\hskip
0.85358pt\mathfrak{n}_{3},\hskip 0.85358pt\mathfrak{n}_{4}\\}=\\{0,0,\pm n,\mp
n\\}\equiv\mathsf{A}\,,$ (53)

where only one of the combinations gives the independent amplitude. Then, we
derive the following four-point gauge boson scattering amplitude for this
reaction:

$\mathcal{T}\big{[}A^{a\hskip 0.85358pt0}_{\pm 1}A^{b\hskip 0.85358pt0}_{\mp
1}A^{c\hskip 0.85358pt\pm n}_{L}A^{d\hskip 0.85358pt\mp
n}_{L}\big{]}=\,g^{2}\\!\left(\\!\frac{\,\mathcal{C}_{s}\hskip
0.85358pt\mathcal{N}_{s}^{\mathsf{A}}\,}{s}+\frac{\mathcal{C}_{t}\hskip
0.85358pt\mathcal{N}_{t}^{\mathsf{A}}}{\,t-\\!M_{n}^{2}\,}+\frac{\mathcal{C}_{u}\hskip
0.85358pt\mathcal{N}_{u}^{\mathsf{A}}}{\,u-\\!M_{n}^{2}\,}\\!\right)\\!,$ (54)

where the kinematic numerators $\,\\{\mathcal{N}_{j}^{\mathsf{A}}\\}\,$ are
given in Eq.(203) of Appendix B. With this, we impose the CK duality and
derive the corresponding four-point scattering amplitude of the longitudinal
KK gravitons with the external states chosen as in Eq.(53):

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\big{[}h_{\pm
2}^{0}h_{\mp 2}^{0}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\big{]}$
$\displaystyle=\,-\frac{~{}\kappa^{2}~{}}{16}\bigg{[}\frac{\,(\mathcal{N}_{s}^{\mathsf{A}})^{2}\,}{s}+\frac{\,(\mathcal{N}_{t}^{\mathsf{A}})^{2}\,}{\,t\\!-\\!M_{n}^{2}\,}+\frac{\,(\mathcal{N}_{u}^{\mathsf{A}})^{2}\,}{\,u\\!-\\!M_{n}^{2}~{}}\bigg{]}$
$\displaystyle=\,\frac{~{}\kappa^{2}M_{n}^{2}\hskip
0.85358pt(\bar{s}\\!+\\!4)^{2}\hskip 0.85358pts^{4}_{\theta}~{}}{~{}{8}\hskip
0.85358pt[(\bar{s}+4)-(\bar{s}-4)\hskip 0.85358ptc_{2\theta}\hskip
0.85358pt]\,}\,,$ (55)

where $\,\bar{s}\\!=\\!s/M_{n}^{2}\,$ and
$\,(s_{k\theta},\,c_{k\theta})\\!\equiv\\!(\sin k\theta,\,\cos k\theta)\hskip
0.85358pt$. In Eq.(3.2), we have made the replacement of couplings
$\,g^{2}\\!\rightarrow\\!-(\kappa/4)^{2}\,$ based on Eq.(22). We see that the
above double-copied inelastic graviton amplitude with the leading-order
longitudinal polarization tensor (52) takes a fairly compact form. Then, we
make the high energy expansion of the amplitude (3.2) and derive the LO
amplitude as follows:

$\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}h_{\pm
2}^{0}h_{\mp 2}^{0}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\big{]}\hskip
0.85358pt=\hskip 0.85358pt\frac{~{}\kappa^{2}\hskip
0.85358pts^{2}_{\theta}\,s~{}}{\,{16}\,}=\frac{~{}\kappa^{2}\hskip
0.85358pt(1\\!-c_{2\theta})\hskip 0.85358pts~{}}{\,{32}\,}\hskip 0.85358pt.$
(56)

We further present the full graviton amplitude in Eq.(218) of Appendix C by
using the exact longitudinal polarization tensor (25b) of the KK gravitons.
Under the high energy expansion, the LO and NLO parts of the full graviton
amplitude (218) are derived in Eqs.(219a) and (219b) respectively and we have
${\mathcal{M}}\big{[}h_{\pm 2}^{0}h_{\mp 2}^{0}\bar{h}_{L}^{\mp
n}\bar{h}_{L}^{\pm n}\big{]}\\!=\\!{\mathcal{M}}_{0}+{\delta\mathcal{M}}\hskip
0.85358pt$. Comparing Eqs. (56) and (219a), we can deduce:

$\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}h_{\pm
2}^{0}h_{\mp 2}^{0}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm
n}\big{]}={\mathcal{M}}_{0}\big{[}h_{\pm 2}^{0}h_{\mp 2}^{0}{h}_{L}^{\mp
n}{h}_{L}^{\pm n}\big{]}.$ (57)

Hence, the scattering amplitude (3.2) which we have computed using the leading
polarization tensor (52) of longitudinal KK gravitions does have an important
physical meaning, namely, its expanded LO amplitude (56) can give the correct
LO contribution (219a) of the full graviton amplitude (218). This is a
striking feature because using the leading polarization tensor (52) of
longitudinal KK gravitions we only need to compute the corresponding gauge
boson amplitude including the external longitudinal polarization state alone
(for the double-copy construction), but in the full graviton amplitude its
external longitudinal KK graviton state contains the exact longitudinal
polarization tensor (25b) which includes the products of the gauge boson
polarization vectors of both the helicities $\pm 1$ and $0\hskip 0.85358pt$,
and is thus much more complicated. In the following, we will demonstrate
explicitly that this feature holds for all the four-point KK graviton
scattering amplitudes studied in this section.

####  $\blacklozenge$ Inelastic KK Scattering Amplitudes of
$\\{n,2n,3n,4n\\}$:

As the second example, we study the inelastic KK scattering process containing
the following KK indices of the external states:

$\\{\mathfrak{n}_{1},\hskip 0.85358pt\mathfrak{n}_{2},\hskip
0.85358pt\mathfrak{n}_{3},\hskip 0.85358pt\mathfrak{n}_{4}\\}=\\{\pm n,\mp
2n,\mp 3n,\pm 4n\\}\equiv\mathsf{A}\,.$ (58)

Thus, we derive the following four-point leading scattering amplitude of
longitudinal KK gauge bosons:

$\mathcal{T}\\!\left[A^{a\hskip 0.85358pt\pm n}_{L}A^{b\hskip 0.85358pt\mp
2n}_{L}A^{c\hskip 0.85358pt\mp 3n}_{L}A^{d\hskip 0.85358pt\pm
4n}_{L}\right]=\,g^{2}\\!\left(\frac{\mathcal{C}_{s}\hskip
0.85358pt\mathcal{N}_{s}^{\mathsf{A}}}{\,s-M_{n}^{2}\,}+\frac{\mathcal{C}_{t}\hskip
0.85358pt\mathcal{N}_{t}^{\mathsf{A}}}{\,t-25M_{n}^{2}\,}+\frac{\mathcal{C}_{u}\hskip
0.85358pt\mathcal{N}_{u}^{\mathsf{A}}}{\,u-4M_{n}^{2}\,}\right)\\!,$ (59)

where the kinematic numerators $\,\\{\mathcal{N}_{j}^{\mathsf{A}}\\}\,$ are
given by Eq.(203). With the longitudinal KK gauge boson amplitude (59) and
using the CK duality, we compute the corresponding four-point leading
longitudinal KK graviton amplitude as follows:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}^{\pm
n}_{L}\bar{h}^{\mp 2n}_{L}\bar{h}^{\mp 3n}_{L}\bar{h}^{\pm
4n}_{L}\right]=-\frac{~{}\kappa^{2}~{}}{16}\\!\left[\frac{(\mathcal{N}_{s}^{\mathsf{A}})^{2}}{~{}s-M_{n}^{2}\,}+\frac{(\mathcal{N}_{t}^{\mathsf{A}})^{2}}{\,t-25M_{n}^{2}\,}+\frac{(\mathcal{N}_{u}^{\mathsf{A}})^{2}}{\,u-4M_{n}^{2}~{}}\right]\hskip
22.76219pt$ $\displaystyle\hskip
14.22636pt=\frac{~{}\kappa^{2}M_{n}^{2}(49\hskip
0.85358pt\bar{s}^{\prime}\\!-\\!1)[(343\hskip 0.85358pt\bar{s}^{\prime
2}\\!-174\hskip 0.85358pt\bar{s}^{\prime}\\!+27)+12\hskip
0.85358pt{\omega}_{-}c_{\theta}+{\omega}_{+}^{2}c_{2\theta}\hskip
0.85358pt]^{2}~{}}{3136\hskip 0.85358pt\bar{s}^{\prime 2}\hskip
0.85358pt[(7\hskip
0.85358pt\bar{s}^{\prime}\\!+\\!3)\\!+{\omega}_{-}c_{\theta}][(7\hskip
0.85358pt\bar{s}^{\prime}\\!-\\!3)\\!-\\!{\omega}_{-}c_{\theta}\hskip
0.85358pt]}\,,\hskip 22.76219pt$ (60)

where we have defined $\,\bar{s}\\!=\\!s/M_{n}^{2}\hskip 0.85358pt$,
$\,\bar{s}^{\prime}\\!=\\!\bar{s}/49\hskip 0.85358pt$, and $\hskip
0.85358pt{\omega}_{\pm}\\!=\\!\sqrt{49\hskip 0.85358pt\bar{s}^{\prime\hskip
0.85358pt2}\\!\pm 58\hskip 0.85358pt\bar{s}^{\prime}\\!+9\,}\hskip 0.85358pt$.
Then, making the high energy expansion for Eq.(3.2), we derive the following
LO KK graviton scattering amplitude:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}^{\pm
n}_{L}\bar{h}^{\mp 2n}_{L}\bar{h}^{\mp 3n}_{L}\bar{h}^{\pm
4n}_{L}\big{]}=\frac{~{}\kappa^{2}\hskip 0.85358pts~{}}{\,64\,}\hskip
0.85358pt(7+c_{2\theta})^{2}\csc^{2}\\!\theta\,,\hskip 22.76219pt$ (61)

which have $O(E^{2})$ and are mass-independent, as expected.

Moreover, we have used the exact longitudinal polarization tensor (25b) to
further derive the corresponding full inelastic KK graviton amplitudes
$\mathcal{M}\\!\left[{h}^{\pm n}_{L}{h}^{\mp 2n}_{L}{h}^{\mp 3n}_{L}{h}^{\pm
4n}_{L}\right]$ in Eq.(C) of Appendix C. Under high energy expansion, the LO
and NLO contributions of the full amplitude (C) are given by Eqs.(222a) and
(222), respectively. Inspecting the two LO amplitudes (61) and (222a), we find
that they are equal:

$\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}^{\pm
n}_{L}\bar{h}^{\mp 2n}_{L}\bar{h}^{\mp 3n}_{L}\bar{h}^{\pm
4n}_{L}\big{]}={\mathcal{M}}_{0}\big{[}{h}^{\pm n}_{L}{h}^{\mp 2n}_{L}{h}^{\mp
3n}_{L}{h}^{\pm 4n}_{L}\big{]}\hskip 0.85358pt.$ (62)

Hence, as we observed earlier, the scattering amplitude (3.2) which we have
computed using the leading polarization tensor (52) of longitudinal KK
gravitions does have an important physical meaning, because its expanded LO
amplitude (61) can give the correct LO contribution (222a) of the full KK
graviton amplitude (C).

####  $\blacklozenge$ Inelastic KK Scattering Amplitudes of $\\{n,n,m,m\\}$:

As the third example of the inelastic KK scattering process, we study the four
longitudinal KK gauge boson scattering amplitude whose external states have
the following KK indices:

$\displaystyle\\{\mathfrak{n}_{1},\hskip 0.85358pt\mathfrak{n}_{2},\hskip
0.85358pt\mathfrak{n}_{3},\hskip
0.85358pt\mathfrak{n}_{4}\\}\,=\,\left\\{\begin{array}[]{r}\\{\pm n,\mp n,\mp
m,\pm m\\}\equiv\mathsf{A}\,,\\\\[1.99168pt] \\{\pm n,\mp n,\pm m,\mp
m\\}\equiv\mathsf{B}\,,\end{array}\right.$ (63)

where the positive integers $(n,\hskip 0.85358ptm)$ are unequal ($\hskip
0.85358ptn\\!\neq\\!m\hskip 0.85358pt$), and the inelastic amplitudes of
type-$\mathsf{A}$ and type-$\mathsf{B}$ are independent. Thus, we compute the
corresponding leading longitudinal KK gauge boson scattering amplitudes as
follows:

$\displaystyle\mathcal{T}\\!\left[A^{a\hskip 0.85358pt\pm n}_{L}A^{b\hskip
0.85358pt\mp n}_{L}A^{c\hskip 0.85358pt\mp m}_{L}A^{\hskip 0.85358pt\pm
m}_{L}\right]$ $\displaystyle=\,g^{2}\\!\left(\frac{\,\mathcal{C}_{s}\hskip
0.85358pt\mathcal{N}_{s}^{\mathsf{A}}\,}{s}+\frac{\,\mathcal{C}_{t}\hskip
0.85358pt\mathcal{N}_{t}^{\mathsf{A}}\,}{t-M_{n+m}^{2}}+\frac{\,\mathcal{C}_{u}\hskip
0.85358pt\mathcal{N}_{u}^{\mathsf{A}}\,}{u-M_{n-m}^{2}}\right)\\!,$ (64a)
$\displaystyle\mathcal{T}\\!\left[A^{a\hskip 0.85358pt\pm n}_{L}A^{b\hskip
0.85358pt\mp n}_{L}A^{c\hskip 0.85358pt\pm m}_{L}A^{d\hskip 0.85358pt\mp
m}_{L}\right]$ $\displaystyle=\,g^{2}\\!\left(\frac{\,\mathcal{C}_{s}\hskip
0.85358pt\mathcal{N}_{s}^{\mathsf{B}}\,}{s}+\frac{\mathcal{C}_{t}\hskip
0.85358pt\mathcal{N}_{t}^{\mathsf{B}}}{\,t-M_{n-m}^{2}\,}+\frac{\mathcal{C}_{u}\hskip
0.85358pt\mathcal{N}_{u}^{\mathsf{B}}}{\,u-M_{n+m}^{2}\,}\right)\\!,$ (64b)

where the kinematic numerators $\\{\mathcal{N}_{j}^{\mathsf{A}},\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{B}}\\}$ are presented in Eqs.(207) and (210)
of Appendix B.

Using the inelastic KK gauge boson scattering amplitudes (64a)-(64b) and the
extended massive CK duality, we can derive the following leading scattering
amplitudes for longitudinal KK gravitons:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp m}\bar{h}_{L}^{\pm
m}\right]=-\frac{\kappa^{2}}{\,16\,}\bigg{[}\,\frac{(\mathcal{N}_{s}^{\mathsf{A}})^{2}}{s}+\frac{(\mathcal{N}_{t}^{\mathsf{A}})^{2}}{\,t-M_{n+m}^{2}\,}+\frac{(\mathcal{N}_{u}^{\mathsf{A}})^{2}}{\,u-M_{n-m}^{2}\,}\,\bigg{]}\hskip
34.1433pt$ $\displaystyle=\frac{~{}\kappa^{2}M_{n}^{2}\hskip
0.85358pt[(7\bar{s}^{2}\\!-\\!12\hskip 0.85358pt\bar{s}\hskip
0.85358ptr_{+}^{2}\\!+\\!48\hskip 0.85358ptr^{2})\\!+\\!64\hskip
0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}rc_{\theta}\\!+\\!(\bar{s}\\!+\\!4)(\bar{s}\\!+\\!4\hskip
0.85358ptr^{2})c_{2\theta}]^{2}~{}}{64\hskip 0.85358pt\bar{s}\hskip
0.85358pt(\bar{s}\\!-\\!4\hskip 0.85358ptr\\!-\\!4\hskip
0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}c_{\theta})(\bar{s}\\!+\\!4\hskip
0.85358ptr\\!+\\!4\hskip 0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}c_{\theta})}\,,$ (65a) $\displaystyle\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm m}\bar{h}_{L}^{\mp
m}\right]=-\frac{\kappa^{2}}{\,16\,}\bigg{[}\,\frac{(\mathcal{N}_{s}^{\mathsf{B}})^{2}}{s}+\frac{(\mathcal{N}_{t}^{\mathsf{B}})^{2}}{\,t-M_{n-m}^{2}\,}+\frac{(\mathcal{N}_{u}^{\mathsf{B}})^{2}}{\,u-M_{n+m}^{2}\,}\,\bigg{]}$
$\displaystyle=\frac{~{}\kappa^{2}M_{n}^{2}\hskip 0.85358pt[(7\hskip
0.85358pt\bar{s}^{2}\\!-\\!12\hskip 0.85358pt\bar{s}\hskip
0.85358ptr_{+}^{2}\\!+\\!48\hskip 0.85358ptr^{2})\\!-\\!64\hskip
0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}rc_{\theta}\\!+\\!(\bar{s}\\!+\\!4)(\bar{s}\\!+\\!4r^{2})c_{2\theta}]^{2}~{}}{64\hskip
0.85358pt\bar{s}\hskip 0.85358pt(\bar{s}\\!-\\!4\hskip
0.85358ptr\\!+\\!4\hskip 0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}c_{\theta})(\bar{s}\\!+\\!4\hskip
0.85358ptr\\!-\\!4\hskip 0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}c_{\theta})}\,,$ (65b)

where we have used the notations,

$\displaystyle r=M_{m}/M_{n}\hskip 0.85358pt,\hskip
11.38109ptr_{+}^{2}=1+r^{2},\hskip
11.38109ptq\\!=\sqrt{E^{2}\\!-\\!M_{n}^{2}\,}\,,\hskip
11.38109ptq^{\prime}\\!=\sqrt{E^{2}\\!-\\!M_{m}^{2}\,}\,,$
$\displaystyle\bar{q}^{2}\\!=q^{2}/M_{n}^{2}=\bar{s}/4-1\hskip
0.85358pt,\hskip 8.53581pt\bar{q}^{\prime 2}\\!=q^{\prime
2}/M_{n}^{2}=\bar{s}/4-r^{2},\hskip 8.53581pt\bar{q}^{2}\bar{q}^{\prime
2}=(\bar{s}-4)(\bar{s}-4\hskip 0.85358ptr^{2})/16\hskip
0.85358pt.~{}~{}~{}\hskip 8.53581pt$

Making the high energy expansion, we can derive the LO amplitudes of
Eqs.(65)-(65) as follows:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp m}\bar{h}_{L}^{\pm m}\big{]}=\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm m}\bar{h}_{L}^{\mp
m}\big{]}=\frac{~{}\kappa^{2}\hskip 0.85358pts~{}}{\,64\,}\hskip
0.85358pt(7+c_{2\theta})^{2}\csc^{2}\\!\theta\,,$ (67)

which are of $O(E^{2})$ and mass-independent, as expected. We note that the LO
amplitudes (61) and (67) coincide.

In addition, using the exact longitudinal polarization tensor (25b) for each
external KK graviton state, we have further derived the full inelastic KK
graviton scattering amplitudes $\mathcal{M}\\!\left[{h}_{L}^{\pm
n}{h}_{L}^{\mp n}{h}_{L}^{\mp m}{h}_{L}^{\pm m}\right]$ and
$\mathcal{M}\\!\left[{h}_{L}^{\pm n}{h}_{L}^{\mp n}{h}_{L}^{\pm m}{h}_{L}^{\mp
m}\right]$ in Eqs.(223)-(224) of Appendix C. Under the high energy expansion,
their LO and NLO scattering amplitudes are given in Eq.(225). Comparing their
LO amplitudes (225a) with our above LO amplitudes (67), we find that they are
equal:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp m}\bar{h}_{L}^{\pm m}\big{]}=\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm m}\bar{h}_{L}^{\mp m}\big{]}$
$\displaystyle=\,\,$ $\displaystyle{\mathcal{M}}_{0}\big{[}{h}_{L}^{\pm
n}{h}_{L}^{\mp n}{h}_{L}^{\mp m}{h}_{L}^{\pm
m}\big{]}={\mathcal{M}}_{0}\big{[}{h}_{L}^{\pm n}{h}_{L}^{\mp n}{h}_{L}^{\pm
m}{h}_{L}^{\mp m}\big{]}.$ (68)

This is similar to Eqs.(57) and (62), which we have demonstrated for other
four-point KK graviton scattering amplitudes at the LO.

####  $\blacklozenge$ Elastic KK Scattering Amplitudes of $\\{n,n,n,n\\}$:

Next, we study the elastic scattering processes for the four longitudinal KK
gauge bosons and KK gravitons. The external states are allowed to have the
following combinations of KK indices:

$\displaystyle\\{\mathfrak{n}_{1},\hskip 0.85358pt\mathfrak{n}_{2},\hskip
0.85358pt\mathfrak{n}_{3},\hskip
0.85358pt\mathfrak{n}_{4}\\}=\left\\{\begin{array}[]{r}\\{\pm n,\pm n,\mp
n,\mp n\\}\equiv\mathsf{A}\,,\\\\[2.27621pt] \\{\pm n,\mp n,\mp n,\pm
n\\}\equiv\mathsf{B}\,,\\\\[2.27621pt] \\{\pm n,\mp n,\pm n,\mp
n\\}\equiv\mathsf{C}\,,\end{array}\right.$ (69)

where only three types of them, denoted by $(\mathsf{A},\hskip
0.85358pt\mathsf{B},\hskip 0.85358pt\mathsf{C})$, are independent. Then, we
compute the four-point elastic scattering amplitudes of longitudinal KK gauge
bosons as follows:

$\displaystyle\mathcal{T}\\!\left[A^{a\hskip 0.85358pt\pm n}_{L}A^{b\hskip
0.85358pt\pm n}_{L}A^{c\hskip 0.85358pt\mp n}_{L}A^{d\hskip 0.85358pt\mp
n}_{L}\right]$ $\displaystyle=\,g^{2}\\!\left(\\!\frac{\mathcal{C}_{s}\hskip
0.85358pt\mathcal{N}_{s}^{\mathsf{A}}}{\,s-4M_{n}^{2}\,}+\frac{\,\mathcal{C}_{t}\hskip
0.85358pt\mathcal{N}_{t}^{\mathsf{A}}\,}{t}+\frac{\,\mathcal{C}_{u}\hskip
0.85358pt\mathcal{N}_{u}^{\mathsf{A}}\,}{u}\\!\right)\\!,$ (70a)
$\displaystyle\mathcal{T}\\!\left[A^{a\hskip 0.85358pt\pm n}_{L}A^{b\hskip
0.85358pt\mp n}_{L}A^{c\hskip 0.85358pt\mp n}_{L}A^{d\hskip 0.85358pt\pm
n}_{L}\right]$ $\displaystyle=\,g^{2}\\!\left(\\!\frac{\,\mathcal{C}_{s}\hskip
0.85358pt\mathcal{N}_{s}^{\mathsf{B}}\,}{s}+\frac{\mathcal{C}_{t}\hskip
0.85358pt\mathcal{N}_{t}^{\mathsf{B}}}{\,t-4M_{n}^{2}\,}+\frac{\,\mathcal{C}_{u}\hskip
0.85358pt\mathcal{N}_{u}^{\mathsf{B}}\,}{u}\\!\right)\\!,$ (70b)
$\displaystyle\mathcal{T}\\!\left[A^{a\hskip 0.85358pt\pm n}_{L}A^{b\hskip
0.85358pt\mp n}_{L}A^{c\hskip 0.85358pt\pm n}_{L}A^{d\hskip 0.85358pt\mp
n}_{L}\right]$ $\displaystyle=\,g^{2}\\!\left(\\!\frac{\,\mathcal{C}_{s}\hskip
0.85358pt\mathcal{N}_{s}^{\mathsf{C}}\,}{s}+\frac{\,\mathcal{C}_{t}\hskip
0.85358pt\mathcal{N}_{t}^{\mathsf{C}}\,}{t}+\frac{\mathcal{C}_{u}\hskip
0.85358pt\mathcal{N}_{u}^{\mathsf{C}}}{\,u-4M_{n}^{2}\,}\\!\right)\\!,$ (70c)

where the kinematic numerators $\\{\mathcal{N}_{j}^{\mathsf{A}},\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{B}},\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{C}}\\}$ are given by Eqs.(212)-(216) of
Appendix B.

Using the above KK gauge boson scattering amplitudes (70a)-(70c) and the
extended massive CK duality, we derive the following leading elastic
scattering amplitudes of longitudinal KK gravitons:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp
n}\big{]}=-\frac{~{}\kappa^{2}~{}}{16}\bigg{[}\frac{(\mathcal{N}_{s}^{\mathsf{A}})^{2}}{\,s-4M_{n}^{2}\,}+\frac{\,(\mathcal{N}_{t}^{\mathsf{A}})^{2}\,}{t}+\frac{\,(\mathcal{N}_{u}^{\mathsf{A}})^{2}\,}{u}\bigg{]},\hskip
14.22636pt$ (71a) $\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm
n}\right]=-\frac{~{}\kappa^{2}~{}}{16}\bigg{[}\frac{\,(\mathcal{N}_{s}^{\mathsf{B}})^{2}\,}{s}+\frac{(\mathcal{N}_{t}^{\mathsf{B}})^{2}}{\,t-4M_{n}^{2}\,}+\frac{\,(\mathcal{N}_{u}^{\mathsf{B}})^{2}\,}{u}\bigg{]},\hskip
14.22636pt$ (71b) $\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp
n}\right]=-\frac{~{}\kappa^{2}~{}}{16}\bigg{[}\frac{\,(\mathcal{N}_{s}^{\mathsf{C}})^{2}\,}{s}+\frac{\,(\mathcal{N}_{t}^{\mathsf{C}})^{2}\,}{t}+\frac{(\mathcal{N}_{u}^{\mathsf{C}})^{2}}{\,u-4M_{n}^{2}~{}}\bigg{]}.\hskip
14.22636pt$ (71c)

After substituting the expressions of the numerators into the above formulas,
we derive the explicit forms of these scattering amplitudes:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\big{]}$
$\displaystyle=\frac{~{}\kappa^{2}M_{n}^{2}~{}}{64}(\bar{s}\\!-\\!4)(7\\!+\\!c_{2\theta})^{2}\csc^{2}\theta\,,$
(72a) $\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\right]$
$\displaystyle=\frac{~{}\kappa^{2}M_{n}^{2}\hskip 0.85358pt[(7\hskip
0.85358pt\bar{s}^{2}\\!-\\!24\hskip 0.85358pt\bar{s}\\!+\\!48)\\!+\\!16\hskip
0.85358pt(\bar{s}-4)c_{\theta}\\!+\\!(\bar{s}+4)^{2}c_{2\theta}\hskip
0.85358pt]^{2}\csc^{2}\\!\frac{{\theta}}{2}~{}}{128\hskip
0.85358pt\bar{s}\hskip
0.85358pt(\bar{s}\\!-\\!4)[(\bar{s}+4)+(\bar{s}-4)c_{\theta}\hskip
0.85358pt]}\hskip 0.85358pt,$ (72b) $\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\right]$
$\displaystyle=\frac{~{}\kappa^{2}M_{n}^{2}\hskip 0.85358pt[(7\hskip
0.85358pt\bar{s}^{2}\\!-\\!24\hskip 0.85358pt\bar{s}+48)\\!-\\!16\hskip
0.85358pt(\bar{s}-4)c_{\theta}+(\bar{s}+4)^{2}c_{2\theta}\hskip
0.85358pt]^{2}\sec^{2}\\!\frac{{\theta}}{2}~{}}{128\hskip
0.85358pt\bar{s}\hskip
0.85358pt(\bar{s}-4)[(\bar{s}+4)-(\bar{s}-4)c_{\theta}\hskip 0.85358pt]}\hskip
0.85358pt.$ (72c)

Then, making the high energy expansion, we derive the following LO scattering
amplitudes:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\big{]}=\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\big{]}=\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\big{]}$
$\displaystyle=\frac{~{}\kappa^{2}\hskip
0.85358pts~{}}{64}(7\\!+\\!c_{2\theta})^{2}\csc^{2}\theta\,.$ (73)

We see that the LO elastic KK graviton amplitudes of the above three helicity
combinations are equal and are mass-independent, as expected.

Moreover, using the exact longitudinal polarization tensor (25b) for each
external KK graviton state, we have further derived the full elastic KK
graviton scattering amplitudes in Eqs.(229)-(230) of Appendix C. The LO and
NLO contributions of these KK graviton scattering amplitudes are given in
Eq.(229). Comparing Eqs.(3.2) and (231a), we find that they are equal:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\big{]}=\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\big{]}=\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\big{]}\hskip
34.1433pt$ $\displaystyle=\hskip 0.85358pt\,$
$\displaystyle{\mathcal{M}}_{0}\big{[}h_{L}^{\pm n}h_{L}^{\pm n}h_{L}^{\mp
n}h_{L}^{\mp n}\big{]}={\mathcal{M}}_{0}\big{[}h_{L}^{\pm n}h_{L}^{\mp
n}h_{L}^{\mp n}h_{L}^{\pm n}\big{]}={\mathcal{M}}_{0}\big{[}h_{L}^{\pm
n}h_{L}^{\mp n}h_{L}^{\pm n}h_{L}^{\mp n}\big{]}.$ (74)

In general, we expect that the equality between the two types of the LO KK
graviton amplitudes, such as those shown in Eqs.(57), (62), (68), and (74),
can hold for any such $N$-point KK graviton scattering amplitudes at the LO
(with some or all external KK graviton states being longitudinally polarized),
as long as these KK amplitudes respect the KK GAET 5DYM2002 KK-ET-He2004  and
KK GRET Hang:2021fmp  in which the residual terms can be supressed and belong
to the NLO. For the current study, considering the four-point KK scattering
amplitudes, we can use the KK GAET and GRET to prove this equality. For any
four longitudinal KK gauge boson scattering amplitude, the KK GAET states
5DYM2002 KK-ET-He2004  that under the high energy expansion its LO scattering
amplitude of longitudinal KK gauge bosons ($A_{L}^{a\mathfrak{n}}$) equals
that of the corresponding KK Goldstone bosons ($A_{5}^{a\mathfrak{n}}$) at the
LO:

$\mathcal{T}_{0}\big{[}A_{L}^{a\mathfrak{n}_{1}}\\!A_{L}^{b\mathfrak{n}_{2}}\\!A_{L}^{c\mathfrak{n}_{3}}\\!A_{L}^{d\mathfrak{n}_{4}}\big{]}=\mathcal{T}_{0}\big{[}A_{5}^{a\mathfrak{n}_{1}}\\!A_{5}^{b\mathfrak{n}_{2}}\\!A_{5}^{c\mathfrak{n}_{3}}\\!A_{5}^{d\mathfrak{n}_{4}}\big{]},$
(75)

which are of $O(g^{2}E^{0})$ and are mass-independent. As we have demonstrated
in this work and Hang:2021fmp , we can make double-copy construction on both
sides of Eq.(75) and thus deduce the corresponding GRET formula which connects
the LO scattering amplitude of longitudinal KK gravitons
($\bar{h}_{L}^{\mathfrak{n}}$) to that of the corresponding KK Goldstone
bosons ($h_{55}^{\mathfrak{n}}$) at the LO:

$\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\mathfrak{n}_{1}}\bar{h}_{L}^{\mathfrak{n}_{2}}\bar{h}_{L}^{\mathfrak{n}_{3}}\bar{h}_{L}^{\mathfrak{n}_{4}}\big{]}=\mathcal{M}_{0}\big{[}h_{55}^{\mathfrak{n}_{1}}h_{55}^{\mathfrak{n}_{2}}h_{55}^{\mathfrak{n}_{3}}h_{55}^{\mathfrak{n}_{4}}\big{]},$
(76)

where on the left-hand-side of the equality each external longitudinal KK
graviton state
$\bar{h}_{L}^{\mathfrak{n}}\\!=\bar{\zeta}_{\mu\nu}h_{\mathfrak{n}}^{\mu\nu}$
has its leading-order longitudinal polarization tensor defined in
Eq.(52).555We note that the double-copy of the LO scattering amplitudes of the
KK Goldstone bosons,
$\,\mathcal{T}_{0}\big{[}A_{5}^{a\mathfrak{n}_{1}}\\!A_{5}^{b\mathfrak{n}_{2}}\\!A_{5}^{c\mathfrak{n}_{3}}\\!A_{5}^{d\mathfrak{n}_{4}}\big{]}\\!\\!\rightarrow\\!\\!\mathcal{M}_{0}\big{[}h_{55}^{\mathfrak{n}_{1}}h_{55}^{\mathfrak{n}_{2}}h_{55}^{\mathfrak{n}_{3}}h_{55}^{\mathfrak{n}_{4}}\big{]}$,
is well justified because at the LO of the high energy expansion the KK
Goldstone bosons $A_{5}^{a\mathfrak{n}}$ and $h_{55}^{\mathfrak{n}}$ behave as
massless states and their LO amplitudes are mass-independent Hang:2021fmp .
Hence, the double-copy of the LO KK Goldstone amplitudes should be guaranteed
by the double-copy of the corresponding massless scattering amplitudes,
$\mathcal{T}\big{[}\hat{A}_{5}^{a}\hat{A}_{5}^{b}\hat{A}_{5}^{c}\hat{A}_{5}^{d}\big{]}\\!\\!\rightarrow\\!\mathcal{M}\big{[}\hat{h}_{55}\hat{h}_{55}\hat{h}_{55}\hat{h}_{55}\big{]}$,
in the non-compactified 5d gauge/gravity theories. This conclusion also holds
for the general $N$-point massless Goldstone boson amplitudes as is obvious.
On the other hand, the KK GRET Hang:2021fmp  gives the following LO equality:

${\mathcal{M}}_{0}\big{[}{h}_{L}^{\mathfrak{n}_{1}}{h}_{L}^{\mathfrak{n}_{2}}{h}_{L}^{\mathfrak{n}_{3}}{h}_{L}^{\mathfrak{n}_{4}}\big{]}=\mathcal{M}_{0}\big{[}h_{55}^{\mathfrak{n}_{1}}h_{55}^{\mathfrak{n}_{2}}h_{55}^{\mathfrak{n}_{3}}h_{55}^{\mathfrak{n}_{4}}\big{]},$
(77)

where according to the original GRET formulation Hang:2021fmp , the LO
scattering amplitude of longitudinal KK gravitons on the left-hand-side of the
equality is computed by using the exact longitudinal polarization tensor (25b)
for each external KK graviton state $\hskip
0.85358pt{h}_{L}^{\mathfrak{n}}\\!={\zeta}_{\mu\nu}h_{\mathfrak{n}}^{\mu\nu}$.
Comparing Eqs.(76) and (77), we deduce the following identity for the two
types of four-point longitudinal KK graviton scattering amplitudes at the LO:

$\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set<EMAIL_ADDRESS>(78)

We can extend the above identity to the general case of the scattering
amplitudes of $N$ longitudinal KK gravitons ($N\\!\geqslant 4\hskip
0.85358pt$):

$\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set<EMAIL_ADDRESS>(79)

We can readily prove the $N$-point identity (79) by repeating the same
reasoning of the above proof for the four-point scattering case, because the
GRET generally holds for $N$ longitudinal KK graviton scattering amplitudes
Hang:2021fmp  and the double-copy also holds for $N$-point massless scalar KK
Goldstone boson amplitudes (as noticed in footnote-5). We have explicitly
verified this identity for a number of typical four-point longitudinal KK
graviton scattering amplitudes as shown in Eqs.(62), (68), and (74). We can
readily extend the above proof to other four-point KK graviton scattering
amplitudes and to the case of $N$-point KK graviton scattering amplitudes with
$N\\!\\!>\\!4\hskip 1.42262pt$. For the extension to other cases, the
scattering amplitudes should contain more than one external longitudinal KK
state. This is because for a given scattering amplitude having only one
external longitudinal KK state, the the residual term of the KK GAET or GRET
has the same order as the LO $A_{5}^{\mathfrak{n}}$ amplitude or the LO
$h_{55}^{\mathfrak{n}}$ amplitude Hang:2021fmp ; hence the LO amplitudes of KK
longitudinal gauge bosons and of KK Goldstone bosons will not equal, unlike
the case of Eq.(75) or Eq.(77). This means that the extension of our
conclusion of Eq.(78) or Eq.(79) to other KK graviton scattering amplitudes
should contain more than one external longitudinal KK state. In addition, we
note that the identities (78)-(79) and their extension should hold for any
consistent 5d KK compactifications including the $S^{1}$ compactification and
the orbifold compactification of $S^{1}\\!/\mathbb{Z}_{2}\hskip 0.85358pt$.

According to the LO double-copy identities (45) and (79), we can translate the
identity (78) into a nontrivial sum rule condition on the LO kinematic
numerators of KK gauge boson scattering amplitudes:

$\sum_{j}\\!\frac{1}{\,s_{j}\,}\\!\left\\{\\!\Big{[}\mathcal{N}_{j}^{0,\mathsf{P}}\\!(L)\Big{]}^{2}\\!-\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\prod_{k}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\,\sigma_{\\!k}}\hskip
0.85358pt{\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda_{k})\hskip
0.85358pt\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda^{\prime}_{k})}\\!\right\\}=\,0\,.$
(80)

In summary, the above identities (78) and (79) demonstrate that under high
energy expansion, each leading-order (LO) scattering amplitude of longitudinal
KK gravitions computed by using the simple effective leading longitudinal
polarization tensor (52) always equals the LO amplitude computed by using the
exact longitudinal polarization tensor (24)-(25). This leads to the above
nontrivial sum rule condition (80), from which we draw an important
conclusion: each $N$-point longitudinal KK graviton scattering amplitude (with
two or more external KK graviton states being longitudinally polarized) can be
constructed by double-copy of a single amplitude of KK gauge bosons (in which
the corresponding KK gauge bosons are longitudinally polarized only) according
to the effective leading longitudinal polarization tensor (52).

###  3.3 KK Amplitudes and Double-Copy under Orbifold Compactification

In this subsection, we further study the massive double-copy construction of
the four-point scattering amplitudes of KK gauge bosons and KK gravitons,
under the 5d orbifold compactification of $S^{1}\\!/\mathbb{Z}_{2}\hskip
0.85358pt$. In this case, the KK gauge boson fields $A^{a\mu}_{n}$ and KK
graviton fields $h^{\mu\nu}_{n}$ are defined as $\mathbb{Z}_{2}$ even
Hang:2021fmp . Thus, the KK amplitudes can be expressed as the sum of the
relevant subamplitudes under the $S^{1}$ toroidal compactification without
orbifold. Similar to Eq.(16), the $\mathbb{Z}_{2}$ even states of KK gauge
bosons and KK gravitons can be defined as follows:

$\displaystyle A_{\lambda}^{a\hskip
0.85358ptn}=\frac{1}{\sqrt{2\,}\,}\\!\left[A_{\lambda}^{a(+n)}\\!+A_{\lambda}^{a(-n)}\right]\\!,~{}~{}~{}~{}h_{\sigma}^{n}=\frac{1}{\sqrt{2\,}\,}\\!\left[h_{\sigma}^{(+n)}\\!+h_{\sigma}^{(-n)}\right]\\!,$
(81)

where we denote the helicity of KK gauge boson by $\hskip
0.85358pt\lambda\,(=\\!0,\pm 1)\hskip 0.85358pt$ and the helicity of KK
graviton by $\hskip 0.85358pt\sigma\,(=\\!0,\pm 1,\pm 2)\hskip 0.85358pt$.
With these, we can derive the four-point scattering amplitudes of the KK gauge
bosons and KK gravitons:

$\displaystyle\mathcal{T}[A_{\lambda_{1}}^{an_{1}}\\!A_{\lambda_{2}}^{bn_{2}}\\!A_{\lambda_{3}}^{cn_{3}}\\!A_{\lambda_{4}}^{dn_{4}}]$
$\displaystyle\,=\,\frac{~{}g^{2}~{}}{\,2^{\scriptsize{\hbox to0.0pt{\hskip
1.66495pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}}}\\!/2\,}}\sum_{\mathsf{P}}\sum_{j}\,\frac{~{}\mathcal{C}_{j}\hskip
0.85358pt\mathcal{N}_{j}^{\mathsf{P}}(\lambda)~{}}{~{}s_{j}\\!-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}~{}}\,,$
(82a) $\displaystyle\mathcal{M}[h_{\sigma_{1}}^{n_{1}}\hskip
0.85358pth_{\sigma_{2}}^{n_{2}}\hskip 0.85358pth_{\sigma_{3}}^{n_{3}}\hskip
0.85358pth_{\sigma_{4}}^{n_{4}}]$
$\displaystyle\,=\,-\frac{~{}\kappa^{2}~{}}{\,2^{\scriptsize{\hbox
to0.0pt{\hskip
1.66495pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}}}/2+4}\,}\sum_{\mathsf{P}}\sum_{j}\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\prod_{k=1}^{4}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\frac{~{}\mathcal{N}^{\mathsf{P}}_{j}(\lambda_{k})\hskip
0.85358pt\mathcal{N}^{\hskip
0.85358pt\mathsf{P}}_{j}(\lambda^{\prime}_{k})~{}}{s_{j}-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}}\,,\hskip
19.91692pt$ (82b)

where the external KK indices are non-negative integers $\hskip
0.85358ptn_{i}\\!\geqslant\\!0\hskip 0.85358pt$, and the sum of $\mathsf{P}$
runs over the possible combinations of the external KK indices which obey the
neutral condition (32). In the above formulas, the external states could
contain possible zero-mode gauge bosons (gravitons), so we denote the number
of external KK states (with $n_{i}\\!>0\hskip 0.85358pt$) as
$\overline{\hbox{}}$$N$ , and thus the number of the external zero-mode states
(with $n_{i}\\!=0\hskip 0.85358pt$) equals $(4-\\!\hbox to0.0pt{\hskip
4.94992pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}})$.
The factor $\,2^{-\scriptsize{\hbox to0.0pt{\hskip
1.66495pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}}}/2}$
arises from the overall coefficient $\hskip 0.85358pt1/\\!\sqrt{2}\,$ of
Eq.(81). Substituting the kinematic numerators into Eqs.(82a)-(82b), we have
verified that the scattering amplitudes (82a)-(82b) agree with our previous
results KKString .

In addition, using the method of Eqs.(3.1)-(45), we can make high energy
expansion for the double-copied KK graviton amplitides (82b). Similar to
Eqs.(3.1)-(45), we derive the LO and NLO scattering amplitudes of four KK
gravitons from Eq.(82b), $\hskip
0.85358pt\mathcal{M}=\mathcal{M}_{0}+\delta\mathcal{M}\hskip 0.85358pt$, under
the $S^{1}\\!/\mathbb{Z}_{2}\hskip 0.85358pt$ orbifold compactification:

$\displaystyle\mathcal{M}_{0}$
$\displaystyle=-\frac{~{}\kappa^{2}~{}}{\,2^{\scriptsize{\hbox to0.0pt{\hskip
1.66495pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}}}/2+4}\,}\sum_{\mathsf{P}}\\!\sum_{j}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}\\!\prod_{k}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\frac{~{}\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda_{k})\hskip
0.85358pt\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda^{\prime}_{k})~{}}{s_{j}}\,,$
(83a) $\displaystyle\delta\mathcal{M}$
$\displaystyle=-\frac{~{}\kappa^{2}~{}}{\,2^{\scriptsize{\hbox to0.0pt{\hskip
1.66495pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$N$}}}/2+4}\,}\\!\\!\sum_{\lambda_{k},\lambda^{\prime}_{k}}^{\mathsf{P},\hskip
0.85358ptj}\\!\\!\prod_{k}\\!C_{\lambda_{k}\lambda^{\prime}_{k}}^{\hskip
0.85358pt\sigma_{k}}\\!\frac{~{}\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda_{k})\delta\mathcal{N}_{j}^{\mathsf{P}}\\!(\lambda^{\prime}_{k})\\!+\\!\mathcal{N}_{j}^{0,\mathsf{P}}\\!(\lambda^{\prime}_{k})\delta\mathcal{N}_{j}^{\mathsf{P}}\\!(\lambda_{k})\\!-\\!N_{j}^{0,\mathsf{P}}\\!(\lambda_{k})N_{j}^{0,\mathsf{P}}\\!(\lambda^{\prime}_{k})M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}/s_{j}\,}{s_{j}},$
(83b)

where the LO and NLO kinematic numerators
$(\mathcal{N}_{j}^{0,\mathsf{P}}\\!,\hskip
0.85358pt\delta\mathcal{N}_{j}^{\mathsf{P}})$ are derived under the high
energy expansion,

$\displaystyle\frac{\mathcal{N}_{j}^{\mathsf{P}}(\lambda)}{~{}1\\!-M^{2}_{\mathfrak{n}\mathfrak{n}_{j}}/s_{j}\,}$
$\displaystyle=\,\mathcal{N}_{j}^{0,\mathsf{P}}(\lambda)+\delta\mathcal{N}_{j}^{\mathsf{P}}(\lambda)\hskip
0.85358pt.$ (84)

For the four-point elastic KK graviton scattering, we verified Hang:2021fmp
explicitly that the above BCJ-type massive double-copy formulas give the
correct results at both the LO and NLO, and they agree with the results of the
extended massive KLT relations KKString  derived from the KK string theory
approach.

Using Eq.(81) and (82a), we can derive relations between the KK scattering
amplitudes under the $\hskip 0.85358ptS^{1}\\!/\mathbb{Z}_{2}\hskip 0.85358pt$
orbifold compactification and the corresponding KK scattering amplitudes under
the $S^{1}$ compactification. For the inelastic KK scattering processes
$\hskip 0.85358pt(0,0)\\!\rightarrow\\!(n,n)$ and
$(n,2n)\\!\rightarrow\\!(3n,4n)$ discussed in the example-I and example-II of
Section 3.2, we derive the following relations:

$\displaystyle{\mathcal{M}}\\!\left[{h}_{\pm 2}^{0}{h}_{\mp
2}^{0}{h}_{L}^{n}{h}_{L}^{n}\right]$
$\displaystyle=\,{\mathcal{M}}\\!\left[{h}_{\pm 2}^{0}{h}_{\mp
2}^{0}{h}_{L}^{\pm n}{h}_{L}^{\mp n}\right]\\!,$ (85a)
$\displaystyle{\mathcal{M}}\\!\left[{h}^{n}_{L}{h}^{2n}_{L}{h}^{3n}_{L}{h}^{4n}_{L}\right]$
$\displaystyle=\,\mbox{$\frac{\,{1}\,}{\,2\,}$}{\mathcal{M}}\\!\left[{h}^{\pm
n}_{L}{h}^{\mp 2n}_{L}{h}^{\mp 3n}_{L}{h}^{\pm 4n}_{L}\right]\\!.$ (85b)

The above relations hold for the external longitudinal KK graviton states with
either the exact longitudinal polarization tensor $\zeta_{L}^{\mu\nu}$ of
Eq.(25b) or the leading-order longitudinal polarization tensor $\hskip
0.85358pt\bar{\zeta}_{L}^{\mu\nu}$ of Eq.(52). To derive the relation (85a),
we have used the fact of $\hskip 0.85358pt\mathcal{M}\big{[}{h}_{\pm
2}^{0}{h}_{\mp 2}^{0}{h}_{L}^{\pm n}{h}_{L}^{\pm n}\big{]}\\!=\\!0\hskip
0.85358pt$ based upon Eq.(53), whereas for deriving the relation (85b), we
have used the fact that Eq.(58) gives all the allowed combinations of the
external KK indices in this scattering process.

For the inelastic KK scattering process $(n,n)\\!\rightarrow\\!(m,m)$
discussed in the example-III of Section 3.2, we can derive the KK scattering
amplitude under $S^{1}\\!/\mathbb{Z}_{2}$ compactification:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{n}\bar{h}_{L}^{n}\bar{h}_{L}^{m}\bar{h}_{L}^{m}\right]=\,\mbox{$\frac{\,{1}\,}{\,2\,}$}\left\\{\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp m}\bar{h}_{L}^{\pm m}\right]+\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\\!\left[\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm m}\bar{h}_{L}^{\mp m}\right]\right\\}$
$\displaystyle=\,\frac{~{}\kappa^{2}M_{n}^{2}\hskip
0.85358pt(X_{0}\\!+\\!X_{2}\hskip 0.85358ptc_{2\theta}\\!+\\!X_{4}\hskip
0.85358ptc_{4\theta}\\!+\\!X_{6}\hskip 0.85358ptc_{6\theta})~{}}{64\hskip
0.85358pt\bar{s}\hskip 1.42262pt(Q_{0}\\!+Q_{2}\hskip
0.85358ptc_{2\theta}\\!+Q_{4}\hskip 0.85358ptc_{4\theta})}\,,$ (86)

where the polynomials $\,\\{X_{i},Q_{j}\hskip 0.85358pt\\}\,$ are given by

$\displaystyle X_{0}$ $\displaystyle=2\hskip 0.85358pt\big{[}\hskip
0.85358pt85\hskip 0.85358pt\bar{s}^{6}+92\hskip
0.85358pt\bar{s}^{5}r_{+}^{2}-16\hskip 0.85358pt\bar{s}^{4}(49\hskip
0.85358ptr^{4}+95\hskip 0.85358ptr^{2}+49)+64\hskip
0.85358pt\bar{s}^{3}(13\hskip 0.85358ptr^{6}+45\hskip 0.85358ptr^{4}+45\hskip
0.85358ptr^{2}+13)$ $\displaystyle\hskip 12.80373pt-256\hskip
0.85358pt\bar{s}^{2}r^{2}(29\hskip 0.85358ptr^{4}+19\hskip
0.85358ptr^{2}+29)+15360\hskip 0.85358pt\bar{s}r^{4}r_{+}^{2}-28672\hskip
0.85358ptr^{6}\hskip 0.85358pt\big{]}\hskip 0.85358pt,$ $\displaystyle X_{2}$
$\displaystyle=-\big{[}\hskip 0.85358pt143\hskip
0.85358pt\bar{s}^{6}\\!-\\!1796\hskip 0.85358pt\bar{s}^{5}r_{+}^{2}+16\hskip
0.85358pt\bar{s}^{4}(193\hskip 0.85358ptr^{4}+335\hskip
0.85358ptr^{2}+193)\\!-320\hskip 0.85358pt\bar{s}^{3}(3\hskip
0.85358ptr^{6}\\!-29\hskip 0.85358ptr^{4}$ $\displaystyle\hskip
12.80373pt-29\hskip 0.85358ptr^{2}\\!+3)\\!-256\hskip
0.85358pt\bar{s}^{2}r^{2}(31\hskip 0.85358ptr^{4}\\!+273\hskip
0.85358ptr^{2}\\!+31)+64512\hskip 0.85358pt\bar{s}\hskip
0.85358ptr^{4}r_{+}^{2}-69632\hskip 0.85358ptr^{6}\hskip
0.85358pt\big{]}\hskip 0.85358pt,$ $\displaystyle X_{4}$
$\displaystyle=-2\hskip 0.85358pt\big{[}13\hskip
0.85358pt\bar{s}^{6}\\!-36\hskip 0.85358pt\bar{s}^{5}r_{+}^{2}\\!-\\!16\hskip
0.85358pt\bar{s}^{4}(17\hskip 0.85358ptr^{4}\\!+7\hskip
0.85358ptr^{2}\\!+17)+64\hskip 0.85358pt\bar{s}^{3}(5\hskip
0.85358ptr^{6}\\!-11\hskip 0.85358ptr^{4}\\!-11\hskip
0.85358ptr^{2}+5)~{}~{}~{}$ (87) $\displaystyle\hskip 12.80373pt+256\hskip
0.85358pt\bar{s}^{2}r^{2}(19\hskip 0.85358ptr^{4}\\!+53\hskip
0.85358ptr^{2}\\!+19)-\\!17408\hskip 0.85358pt\bar{s}\hskip
0.85358ptr^{4}r_{+}^{2}+4096\hskip 0.85358ptr^{6}\big{]}\hskip 0.85358pt,$
$\displaystyle X_{6}$ $\displaystyle=-(\bar{s}^{2}+4\hskip
0.85358pt\bar{s}\hskip 0.85358ptr_{+}^{2}+16\hskip
0.85358ptr^{2})^{2}(\bar{s}^{2}-4\bar{s}r_{+}^{2}+16r^{2})\hskip 0.85358pt,$
$\displaystyle Q_{0}$ $\displaystyle=3\hskip 0.85358pt\bar{s}^{4}\\!+8\hskip
0.85358pt\bar{s}^{3}r_{+}^{2}+16\hskip 0.85358pt\bar{s}^{2}(3\hskip
0.85358ptr^{4}\\!-20\hskip 0.85358ptr^{2}\\!+3)+128\hskip
0.85358pt\bar{s}\hskip 0.85358ptr^{2}r_{+}^{2}+768\hskip 0.85358ptr^{4}\hskip
0.85358pt,$ $\displaystyle Q_{2}$ $\displaystyle=-4\hskip
0.85358pt(\bar{s}^{2}\\!-\\!16)(\bar{s}^{2}\\!-\\!16\hskip
0.85358ptr^{4})\hskip
0.85358pt,~{}~{}~{}~{}~{}Q_{4}=(\bar{s}-4)^{2}(\bar{s}-4\hskip
0.85358ptr^{2})^{2}\hskip 0.85358pt.$

In the above Eq.(86) and hereafter, the external on-shell KK graviton states
such as $\bar{h}_{L}^{n}$ or $\bar{h}_{L}^{m}$ are defined as in Eq.(52) by
using the leading-order longitudinal polarization tensor $\hskip
0.85358pt\bar{\zeta}^{\mu\nu}_{L}$. We also denote their amplitudes by
$\overline{\hbox{}}$$\mathcal{M}$ . Making the high energy expansion for
Eq.(86) and using the LO relation (67), we derive the LO inelastic KK graviton
scattering amplitude as follows:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{n}\bar{h}_{L}^{n}\bar{h}_{L}^{m}\bar{h}_{L}^{m}\big{]}=\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}_{0}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm m}\bar{h}_{L}^{\mp
m}\big{]}=\frac{~{}\kappa^{2}\hskip 0.85358pts~{}}{\,64\,}\hskip
0.85358pt(7+c_{2\theta})^{2}\csc^{2}\\!\theta\,.$ (88)

Next, we inspect the elastic KK scattering amplitudes of
$(n,n)\\!\rightarrow\\!(n,n)$ as discussed in the example-IV of Section 3.2.
Under the $S^{1}\\!/\mathbb{Z}_{2}$ compactification, we derive the elastic
longitudinal KK graviton scattering amplitude in terms of the corresponding KK
graviton amplitudes under the $S^{1}$ compactification:

$\displaystyle\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\big{[}\bar{h}_{L}^{n}\bar{h}_{L}^{n}\bar{h}_{L}^{n}\bar{h}_{L}^{n}\big{]}$
$\displaystyle=\mbox{$\frac{\,{1}\,}{\,2\,}$}\\!\left\\{\hbox to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\big{]}\\!+\\!\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\big{]}\\!+\\!\hbox
to0.0pt{\hskip
4.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\big{[}\bar{h}_{L}^{\pm
n}\bar{h}_{L}^{\mp n}\bar{h}_{L}^{\pm n}\bar{h}_{L}^{\mp n}\big{]}\right\\}$
$\displaystyle=-\frac{~{}\kappa^{2}M_{n}^{2}\hskip 0.85358pt(X_{0}+X_{2}\hskip
0.85358ptc_{2\theta}+X_{4}\hskip 0.85358ptc_{4\theta}+X_{6}\hskip
0.85358ptc_{6\theta})\csc^{2}\theta~{}}{512\hskip 0.85358pt\bar{s}\hskip
0.85358pt(\bar{s}\\!-\\!4)[\hskip 0.85358pt(\bar{s}^{2}\\!+\\!24\hskip
0.85358pt\bar{s}\\!+\\!16)\\!-\\!(\bar{s}\\!-\\!4)^{2}\hskip
0.85358ptc_{2\theta}\hskip 0.85358pt]}\,,$ (89)

where the polynomials $\\{X_{i}\\}$ take the following forms,

$\displaystyle X_{0}$ $\displaystyle=-2\hskip 0.85358pt(255\hskip
0.85358pt\bar{s}^{5}\\!+2856\hskip 0.85358pt\bar{s}^{4}\\!-19168\hskip
0.85358pt\bar{s}^{3}\\!+35840\hskip
0.85358pt\bar{s}^{2}\\!+9984\bar{s}+14336)\hskip 0.85358pt,$ (90)
$\displaystyle X_{2}$ $\displaystyle=429\hskip