GRET of the 5d KK GR theory (GR5):

$\mathrm{KK\ GAET\,(YM5)}\ \Longrightarrow\ \mathrm{KK\ GRET\,(GR5)}\,.$
(2.3cx)

We have demonstrated that the residual term in the GRET (2.3cr) or (2.3p) is
indeed suppressed relative to the leading KK Goldstone $\phi_{n}$-amplitude;
and in the case of four-particle longitudinal KK graviton scattering, the
leading (helicity-zero) longitudinal KK graviton amplitude and KK Goldstone
amplitude scale as $\mathcal{O}(E^{2}M_{n}^{0})$ and are equal to each other;
while the residual term of the GRET is only of $\mathcal{O}(E^{0}M_{n}^{2})$,
as in Eq.(2.3cw), due to a nontrivial energy-cancellation of
$\,\mathcal{O}(E^{2})\\!\rightarrow\\!\mathcal{O}(E^{0})$.  This conclusion
can be readily extended to other longitudinal KK graviton scattering processes
with two or more external longitudinal KK graviton states. As a final remark,
we build the above correspondence (2.3cx) based on our current analyses of the
tree-level scattering amplitudes, and it will be worthwhile to further extend
it to loop orders by invoking the BRST transformations in both the 5d KK YM
gauge theory and the 5d KK GR theory [35]. We also note that our power-
counting method presented in section 3.2 holds for the general $N$-point
amplitudes with $L$ loops ($L\\!\geqslant 0\hskip 0.85358pt$), so our present
power-counting analysis can be extended up to loop orders in a straightforward
way.

##  Conclusions

Studying the structure of scattering amplitudes of Kaluza-Klein (KK) gravitons
and that of the KK gauge bosons is important for understanding the dynamics of
KK theories and the deep gauge-gravity connection. The KK gravitons and KK
gauge bosons serve as the key ingredients in all extra dimensional models
[3]-[6] and string theories [2] which attempt to resolve the naturalness
problem, the quantum gravity, and the gauge-gravity unification.

In this work, we studied the structure of the scattering amplitudes of the KK
gravitons and their KK Goldstone bosons (radions) with compactified fifth
dimension. In section 2, using a general $R_{\xi}$ gauge-fixing (2.3p) for the
quantization of 5d KK General Relativity (GR), we derived the massive KK
graviton propagator and the corresponding Goldstone boson propagators in
Eq.(2.3u). These propagators take particularly simple forms of Eq.(2.3v) under
the Feynman-’t Hooft gauge ($\,\xi_{n}\\!=\\!1\,$). We proved that the KK
graviton propagator is naturally free from the longstanding puzzle of the vDVZ
discontinuity [23], in contrast to that of the Fierz-Pauli gravity [18] and
alike [19]. Hence, the KK gravity theories provide a consistent realization of
mass-generations for the spin-2 KK gravitons.

With these, we presented in section 3.1 the formulation of the Gravitational
Equivalence Theorem (GRET) to connect the scattering amplitudes of
longitudinally-polarized (helicity-zero) KK gravitons $\,h_{L}^{n}\,$ to that
of the corresponding gravitational KK scalar Goldstone bosons
$\phi_{n}\,(\equiv\\!h^{55}_{n})$. The GRET is a manifestation of the
geometric “Higgs” mechanism at the $S$-matrix level. Starting from the general
Slavnov-Taylor-type identity (2.3c) for the gravitational gauge-fixing
functions, we derived its LSZ amputated form (2.3l) under the Feynman-’t Hooft
gauge at tree level, which suffices for the present study. From this we
derived the key GRET identity (2.3oa) and gave the GRET formulation in
Eqs.(2.3o) and (2.3p). Then, extending Weinberg’s power counting rule for the
low energy QCD [30] to the KK theories, we derived general energy-power
counting rule (2.3t) for the 5d KK GR theory. With this we derived the leading
energy-dependence of the $N$-particle longitudinal KK graviton scattering
amplitudes and of $N$-particle KK Goldstone scattering amplitudes in
Eqs.(2.3u)-(2.3v), namely, $\,D_{E}(Nh^{L}_{n})\\!=2(N\\!+\\!1)\\!+2L\,$ and
$\,D_{E}(N\phi_{n})\\!=\\!2+2L\hskip 0.85358pt$. We further counted the
superficial leading energy-dependence of the residual term
$\mathcal{M}_{\Delta}$ as in Eq.(2.3x), which gives
$D_{E}(N\tilde{v}_{n})\\!=\\!2+2L\,$. Using the GRET identity (2.3o), we
established the nontrivial energy cancellations in the $N$-particle
longitudinal KK graviton scattering amplitudes by $\hskip 0.85358ptE^{2N}$ as
in Eq.(2.3w), where the number of external KK states $N\\!\\!\geqslant\\!4\,$.
For the scattering amplitudes of $N$ longitudinal KK gravitons at tree level
($L\\!=\\!0$), this proves the large energy cancellations of
$\,E^{2N+2}\\!\\!\rightarrow\\!E^{2}\hskip 0.85358pt$. In the case of the four
longitudinal KK graviton scattering amplitudes ($N\\!\\!=\\!4$), this
establishes the energy cancellations of $\,E^{10}\\!\rightarrow\\!E^{2}\,$,
which we further demonstrated by explicit analyses in sections 4.2 and 5.2.
Hence, the GRET identity (2.3o) provides a general mechanism for guaranteeing
the nontrivial large energy-cancellations in the $N$-particle longitudinal KK
graviton amplitudes by $\,E^{2N}$, where $N\\!\geqslant 4$ . This conclusion
holds up to loop levels because the radiative multiplicative modification
factor $\,C_{\text{mod}}\,$ associated with each external Goldstone state is
energy-independent. Our present GRET formulation is highly nontrivial because
its residual term does not appear superficially suppressed relative to the
leading KK Goldstone amplitude in high energy limit by the naive power
counting. The suppression of the residual term was further established in the
following sections 4-5.

In section 4, we performed systematically a direct computation of the
gravitational KK Goldstone boson scattering amplitudes at tree level. In
section 4.1, we took a simple model of 5d gravitational scalar QED (GSQED5) as
an example and explicitly verified the GRET identity (2.3k) or (2.3q) for the
case of including a single external KK graviton field. Our analysis showed
that the GRET identity in this case holds exactly. Then, in section 4.2, we
derived the exact four-particle KK Goldstone boson scattering amplitude, and
expanded it to the leading order (LO) and the next-to-leading order (NLO)
under the high energy expansion, which are given in Eq.(2.3x) and
Eqs.(2.3ad)-(2.3af). The leading energy-dependence in these KK Goldstone
amplitudes is manifestly of $\mathcal{O}(E^{2})$ without any extra energy
cancellations among the individual diagrams. Hence, they are substantially
simpler than those of the longitudinal KK graviton amplitudes in the
literatures [14][15] since the latter involve various intricate energy
cancellations among individual diagrams from $\mathcal{O}(E^{10})$ down to
$\mathcal{O}(E^{2})$. With these we proved explicitly the equivalence between
the leading $h_{L}^{n}$-amplitudes and $\phi_{n}$-amplitudes at
$\mathcal{O}(E^{2})$, which supports the GRET (2.3p). Hence, the longitudinal-
Goldstone equivalence of the GRET guarantees the nontrivial large energy-power
cancellations in the longitudinal KK graviton amplitudes. We further computed
the difference between the exact $h_{L}^{n}$-amplitude and
$\phi_{n}$-amplitude as in Eq.(F.8), which has $\mathcal{O}(M_{n}^{2}E^{0})$
and determines the size of the residual term of the GRET to be of the NLO.

In section 5, we studied systematically the double-copy construction of the
gravitational KK scattering amplitudes by using the corresponding KK gauge
(Goldstone) boson scattering amplitudes in the 5d KK YM gauge theory, under
the high energy expansion. The conventional BCJ-type double-copy approach
[24][25]) is given for massless gauge theories and massless GR. Because the KK
gauge theories and KK GR can consistently generate masses for KK gauge bosons
and KK gravitons by geometric “Higgs” mechanism under compactification, we
expect that extending the conventional double-copy method to the KK theories
should be truly promising even though highly challenging due to the nontrivial
KK-mass-poles in the scattering amplitudes and the nontrivial polarization
tensors for the external longitudinal KK gravition states. Unlike the
conventional double-copy approaches, we proposed to realize the double-copy
construction by using the high energy expansion order by order. With this, we
demonstrated explicitly how such a double-copy construction can work at the LO
and the NLO, as in sections 5.2-5.3. This high energy expansion approach for
realizing our double-copy construction also perfectly matches our KK GAET and
GRET formulations.

In section 5.2, under the high energy expansion, we found that the LO KK gauge
boson (Goldstone) amplitudes have $\mathcal{O}(E^{0}M_{n}^{0})$ and the LO KK
graviton (Goldstone) amplitudes have $\mathcal{O}(E^{2}M_{n}^{0})$, which are
both mass-independent. Thus, we made an extended BCJ double-copy construction
from our LO KK gauge boson (Goldstone) amplitudes and fully reconstructed the
correct KK graviton (Goldstone) amplitudes at the LO, as shown in Eqs.(2.3ai)
and (2.3aq). Then, in section 5.3, we studied the extended double-copy
constructions of the NLO KK gauge/gravity amplitudes by making two types of
high energy expansions, under $\hskip 0.85358pt1/s_{0}\hskip 0.85358pt$
expansion (section 5.3.1) and under $\hskip 0.85358pt1/s\hskip 0.85358pt$
expansion (section 5.3.2), respectively. We showed that the NLO KK gauge
(Goldstone) boson amplitudes have $\mathcal{O}(M_{n}^{2}/E^{2})$ and the NLO
KK graviton (Goldstone) amplitudes have $\mathcal{O}(E^{0}M_{n}^{2})$, which
are both mass-dependent. We demonstrated that the double-copy construction for
the mass-dependent NLO KK-amplitudes is highly nontrivial, where the
conventional double-copy method could not fully work. We found that the reason
for this problem is due to violations of the kinematic Jacobi identities
(2.3t) at the NLO of the $\hskip 0.85358pt1/s_{0}\hskip 0.85358pt$ expansion,
where the generalized gauge-transformations (2.3p) cannot recover the
kinematic Jacobi identities as we explained below Eq.(2.3t). But, for our
extended double-copy construction under the $\hskip 0.85358pt1/s\hskip
0.85358pt$ expansion in section 5.3.2, we successfully recovered the kinematic
Jacobi identities as in Eq.(2.3bp) by using the generalized gauge-
transformations (2.3bo) and the solutions (2.3bq)-(2.3br). With these, we
derived the NLO double-copied KK graviton/Goldstone amplitudes (2.3ca) and
their difference (2.3cb), which predict the correct angular structures as in
the original exact KK graviton (Goldstone) amplitudes (F.6b)-(F.6c) and (F.8).
Finally, in section 5.3.3, we constructed an exact double-copy of the KK
graviton scattering amplitudes at the NLO based on our recent first principle
approach of the KK string theory [57]. We presented the exact double-copy
formulas for the LO and NLO KK graviton amplitudes in Eqs.(5.3.3)-(2.3ck),
respectively. Applying these formulas to the four-point elastic longitudinal
KK graviton scattering, we explicitly computed the double-copied LO amplitude
[which fully agrees to Eq.(2.3bz)] and the double-copied NLO amplitude (2.3cl)
[which exactly agrees to Eq.(F.6b)]. The above analyses and findings are truly
encouraging and we will pursue this direction in future works.

In passing, we recently studied [59] the extended double-copy construction of
the scattering amplitudes of massive gauge bosons/grvitons in the 3d
topologically massive Yang-Mills theory (TMYM) and the 3d topologically
massive gravity (TMG) theory [58]. There we proposed [59] a brand-new
Topological Equivalence Theorem (TET) to formulate the topological mass-
generation and to uncover the nontrivial energy cancellations in the massive
Chern-Simons gauge boson scattering amplitudes of the TMYM. Using the TET and
the double-copy construction, we further discovered [59] the strikingly large
energy cancellations in the massive Chern-Simons graviton scattering
amplitudes of the TMG.

Finally, in section 5.4, based upon the KK GAET identity (2.3cm) in the 5d KK
YM theory, we used double-copy approach to reconstruct the KK GRET identity
(2.3cr), and demonstrated a new correspondence of KK GAET $\Rightarrow$ KK
GRET in Eq.(2.3cx). Especially, we analyzed the (energy, mass)-dependence of
the residual term $\mathcal{M}_{\\!\Delta}$ in the GRET and deduced
$\,\mathcal{M}_{\\!\Delta}\\!=\mathcal{O}(E^{0}M_{n}^{2})\,$ in Eq.(2.3cw).
This justifies that even though the amplitudes in the GRET residual term
$\,\mathcal{M}_{\Delta}\,$ contain individual contributions having superficial
energy-dependence of $\,\mathcal{O}(E^{2})\,$ by naive power counting, they
are ensured to cancel down to $\mathcal{O}(E^{0}M_{n}^{2})$, in agreement with
our explicit computation of
$\,\mathcal{M}_{\\!\Delta}\\!=\delta\mathcal{M}\\!-\delta\widetilde{\mathcal{M}}\,$
in Eq.(F.8).

In summary, it is impressive that using the double-copy approach, we
established a new correspondence between the two energy-cancellations in the
four-particle longitudinal KK scattering amplitudes:
$\,E^{4}\\!\\!\rightarrow\\!E^{0}\,$ in the 5d KK YM gauge theory and
$\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$ in the 5d KK GR theory. This was
presented schematically in Eq.(2.3al). Furthermore, using the double-copy
approach, we analyzed the structure of the residual term
$\,\mathcal{M}_{\\!\Delta}$ in the GRET and further uncovered a new energy-
cancellation mechanism of $\,E^{2}\\!\rightarrow\\!E^{0}\,$ therein. Some of
our key analyses and findings of this substantial work are summarized in the
companion letter paper [60].

Acknowledgements We would like to thank Song He, Henry Tye, and Yang Zhang for
discussing the double-copy approaches and KLT relations, and thank Huan-Hang
Chi for discussing the conventional BCJ method and joining this study at an
early stage. We thank Sekhar Chivukula and Elizabeth Simmons for
correspondence on this subject. We also thank Tony Zee for discussing the
geometric mass-generation of Kaluza-Klein compactification. This research was
supported in part by National NSF of China (under grants Nos. 11835005,
12175136 and 11675086), by National Key R & D Program of China (under grant
No. 2017YFA0402204).

Appendix:

##  Appendix A Kinematics of KK Particle Scattering

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

Figure 7: Kinematics of the $\,2\\!\rightarrow\\!2\,$ scattering process in
the center-of-mass frame.

In the center-of-mass frame (Fig.7), we define the momenta as follows:

$\displaystyle p_{1}^{\mu}=$ $\displaystyle-\left(E,0,0,k\right),$
$\displaystyle\quad\quad p_{2}^{\mu}=-\left(E,0,0,-k\right),$ $\displaystyle
p_{3}^{\mu}=$ $\displaystyle\left(E,ks_{\theta},0,kc_{\theta}\right),$
$\displaystyle\quad\quad
p_{4}^{\mu}=\left(E,-ks_{\theta},0,-kc_{\theta}\right),$ (A.1)

where $k=|\vec{p}\,|$. Then, the Mandelstam variables $(s,t,u)$ take the
following form:

$\displaystyle s\,$ $\displaystyle=-\left(p_{1}+p_{2}\right)^{2}=4E^{2}\,,$
(A.2a) $\displaystyle t\,$
$\displaystyle=-\left(p_{1}+p_{4}\right)^{2}=-\frac{s-4M_{n}^{2}}{2}(1+c_{\theta})\,,$
(A.2b) $\displaystyle u\,$
$\displaystyle=-\left(p_{1}+p_{3}\right)^{2}=-\frac{s-4M_{n}^{2}}{2}(1-c_{\theta})\,.$
(A.2c)

Optionally, with Eq.(A.2), we can also use the relation
$\,E^{2}\\!=k^{2}\\!+\\!M_{n}^{2}\,$ to define another set of Mandelstam
variables $(s_{0},t_{0},u_{0})$:

$\displaystyle s_{0}$ $\displaystyle\,=\,4k^{2}\,,$ (A.3a) $\displaystyle
t_{0}$ $\displaystyle\,=\,-\frac{\,s_{0}\,}{2}(1+c_{\theta})\,,$ (A.3b)
$\displaystyle u_{0}$
$\displaystyle\,=\,-\frac{\,s_{0}\,}{2}(1-c_{\theta})\,.$ (A.3c)

The summations of the Mandelstam variables (A.2) and (A.3) obey the identities
$\,s+t+u\\!=\\!4M_{n}^{2}$ and $\,s_{0}\\!+t_{0}\\!+u_{0}\\!=0$ ,
respectively.

Moreover, the above formulas can be extended to the general scattering process
$\,X_{n}X_{k}\\!\\!\rightarrow\\!X_{m}X_{\ell}\,$, where
$\,n,k,m,\ell\\!\geqslant 0$ . Thus, the sum of these Mandelstam variables
$(s,t,u)$ satisfies
$\,s+t+u=M_{n}^{2}\\!+\\!M_{k}^{2}\\!+\\!M_{m}^{2}\\!+\\!M_{\ell}^{2}$ . The
incident and outgoing states have the following momenta in the center of mass
frame,

$\displaystyle p_{1}^{\mu}=$ $\displaystyle-\\!(E_{1},0,0,k),$
$\displaystyle\hskip 28.45274ptp_{2}^{\mu}=-(E_{2},0,0,-k),$ $\displaystyle
p_{3}^{\mu}=$
$\displaystyle\,(E_{3},k^{\prime}s_{\theta},0,k^{\prime}c_{\theta}),$
$\displaystyle\hskip
28.45274ptp_{4}^{\mu}=(E_{4},-k^{\prime}s_{\theta},0,-k^{\prime}c_{\theta}),$
(A.4)

where the energy conservation condition,
$\sqrt{s\,}\\!=E_{1}\\!+\\!E_{2}=E_{3}\\!+\\!E_{4}$ , determines the momenta
$k$ and $k^{\prime}$ as follows:

$\displaystyle k$
$\displaystyle=\frac{1}{\,2\sqrt{s\,}\,}\left([s-(M_{1}\\!+\\!M_{2})^{2}][s-(M_{1}\\!-\\!M_{2})^{2}]\right)^{\\!{1}/{2}},$
$\displaystyle k^{\prime}$
$\displaystyle=\frac{1}{\,2\sqrt{s\,}\,}\left([s-(M_{3}\\!+\\!M_{4})^{2}][s-(M_{3}\\!-\\!M_{4})^{2}]\right)^{\\!{1}/{2}}.$
(A.5)

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

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

where $(\epsilon_{\pm}^{\mu},\,\epsilon_{L}^{\mu})$ denote the (transverse,
longitudinal) polarization vectors of a vector boson with the same 4-momentum
$p^{\mu}$. These polarization tensors satisfy the traceless and orthonormal
conditions. They are also orthogonal to the 4-momentum $p^{\mu}$ of the KK
graviton. Thus, the following conditions hold:

$\eta_{\mu\nu}\varepsilon^{\mu\nu}=0\,,\qquad\varepsilon_{\lambda}^{\mu\nu}\varepsilon_{\lambda^{\prime}\\!,\,\mu\nu}^{*}=\delta_{\lambda\lambda^{\prime}}\,,\qquad
p_{\mu}\varepsilon^{\mu\nu}=0\,,$ (A.7)

where $\,\lambda,\lambda^{\prime}\,(=\pm 2,\,\pm 1,\,0)\,$ are the helicity
indices of the KK graviton.

##  Appendix B From $R_{\xi}$ Gauge to Unitary Gauge

We note that the KK graviton propagator (2.3ua) in the general $R_{\xi}$ gauge
can be decomposed into the unitary gauge propagator (2.3w) plus the
$\xi_{n}$-dependent part. In momentum space, we present this decomposition in
the following form:

$\displaystyle\mathcal{D}^{\mu\nu\alpha\beta}_{nm}(p)\ =$ $\displaystyle\
\mathcal{D}_{nm,{\mathrm{UG}}}^{\mu\nu\alpha\beta}(p)+\mathcal{D}_{nm,\,\xi}^{\mu\nu\alpha\beta}(p)\,,$
(B.1a) $\displaystyle\mathcal{D}_{nm,{\mathrm{UG}}}^{\mu\nu\alpha\beta}(p)\ =$
$\displaystyle\ -\frac{\text{i}\delta_{nm}}{2}\frac{\leavevmode\nobreak\
\bar{\eta}^{\mu\alpha}\bar{\eta}^{\nu\beta}\\!+\\!\bar{\eta}^{\mu\beta}\bar{\eta}^{\nu\alpha}\\!-\\!\frac{2}{3}\bar{\eta}^{\mu\nu}\bar{\eta}^{\alpha\beta}\leavevmode\nobreak\
}{\,p^{2}\\!+\\!M_{n}^{2}\,}\,,$ (B.1b)
$\displaystyle\mathcal{D}_{nm,\,\xi}^{\mu\nu\alpha\beta}(p)\ =$
$\displaystyle\
\frac{\text{i}\delta_{nm}/2}{\,p^{2}+\xi_{n}M_{n}^{2}\,}\biggl{[}\\!\left(\\!\eta^{\mu\alpha}\\!+\\!\frac{p^{\mu}p^{\alpha}}{\xi_{n}M_{n}^{2}\,}\\!\right)\\!\frac{p^{\nu}p^{\beta}}{\,\xi_{n}M_{n}^{2}\,}\\!+\\!\left(\\!\eta^{\mu\beta}\\!+\\!\frac{p^{\mu}p^{\beta}}{\xi_{n}M_{n}^{2}\,}\\!\right)\\!\frac{p^{\nu}p^{\alpha}}{\,\xi_{n}M_{n}^{2}\,}$
$\displaystyle\
+\\!\left(\\!\eta^{\nu\alpha}\\!+\\!\frac{p^{\nu}p^{\alpha}}{\xi_{n}M_{n}^{2}\,}\\!\right)\\!\\!\frac{p^{\mu}p^{\beta}}{\,\xi_{n}M_{n}^{2}\,}\\!+\\!\left(\\!\eta^{\nu\beta}\\!+\\!\frac{p^{\nu}p^{\beta}}{\,\xi_{n}M_{n}^{2}\,}\\!\right)\\!\frac{p^{\mu}p^{\alpha}}{\,\xi_{n}M_{n}^{2}\,}\\!\biggr{]}-\frac{\text{i}\delta_{nm}2p^{\mu}p^{\nu}p^{\alpha}p^{\beta}}{\,(p^{2}\\!+\\!\xi_{n}^{2}M_{n}^{2})\,\xi_{n}M_{n}^{4}\,}$
$\displaystyle\
+\frac{\text{i}\delta_{nm}/6}{\,p^{2}\\!+\\!(3\xi_{n}\\!-\\!2)M_{n}^{2}\,}\\!\left(\\!\eta^{\mu\nu}\\!\\!-\\!\frac{\,2p^{\mu}p^{\nu}\,}{M_{n}^{2}}\\!\right)\\!\\!\left(\\!\eta^{\alpha\beta}\\!-\\!\frac{\,2p^{\alpha}p^{\beta}\,}{M_{n}^{2}}\\!\right)\\!,$
(B.1c)

where
$\,\bar{\eta}^{\mu\nu}\\!=\\!\eta^{\mu\nu}\\!+p^{\mu}p^{\nu}\\!/\\!M^{2}$. We
see that the $\xi_{n}$-dependent part
$\mathcal{D}_{nm,\,\xi}^{\mu\nu\alpha\beta}(p)$ vanishes under
$\,\xi_{n}\\!\\!\rightarrow\\!\infty$. So, the propagator
$\,\mathcal{D}^{\mu\nu\alpha\beta}_{nm}(p)\,$ will reduce to the unitary gauge
form $\,\mathcal{D}_{nm,{\mathrm{UG}}}^{\mu\nu\alpha\beta}(p)\,$ in this
limit. Also, the gravitational KK Goldstone propagators
$\mathcal{D}_{nm}^{\mu\nu}(p)$ and $\mathcal{D}_{nm}(p)$ in
Eqs.(2.3ub)-(2.3uc) vanish in this limit
$\,\xi_{n}\\!\\!\rightarrow\\!\infty\,$, which removes the unphysical KK
Goldstone bosons in the unitary gauge as expected. For the Feynman-’t Hooft
gauge ($\hskip 0.85358pt\xi_{n}\\!=1$), we find that the $\xi_{n}$-dependent
part of the KK graviton propagator takes a much simpler form:

$\displaystyle\mathcal{D}_{nm,\,\xi}^{\mu\nu\alpha\beta}(p)\Big{|}_{\xi=1}=$
$\displaystyle\ \frac{\text{i}\delta_{nm}/6}{\leavevmode\nobreak\
p^{2}\\!+\\!M_{n}^{2}\leavevmode\nobreak\
}\\!\left(\\!\eta^{\mu\nu}\\!\\!-\\!\frac{\,2p^{\mu}p^{\nu}\,}{M_{n}^{2}}\\!\right)\\!\\!\left(\\!\eta^{\alpha\beta}\\!-\\!\frac{\,2p^{\alpha}p^{\beta}\,}{M_{n}^{2}}\\!\right)$
$\displaystyle\
+\frac{\text{i}\delta_{nm}/2}{\,(p^{2}\\!+\\!M_{n}^{2})M_{n}^{2}\,}\\!\left(p^{\mu}p^{\alpha}\eta^{\nu\beta}\\!+p^{\nu}p^{\alpha}\eta^{\mu\beta}\\!+p^{\mu}p^{\beta}\eta^{\nu\alpha}\\!+p^{\nu}p^{\beta}\eta^{\mu\alpha}\right)\\!.\hskip
28.45274pt$ (B.2)

In passing, a $R_{\xi}$ gauge-fixing was considered [48] in the Randall-
Sundrum model with warped 5d which contains additional terms related to warp
parameter; we note that their KK graviton propagator was written in a rather
different form, but can be converted into the form consistent with our (B.1).

We note that the Lagrangian (2.3m) is invariant under the general coordinate
transformation (gauge transformation):

$\hat{h}_{AB}\,\rightarrow\leavevmode\nobreak\
\hat{h}_{AB}^{\prime}=\hat{h}_{AB}-2\partial_{(A}\hat{\chi}_{B)}\,,$ (B.3)

where $\hat{\chi}_{A}(x)$ is an infinitesimal translation which refers to a
vector field generating a one-parameter diffeomorphism group in the background
spacetime.

Taking Eq.(2.3h) under the KK expansion (2.3l), we derive the following gauge
transformations for the KK fields:

$\displaystyle h^{\mu\nu}_{n}$ $\displaystyle\,\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
h^{\prime\,\mu\nu}_{n}=\,h^{\mu\nu}_{n}-2\partial^{(\mu}\chi^{\nu)}_{n}-M_{n}\eta^{\mu\nu}\chi^{5}_{n}\,,$
(B.4a) $\displaystyle\mathcal{A}^{\mu}_{n}$
$\displaystyle\,\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
\leavevmode\nobreak\
\mathcal{A}^{\prime\,\mu}_{n}=\,\mathcal{A}^{\mu}_{n}-\partial^{\mu}\chi^{5}_{n}+M_{n}\chi^{\mu}_{n}\,,$
(B.4b) $\displaystyle\phi_{n}$ $\displaystyle\,\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
\phi^{\prime}_{n}=\,\phi_{n}-2M_{n}\chi^{5}_{n}\,.$ (B.4c)

In the above the group parameters $(\chi_{n}^{\mu},\,\chi_{n}^{5})$ arise from
the following KK expansions of the corresponding 5d parameters
$(\hat{\chi}^{\mu},\,\hat{\chi}^{5})$:

$\displaystyle\hat{\chi}^{\mu}(x^{\nu},x^{5})$
$\displaystyle\,=\,\frac{1}{\sqrt{L}}\left[\,\chi^{\mu}_{0}(x^{\nu})+\sqrt{2}\sum_{n=1}^{\infty}\chi^{\mu}_{n}(x^{\nu})\cos\\!\frac{\,n\pi
x^{5}\,}{L}\,\right]\\!,$ (B.5a) $\displaystyle\hat{\chi}^{5}(x^{\nu},x^{5})$
$\displaystyle\,=\,\sqrt{\frac{2}{L}}\,\sum_{n=1}^{\infty}\chi^{5}_{n}(x^{\nu})\sin\\!\frac{\,n\pi
x^{5}\,}{L}\,,$ (B.5b)

where we set $\,(\hat{\chi}^{\mu},\,\hat{\chi}^{5})$ as (even, odd) under the
$\mathbb{Z}_{2}$ reflection of 5d orbifold.

To transform into unitary gauge, we choose the gauge parameters as follows:

$\displaystyle\chi^{\mu}_{n}=-\frac{1}{M_{n}}\\!\left(\\!\mathcal{A}^{\mu}_{n}-\frac{\,\partial^{\mu}\phi_{n}\,}{2M_{n}}\right)\\!,\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\chi^{5}_{n}=\frac{\phi_{n}}{2M_{n}}\,.$ (B.6)

Then, we derive the field transformations to the unitary gauge:

$\displaystyle h^{\mu\nu}_{n}$ $\displaystyle\,\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
h^{\prime\,\mu\nu}_{n}\\!=\,h^{\mu\nu}_{n}+\frac{2}{M_{n}}\,\partial^{(\mu}\\!\mathcal{A}^{\nu)}_{n}-\frac{1}{2}\\!\left(\\!\eta^{\mu\nu}\\!+\\!\frac{2\partial^{\mu}\partial^{\nu}}{M_{n}^{2}}\right)\\!\phi_{n}\,,$
(B.7a) $\displaystyle\mathcal{A}^{\mu}_{n}$
$\displaystyle\,\leavevmode\nobreak\ \rightarrow\leavevmode\nobreak\
\leavevmode\nobreak\ \mathcal{A}^{\prime\,\mu}_{n}=\,0\,,$ (B.7b)
$\displaystyle\phi_{n}$ $\displaystyle\,\leavevmode\nobreak\
\rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \phi_{n}^{\prime}=\,0\,.$
(B.7c)

Thus, under the unitary gauge, both the KK Goldstone states
$\,\mathcal{A}^{\mu}_{n}\,$ and $\,\phi_{n}$ ($n\\!>\\!0$) are gauged away, so
the 4d action of the Lagrangian (2.2) becomes:

$\displaystyle S_{\mathrm{eff}}$
$\displaystyle\,=\int\\!\text{d}^{4}x\\!\sum_{n=0}^{\infty}\bigg{\\{}\\!\\!-\mbox{$\frac{\,{1}\,}{2}$}(\partial^{\mu}h_{n})^{2}+\mbox{$\frac{\,{1}\,}{2}$}(\partial^{\rho}h_{n}^{\mu\nu})^{2}+\partial_{\mu}h^{\mu\nu}_{n}\partial_{\nu}h_{n}-\partial_{\mu}h^{\mu\rho}_{n}\partial^{\nu}h_{\nu\rho,n}$
$\displaystyle\hskip
60.00009pt-\mbox{$\frac{\,{1}\,}{2}$}M^{2}_{n}\left[\,h_{n}^{2}\\!-\\!(h^{\mu\nu}_{n})^{2}\,\,\right]\\!\bigg{\\}}+\mbox{$\frac{\,{3}\,}{4}$}(\partial_{\mu}\phi_{0})^{2}\,.$
(B.8)

##  Appendix C Feynman Rules for KK Graviton Interaction with Matter

In this Appendix, we present the relevant Feynman rules of the 5d
gravitational scalar QED (GSQED5) as studied in section 4.1, including the
propagators of the matter fields and the vertices for the KK graviton
(Goldstone) interaction with the matter fields. All the Feynman Rules are
derived in the Feynman-’t Hooft gauge ($\zeta_{n}=1$).

We first present the photon propagator and scalar propagator as follows:

$\mathcal{D}_{nm}^{\mu\nu}(p)=\frac{-\text{i}\eta^{\mu\nu}\delta_{nm}}{p^{2}+M_{n}^{2}}\,,\qquad\mathcal{D}_{nm}(p)=\frac{\text{i}\delta_{nm}}{p^{2}+m_{n}^{2}}\,,$
(C.1)

where the KK number $n\\!\geqslant\\!0$ and the KK mass for the scalar field
is $\,m_{n}\\!=\\!\sqrt{m_{0}^{2}\\!+\\!M_{n}^{2}\,}\,$.

Then, with the relations between the 4d coupling constants and 5d couplings
$\,e=\hat{e}/\\!\sqrt{L\,}$ and $\,\kappa=\hat{\kappa}/\\!\sqrt{L\,}$ , we
derive the cubic and quartic interaction vertices for the KK graviton
(Goldstone) interactions with matter, which are presented in Fig. 8.

Figure 8: Feynman rules for the cubic and quartic interaction vertices between
the KK graviton (KK Goldstone boson) and matter fields.

##  Appendix D Gravitational KK Goldstone Amplitudes from Goldstone Exchanges
and Contact Interactions

In this Appendix, we derive the Feynman rules of the KK Goldstone self-
interactions which we will use to compute the subleading diagrams in Fig. 5,
for the analysis of section 4.2.

$\hat{\mathcal{L}}_{1}[\hat{\mathcal{A}}\hat{\phi}^{2}]$ | $\hat{\mathcal{A}}^{\mu}\partial_{\mu}\hat{\phi}\,\partial_{5}\hat{\phi}$ | $\hat{\mathcal{A}}^{\mu}(\partial_{\mu}\partial_{5}\hat{\phi})\hat{\phi}$ | $\partial_{\mu}\hat{\mathcal{A}}^{\mu}\partial_{5}\hat{\phi}\,\hat{\phi}$ | $\partial_{5}\hat{\mathcal{A}}^{\mu}\partial_{\mu}\hat{\phi}\,\hat{\phi}$ | $\partial_{\mu}\partial_{5}\hat{\mathcal{A}}^{\mu}\hat{\phi}^{2}$
---|---|---|---|---|---
$\hat{\mathcal{L}}_{1}[\hat{\phi}^{3}]$ | $\hat{\phi}(\partial_{\mu}\hat{\phi})^{2}$ | $\hat{\phi}(\partial_{5}\hat{\phi})^{2}$ | $\hat{\phi}^{2}\partial_{\mu}^{2}\hat{\phi}$ | $\hat{\phi}^{2}\partial_{5}^{2}\hat{\phi}$ |
$\hat{\mathcal{L}}_{2}[\hat{\phi}^{4}]$ | $\hat{\phi}^{2}(\partial_{\mu}\hat{\phi})^{2}$ | $\hat{\phi}^{2}(\partial_{5}\hat{\phi})^{2}$ | $\hat{\phi}^{3}\partial_{\mu}^{2}\hat{\phi}$ | $\hat{\phi}^{3}\partial_{5}^{2}\hat{\phi}$
Table 2: Lorentz-invariant vertices in the 5d Lagrangian terms
$\hat{\mathcal{L}}_{1}[\hat{\mathcal{A}}\hat{\phi}^{2}]$,
$\hat{\mathcal{L}}_{1}[\hat{\phi}^{3}]$, and
$\,\hat{\mathcal{L}}_{2}[\hat{\phi}^{4}]$.

Under the basis defined in Table 2, we can expand the Lorentz-invariant
structure of 5d Lagrangian terms
$\hat{\mathcal{L}}_{1}[\hat{\mathcal{A}}\hat{\phi}^{2}]$,
$\hat{\mathcal{L}}_{1}[\hat{\phi}^{3}]$, and
$\hat{\mathcal{L}}_{2}[\hat{\phi}^{4}]$ as follows:

$\displaystyle\hat{\mathcal{L}}_{1}[\hat{\mathcal{A}}\hat{\phi}^{2}]$
$\displaystyle\,=\,b_{1}\,\hat{\mathcal{A}}^{\mu}\partial_{\mu}\hat{\phi}\,\partial_{5}\hat{\phi}+b_{2}\,\hat{\mathcal{A}}^{\mu}\hat{\phi}\,\partial_{\mu}\partial_{5}\hat{\phi}\,,$
(D.4a) $\displaystyle\hat{\mathcal{L}}_{1}[\hat{\phi}^{3}]$
$\displaystyle\,=\,c_{1}\,\hat{\phi}\,(\partial_{\mu}\hat{\phi})^{2}+c_{2}\,\hat{\phi}\,(\partial_{5}\hat{\phi})^{2}\,,$
(D.4b) $\displaystyle\hat{\mathcal{L}}_{2}[\hat{\phi}^{4}]$
$\displaystyle\,=\,d_{1}\,\hat{\phi}^{2}(\partial_{\mu}\hat{\phi})^{2}+d_{2}\,\hat{\phi}^{2}(\partial_{5}\hat{\phi})^{2}\,,$
(D.4c)

where we have computed systematically the coefficients
$(b_{1},\,b_{2},\,c_{1},\,c_{2},\,d_{1},\,d_{2})$ as follows:

$\displaystyle(b_{1},\,b_{2},\,c_{1},\,c_{2},\,d_{1},\,d_{2})\,=\left(0,\,\frac{1}{\sqrt{2\,}\,},\,\sqrt{\frac{2}{3}\,},\,-\frac{1}{\sqrt{6\,}\,}\,,-\frac{1}{\,8\,},\,\frac{1}{\,4\,}\right)\\!.$
(D.5)

Then, by integrating over $x^{5}$ on the 5d interval $[0,\,L]$, we derive the
4d effective KK Lagrangian terms as follows:

$\displaystyle\mathcal{L}_{1}[\mathcal{A}\phi^{2}]$
$\displaystyle=-\frac{\kappa}{\sqrt{2\,}}\sum_{n,m,\ell=1}^{\infty}\\!\biggl{\\{}b_{1}M_{\ell}[\sqrt{2}\mathcal{A}^{\mu}_{n}\partial_{\mu}\phi_{0}\phi_{\ell}\delta_{n\ell}+\mathcal{A}^{\mu}_{n}\partial_{\mu}\phi_{m}\phi_{\ell}\widetilde{\Delta}_{3}(m,n,\ell)]$
$\displaystyle+b_{2}M_{\ell}[\sqrt{2}\mathcal{A}^{\mu}_{n}\phi_{0}\partial_{\mu}\phi_{\ell}\delta_{n\ell}+\mathcal{A}^{\mu}_{n}\phi_{m}\partial_{\mu}\phi_{\ell}\widetilde{\Delta}_{3}(m,n,\ell)]\biggr{\\}},$
(D.6a) $\displaystyle\mathcal{L}_{1}[\phi^{3}]$
$\displaystyle=\frac{\kappa}{\sqrt{2\,}}\sum_{n,m,\ell=1}^{\infty}\bigg{\\{}c_{1}[\sqrt{2}(\phi_{0}(\partial_{\mu}\phi_{0})^{2}+\phi_{0}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{\ell}\,\delta_{m\ell}+\phi_{n}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{m}\delta_{nm}$
$\displaystyle+\phi_{n}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{\ell}\,\delta_{n\ell})+\phi_{n}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{\ell}\Delta_{3}(n,m,\ell)]+c_{2}M_{m}M_{\ell}[\sqrt{2}\,\phi_{0}\phi_{m}\phi_{\ell}\delta_{m\ell}$
$\displaystyle+\phi_{n}\phi_{m}\phi_{\ell}\widetilde{\Delta}_{3}(n,m,\ell)]\bigg{\\}},$
(D.6b) $\displaystyle\mathcal{L}_{2}[\phi^{4}]$
$\displaystyle=\frac{\kappa^{2}}{2\,}\\!\sum_{n,m,\ell,k=1}^{\infty}\biggl{\\{}d_{1}\\{2\left(\phi_{0}\partial_{\mu}\phi_{0}\right)^{2}+2[\left(\partial_{\mu}\phi_{0}\right)^{2}\phi_{n}\phi_{m}\delta_{nm}+\phi_{0}\partial_{\mu}\phi_{0}\phi_{n}\partial^{\mu}\phi_{\ell}\delta_{n\ell}$
$\displaystyle+\phi_{0}\partial_{\mu}\phi_{0}\phi_{n}\partial^{\mu}\phi_{k}\delta_{nk}+\phi_{0}\partial_{\mu}\phi_{0}\phi_{m}\partial^{\mu}\phi_{k}\delta_{mk}+\phi_{0}\partial_{\mu}\phi_{0}\phi_{m}\partial^{\mu}\phi_{\ell}\delta_{m\ell}+\phi_{0}^{2}\partial_{\mu}\phi_{\ell}\partial^{\mu}\phi_{k}\delta_{\ell
k}]$
$\displaystyle+\sqrt{2}[\partial_{\mu}\phi_{0}\phi_{n}\phi_{m}\partial^{\mu}\phi_{\ell}\Delta_{3}(n,m,\ell)+\partial_{\mu}\phi_{0}\phi_{n}\phi_{m}\partial^{\mu}\phi_{k}\Delta_{3}(n,m,k)+\partial_{\mu}\phi_{0}\phi_{n}\phi_{\ell}\partial^{\mu}\phi_{k}\Delta_{3}(n,\ell,k)$
$\displaystyle+\phi_{0}\phi_{m}\partial_{\mu}\phi_{\ell}\partial^{\mu}\phi_{k}\Delta_{3}(m,\ell,k)]+\phi_{n}\phi_{m}\partial_{\mu}\phi_{\ell}\partial^{\mu}\phi_{k}\Delta_{4}(n,m,\ell,k)\\}+d_{2}M_{\ell}M_{k}[2(\phi_{0})^{2}\phi_{\ell}\phi_{k}\delta_{\ell
k}$
$\displaystyle+\sqrt{2}\phi_{0}\phi_{m}\phi_{\ell}\phi_{k}\widetilde{\Delta}_{3}(m,\ell,k)+\sqrt{2}\phi_{0}\phi_{n}\phi_{\ell}\phi_{k}\widetilde{\Delta}_{3}(n,\ell,k)+\phi_{n}\phi_{m}\phi_{\ell}\phi_{k}\widetilde{\Delta}_{4}(n,m,\ell,k)]\biggr{\\}},$
(D.6c)

where

$\displaystyle\Delta_{4}(n,m,\ell,k)=$
$\displaystyle\,\,\delta(n\\!+\\!m\\!+\\!\ell\\!-\\!k)+\delta(n\\!+\\!m\\!-\\!\ell\\!-\\!k)+\delta(n\\!-\\!m\\!+\\!\ell\\!-\\!k)+\delta(n\\!-\\!m\\!-\\!\ell\\!-\\!k)\hskip
36.98857pt$
$\displaystyle+\delta(n\\!-\\!m\\!-\\!\ell\\!+\\!k)+\delta(n\\!+\\!m\\!-\\!\ell\\!+\\!k)+\delta(n\\!-\\!m\\!+\\!\ell\\!+\\!k)\,,$
(D.7a) $\displaystyle\widetilde{\Delta}_{4}(n,m,\ell,k)=$
$\displaystyle\,\,\delta(n\\!+\\!m\\!+\\!\ell\\!-\\!k)-\delta(n\\!+\\!m\\!-\\!\ell\\!-\\!k)+\delta(n\\!-\\!m\\!+\\!\ell\\!-\\!k)-\delta(n\\!-\\!m\\!-\\!\ell\\!-\\!k)$
$\displaystyle+\delta(n\\!-\\!m\\!-\\!\ell\\!+\\!k)-\delta(n\\!+\\!m\\!-\\!\ell\\!+\\!k)+\delta(n\\!-\\!m\\!+\\!\ell\\!+\\!k)\,.$
(D.7b)

With these, we derive the cubic and quartic interaction vertices of
$\mathcal{A}\phi^{2}$, $\phi^{3}$, and $\phi^{4}$, which are presented in Fig.
9.

Figure 9: Feynman rules for the cubic and quartic interaction vertices among
the KK gravitational Goldstone bosons $\phi_{n}$ and $\mathcal{A}^{\mu}_{n}$ .

By explicit calculations, we find that the scattering amplitudes of
$\,\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}\,$ from the $\phi_{0}$
($\phi_{2n}$) exchanges and from the contact interaction are all of
$\,\mathcal{O}(M_{n}^{2})\,$ under the high energy expansion, which do not
contribute to the LO Goldstone boson amplitude of $\mathcal{O}(E^{2})$ . This
conclusion still holds for the inelastic scattering process
$\phi_{n}\phi_{k}\\!\rightarrow\\!\phi_{m}\phi_{\ell}$ .

##  Appendix E Massless Graviton Scattering and Double-Copy in 4d

For the sake of comparison, in this Appendix we compute the scattering
amplitudes of four gluons and of four gravitons at tree level in 4d, by using
the conventional Feynman techniques and the reduced super-string amplitudes,
respectively. We also verify the double-copy construction of massless graviton
amplitudes from the massless gluon amplitudes by using the color-kinematics
(CK) duality. We find that the conversion constant between the gauge boson
coupling and the graviton coupling in 4d differs from what we have obtained in
the 5d KK theory analysis (section 5.2).

###  Massless Graviton Scattering from Double-Copy Construction in 4d

For an SU(N) non-Abelian gauge theory, we can express the four-gluon
scattering amplitudes at tree level as follows:

$\mathcal{T}[gg\rightarrow
gg]\,=\,-g^{2}\,(\mathcal{T}_{c}+\mathcal{T}_{s}+\mathcal{T}_{t}+\mathcal{T}_{u})\,,$
(E.1)

where the amplitude contains the contributions from a contact interaction
diagram and the $(s,t,u)$-channel pole-diagrams whose amplitudes are given by

$\displaystyle\mathcal{T}_{c}\ =$ $\displaystyle\ \
\mathcal{C}_{s}\left[\,(\epsilon_{1}\\!\cdot\epsilon_{3})(\epsilon_{2}\\!\cdot\epsilon_{4})-(\epsilon_{1}\\!\cdot\epsilon_{4})(\epsilon_{2}\\!\cdot\epsilon_{3})\,\right]+\mathcal{C}_{t}\left[\,(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot\epsilon_{4})-(\epsilon_{1}\\!\cdot\epsilon_{3})(\epsilon_{2}\\!\cdot\epsilon_{4})\,\right]\hskip
22.76219pt$ $\displaystyle\
+\mathcal{C}_{u}\left[\,(\epsilon_{1}\\!\cdot\epsilon_{4})(\epsilon_{2}\\!\cdot\epsilon_{3})-(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot\epsilon_{4})\,\right],$
(E.2a) $\displaystyle\mathcal{T}_{s}\ =$ $\displaystyle\ \
\frac{\,\mathcal{C}_{s}\,}{s}\left[\,\left(p_{1}\\!-p_{2}\right)(\epsilon_{1}\\!\cdot\epsilon_{2})+2(p_{2}\\!\cdot\epsilon_{1})\epsilon_{2}-2(p_{1}\\!\cdot\epsilon_{2})\epsilon_{1}\,\right]$
$\displaystyle\hskip
20.00003pt\cdot\left[\,\left(p_{4}\\!-p_{3}\right)(\epsilon_{3}\\!\cdot\epsilon_{4})+2(p_{3}\\!\cdot\epsilon_{4})\epsilon_{3}-2(p_{4}\\!\cdot\epsilon_{3})\epsilon_{4}\,\right],$
(E.2b) $\displaystyle\mathcal{T}_{t}\ =$ $\displaystyle\ \
-\frac{\leavevmode\nobreak\
\mathcal{C}_{t}\,}{t}\left[\,\left(-p_{1}\\!+p_{4}\right)(\epsilon_{1}\\!\cdot\epsilon_{4})-2(p_{4}\\!\cdot\epsilon_{1})\epsilon_{4}+2(p_{1}\\!\cdot\epsilon_{4})\epsilon_{1}\,\right]$
$\displaystyle\hskip
20.00003pt\cdot\left[\,\left(-p_{2}\\!+p_{3}\right)(\epsilon_{2}\\!\cdot\epsilon_{3})-2(p_{3}\\!\cdot\epsilon_{2})\epsilon_{3}+2(p_{2}\\!\cdot\epsilon_{3})\epsilon_{2}\,\right],$
(E.2c) $\displaystyle\mathcal{T}_{u}\ =$ $\displaystyle\ \
\frac{\leavevmode\nobreak\
\mathcal{C}_{u}\,}{u}\left[\,\left(-p_{1}\\!+p_{3}\right)(\epsilon_{1}\\!\cdot\epsilon_{3})-2(p_{3}\\!\cdot\epsilon_{1})\epsilon_{3}+2(p_{1}\\!\cdot\epsilon_{3})\epsilon_{1}\,\right]$
$\displaystyle\hskip
20.00003pt\cdot\left[\,\left(-p_{2}\\!+p_{4}\right)(\epsilon_{2}\\!\cdot\epsilon_{4})-2(p_{4}\\!\cdot\epsilon_{2})\epsilon_{4}+2(p_{2}\\!\cdot\epsilon_{4})\epsilon_{2}\,\right].$
(E.2d)

Here each external massless gauge boson (gluon) has two helicity states, as
described by its two transverse polarization vectors
$\,\epsilon_{j\pm}^{\mu}\,$ ($j=1,2,3,4$):

$\displaystyle\epsilon^{\mu}_{1+}=\epsilon^{\mu}_{2-}$
$\displaystyle=\frac{1}{\sqrt{2\,}\,}(0,1,\text{i},0),$
$\displaystyle\qquad\epsilon^{\mu}_{1-}=\epsilon^{\mu}_{2+}=\frac{1}{\sqrt{2\,}\,}(0,-1,\text{i},0),$
$\displaystyle\epsilon^{\mu}_{3+}=\epsilon^{\mu}_{4-}$
$\displaystyle=\frac{1}{\sqrt{2\,}\,}(0,\text{i}c_{\theta},1,-\text{i}s_{\theta}),$
$\displaystyle\qquad\epsilon^{\mu}_{3-}=\epsilon^{\mu}_{4+}=\frac{1}{\sqrt{2\,}\,}(0,\text{i}c_{\theta},-1,-\text{i}s_{\theta}).$
(E.3)

Then, we compute the helicity amplitudes of the gauge boson scattering:

$\displaystyle\mathcal{T}[++++]=\mathcal{T}[----]$ $\displaystyle\hskip
58.32823pt=g^{2}\\!\left\\{\mathcal{C}_{s}\left(-2c_{\theta}\right)+\mathcal{C}_{t}\left[\,\\!\frac{3-2c_{\theta}-c_{2\theta}}{1-c_{\theta}}\\!\,\right]+\mathcal{C}_{u}\left[\,\\!\frac{-3-2c_{\theta}+c_{2\theta}}{1+c_{\theta}}\\!\,\right]\right\\}\\!,$
(E.4a) $\displaystyle\mathcal{T}[+-+-]=\mathcal{T}[-+-+]$ $\displaystyle\hskip
58.32823pt=g^{2}\\!\left\\{\mathcal{C}_{t}\left(-1-c_{\theta}\right)+\mathcal{C}_{u}\left[\,\\!\frac{3+4c_{\theta}+c_{2\theta}}{2(1-c_{\theta})}\\!\,\right]\right\\}\\!,$
(E.4b) $\displaystyle\mathcal{T}[+--+]=\mathcal{T}[-++-]$ $\displaystyle\hskip
58.32823pt=g^{2}\\!\left\\{\mathcal{C}_{t}\left[\,\\!\frac{-3+4c_{\theta}-c_{2\theta}}{2(1+c_{\theta})}\\!\,\right]+\mathcal{C}_{u}\left(1-c_{\theta}\right)\right\\},$
(E.4c)

where $\,c_{2\theta}\\!=\\!\cos 2\theta$, and all the helicity-flipped
amplitudes vanish, which include the amplitudes like $\mathcal{T}[+++-]$,
$\mathcal{T}[++--]$, and so on. For convenience, we rewrite the above
amplitudes (E.4)-(E.4) as follows:

$\displaystyle\mathcal{T}[++++]\,=\,\mathcal{T}[----]$
$\displaystyle\,=\,g^{2}\\!\left[\,\\!\frac{\mathcal{C}_{s}\mathcal{N}_{s}}{s}+\frac{\mathcal{C}_{t}\mathcal{N}_{t}}{t}+\frac{\mathcal{C}_{u}\mathcal{N}_{u}}{u}\\!\,\right]\\!,$
(E.5a) $\displaystyle\mathcal{T}[+-+-]\,=\,\mathcal{T}[-+-+]$
$\displaystyle\,=\,g^{2}\\!\left[\,\\!\frac{\mathcal{C}_{s}\mathcal{N}^{\prime}_{s}}{s}+\frac{\mathcal{C}_{t}\mathcal{N}^{\prime}_{t}}{t}+\frac{\mathcal{C}_{u}\mathcal{N}^{\prime}_{u}}{u}\\!\,\right]\\!,$
(E.5b) $\displaystyle\mathcal{T}[+--+]\,=\,\mathcal{T}[-++-]$
$\displaystyle\,=\,g^{2}\\!\left[\,\\!\frac{\mathcal{C}_{s}\mathcal{N}^{\prime\prime}_{s}}{s}+\frac{\mathcal{C}_{t}\mathcal{N}^{\prime\prime}_{t}}{t}+\frac{\mathcal{C}_{u}\mathcal{N}^{\prime\prime}_{u}}{u}\\!\,\right]\\!,$
(E.5c)

where the numerator parameters
$(\mathcal{N}_{j},\,\mathcal{N}_{j}^{\prime},\,\mathcal{N}_{j}^{\prime\prime})$
are given by

$\displaystyle\mathcal{N}_{s}$ $\displaystyle=-2sc_{\theta}\,,\hskip
14.22636pt\mathcal{N}_{t}$
$\displaystyle=\frac{\,s\,}{2}(-3+\\!2c_{\theta}\\!+c_{2\theta})\,,\hskip
14.22636pt\hskip 19.91692pt\mathcal{N}_{u}$
$\displaystyle=\frac{\,s\,}{2}(3+\\!2c_{\theta}\\!-c_{2\theta})\,,$ (E.6a)
$\displaystyle\mathcal{N}^{\prime}_{s}$ $\displaystyle=0\,,\qquad\hskip
14.22636pt\mathcal{N}^{\prime}_{t}$
$\displaystyle=\frac{\,s(2\\!+\\!c_{\theta}\\!-2c_{2\theta}\\!-c_{3\theta})\,}{8(1\\!-\\!c_{\theta})}\,,\hskip
14.22636pt\hskip 3.69885pt\mathcal{N}^{\prime}_{u}$
$\displaystyle=\frac{\,s(-2\\!-c_{\theta}\\!+2c_{2\theta}\\!+\\!c_{3\theta})\,}{8(1\\!-\\!c_{\theta})}\,,$
(E.6b) $\displaystyle\mathcal{N}^{\prime\prime}_{s}$
$\displaystyle=0\,,\qquad\hskip 14.22636pt\mathcal{N}^{\prime\prime}_{t}$
$\displaystyle=\frac{\,s(2\\!-\\!c_{\theta}\\!-\\!2c_{2\theta}\\!+\\!c_{3\theta})\,}{8(1\\!+\\!c_{\theta})}\,,\hskip
14.22636pt\hskip 3.69885pt\mathcal{N}^{\prime\prime}_{u}$
$\displaystyle=\frac{\,s(-2\\!+\\!c_{\theta}\\!+\\!2c_{2\theta}\\!-\\!c_{3\theta})\,}{8(1\\!+\\!c_{\theta})}\,.$
(E.6c)

Hence, we can readily verify that the numerators in Eq.(E.6) obey the
kinematic Jacobi identity:

$\displaystyle\mathcal{N}_{s}+\mathcal{N}_{t}+\mathcal{N}_{u}$
$\displaystyle\,=\,0\,,$ (E.7a)
$\displaystyle\mathcal{N}^{\prime}_{s}+\mathcal{N}^{\prime}_{t}+\mathcal{N}^{\prime}_{u}$
$\displaystyle\,=\,0\,,$ (E.7b)
$\displaystyle\mathcal{N}^{\prime\prime}_{s}+\mathcal{N}^{\prime\prime}_{t}+\mathcal{N}^{\prime\prime}_{u}$
$\displaystyle\,=\,0\,.$ (E.7c)

Next, by using the double-copy approach with CK duality, we reconstruct the
massless graviton scattering amplitudes as follows:

$\displaystyle\mathcal{T}_{\mathrm{DC}}[++++]\,=\,\mathcal{T}_{\mathrm{DC}}[----]$
$\displaystyle\,=\,-16\tilde{c}_{0}g^{2}s\csc^{2}\\!\theta$
$\displaystyle=\,-\frac{\kappa^{2}}{4}\frac{s^{3}}{\,t\hskip 1.3ptu\,}\,,$
(E.8a)
$\displaystyle\mathcal{T}_{\mathrm{DC}}[+-+-]\,=\,\mathcal{T}_{\mathrm{DC}}[-+-+]$
$\displaystyle\,=\,-\,\tilde{c}_{0}g^{2}\frac{s\,s^{6}_{\theta}}{\,(1\\!-\\!c_{\theta})^{4}\,}$
$\displaystyle=\,-\frac{\kappa^{2}}{4}\frac{t^{3}}{\,s\hskip 1.3ptu\,}\,,$
(E.8b)
$\displaystyle\mathcal{T}_{\mathrm{DC}}[+--+]\,=\,\mathcal{T}_{\mathrm{DC}}[-++-]$
$\displaystyle\,=\,-\,\tilde{c}_{0}g^{2}\frac{s\left(1-c_{\theta}\right)^{4}}{s^{2}_{\theta}}$
$\displaystyle=\,-\frac{\kappa^{2}}{4}\frac{u^{3}}{\,s\hskip 1.3ptt\,}\,,$
(E.8c)

where we have applied the conversion constant (2.3an) for the amplitudes in
the last equality of Eqs.(E.8a)-(E.8c) and $(+,-)\equiv(+2,-2)$. The above
reconstructed massless graviton scattering amplitudes agree with the results
of Refs. [49][50] which computed directly the graviton amplitudes by the
conventional Feynman techniques.

###  Massless Graviton Scattering from Type-II Superstring Theory

The massless graviton scattering amplitudes can be computed by the
conventional Feynman technique in quantum field theory. The 3-point and
4-point graviton interaction vertices were derived by DeWitt [51]. It is clear
that by using the conventional Feynman diagram approach, the calculations of
graviton scattering amplitudes are extremely complicated and tedious. Hence,
we will use the amplitudes as computed within the type-II superstring theory
(SST-II) [2][52][53]. The massless gravitons are described by closed strings
and their 4-point scattering amplitude at tree level in the SST-II is given by

$\mathcal{M}[\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}]\,=\,\hat{\kappa}^{2}C(\hat{s},\hat{t},\hat{u})\,\varepsilon_{\lambda_{1}}^{\hat{\mu}\hat{\alpha}}\varepsilon_{\lambda_{2}}^{\hat{\nu}\hat{\beta}}\varepsilon_{\lambda_{3}}^{\hat{\rho}\hat{\gamma}}\varepsilon_{\lambda_{4}}^{\hat{\sigma}\hat{\lambda}}\widehat{K}_{\hat{\mu}\hat{\nu}\hat{\rho}\hat{\sigma}}\widehat{K}_{\hat{\alpha}\hat{\beta}\hat{\gamma}\hat{\lambda}}\,,$
(E.9)

where $\,\hat{\kappa}\,$ is the 10-dimensional gravitational coupling
constant, and the Mandelstam variables $(\hat{s},\,\hat{t},\,\hat{u})$ and the
Lorentz indices $(\hat{\mu},\hat{\nu},\cdots)$ are defined in 10d. In
Eq.(E.9), the $C$ function takes the following form:

$C(\hat{s},\hat{t},\hat{u})\,=\,\frac{1}{\,128\,}\frac{\leavevmode\nobreak\
\Gamma(-\alpha^{\prime}\hat{s}/2)\Gamma(-\alpha^{\prime}\hat{t}/2)\Gamma(-\alpha^{\prime}\hat{u}/2)\leavevmode\nobreak\
}{\,\Gamma(1\\!+\\!\alpha^{\prime}\hat{s}/2)\Gamma(1\\!+\\!\alpha^{\prime}\hat{t}/2)\Gamma(1\\!+\\!\alpha^{\prime}\hat{u}/2)\,}\,,$
(E.10)

where we have set the Regge slope for the closed string to be
$\,\alpha^{\prime}\\!=\\!\mbox{$\frac{\,{1}\,}{4}$}\,$.

Using the relation of gamma functions
$\,\Gamma(1\\!+\\!z)\\!=\\!z\Gamma(z)\,$, we can rewrite the function
$C(\hat{s},\hat{t},\hat{u})$ as follows:

$C(\hat{s},\hat{t},\hat{u})\,=\,-\frac{4}{\,\hat{s}\,\hat{t}\,\hat{u}\,}\frac{\leavevmode\nobreak\
\Gamma(1\\!-\\!\alpha^{\prime}\hat{s}/2)\,\Gamma(1\\!-\\!\alpha^{\prime}\hat{t}/2)\,\Gamma(1\\!-\\!\alpha^{\prime}\hat{u}/2)\leavevmode\nobreak\
}{\leavevmode\nobreak\
\Gamma(1\\!+\\!\alpha^{\prime}\hat{s}/2)\,\Gamma(1\\!+\\!\alpha^{\prime}\hat{t}/2)\,\Gamma(1\\!+\\!\alpha^{\prime}\hat{u}/2)\,}\,.$
(E.11)

With Ref. [52], we note that by imposing compactification of $(10\\!-\\!d)$
spatial dimensions, the amplitude defined in $d$-dimension has the same
structure as that of the original 10-dimension case at tree level. In order to
reduce the SST-II amplitude (E.9) to the amplitude in 4d , we take the limit
for Regge slope $\,\alpha^{\prime}\\!\rightarrow\\!0$ and derive the reduced
4d graviton scattering amplitude:

$\mathcal{M}[\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}]\,=\,-\frac{\,4\kappa^{2}\,}{\,s\hskip
1.2ptt\hskip
1.0ptu\,}K_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}\lambda_{3}^{\prime}\lambda_{4}^{\prime}}^{2},$
(E.12)

where the
$K_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}\lambda_{3}^{\prime}\lambda_{4}^{\prime}}$
factor is given as follows [2][52][53]:

$\displaystyle
K_{\lambda_{1}^{\prime}\lambda_{2}^{\prime}\lambda_{3}^{\prime}\lambda_{4}^{\prime}}\\!=$
$\displaystyle-\frac{1}{4}\big{\\{}[st(\epsilon_{1}\\!\cdot\epsilon_{3})(\epsilon_{2}\\!\cdot\epsilon_{4})+su(\epsilon_{1}\\!\cdot\epsilon_{4})(\epsilon_{2}\\!\cdot\epsilon_{3})+tu(\epsilon_{1}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot\epsilon_{4})]$
$\displaystyle-2s\,[(p_{1}\\!\cdot\epsilon_{4})(p_{3}\\!\cdot\epsilon_{2})(\epsilon_{1}\\!\cdot\epsilon_{3})+(p_{1}\\!\cdot\epsilon_{3})(p_{4}\\!\cdot\epsilon_{2})(\epsilon_{1}\\!\cdot\epsilon_{4})+(p_{2}\\!\cdot\epsilon_{3})(p_{4}\\!\cdot\epsilon_{1})(\epsilon_{2}\\!\cdot\epsilon_{4})$
$\displaystyle+(p_{2}\\!\cdot\epsilon_{4})(p_{3}\\!\cdot\epsilon_{1})(\epsilon_{2}\\!\cdot\epsilon_{3})]-2t\,[(p_{1}\\!\cdot\epsilon_{2})(p_{3}\\!\cdot\epsilon_{4})(\epsilon_{1}\\!\cdot\epsilon_{3})+(p_{1}\\!\cdot\epsilon_{3})(p_{2}\\!\cdot\epsilon_{4})(\epsilon_{1}\\!\cdot\epsilon_{2})$
$\displaystyle+(p_{2}\\!\cdot\epsilon_{1})(p_{4}\\!\cdot\epsilon_{3})(\epsilon_{2}\\!\cdot\epsilon_{4})+(p_{3}\\!\cdot\epsilon_{1})(p_{4}\\!\cdot\epsilon_{2})(\epsilon_{3}\\!\cdot\epsilon_{4})]-2u\,[(p_{1}\\!\cdot\epsilon_{2})(p_{4}\\!\cdot\epsilon_{3})(\epsilon_{1}\\!\cdot\epsilon_{4})$
$\displaystyle+(p_{1}\\!\cdot\epsilon_{4})(p_{2}\\!\cdot\epsilon_{3})(\epsilon_{1}\\!\cdot\epsilon_{2})+(p_{3}\\!\cdot\epsilon_{2})(p_{4}\\!\cdot\epsilon_{1})(\epsilon_{3}\\!\cdot\epsilon_{4})+(p_{3}\\!\cdot\epsilon_{4})(p_{2}\\!\cdot\epsilon_{1})(\epsilon_{2}\\!\cdot\epsilon_{3})]\big{\\}}\,,$
(E.13)

where we denote the polarization vectors as
$\,\epsilon_{j}^{\mu}\equiv\epsilon_{\lambda^{\prime}_{j}}^{\mu}$ . We
substitute the polarizations (E.3) into the above kinematic factor (E.2).
Thus, we can readily deduce the 4d graviton scattering amplitudes at tree
level:

$\displaystyle\mathcal{M}[++++]$
$\displaystyle=\mathcal{M}[----]=-\frac{\,\kappa^{2}\,}{4}\,\frac{s^{3}}{\,t\hskip
1.0ptu\,}\,,$ (E.14a) $\displaystyle\mathcal{M}[+-+-]$
$\displaystyle=\mathcal{M}[-+-+]=-\frac{\,\kappa^{2}\,}{4}\frac{t^{3}}{\,s\hskip
1.0ptu\,}\,,$ (E.14b) $\displaystyle\mathcal{M}[+--+]$
$\displaystyle=\mathcal{M}[-++-]=-\frac{\,\kappa^{2}\,}{4}\frac{u^{3}}{\,s\hskip
1.5ptt\,}\,.$ (E.14c)

These fully agree with the amplitudes (E.8) by the double-copy construction
from the corresponding gauge boson amplitudes.

##  Appendix F Full Amplitudes of KK Gravitons and Goldstones in 5d GR

For completeness, we summarize the full elastic amplitudes of the four
longitudinal KK graviton scattering [15] and the four gravitational KK
Goldstone boson scattering (section 4.2) as follows:

$\displaystyle\mathcal{M}[4h_{L}^{n}]$
$\displaystyle=-\frac{\kappa^{2}M_{n}^{2}(X_{0}+X_{2}c_{2\theta}+X_{4}c_{4\theta}+X_{6}c_{6\theta})\csc^{2}\\!\theta}{512\bar{s}(\bar{s}-4)[\bar{s}^{2}-(\bar{s}-4)^{2}c_{2\theta}+24\bar{s}+16]}\,,$
(F.1a) $\displaystyle\widetilde{\mathcal{M}}[4\phi_{n}]$
$\displaystyle=-\frac{\kappa^{2}M_{n}^{2}(\widetilde{X}_{0}+\widetilde{X}_{2}c_{2\theta}+\widetilde{X}_{4}c_{4\theta}+\widetilde{X}_{6}c_{6\theta})\csc^{2}\\!\theta}{512\bar{s}(\bar{s}-4)[\bar{s}^{2}-(\bar{s}-4)^{2}c_{2\theta}+24\bar{s}+16]}\,,$
(F.1b)

where parameters $X_{j}$ and $\widetilde{X}_{j}$ are defined as

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

For our analyses in sections 4-5, it is also useful to express the above
amplitudes in terms of the variable $\bar{s}_{0}$ :

$\displaystyle\mathcal{M}[4h_{L}^{n}]$
$\displaystyle=\frac{\,\kappa^{2}M_{n}^{2}(X^{0}_{0}\\!+\\!X^{0}_{2}c_{2\theta}\\!+\\!X^{0}_{4}c_{4\theta}\\!+\\!X^{0}_{6}c_{6\theta})\csc^{2}\\!\theta}{\leavevmode\nobreak\
512\bar{s}_{0}(\bar{s}_{0}\\!+\\!4)\left[\,2\bar{s}_{0}^{2}s_{\theta}^{2}\\!+\\!32\bar{s}_{0}\\!+\\!128\,\right]}\,,$
(F.3a) $\displaystyle\widetilde{\mathcal{M}}[4\phi_{n}]$
$\displaystyle=\frac{\,\kappa^{2}M_{n}^{2}(\widetilde{X}^{0}_{0}\\!+\\!\widetilde{X}^{0}_{2}c_{2\theta}\\!+\\!\widetilde{X}^{0}_{4}c_{4\theta}\\!+\\!\widetilde{X}^{0}_{6}c_{6\theta})\csc^{2}\\!\theta}{\leavevmode\nobreak\
512\bar{s}_{0}(\bar{s}_{0}\\!+\\!4)\left[\,2\bar{s}_{0}^{2}s_{\theta}^{2}\\!+\\!32\bar{s}_{0}\\!+\\!128\,\right]}\,,$
(F.3b)

where we have the following coefficients:

$\displaystyle X^{0}_{0}$ $\displaystyle=2(255\hskip
0.85358pt\bar{s}_{0}^{5}+7924\hskip 0.85358pt\bar{s}_{0}^{4}+66048\hskip
0.85358pt\bar{s}_{0}^{3}+235008\hskip 0.85358pt\bar{s}_{0}^{2}+411648\hskip
0.85358pt\bar{s}_{0}+360448)\hskip 0.85358pt,$ (F.4a) $\displaystyle
X^{0}_{2}$ $\displaystyle=-429\hskip 0.85358pt\bar{s}_{0}^{5}+1572\hskip
0.85358pt\bar{s}_{0}^{4}+62976\hskip 0.85358pt\bar{s}_{0}^{3}+357376\hskip
0.85358pt\bar{s}_{0}^{2}+837632\hskip 0.85358pt\bar{s}_{0}+786432\hskip
0.85358pt,$ (F.4b) $\displaystyle X^{0}_{4}$ $\displaystyle=2(-39\hskip
0.85358pt\bar{s}_{0}^{5}-468\hskip 0.85358pt\bar{s}_{0}^{4}+1536\hskip
0.85358pt\bar{s}_{0}^{3}+49664\hskip 0.85358pt\bar{s}_{0}^{2}+227328\hskip
0.85358pt\bar{s}_{0}+294912),$ (F.4c) $\displaystyle X^{0}_{6}$
$\displaystyle=-(3\hskip 0.85358pt\bar{s}_{0}^{5}+100\hskip
0.85358pt\bar{s}_{0}^{4}+1536\hskip 0.85358pt\bar{s}_{0}^{3}+9216\hskip
0.85358pt\bar{s}_{0}^{2}+18432\hskip 0.85358pt\bar{s}_{0})\hskip 0.85358pt,$
(F.4d) $\displaystyle\widetilde{X}^{0}_{0}$ $\displaystyle=2(255\hskip
0.85358pt\bar{s}_{0}^{5}+13348\hskip 0.85358pt\bar{s}_{0}^{4}+168624\hskip
0.85358pt\bar{s}_{0}^{3}+984384\hskip 0.85358pt\bar{s}_{0}^{2}+3514368\hskip
0.85358pt\bar{s}_{0}+6012928)\hskip 0.85358pt,$ (F.4e)
$\displaystyle\widetilde{X}^{0}_{2}$ $\displaystyle=-429\hskip
0.85358pt\bar{s}_{0}^{5}-12732\hskip 0.85358pt\bar{s}_{0}^{4}-156288\hskip
0.85358pt\bar{s}_{0}^{3}-777728\hskip 0.85358pt\bar{s}_{0}^{2}-2572288\hskip
0.85358pt\bar{s}_{0}-5210112\hskip 0.85358pt,$ (F.4f)
$\displaystyle\widetilde{X}^{0}_{4}$
$\displaystyle=-2(39\bar{s}_{0}^{5}-1212\hskip
0.85358pt\bar{s}_{0}^{4}-7824\hskip 0.85358pt\bar{s}_{0}^{3}-10688\hskip
0.85358pt\bar{s}_{0}^{2})\hskip 0.85358pt,$ (F.4g)
$\displaystyle\widetilde{X}^{0}_{6}$
$\displaystyle=-(3\bar{s}_{0}^{5}+4\bar{s}_{0}^{4})\hskip 0.85358pt.$ (F.4h)

Then, we make high energy expansion for the KK graviton (Goldstone) scattering
amplitudes (F.1a)-(F.1b) or (F.3a)-(F.3b) as follows:

$\displaystyle\mathcal{M}[4h_{L}^{n}]$
$\displaystyle\,=\,\mathcal{M}_{0}[4h_{L}^{n}]+\delta\mathcal{M}[4h_{L}^{n}]\,,$
(F.5a) $\displaystyle\widetilde{\mathcal{M}}[4\phi_{n}]$
$\displaystyle\,=\,\widetilde{\mathcal{M}}_{0}[4\phi_{n}]+\delta\widetilde{\mathcal{M}}[4\phi_{n}]\,,$
(F.5b)

where the LO and NLO amplitudes are given by
$(\mathcal{M}_{0},\,\widetilde{\mathcal{M}}_{0})$ and
$(\delta\mathcal{M},\,\delta\widetilde{\mathcal{M}})$, respectively. For the
high energy expansion in terms $\,1/\bar{s}\,$, we derive from expanding
Eqs.(F.1a)-(F.1b) the following KK graviton (Goldstone) scattering amplitudes
at the LO and NLO:

$\displaystyle\mathcal{M}_{0}[4h_{L}^{n}]$
$\displaystyle\,=\,\widetilde{\mathcal{M}}_{0}[4\phi_{n}]=\frac{\leavevmode\nobreak\
3\kappa^{2}\,}{\leavevmode\nobreak\ 128\leavevmode\nobreak\ }\hskip
0.85358pts\hskip 0.85358pt(7+c_{2\theta})^{2}\csc^{2}\\!\theta\,,$ (F.6a)
$\displaystyle\delta\mathcal{M}[4h_{L}^{n}]$
$\displaystyle\,=\,-\frac{\leavevmode\nobreak\
\kappa^{2}M_{n}^{2}\leavevmode\nobreak\ }{256}(1810+93\hskip
0.85358ptc_{2\theta}+126\hskip 0.85358ptc_{4\theta}+19\hskip
0.85358ptc_{6\theta})\csc^{4}\theta\,,$ (F.6b)
$\displaystyle\delta\widetilde{\mathcal{M}}[4\phi_{n}]$
$\displaystyle\,=\,-\frac{\,\kappa^{2}M_{n}^{2}\,}{256}(-902+3669\hskip
0.85358ptc_{2\theta}-714\hskip 0.85358ptc_{4\theta}-5\hskip
0.85358ptc_{6\theta})\csc^{4}\\!\theta\,.$ (F.6c)

For the high energy expansion in terms $\,1/\bar{s}_{0}\,$, we can expand
Eqs.(F.3a)-(F.3b) and derive the following KK graviton (Goldstone) scattering
amplitudes at the LO and NLO:

$\displaystyle\mathcal{M}_{0}[4h_{L}^{n}]$
$\displaystyle\,=\,\widetilde{\mathcal{M}}_{0}[4\phi_{n}]\,=\,\frac{\,3\kappa^{2}\,}{\,128\,}s_{0}\left(7+c_{2\theta}\right)^{2}\csc^{2}\\!\theta\hskip
0.85358pt,$ (F.7a) $\displaystyle\delta\mathcal{M}[4h_{L}^{n}]$
$\displaystyle\,=\,-\frac{\,\kappa^{2}M_{n}^{2}\,}{128}(650+261c_{2\theta}\\!+102\hskip
0.85358ptc_{4\theta}\\!+11c_{6\theta})\csc^{4}\\!\theta\hskip 0.85358pt,$
(F.7b) $\displaystyle\delta\widetilde{\mathcal{M}}[4\phi_{n}]$
$\displaystyle\,=\,-\frac{\,\kappa^{2}M_{n}^{2}\,}{128}(-706+2049\hskip
0.85358ptc_{2\theta}-318\hskip
0.85358ptc_{4\theta}-c_{6\theta})\csc^{4}\\!\theta\hskip 0.85358pt,$ (F.7c)

where $\,s_{0}=s-4M_{n}^{2}\hskip 0.85358pt$. We note that the $\hskip
0.85358pt1/s_{0}\hskip 0.85358pt$ expansion has shifted a hidden
$\mathcal{O}(M_{n}^{2})$ subleading term (contained in $\hskip
0.85358pts=s_{0}+4M_{n}^{2}\hskip 0.85358pt)\hskip 0.85358pt$ from the LO
amplitudes (F.6a) into the NLO amplitudes (F.7b)-(F.7c). But the difference
between the above two NLO amplitudes remains unchanged under this
rearrangement in Eqs.(F.7a)-(F.7c). Hence, we can derive the contribution of
the residual terms by computing the following amplitude-difference either from
Eqs.(F.6b)-(F.6c) or from Eqs.(F.7b)-(F.7c):

$\displaystyle\delta\mathcal{M}[4h_{L}^{n}]-\delta\widetilde{\mathcal{M}}[4\phi_{n}]\leavevmode\nobreak\
=\leavevmode\nobreak\
-\frac{\,3\kappa^{2}M_{n}^{2}\,}{2}\\!\left(\\!\frac{\,39\,}{2}+c_{2\theta}\right)\\!.$
(F.8)

##  Appendix G Extending KLT Construction to KK Amplitudes

In this Appendix, we extend the KLT [26] relation to studying the double-copy
construction of the KK amplitudes, in comparison with the extended BCJ
approach used in sections 5.2-5.3. The KLT relation was derived to connect the
product of the scattering amplitudes of two open strings to that of the closed
string at tree level. The KLT kernal may be further reinterpreted as the
inverse amplitude of a bi-adjoint scalar theory in QFT (à la CHY) [47].

We summarize the LO and NLO amplitudes for $A^{n}_{L}$ and $A^{n}_{5}$ as well
as their difference:

$\displaystyle\mathcal{T}_{0L}$
$\displaystyle=\,g^{2}\sum_{j}\\!\frac{\,\mathcal{C}_{j}\mathcal{N}_{j}^{0}\,}{s_{0j}}\,,\qquad\delta\mathcal{T}_{L}=\,g^{2}\sum_{j}\\!\frac{\,\mathcal{C}_{j}\delta\mathcal{N}_{j}\,}{s_{0j}}\,,$
(G.1a) $\displaystyle\widetilde{\mathcal{T}}_{05}$
$\displaystyle=\,g^{2}\sum_{j}\\!\frac{\,\mathcal{C}_{j}\widetilde{\mathcal{N}}_{j}^{0}\,}{s_{0j}}\,,\qquad\delta\widetilde{\mathcal{T}}_{5}=\,g^{2}\sum_{j}\\!\frac{\,\mathcal{C}_{j}\delta\widetilde{\mathcal{N}}_{j}\,}{s_{0j}}\,,$
(G.1b) $\displaystyle\Delta\mathcal{T}$
$\displaystyle\equiv\delta\mathcal{T}_{L}\\!-\delta\widetilde{\mathcal{T}}_{5}\,=\,g^{2}\sum_{j}\\!\frac{\,\mathcal{C}_{j}(\delta\mathcal{N}_{j}\\!-\\!\delta\widetilde{\mathcal{N}}_{j})\,}{s_{0j}}\,,$
(G.1c)

where $j\in(s,t,u)$ and their numerators satisfy the kinematic Jacobi
identities:

$\sum_{j}\mathcal{N}_{j}^{0}=\sum_{j}\widetilde{\mathcal{N}}_{j}^{0}=0\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{j}(\delta\mathcal{N}_{j}\\!-\\!\delta\widetilde{\mathcal{N}}_{j})=0\,.$
(G.2)

Then, we expand the color factors ($\mathcal{C}_{j}$) in terms of traces of
group generators:

$\displaystyle\mathcal{C}_{s}$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}\\!\left(-\text{Tr}[1234]+\text{Tr}[1243]+\text{Tr}[2134]-\text{Tr}[2143]\right)\,,$
(G.3a) $\displaystyle\mathcal{C}_{t}$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}\\!\left(-\text{Tr}[1423]+\text{Tr}[1432]+\text{Tr}[4123]-\text{Tr}[4132]\right)\,,$
(G.3b) $\displaystyle\mathcal{C}_{u}$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}\\!\left(-\text{Tr}[1342]+\text{Tr}[1324]+\text{Tr}[3142]-\text{Tr}[3124]\right)\,,$
(G.3c)

where the abbreviation $\,\\{1,2,3,4\\}\\!=\\!\\{T^{a},T^{b},T^{c},T^{d}\\}$
is used. Thus, each full four-particle scattering amplitude
$\,\mathcal{T}_{4}\,$ can be decomposed into the sum of color-ordered partial
amplitudes in terms of the trace of group factors:

$\mathcal{T}_{4}\leavevmode\nobreak\
=\,g^{2}\\!\\!\sum_{\mathcal{P}(234)}\\!\mathcal{A}_{4}[1234]\,\text{Tr}\\!\left[\,\\!T^{a}T^{b}T^{c}T^{d}\\!\,\right]\\!.$
(G.4)

We may further write the $n$-point color-ordered partial amplitudes in the
following general form [28][54],

$\mathcal{A}_{i}\leavevmode\nobreak\ =\leavevmode\nobreak\
g^{2}\sum_{j=1}^{(n-2)!}\Theta_{ij}\,\hat{n}_{j}\,,$ (G.5)

where the quantities $\,\\{\Theta_{ij}\\}\,$ form a
$\,(n\\!-\\!2)!\\!\times\\!(n\\!-\\!2)!$ matrix containing massless scalar
propagators, and the numerators $\,\hat{n}_{j}\,$ include the kinematic
information.

For the case of four-particle scattering ($n=4$), we choose the partial
amplitudes with ordering $\mathcal{A}[1234]$ and $\mathcal{A}[1243]$ in the
Kleiss-Kuijf basis [55][56].141414Alternatively, one may choose
$\mathcal{A}[1324]$ instead of $\mathcal{A}[1243]$ in the basis, because the
U(1) decoupling identity gives
$\mathcal{A}[1234]+\mathcal{A}[1243]+\mathcal{A}[1324]=0$ . Thus, the
amplitudes in Eq.(G.1) can be reexpressed as follows:

$\left(\\!\begin{aligned} \mathcal{A}[1234]\\\\[-2.84526pt]
\mathcal{A}[1243]\end{aligned}\right)=\leavevmode\nobreak\
g^{2}\,\Theta\\!\times\\!\left(\\!\begin{aligned} \hat{n}_{s}\\\\[-2.84526pt]
\hat{n}_{t}\end{aligned}\right)\\!,$ (G.6)

where
$\,\mathcal{A}=\mathcal{T}_{0L}\,,\,\widetilde{\mathcal{T}}_{05}\,,\,\Delta\mathcal{T}$
and
$\,\hat{n}_{j}=\mathcal{N}^{0}_{j},\,\widetilde{\mathcal{N}}^{0}_{j},\,\delta\mathcal{N}_{j}\\!-\\!\delta\widetilde{\mathcal{N}}_{j}\,$.
The above propagator matrix $\,\Theta\,$ takes the form:

$\Theta\leavevmode\nobreak\ =\left(\\!\begin{aligned} -\frac{1}{\,s_{0}}\hskip
12.80365pt&\hskip 14.22636pt\frac{1}{\,t_{0}}\\\\[2.84526pt]
\frac{1}{\,s_{0}}\\!+\\!\frac{1}{\,u_{0}}&\hskip
14.22636pt\frac{1}{\,u_{0}}\end{aligned}\,\right)\\!,$ (G.7)

where we can readily check $\,\det\Theta=0\,$. Then, we can derive the color-
ordered LO amplitudes:

$\displaystyle\mathcal{T}_{0L}[1234]$
$\displaystyle\,=\,g^{2}\\!\left(\\!-\frac{\mathcal{N}_{s}^{0}}{s_{0}}+\frac{\mathcal{N}_{t}^{0}}{t_{0}}\right)\\!,$
$\displaystyle\mathcal{T}_{0L}[1243]$
$\displaystyle\,=\,g^{2}\\!\left(\\!\frac{\mathcal{N}_{s}^{0}}{s_{0}}-\frac{\mathcal{N}_{u}^{0}}{u_{0}}\right)\\!,$
(G.8a) $\displaystyle\widetilde{\mathcal{T}}_{05}[1234]$
$\displaystyle\,=\,g^{2}\\!\left(\\!-\frac{\widetilde{\mathcal{N}}_{s}^{0}}{s_{0}}+\frac{\widetilde{\mathcal{N}}_{t}^{0}}{t_{0}}\right)\\!,$
$\displaystyle\widetilde{\mathcal{T}}_{05}[1243]$
$\displaystyle\,=\,g^{2}\\!\left(\\!\frac{\widetilde{\mathcal{N}}_{s}^{0}}{s_{0}}-\frac{\widetilde{\mathcal{N}}_{u}^{0}}{u_{0}}\right)\\!,$
(G.8b)

and the color-ordered NLO amplitudes:

$\displaystyle\Delta\mathcal{T}[1234]$
$\displaystyle\,=\,g^{2}\\!\left(\\!-\frac{\,\delta\mathcal{N}_{s}\\!-\\!\delta\widetilde{\mathcal{N}}_{s}\,}{s_{0}}+\frac{\,\delta\mathcal{N}_{t}\\!-\\!\delta\widetilde{\mathcal{N}}_{t}\,}{t_{0}}\right)\\!,$
(G.9a) $\displaystyle\Delta\mathcal{T}[1243]$
$\displaystyle\,=\,g^{2}\\!\left(\\!\frac{\,\delta\mathcal{N}_{s}\\!-\\!\delta\widetilde{\mathcal{N}}_{s}\,}{s_{0}}-\frac{\,\delta\mathcal{N}_{u}\\!-\\!\delta\widetilde{\mathcal{N}}_{u}\,}{u_{0}}\right)\\!.$
(G.9b)

With the above, we extend the KLT double-copy construction and compute the KK
graviton scattering amplitudes at the LO:

$\displaystyle\mathcal{M}_{0}[1234]$
$\displaystyle\,=\,\frac{\kappa^{2}}{\,24\,}\,s_{0}\,\mathcal{T}_{0L}[1234]\,\mathcal{T}_{0L}[1243]\,=\,\frac{\,3\kappa^{2}}{\,128\,}\,s_{0}\,(7\\!+c_{2\theta})^{2}\csc^{2}\\!\theta\,,$
(G.10a) $\displaystyle\widetilde{\mathcal{M}}_{0}[1234]$
$\displaystyle\,=\,\frac{\kappa^{2}}{\,24\,}\,s_{0}\,\widetilde{\mathcal{T}}_{05}[1234]\,\widetilde{\mathcal{T}}_{05}[1243]\,=\,\frac{\,3\kappa^{2}}{\,128\,}\,s_{0}\,(7\\!+c_{2\theta})^{2}\csc^{2}\\!\theta\,.$
(G.10b)

Then, with the definitions of
$(\Delta\mathcal{M}_{1},\,\Delta\mathcal{M}_{2})$ in Eqs.(2.3aya)-(2.3ayb), we
construct the KK graviton amplitudes at the NLO:

$\displaystyle\Delta\mathcal{M}_{1}[1234]$
$\displaystyle\,=\,\frac{\kappa^{2}}{\,12\,}s_{0}\,\widetilde{\mathcal{T}}_{05}[1234]\,\Delta\mathcal{T}[1243]=\frac{\,\kappa^{2}}{\,12\,}\,s_{0}\,\Delta\mathcal{T}[1234]\,\widetilde{\mathcal{T}}_{05}[1243]$
$\displaystyle\,=\,-\kappa^{2}M_{n}^{2}\,(7\\!+c_{2\theta})\,,$ (G.11a)
$\displaystyle\Delta\mathcal{M}_{2}[1234]$
$\displaystyle=\frac{\,\kappa^{2}}{\,12\,}\,s_{0}\,\mathcal{T}_{0L}[1234]\,\Delta\mathcal{T}[1243]\,=\,\frac{\,\kappa^{2}}{\,12\,}\,s_{0}\,\Delta\mathcal{T}[1234]\,\mathcal{T}_{0L}[1243]$
$\displaystyle\,=\,-\kappa^{2}M_{n}^{2}\,(7\\!+c_{2\theta})\,.$ (G.11b)

From the above, we see that the amplitudes (G.10a)-(G.10b) and (G.11a)-(G.11b)
agree with the amplitudes (2.3aib) and (2.3aya)-(2.3ayb) which we derived by
using the improved BCJ construction in the case of four-point amplitudes. We
will consider generalizing the present analysis to the KK graviton (Goldstone)
amplitudes with five or more external lines in the future work.

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