\frac{p_{a}\\!\cdot p_{b}}{\,z_{a}\\!-\\!z_{b}\,}\hskip 0.85358pt,&\hskip
11.38109pt(\mathrm{for}~{}a\\!\neq\\!b),\\\\[2.84526pt] 0\hskip
0.85358pt,~{}\hskip 8.53581pt&\hskip
11.38109pt(\mathrm{for}~{}a\\!=\\!b).\end{aligned}\right.$ (67jf)

In Eq.(67jea), the color factor $\hskip 0.85358pt\mathcal{C}[\alpha]$ is
decomposed into the DDM basis and contains $(N\\!-2)!\hskip 0.85358pt$
elements, as defined below:

$\mathcal{C}[\alpha]\,=\sum_{e_{1},\cdots,\hskip
0.85358pte_{N-3}}f^{a_{1}a_{\alpha(2)}e_{1}}f^{e_{1}a_{\alpha(3)}e_{2}}\cdots
f^{e_{N-3}a_{\alpha(N-1)}a_{N}}\,,$ (67jg)

where the the 1st and $N$-th labels are fixed in the color ordering
$\alpha\hskip 0.85358pt$, i.e.,
$\alpha\\!=\\![1,\alpha(2),\cdots\\!,\alpha(N\\!-\\!1),N]\\!\in\\!S_{N-2}\hskip
0.85358pt$. The symbol $\mathrm{PT}[\alpha]$ stands for the Parke-Taylor
factor and is given by

$\mathrm{PT}[\alpha]=\frac{1}{~{}(z_{1}\\!-\\!z_{\alpha(2)})\cdots(z_{\alpha(N-1)}\\!-\\!z_{N})(z_{N}\\!-\\!z_{1})~{}}\hskip
0.85358pt.$ (67jh)

The above CHY formulation works for even number $N$, whereas for the odd
number $N$ the Pfaffian is trivially zero. This is consistent with the
conclusions we reached earlier in this subsection. Namely, in the case of
$N$-point scattering (with even number $N$), the leading-order double-copy
holds. But, for the $N$-point scattering (with odd number $N$), the KK GAET
and GRET (which are connected by double-copy) will take the trivial form of
$0=0\hskip 0.85358pt$ at the leading order of high energy expansion. Note that
the scattering equations are the same for both the KK YMS theory and the KK
EMS theory. So the above LO double-copy relation (67jeb) holds under the
replacement $\,c_{N}\tilde{a}_{n_{1}\cdots\hskip
0.85358ptn_{N}}^{2}\\!\\!\\!\\!\longrightarrow\\!\tilde{\beta}_{n_{1}\cdots\hskip
0.85358ptn_{N}}$. Using Eqs.(67jd)-(67je) and the fundamental BCJ relations,
we derive the extended KLT-type double-copy formula for the $N$-point LO
scattering amplitude of gravitational KK Goldstone bosons:

$\displaystyle\widetilde{\mathcal{M}}_{0}^{\mathrm{dc}}\big{[}\phi_{n_{1}}\\!,\cdots\\!,\phi_{n_{N}}\big{]}=-\,c_{N}\tilde{a}_{n_{1}\\!\cdots\hskip
0.85358ptn_{N}}^{2}\\!\left(\\!\frac{\,\kappa\,}{4}\right)^{\\!\\!N-2}\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\\{\alpha,\beta\\}\in
S_{N\\!-\\!3}}\widetilde{\mathcal{T}}_{0}\big{[}A^{a_{\alpha(i)}n_{\alpha(i)}}_{5}\big{]}\mathcal{K}[\alpha|\beta]\,\widetilde{\mathcal{T}}_{0}\big{[}A^{a_{\beta(i)}n_{\beta(i)}}_{5}\big{]}\hskip
0.85358pt,$ (67ji)

where $\mathcal{K}[\alpha|\beta]$ is the KLT kernel. Finally, we also note
that the CHY double-copy construction of the $N$-point LO scattering amplitude
of the vector-type gravitational KK Goldstone bosons
$\widetilde{\mathcal{M}}_{0}^{\mathrm{dc}}\big{[}\mathcal{V}_{n_{1}}^{\pm
1},\\!\cdots\\!,\mathcal{V}_{n_{N}}^{\pm 1}\big{]}$ can be similarly done.

###  4.4 Prescription for Warped Double-Copy Construction at Leading Order

Our previous work Li:2022rel  proved that for the compactified KK
gauge/gravity theories, only the toroidal compactification of flat extra
dimensions can satisfy the mass spectrum condition and directly realize
extended double-copy of massive KK gauge/gravity scattering amplitudes. It was
also proven Li:2021yfk  that under the toroidal compactification of flat extra
dimensions such massive KK double-copies can be derived as the field theory
limit of the massive KLT relations of KK open/closed string amplitudes. But we
note that the compactified warped 5d gauge/gravity theories with orbifold
$S^{1}\\!/\mathbb{Z}_{2}$ (such as the RS1 model RS1 ) clearly do not meet
this criterion. This is because such warped KK gauge/gravity theories have
highly nonlinear mass spectra and the mass spectrum of KK gauge bosons differ
from that of the KK gravitons due to the 5d warped metric, whereas coupling
coefficients of the KK gauge bosons and of the KK gravitons also differ
significantly. Moreover, the $S^{1}/\mathbb{Z}_{2}$ orbifold compactification
spoils the KK number conservation and the KK scattering amplitudes do not
exhibit single-pole structure in each channel. Nevertheless, as demonstrated
in Sections 4.1-4.2, for tree-level KK gauge/gravity scattering amplitudes, we
find proper ways (or restriction) to evade these problems. These include: (i)
the 3-point double-copied KK graviton scattering amplitudes (from that of the
KK gauge bosons) exhibit exactly the same kinematic structure as the original
3-point KK grviton amplitudes even without high energy expansion except that
we need to set up a proper prescription for the correspondence between the
double-copied trilinear KK gauge couplings and the original trilinear KK
graviton couplings, whereas under such prescription the double-copy works for
the corresponding 3-point gravitational KK Goldstone amplitudes only at the
leading-order of high energy expansion;999We found previously Li:2022rel  that
for the gauge/gravity KK Goldstone scattering amplitudes with toroidal
compactification of flat extra dimensions, the double-copy construction also
works only at the leading order of high energy expansion. (ii) at the leading-
order of high energy expansion, the numerators of 4-point scattering
amplitudes of KK gauge bosons and of KK Goldstone bosons obey the kinematic
Jacobi identities; (iii) the double-copied leading-order KK graviton
scattering amplitudes (constructed from that of the KK gauge bosons) exhibit
exactly the same kinematic structure as the original leading-order amplitudes
of KK grvitons except that we need to set up a proper prescription for the
correspondence between double-copied KK gauge couplings and the original KK
graviton couplings.

Based upon the above observation and the double-copy analysis shown in
Sections 4.1-4.2, we summarize the prescriptions for successful double-copy
constructions of the 3-point and 4-point KK graviton scattering amplitudes in
the warped 5d gauge/gravity theories under orbifold compactification of
$S^{1}\\!/\mathbb{Z}_{2}\hskip 0.85358pt$:

  1. 1).

For the present double-copy construction of $N$-point warped KK graviton
scattering amplitudes from the corresponding $N$-point warped KK gauge boson
scattering amplitudes, we first replace the 4d gauge coupling $g$ by the 4d
gravitational coupling $\kappa$ as follows:

$g^{N-2}~{}\longrightarrow~{}\\!-\left(\frac{\kappa}{4}\right)^{\\!\\!N-2}\,.$
(67jj)

For instance, this gives $\hskip 0.85358ptg\\!\rightarrow\\!-\kappa/4\,$ for
$N\\!=\\!3$ and $\hskip 0.85358ptg^{2}\\!\rightarrow\\!-\kappa^{2}/16$ for
$N\\!=\\!4\hskip 0.85358pt$. This double-copy replacement (67jj) is the
similar to what we established for the compactified flat 5d gauge/gravity
theories Li:2021yfk Li:2022rel . The difference is that in the present
analysis, due to an overall factor of i of the $N$-point (for $N\\!=\hskip
0.85358pt$odd) gauge amplitude, the corresponding double-copied gravitational
amplitude has an overall factor of $-1\hskip 0.85358pt$. Thus, the factor
$(-)^{N+1}$ in Refs.Li:2021yfk Li:2022rel  will be replaced by $-1$ in this
study.

  2. 2).

Then, for the scattering amplitudes of KK gauge bosons (KK Goldstone bosons),
we apply the extended massive color-kinematics (CK) duality and make the
following group factor replacements as in Eq.(67gb) for the 3-point KK
amplitudes, or as in Eqs.(67gw)(67hj) for the 4-point KK amplitudes:

$\begin{array}[]{rlrll}f^{abc}&{\,\longrightarrow\,}\mathcal{N}[\\{\epsilon_{j}\\}]\hskip
0.85358pt,&f^{abc}&{\,\longrightarrow\,}\widetilde{\mathcal{N}}[\epsilon_{3}]\hskip
0.85358pt,&\text{(3-point~{}amplitudes),}\\\\[2.84526pt]
\mathcal{C}_{j}&{\,\longrightarrow\,}\mathcal{N}_{j}^{0}\hskip
0.85358pt,&\mathcal{C}_{j}&{\,\longrightarrow\,}\widetilde{\mathcal{N}}_{j}^{0}\hskip
0.85358pt,&\text{(4-point~{}elastic~{}amplitudes),}\\\\[2.84526pt]
\mathcal{C}_{j}&{\,\longrightarrow\,}\mathcal{N}_{j}^{\mathrm{in}\hskip
0.85358pt0}\hskip
0.85358pt,&\mathcal{C}_{j}&{\,\longrightarrow\,}\widetilde{\mathcal{N}}_{j}^{\mathrm{in}\hskip
0.85358pt0}\hskip
0.85358pt,&\text{(4-point~{}inelastic~{}amplitudes).}\end{array}$ (67jn)

  3. 3).

At each KK level-$n$, we replace the mass-eigenvalue $M_{n}$ of KK gauge
bosons [determined by Eq.(20)] by mass-eigenvalue $\mathbb{M}_{n}$ of KK
gravitons [determined by Eq.(57)], i.e.,
$M_{n}\\!\rightarrow\mathbb{M}_{n}\hskip 0.85358pt$.

  4. 4).

At each KK level-$n$, we further replace the involved KK gauge couplings by
the corresponding KK gravitational couplings as in Eq.(67gk) for the 3-point
KK amplitudes and as in Eqs.(67hd)(67hp)(67id) for the 4-point KK amplitudes:

$\displaystyle
c_{3}a_{n_{1}n_{2}n_{3}}^{2}\\!\\!\rightarrow\alpha_{n_{1}n_{2}n_{3}},\quad
c_{3}a_{n_{1}n_{2}n_{3}}\tilde{a}_{n_{1}n_{2}n_{3}}\\!\\!\rightarrow\tilde{\alpha}_{n_{1}n_{2}n_{3}},\quad
c_{3}\tilde{a}_{n_{1}n_{2}n_{3}}^{2}\\!\\!\rightarrow\tilde{\beta}_{n_{1}n_{2}n_{3}},$
$\displaystyle
c_{4}\tilde{a}_{nnnn}^{2}\\!\rightarrow\tilde{\beta}_{nnnn}\hskip
0.85358pt,~{}~{}~{}~{}c_{4}^{\mathrm{in}}\tilde{a}_{nnmm}^{2}\\!\rightarrow\tilde{\beta}_{nnmm}\hskip
0.85358pt,~{}~{}~{}~{}c_{4}^{\mathrm{in}}\hskip
0.85358pta_{0nn}^{4}\\!\rightarrow\alpha_{0nn}^{2}\hskip 0.85358pt,$
$\displaystyle
c_{4}^{\mathrm{in}}a_{000}^{2}a_{nn0}^{2}\\!\rightarrow\alpha_{000}\alpha_{nn0}\hskip
0.85358pt,~{}~{}~{}~{}c_{4}^{\mathrm{in}}a_{000}^{2}a_{nn0}\tilde{a}_{nn0}\\!\rightarrow\alpha_{000}\tilde{\alpha}_{nn0}\hskip
0.85358pt,$ (67jo)

where the overall coefficient $c_{3}$ or $c_{4}$ is the relevant normalization
factor and its determination is given in Sections 4.1-4.2. Finally, in Section
4.3, we studied the double-copy construction for the $N$-point
($N\\!\\!\geqslant\\!4$) LO KK Goldstone boson amplitudes. Thus for the
$N$-point LO double-copy, we impose the following correspondence (replacement)
between the KK gauge and gravity coupling coefficients:

$\displaystyle c_{N}\hskip 0.85358pt\llbracket\hskip
0.85358pt\cdots\mathsf{f}_{n_{k}}\\!\\!\\!\cdots\tilde{\mathsf{f}}_{n_{k^{\prime}}}\\!\\!\\!\cdots\hskip
0.85358pt\rrbracket\hskip 0.85358pt\hskip 0.85358pt\llbracket\hskip
0.85358pt\cdots\tilde{\mathsf{f}}_{n_{k}}\\!\\!\\!\cdots\mathsf{f}_{n_{k^{\prime}}}\\!\\!\\!\cdots\hskip
0.85358pt\rrbracket\hskip
0.85358pt\,\longrightarrow~{}\tilde{\alpha}_{n_{1}\cdots\,n_{N}}\hskip
0.85358pt,$ (67jpa) $\displaystyle c_{N}\hskip
0.85358pt\tilde{a}_{n_{1}\\!\cdots\,n_{N}}^{2}\,\longrightarrow~{}\tilde{\beta}_{n_{1}\cdots
n_{N}}.$ (67jpb)

##  5 Conclusions

In this work, we conducted comprehensive analyses on the structure of
scattering amplitudes of massive Kaluza-Klein (KK) states in the compactified
5-dimensional warped gauge and gravity theories. We presented systematic
formulations of the gauge theory equivalence theorem (GAET) and the
gravitational equivalence theorem (GRET) within the $R_{\xi}$ gauge and up to
loop level. Under the high energy expansion, the GAET quantitatively connects
the scattering amplitudes of longitudinal KK gauge bosons to that of the
corresponding KK Goldstone bosons, whereas the GRET connects the scattering
amplitudes of massive KK gravitons of helicity-0 (or helicity-1) to that of
the corresponding gravitational KK Goldstone bosons. A key point of our work
is to take the GAET of KK Yang-Mills gauge theories as the truly fundamental
ET formulation, from which the GRET of the corresponding KK gravity theories
can be reconstructed by using the double-copy at the leading order (LO) of
high energy expansion. We systematically studied the double-copy construction
of 3-point and 4-point KK gauge/gravity scattering amplitudes at tree level.
We proved that under proper color-kinematics correspondence and gauge-gravity
coupling correspondence, the double-copy of 3-point physical KK gauge-
boson/graviton scattering amplitudes can be realized without high energy
expansion, and for the corresponding 3-point gauge/gravity KK Goldstone
scattering amplitudes the double-copy can be achieved at the leading order
(LO) of high energy expansion. Moreover, we demonstrated that the double-copy
can be realized for 4-point massive scattering amplitudes of KK gauge bosons
and of KK gravitons (as well as their corresponding KK Goldstone bosons) to
the leading order of high energy expansion. In addition, we can reduce the
validity of 4-point-level GAET and GRET to the validity of 3-point-level GAET
and GRET. Hence, the GAET and GRET formulations at the level of 3-point KK
scattering amplitudes are the most fundamental formulations, from which the
GAET and GRET formulations at 4-point-level of KK scattering amplitudes can be
inferred. We further derived the GRET from GAET by LO double-copy
constructions for 3-point and 4-point KK scattering amplitudes. A more
elaborated summary of our results and conclusions in each sections are
presented as below, which is followed by a schematic summary as shown in the
last paragraph of this section together with Fig. 2.

In Section 2, we proved the formulations of the GAET and GRET within the
$R_{\xi}$ gauge and up to loop level. The GAET formulation was presented in
Eqs.(36)-(37) and Eq.(38), whereas the GRET formulations were given in
Eqs.(67cu)(67cy) for KK gravitions with helicity-0 (called type-I) and in
Eqs.(67cf)(67ch) for KK gravitions with helicity-1 (called type-II). In
essence, the GAET and GRET reflect the geometric “Higgs” mechanism through the
KK compactifications, which hold not only for the compactified flat extra
dimensions Chivukula:2001esy ; Chivukula:2002ej Hang:2021fmp ; Hang:2022rjp ,
but also for the compactified warped extra dimensions as proved in Sections
2.2-2.3. They determine the high energy behaviors of massive KK graviton
scattering amplitudes and ensure the interlancing cancellations among
contributions of different KK levels as happened in the scattering amplitudes
of KK gauge bosons and of KK gravitons.

In Section 3, we analyzed the structure of 3-point and 4-point massive
scattering amplitudes of KK gauge bosons and of KK gravitons (as well as the
corresponding scattering amplitudes of KK Goldstone bosons), including the
interconnections between the 3-point and 4-point KK amplitudes. In Section
3.1, we explicitly proved the warped GAET and GRET for the fundamental 3-point
KK gauge/gravity scattering amplitudes. We found that the nontrivial
realization of GAET is given by the 3-point KK gauge boson amplitude with two
longitudinal external states and one transverse external state (of $LLT$
type), as shown in Eq.(67df). We proved that to hold this 3-point-level GAET
(67df) requires the condition (67dg) which is directly proved in Eq.(67mi) of
Appendix D. Then, we computed the 3-point KK graviton amplitude101010We
derived the most general 3-point scattering amplitude of KK gravitons in
Eq.(C.2) of Appendix C. of helicities $(\pm 1,\pm 1,\pm 2)$ and its
corresponding KK Goldstone amplitude in Eqs.(67do)-(67do) and
Eqs.(67dp)-(67dp). To hold the 3-point-level GRET (67dq) imposes the condition
(67dr) which we proved in Eq.(67mj) of Appdenix D. We further computed the
3-point KK graviton amplitude of helicities $(0,\hskip 0.85358pt0,\hskip
0.85358pt\pm 2)$ and its corresponding KK Goldstone amplitude in
Eqs.(67dua)-(67dub). To hold the 3-point-level GRET (67dv) leads to the
nontrivial condition (67dw) which we proved in Eq.(67mw) of Appdenix D. We
also computed the mixed 3-point KK graviton amplitude of helicities $(\pm
1,\pm 1,\hskip 0.85358pt0)$ and its corresponding KK Goldstone amplitude in
Eq.(67dy). To hold the GRET (67ea) imposes the condition (67eb) which was
proved in (67my) of Appendix D.

In Section 3.2, we further demonstrated that the validity of the warped GAET
and GRET for 4-point KK scattering amplitudes can be effectively reduced to
the validity of these theorems for the 3-point KK scattering amplitudes. For
the 4-point elastic and inelastic scattering amplitudes of KK gauge bosons and
KK Goldstone bosons, we explicitly established the warped GAET as in
Eqs.(67eq), (67ez), and (67fh) respectively. We proved that the validity of
the 4-point-level GAET in these cases is ensured by the validity of the
fundamental 3-point-level GAET (67df). Then, we analyzed the 4-point
scattering amplitudes of KK gravitons and gravitational KK Goldstone bosons.
We explicitly established the warped GRET as in Eq.(67fq) for the 4-body
elastic scattering channel. The validity of the 4-point-level GRET (67fq)
relies on the sum rule condition (67fr), while the proof of Eq.(67fr) requires
the sum rule condition (67dx) to play the key role, where the condition (67dx)
just ensures the validity of the 3-point-level GRET (67dv) in the case of
$(n_{1}{},n_{2},n_{3})=(n,\hskip 0.85358ptn,\hskip 0.85358ptj)$. This shows
that validity of the 4-point-level GRET (67fq) is reduced to the validity of
the 3-point-level GRET (67dv). We further computed the 4-point inelastic
scattering amplitudes of KK gravitons and of KK Goldstone bosons as in
Eqs.(67fwa)-(67fwb). Under high energy expansion, we found that the leading-
order amplitudes (67fxa)-(67fxb) are simple enough in this case. Thus, we
explicitly established the 4-point-level GRET (67fy) without a need of
additional sum rule condition.

In Section 4, we studied the double-copy construction of the massive
gravitational KK scattering amplitudes from the corresponding massive KK gauge
scattering amplitudes for the warped 5d gauge and gravity theories with the
orbifold compactification of $S^{1}\\!/\mathbb{Z}_{2}\hskip 0.85358pt$. This
is nontrivial and challenging since it was proved Li:2022rel  that the direct
construction of double-copy for $N$-point massive gauge/gravity KK scattering
amplitudes (with $N\\!\\!\geqslant\\!4$) can directly work out only for
toroidal compactifications with flat extra dimensions. Nevertheless, we could
realize the double-copy construction for warped gauge/gravity theories with
proper restrictions and prescriptions so as to evade the previous conditions
Li:2022rel . In Section 4.1, we newly proved that the double-copy can be
constructed for the 3-point full scattering amplitudes of KK gravitons at tree
level for warped gauge/gravity theories. We set up the 3-point color-
kinematics (CK) correspondence and the gauge/gravity coupling correspondence
as in Eq.(67gb), and the KK mass replacement
$M_{n}\\!\rightarrow\mathbb{M}_{n}\hskip 0.85358pt$ at each KK level. With
these, we first presented the general 3-point doubel-copy formulas (with any
polarization tensors of physical KK graviton states) as in Eqs.(4.1)-(4.1).
Then, we explicitly constructed the 3-point KK graviton amplitudes with
helicities $(\pm 1,\pm 1,\pm 2)$ and $(0,\hskip 0.85358pt0,\hskip 0.85358pt\pm
2)$ in Eq.(67gh) and Eq.(67gi) respectively. We further derived their
corresponding gravitational KK Goldstone boson amplitudes in
Eqs.(67gja)-(67gjb). The required conversions of gauge-gravity coupling
constants are given in Eq.(67gk). With these we established successful double-
copy constructions of the GRET (67gpa) and (67gpb) (in warped KK gravity
theory) from the GAET (67df) (in the warped KK gauge theory) at the level of
3-point KK scattering amplitudes.

In Section 4.2, we demonstrated that the double-copy construction can be
achieved for the 4-point KK gauge/gravity scatttering amplitudes at the
leading-order (LO) of high energy expansion. For the 4-point elastic
scattering $(n,n)\\!\rightarrow\\!(n,n)\hskip 0.85358pt$, the LO amplitudes of
KK gauge bosons and of KK Goldstone bosons have their effective numerators
connected by the generalized gauge transformations (67gu) and obey the
kinematic Jacobi identities (67gv). The double-copy of these two LO amplitudes
gives the LO amplitudes of KK gravitons and of the corresponding gravitational
KK Goldstone bosons as in Eqs.(67gx) and (67gz). Using the coupling conversion
(67hd), we derived the double-copied LO gravitational KK amplitudes (67he)
which agree with the same amplitudes (67ha) (by explicit Feynman diagram
approach). In parallel, we studied the double-copy constructions for the LO
inelastic KK gauge/gravity scattering amplitudes, including the processes
$(n,n)\\!\rightarrow\\!(m,m)$ (with $n\\!\neq\\!m$),
$(n,m)\\!\rightarrow\\!(\ell,q)$ (with
$n\\!\neq\\!m\\!\neq\\!\ell\\!\neq\\!q$), and $(0,0)\\!\rightarrow\\!(n,n)$
(with $n\\!>\\!0$). We found that their effective numerators satisfy the
kinematic Jacobi identity respectively, as shown in Eqs.(67hi), (67hu)-(67hv),
and so on. For these inelastic processes, we presented the double-copied
inelastic amplitudes of KK gravitons and of gravitational KK Goldstone bosons
as in Eqs.(67hq), (4.2), and (67ia)(67id). In Section 4.3, we further
established that this LO double-copy construction can be extended to the
general $N$-point KK scattering amplitudes with $N\\!\geqslant\\!4\hskip
0.85358pt$. In Section 4.4, we summarized a set of well-defined prescriptions
for the successful tree-level double-copy constructions in the warped massive
KK gauge/gravity theories, which include the double-copy constructions for
3-point full KK gauge/gravity amplitudes and the double-copy constructions for
the 4-point leading-order KK gauge/gravity amplitudes.

Figure 2: Schematic Summary of the present analyses: Equivalence Theorem and
Double-Copy Correspondences from 3-point scattering amplitudes to 4-point
scattering amplitudes and from massive KK gauge scattering amplitudes to
massive KK gravitational scattering amplitudes at the leading order of high
energy expansion.

Finally, we present a schematic summary of the present analyses as in Fig. 2.
In this figure, we start from the horizontal bottom row in which all entries
describe the basic 3-point KK scattering amplitudes. From the left to right,
the long equality sign linking the first two entries gives the “GAET(3)” which
connects the 3-point LO longitudinal KK gauge boson amplitude
$\mathcal{T}_{0}^{(3)}[A_{L}^{n}]$ to the corresponding LO KK Goldstone boson
amplitude $\widetilde{\mathcal{T}}_{0}^{(3)}[A_{5}^{n}]$; then the long
equality sign linking the last two entries gives the “GRET(3)” which connects
the 3-point LO longitudinal KK graviton amplitude
$\mathcal{M}_{0}^{(3)}[h_{L}^{n}]$ to the corresponding LO gravitational KK
Goldstone boson amplitude $\widetilde{\mathcal{M}}_{0}^{(3)}[\phi_{n}]$; in
the middle, the horizontal arrow indicates the double-copy “DC3” which
constructs the 3-point LO longitudinal KK graviton (KK Goldstone) amplitude
from the 3-point LO longitudinal KK gauge boson (KK Goldstone) amplitude.
After this, we see that all the entries and equality signs (or arrow) in the
top row are doing the same jobs as those in the bottom row except that all
entries in the top row deal with the 4-point KK gauge/gravity scattering
amplitudes. Finally, the vertical arrows from bottom to top indicate the
4-point-level “GAET(4)” and “GRET(4)” can be reduced to (reconstructed from)
the fundamental “GAET(3)” and “GRET(3)” for the basic 3-point KK gauge
(gravity) amplitudes; and furthermore, we can construct the GRET(3) from
GAET(3) through double-copy.

Acknowledgments This research was supported in part by the National NSF of
China under grants 12175136 and 11835005. YH is supported in part by the
Northwestern University Amplitudes and Insight group, the Department of
Physics and Astronomy, and Weinberg College.

Appendix

##  Appendix A Kinematics of KK Particle Scattering

In this Appendix we present the kinematics of 3 and 4 KK particle scattering
processes in the (3+1)d spacetime. We choose the 4d Minkowski metric tensor
$\,\eta^{\mu\nu}\\!=\\!\eta_{\mu\nu}\\!=\\!\text{diag}(-1,1,1,1)$.

For the 3 KK particle scattering, we define the 4-momenta of the external
particles as follows:

$\displaystyle\begin{aligned} p_{1}^{\mu}&=(E_{1},\hskip
0.85358ptks_{\theta},\hskip 0.85358pt0,\hskip 0.85358ptkc_{\theta})\hskip
0.85358pt,\qquad&&E_{1}=\sqrt{M_{1}^{2}+k^{2}\,}\,,\\\
p_{2}^{\mu}&=(E_{2},\hskip 0.85358pt-ks_{\theta},\hskip 0.85358pt0,\hskip
0.85358ptkc_{\theta})\hskip
0.85358pt,\qquad&&E_{2}=\sqrt{M_{2}^{2}+k^{2}\,}\,,\\\
p_{3}^{\mu}&=-(E_{3},\hskip 0.85358pt0,\hskip 0.85358pt0,\hskip
0.85358pt2kc_{\theta})\hskip
0.85358pt,\qquad&&E_{3}=\sqrt{M_{3}^{2}+4k^{2}c_{\theta}^{2}\,}\,,\end{aligned}$
(67jq)

where $k\\!=\\!|\vec{p}\hskip 0.85358pt|$, $(s_{\theta},\hskip
0.85358ptc_{\theta})=(\sin\theta,\hskip 0.85358pt\cos\theta)$,
$p_{j}^{2}\\!=\\!-M_{j}^{2}$ (with $j\\!=\\!1,2,3\hskip 0.85358pt$), and all
momenta are outgoing by convention.

Using the energy conservation condition $E_{1}\\!+\\!E_{2}\\!=\\!E_{3}$, we
can solve the magnitude of 3-momentum $k\\!=\\!|\vec{p}\hskip 0.85358pt|$ as a
function of the scattering angle $\theta\,$:

$\displaystyle k$
$\displaystyle=\bigg{\\{}\frac{1}{2}\sqrt{\\!M_{3}^{4}\\!+\\!4\cos^{2}\\!\theta\big{[}M_{1}^{4}\\!+\\!M_{2}^{4}\\!-\\!(M_{1}^{2}\\!+\\!M_{2}^{2})M_{3}^{2}\\!+\\!2M_{1}^{2}M_{2}^{2}\cos
2\theta\hskip 0.85358pt\big{]}\,}$ $\displaystyle\quad~{}~{}+\frac{\,\csc
2\theta\,}{\,2\,}\Big{[}M_{3}^{2}\cot
2\theta\\!-\\!(M_{1}^{2}\\!+\\!M_{2}^{2})\cot\theta\Big{]}\\!\bigg{\\}}^{\\!\frac{1}{2}}\\!.$
(67jr)

Alternatively, we can express $\theta$ as a function of $k$ :

$\cos\theta=\frac{1}{\,2k\,}\\!\left[2k^{2}\\!+\\!M_{1}^{2}\\!+\\!M_{2}^{2}\\!+\\!2\sqrt{\\!(k^{2}\\!+\\!M_{1}^{2})(k^{2}\\!+\\!M_{2}^{2})\,}\\!-\\!M_{3}^{2}\right]^{\\!\frac{1}{2}}\\!.$
(67js)

In high energy limit, we set $\hskip 0.85358ptk\\!\rightarrow\\!\infty$ and
$\theta\\!\rightarrow\\!0\hskip 0.85358pt$. Thus, we further expand $k$ and
$\theta$ as follows:

$\displaystyle k$
$\displaystyle=\frac{~{}M_{3}^{2}\\!-\\!2(M_{1}^{2}\\!+\\!M_{2}^{2})~{}}{4\sin^{2}\\!\theta}+\frac{~{}(M_{1}^{2}\\!-\\!M_{2}^{2})^{2}}{~{}4\big{[}M_{3}^{2}\\!-\\!2(M_{1}^{2}\\!+\\!M_{2}^{2})\big{]}~{}}+O(\theta^{2})\hskip
0.85358pt,$ (67jta) $\displaystyle\cos\theta$
$\displaystyle=1-\frac{1}{\,8k^{2}\,}\\!\left[M_{3}^{2}\\!-\\!2(M_{1}^{2}\\!+\\!2M_{2}^{2})\right]\\!+O(k^{-4})\hskip
0.85358pt,$ (67jtb) $\displaystyle\sin\theta$
$\displaystyle=\frac{1}{2k}\sqrt{M_{3}^{2}\\!-\\!2(M_{1}^{2}\\!+\\!M_{2}^{2})\,}+O(k^{-4})\hskip
0.85358pt.$ (67jtc)

From the above formulas, we see that to have real solutions of $k$ and
$\theta$ requires $M_{3}\\!\geqslant\\!\sqrt{2(M_{1}^{2}\\!+\\!M_{2}^{2})\,}$.

For the four-body scattering of KK states
$\,X_{1}X_{2}\\!\rightarrow\\!X_{3}X_{4}\,$, the 4-momenta in the center-of-
mass frame are defined as follows:

$\displaystyle\begin{aligned} p_{1}^{\mu}=&-\\!(E_{1},0,0,k),&&\hskip
28.45274ptp_{2}^{\mu}=-(E_{2},0,0,-k),\\\
p_{3}^{\mu}=&\,(E_{3},k^{\prime}s_{\theta},0,k^{\prime}c_{\theta}),&&\hskip
28.45274ptp_{4}^{\mu}=(E_{4},-k^{\prime}s_{\theta},0,-k^{\prime}c_{\theta}),\end{aligned}$
(67ju)

where we define the following Mandelstam variables:

$s=-(p_{1}\\!+p_{2})^{2},\qquad t=-(p_{1}\\!+p_{4})^{2},\qquad
u=-(p_{1}\\!+p_{3})^{2},$ (67jv)

from which we have $\hskip
0.85358pts+t+u=\\!\displaystyle\sum_{j=1}^{4}\\!M_{j}^{2}\hskip 0.85358pt$. In
addition, the momenta $k$ and $k^{\prime}$ in Eq.(67ju) are defined as
follows:

$\displaystyle\begin{aligned}
k&=\frac{1}{\,2\sqrt{s\,}\,}\\!\left(\big{[}s-\\!(M_{1}\\!+\\!M_{2})^{2}\big{]}\big{[}s-\\!(M_{1}\\!-\\!M_{2})^{2}\big{]}\right)^{\\!\frac{1}{2}}\\!,\\\
k^{\prime}&=\frac{1}{\,2\sqrt{s\,}\,}\\!\left([s-\\!(M_{3}\\!+\\!M_{4})^{2}][s-\\!(M_{3}\\!-\\!M_{4})^{2}]\right)^{\\!\frac{1}{2}}\\!.\end{aligned}$
(67jw)

We further define the massless Mandelstam variables as
$(s_{0},t_{0},u_{0})\\!=\\!(s,t,u)|_{M_{j}=0}$, and thus we have:

$s_{0}=4k^{2}\hskip
0.85358pt,\quad~{}~{}t_{0}=-\frac{\,s_{0}\,}{2}(1\\!+c_{\theta})\hskip
0.85358pt,\quad~{}~{}u_{0}=-\frac{\,s_{0}\,}{2}(1\\!-c_{\theta})\hskip
0.85358pt,$ (67jx)

where $j\\!=\\!1,2,3,4$ and the sum of these Mandelstam variables is given by
$\hskip 0.85358pts_{0}+t_{0}+u_{0}=0\hskip 0.85358pt$.

A massive KK graviton $h_{n}^{\mu\nu}$ in 4d has 5 physical helicity states
and their polarization tensors are represented by

$\varepsilon^{\mu\nu}_{\pm 2}=\epsilon_{\pm}^{\mu}\epsilon_{\pm}^{\nu}\hskip
0.85358pt,\quad\varepsilon_{\pm
1}^{\mu\nu}=\\!\frac{1}{\sqrt{2\,}\,}(\epsilon_{\pm}^{\mu}\epsilon_{L}^{\nu}\\!+\\!\epsilon_{L}^{\mu}\epsilon_{\pm}^{\nu})\hskip
0.85358pt,\quad\varepsilon^{\mu\nu}_{L}\\!=\frac{1}{\sqrt{6\,}\,}(\epsilon^{\mu}_{+}\epsilon^{\nu}_{-}\\!+\epsilon^{\mu}_{-}\epsilon^{\nu}_{+}\\!+2\hskip
0.85358pt\epsilon^{\mu}_{L}\epsilon^{\nu}_{L})\hskip 0.85358pt,$ (67jy)

where $\epsilon_{\pm}^{\mu}$ and $\epsilon_{L}^{\mu}$ are the spin-1
polarization vectors:

$\epsilon_{\pm}^{\mu}=\pm\frac{e^{\mp\text{i}\phi}}{\sqrt{2\,}\,}(0,c_{\theta}c_{\phi}\pm\\!\text{i}\hskip
0.85358pts_{\phi},\hskip 0.85358ptc_{\theta}s_{\phi}\\!\mp\\!\text{i}\hskip
0.85358ptc_{\phi},-s_{\theta})\hskip
0.85358pt,\quad~{}\epsilon_{L}^{\mu}\\!=\frac{1}{\,\mathbb{M}_{n}\,}(k,E_{n}s_{\theta}c_{\phi},E_{n}s_{\theta}s_{\phi},E_{n}c_{\theta})\,.$
(67jz)

In the above, the KK graviton $h_{n}^{\mu\nu}$ moves in an arbitrary direction
with polar and azimuthal angles $(\theta,\hskip 0.85358pt\phi)$. For instance,
consider the 4-body elastic scattering at the KK level-$n$, we have
$\,E_{n_{j}}\\!\\!=E\,$ and $\mathbb{M}_{n_{j}}\\!\\!=M\hskip 0.85358pt$.
Then, we can define the following polarization vectors in the center-of-mass
frame:

$\displaystyle\epsilon^{\mu}_{1,\hskip 0.85358pt\pm}$
$\displaystyle=\frac{1}{\sqrt{2\,}\,}(0,\mp 1,\text{i},0)\hskip
0.85358pt,\hskip 22.76219pt$
$\displaystyle\epsilon^{\mu}_{1,L}=-\frac{E}{M}(\beta,0,0,1)\hskip 0.85358pt,$
(67ka) $\displaystyle\epsilon^{\mu}_{2,\hskip 0.85358pt\pm}$
$\displaystyle=\frac{1}{\sqrt{2\,}\,}(0,\pm 1,\text{i},0)\hskip
0.85358pt,\hskip 22.76219pt$
$\displaystyle\epsilon^{\mu}_{2,L}=-\frac{E}{M}(\beta,0,0,-1)\hskip
0.85358pt,$ $\displaystyle\epsilon^{\mu}_{3,\hskip 0.85358pt\pm}$
$\displaystyle=\frac{1}{\sqrt{2\,}\,}(0,\pm c_{\theta},-\text{i},\mp
s_{\theta})\hskip 0.85358pt,\hskip 22.76219pt$
$\displaystyle\epsilon^{\mu}_{3,L}=\frac{E}{M}(\beta,s_{\theta},0,c_{\theta})\hskip
0.85358pt,$ $\displaystyle\epsilon^{\mu}_{4,\hskip 0.85358pt\pm}$
$\displaystyle=\frac{1}{\sqrt{2\,}\,}(0,\mp\text{i}\hskip
0.85358ptc_{\theta},-\text{i},\pm s_{\theta})\hskip 0.85358pt,\hskip
22.76219pt$
$\displaystyle\epsilon^{\mu}_{4,L}=\frac{E}{M}(\beta,-s_{\theta},0,-c_{\theta})\hskip
0.85358pt,$

where $\hskip 0.85358pt\beta\\!=\\!(1\\!-\\!M^{2}/E^{2})^{1/2}$. The
polarizations for the inelastic scattering processes can be derived in a
similar way.

##  Appendix B BRST Quantization for GRET Formulation

In this Appendix, we provide more detailed derivations to support the
formulation of warped GRET in the main text of Section 2.3. This includes the
BRST quantization used in Section 2.3.1, the formulation of the warped GRET
type-I as presented in Section 2.3.3, and the formulation of the warped GRET
type-II as given in Section 2.3.2.

###  B.1 BRST Quantization for Warped 5d Gravity

For the BRST quantization approach, the ghost fields are introduced in the
path intergral formulation. The 5d Faddeev-Popov ghost Lagrangian for the 5d
GR with warped metric takes the following form:

$\hat{\mathcal{L}}_{\mathrm{FP}}\,=\,e^{3A(z)}\,\hat{\bar{c}}^{M}\widehat{\tt
s}\hat{\mathcal{F}}_{M},$ (67kb)

where the 5d gauge-fixing functions
$\hat{\mathcal{F}}_{M}\\!\\!=\\!(\hat{\mathcal{F}}_{\mu},\hat{\mathcal{F}}_{5})$
are given by Eq.(44). The BRST transformations for 5d graviton, ghost and
anti-ghost fields take the following form:

$\displaystyle\widehat{\tt s}\hat{h}_{MN}$
$\displaystyle=-\partial_{M}\hat{c}_{N}-\partial_{N}\hat{c}_{M}+\hat{\kappa}\\!\left(\hat{h}_{MN}\partial_{P}-\hat{h}_{MP}\partial_{N}-\hat{h}_{NP}\partial_{M}\right)\\!\hat{c}^{P},$
$\displaystyle\widehat{\tt s}\hat{c}_{M}$ $\displaystyle=\hat{\kappa}\hskip
0.85358pt\hat{c}_{N}\partial^{N}\hat{c}_{M}\,,\qquad\widehat{\tt
s}\hat{\bar{c}}_{M}=-2\hskip 0.85358pt\xi^{-1}\hat{\mathcal{F}}_{M}\,.$ (67kc)

The BRST transformations exhibit nilpotency and obey $\widehat{\tt
s}^{2}\\!=0\hskip 0.85358pt$.

Next, we make KK expansions for ghost and anti-ghost fields as follows:

$\displaystyle\hat{c}^{\mu}(x,z)$
$\displaystyle=\frac{1}{\sqrt{L\,}\,}\\!\sum_{n=0}^{\infty}\\!c^{\mu}_{n}(x)\hskip
0.85358pt\mathsf{u}_{n}(z)\hskip
0.85358pt,\qquad\hat{\bar{c}}^{\mu}(x,z)=\frac{1}{\sqrt{L\,}\,}\\!\sum_{n=0}^{\infty}\\!\bar{c}_{n}^{\mu}(x)\hskip
0.85358pt\mathsf{u}_{n}(z)\hskip 0.85358pt,$ (67kda)
$\displaystyle\hat{c}^{5}(x,z)$
$\displaystyle=\frac{1}{\sqrt{L\,}\,}\\!\sum_{n=1}^{\infty}\\!c_{n}^{5}(x)\hskip
0.85358pt\mathsf{v}_{n}(z)\hskip
0.85358pt,\qquad\hat{\bar{c}}^{5}(x,z)=\frac{1}{\sqrt{L\,}\,}\\!\sum_{n=1}^{\infty}\\!\bar{c}_{n}^{5}(x)\hskip
0.85358pt\mathsf{v}_{n}(z)\hskip 0.85358pt.$ (67kdb)

Using the KK expansions (48), we derive the BRST transformations for the KK
fields in 4d spacetime:

$\displaystyle\widehat{\tt s}h_{n}^{\mu\nu}$
$\displaystyle=-\,\partial^{\mu}c^{\nu}_{n}\\!-\\!\frac{1}{2}\eta^{\mu\nu}\mathbb{M}_{n}c^{5}_{n}\\!+\\!\frac{\kappa}{2}\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{u}_{n}\mathsf{u}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt(h^{\mu\nu}_{m}\partial^{\alpha}\\!-\\!2h^{\mu\alpha}_{m}\partial^{\nu})c_{\alpha}^{\ell}\\!+\\!\frac{\kappa}{2}\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{u}_{n}\mathsf{u}_{m}\mathsf{v}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\mathbb{M}_{\ell}h^{\mu\nu}_{m}c^{5}_{\ell}$
$\displaystyle\quad\,+\\!\frac{\kappa}{\,2\sqrt{2\,}\,}\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{u}_{n}\mathsf{v}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\eta^{\mu\nu}\mathbb{M}_{\ell}\mathcal{V}^{\alpha}_{m}c_{\alpha}^{\ell}\\!-\\!\frac{\kappa}{\sqrt{2\,}\,}\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{u}_{n}\mathsf{v}_{m}\mathsf{v}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\mathcal{V}^{\mu}_{m}\partial^{\nu}c^{5}_{\ell}\\!+\\!\frac{\kappa}{\sqrt{6\,}\,}\big{(}\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{u}_{n}\mathsf{w}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt\phi_{m}\partial^{\mu}c^{\nu}_{\ell}$
$\displaystyle\quad\,-\hskip 0.85358pt\llbracket\hskip
0.85358pt\mathsf{u}_{n}\mathsf{w}_{m}\mathsf{v}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\eta^{\mu\nu}\mathbb{M}_{\ell}\phi_{m}c^{5}_{\ell}\big{)}+(\mu\leftrightarrow\nu)\hskip
0.85358pt,$ (67kea) $\displaystyle\widehat{\tt s}\mathcal{V}_{n}^{\mu}$
$\displaystyle=-\sqrt{2}\hskip
0.85358pt\partial^{\mu}c^{5}_{n}+\sqrt{2\,}\hskip
0.85358pt\mathbb{M}_{n}c_{n}^{\mu}\\!+\sqrt{2\,}\hskip 0.85358pt\kappa\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{v}_{n}\mathsf{u}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\mathbb{M}_{\ell}h^{\mu\nu}_{m}c_{\nu}^{\ell}\\!+\kappa\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{v}_{n}\mathsf{v}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\big{(}\mathcal{V}_{m}^{\mu}\partial^{\nu}\\!-\mathcal{V}_{m}^{\nu}\partial^{\mu}\big{)}c_{\nu}^{\ell}$
$\displaystyle\quad\,-\frac{\kappa}{\sqrt{3\,}\,}(\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{v}_{n}\mathsf{w}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\mathbb{M}_{\ell}\phi_{m}c^{\mu}_{\ell}\\!+2\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{v}_{n}\mathsf{w}_{m}\mathsf{v}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt\phi_{m}\partial^{\mu}c^{5}_{\ell})\hskip
0.85358pt,$ (67keb) $\displaystyle\widehat{\tt s}\phi_{n}$
$\displaystyle=-\sqrt{6\,}\hskip
0.85358pt\mathbb{M}_{n}c^{5}_{n}\\!+\\!\sqrt{3\,}\hskip 0.85358pt\kappa\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{w}_{n}\mathsf{v}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\mathbb{M}_{\ell}\mathcal{V}_{m}^{\mu}c_{\mu}^{\ell}\\!+\kappa\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{w}_{n}\mathsf{w}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt\phi_{m}\partial_{\mu}c^{\mu}_{\ell}$
$\displaystyle\quad\,-\kappa\hskip 0.85358pt\llbracket\hskip
0.85358pt\mathsf{w}_{n}\mathsf{w}_{m}\mathsf{v}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt\mathbb{M}_{\ell}\phi_{m}c^{5}_{\ell}\,,$
(67kec) $\displaystyle\widehat{\tt s}c^{\mu}_{n}$ $\displaystyle=\kappa\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{u}_{n}\mathsf{u}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358ptc^{\nu}_{m}\partial_{\nu}c^{\mu}_{\ell}-\kappa\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{u}_{n}\mathsf{v}_{m}\mathsf{u}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358pt\mathbb{M}_{\ell}c^{5}_{m}c^{\mu}_{\ell}\,,$ (67ked)
$\displaystyle\widehat{\tt s}c^{5}_{n}$ $\displaystyle=\kappa\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{v}_{n}\mathsf{u}_{m}\mathsf{v}_{\ell}\hskip
0.85358pt\rrbracket\hskip
0.85358ptc^{\mu}_{m}\partial_{\mu}c^{5}_{\ell}+\kappa\hskip
0.85358pt\llbracket\hskip
0.85358pt\mathsf{v}_{n}\mathsf{v}_{m}\mathsf{v}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt\mathbb{M}_{\ell}c^{5}_{m}c^{5}_{\ell}\,,$
(67kee) $\displaystyle\widehat{\tt s}\bar{c}^{\mu}_{n}$
$\displaystyle=-\frac{2}{\,\xi_{n}\,}\mathcal{F}_{n}^{\mu}\,,\qquad\widehat{\tt
s}\bar{c}^{5}_{n}=-\frac{2}{\,\xi_{n}\,}\mathcal{F}_{n}^{5}\,,$ (67kef)

where $\hskip 0.85358pt\kappa=\hat{\kappa}/\sqrt{L}$ and the brackets $\hskip
0.85358pt\llbracket\hskip 0.85358pt\cdots\hskip 0.85358pt\rrbracket\hskip
0.85358pt$ are defined in Appendix D.

###  B.2 Gravitational ET of Type-I

In the Sections 2.3.2-2.3.3 of the main text, we have formulated the GRET
which quantitatively connects the scattering amplitude of KK gravitons
$h^{\mu\nu}_{n}$ with helicities $(0,\hskip 0.85358pt\pm 1)$ to that of the
corresponding gravitational KK Goldstone bosons $(\phi_{n},\hskip
0.85358pt\mathcal{V}_{n}^{\pm 1})$ respectively. In this and next sub-
Appendices, we will provide more detailed derivations to support the results
in the main text.

We study the combination of gauge-fixing functions
$\partial_{\mu}\mathcal{F}_{n}^{\mu}\\!-\xi_{n}\mathbb{M}_{n}\mathcal{F}_{n}^{5}\,$
which eliminates the KK vector-Goldstone field $\,\mathcal{V}_{n}^{\mu}\,$.
With this we derive the following formula in the momentum space:

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

where the rescaling
$\phi_{n}\\!\rightarrow\\!\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}}\phi_{n}$ is
made according to Eq.(63). Applying the on-shell condition
$k^{2}\\!=\\!-\mathbb{M}_{n}^{2}\hskip 0.85358pt$, we further derive Eq.(67kf)
in the following form:

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

We then derive a BRST identity involving the gauge-fixing function
$\widetilde{\mathbb{F}}_{n}$ in Eq.(67kgb):

$\big{\langle}0\big{|}T\,\widetilde{\mathbb{F}}_{n}\mathbf{H}_{m}^{T}\big{|}0\big{\rangle}(k)=-\frac{\xi_{n}}{\sqrt{6\,}\hskip
0.85358pt\mathbb{M}_{n}^{2}}\big{\langle}0\big{|}T\,\widehat{\tt
s}\mathbf{H}_{m}^{T}(\text{i}k_{\mu}\bar{c}^{\mu}_{n}+\xi_{n}\mathbb{M}_{n}\bar{c}_{n}^{5})\big{|}0\big{\rangle}(k)\,,$
(67kh)

and it can be further rewritten in the following form by utilizing Eq.(67kg):

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

In the above, we adopt for simplicity the ’t Hooft-Feynman gauge condition
$(\xi_{n}\\!=\\!1)$. Thus, in Eq.(67ki), we define the following notations:

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

with
$\,{\mathsf{c}}_{n}\\!\equiv\text{i}\epsilon_{\mu}^{S}{c}^{\mu}_{n}+{c}_{n}^{5}\hskip
0.85358pt$, $\hskip
0.85358pt\bar{\mathsf{c}}_{n}\\!\equiv\text{i}\epsilon_{\mu}^{S}\bar{c}^{\mu}_{n}+\bar{c}_{n}^{5}\hskip
0.85358pt$, and $\epsilon_{\mu}^{S}\\!=\\!k_{\mu}/\mathbb{M}_{n}\hskip
0.85358pt$. In the above the external momentum is chosen as incoming. The
formulas (67ki)-(67kj) just give Eqs.(67cm)-(67cn) in the text. In addition,
we see that the quantities $\Delta^{\\!(3)}_{mj}(k^{2})$ and
$\widetilde{\Delta}^{\\!(4)}_{mj}(k^{2})$ are of loop order and are generated
by the non-linear terms of the BRST transformations (67ke) and (67ke) of
$h_{n}^{\mu\nu}$ and $\phi_{n}\hskip 0.85358pt$.

Next, we use the identity (66) and Eq.(67kg) to deduce an identity containing
the external state $\widetilde{\mathbb{F}}_{n}(k)\hskip 0.85358pt$:

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

which leads to the following identity,

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

with the amputated external state $\underline{\widetilde{\mathbb{F}}}_{n}(k)$
given by 111111We note that after the LSZ amputation in general $R_{\xi}$
gauge at tree-level, the coefficient of $h_{n}$ in
$\underline{\widetilde{\mathbb{F}}}$ should be $-1/2$ and corrects the
coefficient of $h_{n}$ as in our previous works Hang:2021fmp ; Hang:2022rjp ,
but this does not affect all the conclusions therein Hang:2021fmp ;
Hang:2022rjp .

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

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

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

where the matrix form is used such that
$(\bm{\Delta}^{\\!(3)})_{jj^{\prime}}\\!\\!=\\!{\Delta}^{\\!(3)}_{jj^{\prime}}$
and
$(\bm{\widetilde{\Delta}}^{\\!(4)})_{jj^{\prime}}\\!\\!=\\!\\!\widetilde{\Delta}^{\\!(4)}_{jj^{\prime}}$
with the matrix elements $({\Delta}^{\\!(3)}_{jj^{\prime}},\hskip
0.85358pt\widetilde{\Delta}^{\\!(4)}_{jj^{\prime}})$ from Eq.(67kj). The above
Eqs.(B.2)-(67kn) just reproduce the formulas (67co)-(67cr) in the main text.

Thus, from Eq.(67kl) we deduce another general identity for gravitational
equivalence theorem (GRET):

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

From this, we further derive the following GRET identity:

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

We can directly prove this identity by expanding the right-hand side of
Eq.(67kp) with each external state replacded by
$\Omega_{n}\\!=\\!h_{n}^{L}\\!-\underline{\widetilde{\mathbb{F}}}_{n}$ and
further using Eq.(67ko). This just reproduces the GRET identity (67ct) in the
main text.

We note that at tree level the LSZ reduction may be implemented directly.
Adopting the ’t Hooft-Feynman gauge $(\xi_{n}\\!=\\!1)$ and using the identity
(66), we can derive a new identity involving the external line
$\widetilde{\mathbb{F}}_{n}(k)\hskip 0.85358pt$ as follows:

$\displaystyle 0\,$ $\displaystyle=\big{\langle}0\hskip
0.85358pt|\,\widetilde{\mathbb{F}}_{n}(k)\cdots\overline{\Phi}\,|\hskip
0.85358pt0\big{\rangle}=\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}\big{(}\varepsilon_{\mu\nu}^{S}\\!+\\!\eta_{\mu\nu}\big{)}\big{\langle}0\hskip
0.85358pt|\,h_{n}^{\mu\nu}(k)\cdots\Phi\,|\hskip
0.85358pt0\big{\rangle}-\big{\langle}0\hskip
0.85358pt|\,\phi_{n}(k)\cdots\Phi\,|\hskip 0.85358pt0\big{\rangle}$
$\displaystyle=\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}\big{(}\varepsilon_{\mu\nu}^{S}\\!+\\!\eta_{\mu\nu}\big{)}\mathcal{D}^{\mu\nu\alpha\beta}_{h,nm}(k)\hskip
0.85358pt\mathcal{M}\big{[}h_{m}^{\alpha\beta}(k),\cdots\\!,\Phi\hskip
0.85358pt\big{]}-\mathcal{D}_{\phi,nm}(k)\hskip
0.85358pt\mathcal{M}\big{[}\phi_{m}(k),\cdots\\!,\Phi\hskip 0.85358pt\big{]}$
$\displaystyle=\mathcal{D}_{\phi,nm}(k)\hskip
0.85358pt\mathcal{M}\big{[}\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}(h_{m}^{S}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}h_{m})-\phi_{m},\cdots\\!,\Phi\hskip
0.85358pt\big{]}$ $\displaystyle\equiv\overline{\mathcal{D}}_{\phi}(k)\hskip
0.85358pt\mathcal{M}\big{[}\underline{\widetilde{\mathbb{F}}}_{n}(k),\cdots\\!,\Phi\hskip
0.85358pt\big{]},$ (67kq)

where
$\mathcal{D}_{\phi,nm}(k)\\!=\\!\delta_{nm}\overline{\mathcal{D}}_{\phi}(k)$
and the LSZ-amputated external state
$\underline{\widetilde{\mathbb{F}}}_{n}(k)$ is given by Eq.(67km) with
$C_{nm}\\!=\\!1\hskip 0.85358pt$. We have also used the tree-level propagators
(65a) and (65c) for $\xi_{n}\\!=\\!1\hskip 0.85358pt$. In addition, we can
extend the above derivation to the general $R_{\xi}$ gauge. Using the
$h_{n}^{\mu\nu}$ and $\phi_{n}$ propagators in the $R_{\xi}$ gauge as given by
Eqs.(67lk)-(67lkb), we derive the following identity of propagators at tree
level:

$\displaystyle\big{[}\varepsilon_{\mu\nu}^{S}\\!+\\!\mbox{$\frac{\,{1}\,}{2}$}(\xi_{n}\\!-\xi_{n}^{-1})\eta_{\mu\nu}\big{]}\mathcal{D}^{\mu\nu\alpha\beta}_{h,nm}(k)=\xi_{n}^{2}\mathcal{D}_{nm}(k)\big{(}\varepsilon^{\alpha\beta}_{S}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}\eta^{\alpha\beta}\big{)}\hskip
0.85358pt,$ (67kr)

we derive the following tree-level identity with amputated external state
$\widetilde{\mathbb{F}}_{n}(k)\hskip 0.85358pt$:

$\displaystyle 0\,$ $\displaystyle=\big{\langle}0\hskip
0.85358pt|\,\widetilde{\mathbb{F}}_{n}(k)\cdots\overline{\Phi}\,|\hskip
0.85358pt0\big{\rangle}$
$\displaystyle=\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}\big{[}\varepsilon_{\mu\nu}^{S}\\!+\\!\mbox{$\frac{\,{1}\,}{2}$}(\xi_{n}\\!-\xi_{n}^{-1})\eta_{\mu\nu}\big{]}\mathcal{D}^{\mu\nu\alpha\beta}_{h,nm}(k)\hskip
0.85358pt\mathcal{M}\big{[}h_{m}^{\alpha\beta}(k),\cdots\\!,\Phi\hskip
0.85358pt\big{]}\\!-\xi_{n}^{2}\mathcal{D}_{\phi,nm}(k)\hskip
0.85358pt\mathcal{M}\big{[}\phi_{m}(k),\cdots\\!,\Phi\hskip 0.85358pt\big{]}$
$\displaystyle=\xi_{n}^{2}\mathcal{D}_{nm}(k)\mathcal{M}\Big{[}\\!\sqrt{\\!\mbox{$\frac{\,{2}\,}{3}$}\,}\\!\big{(}h_{m}^{S}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}h_{m}\big{)}\\!-\phi_{m},\cdots\\!,\Phi\Big{]}.$
(67ks)

Thus, we have amputated external state
$\underline{\widetilde{\mathbb{F}}}_{n}(k)$ obey the identity:

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

where we have defined the following quantities at tree level,

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

We can readily extend the identity (67kt) to the case with $N$ external states
$\underline{\widetilde{\mathbb{F}}}_{n}$, which then reproduces the form of
the GRET (67ko) for $R_{\xi}$ gauge and at tree level.

###  B.3 Gravitational ET of Type-II

In Section 2.3.2 of the main text, we have formulated the KK GRET of type-II
which connects the scattering amplitude of KK gravitons $h^{\mu\nu}_{n}$ with
helicity $\pm 1$ to that of the corresponding gravitational KK vector
Goldstone bosons $\mathcal{V}_{n}^{\mu}$ with the same helicity. In this sub-
Appendix, we will provide detailed derivations to support the main text.

We start with the general Slavnov-Taylor (ST) type identity (66) for the
gravitational gauge-fixing functions given in Eq.(64a)-(64b). Then, we
reexpress the gauge-fixing function (64a) in the following matrix notation:

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

where we have also assumed the external momentum $k^{\mu}$ to be incoming.
Therefore, the ST-type identity associated with the gauge function
$\mathbb{F}_{n}^{\mu}$ is given by

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

By taking Eq.(67kv), we can further rewrite Eq.(67kw) as follows:

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

where we have defined the following notations:

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

which $\Delta^{(1)}_{mj}(k^{2})$ and $\widetilde{\Delta}^{(2)}_{mj}(k^{2})$
are the loop-level quantities. The above Eqs.(67kx)-(B.3) just give
Eqs.(67bq)-(67br) in the text. Moreover, according to the BRST transformations
(67ke)-(67ke), each loop-level quantity incorporates distinct contributions
from the non-linear terms of the BRST transformations (67ke)-(67ke):

$\displaystyle\Delta^{(1)}_{mj}(k^{2})$
$\displaystyle=\big{[}\Delta_{h,h}^{(1)}(k^{2})+\Delta_{h,\mathcal{V}}^{(1)}(k^{2})+\Delta_{h,\phi}^{(1)}(k^{2})\big{]}_{mj}\,,$
(67kza) $\displaystyle\widetilde{\Delta}^{(2)}_{mj}(k^{2})$
$\displaystyle=\big{[}\widetilde{\Delta}_{\mathcal{V},h}^{(2)}(k^{2})+\widetilde{\Delta}_{\mathcal{V},\mathcal{V}}^{(2)}(k^{2})+\widetilde{\Delta}_{\mathcal{V},\phi}^{(2)}(k^{2})\big{]}_{mj}\,.$
(67kzb)

Next, we consider the identity (66) with an external state given by the
combination of gauge-fixing functions as in Eq.(67kv). With this, we can
directly amputate the external state as follows:

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

which leads to the following identity:

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

where we have used the BRST identity (67kx) and $\mathcal{M}[\cdots]$ denotes
the amputated scattering amplitude. In Eq.(67lb), we derive the following
amputated external state $\underline{\mathbb{F}}_{n}^{\hskip
0.85358pt\mu}\hskip 0.85358pt$:

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

with the loop-induced modification factor given by

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

The matrix form presented above represents loop-level quantities
$(\bm{\Delta}^{\\!(1)})_{jj^{\prime}}\\!={\Delta}^{\\!(1)}_{jj^{\prime}}$ and
$(\bm{\widetilde{\Delta}}^{\\!(2)})_{jj^{\prime}}\\!=\\!\widetilde{\Delta}^{\\!(2)}_{jj^{\prime}}\hskip
0.85358pt$. We provide the above loop-level formulation for completeness,
since our present focus in the main text is to analyze the KK scattering
amplitudes at tree level. The above Eqs.(B.3)-(67ld) just give the
Eqs.(67bs)-(67bv) in the main text.

By utilizing a transverse polarization vector $\epsilon^{\mu}_{\pm}$, we
contract it with $\underline{\mathbb{F}}_{n}^{\hskip 0.85358pt\mu}(k)$ in
Eq.(67lc), yielding the formula:

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

where $v_{\pm 1}^{\mu\nu}\\!=O(\mathbb{M}_{n}/E_{n})$. Then, we can generalize
Eq.(67lb) to incorporate $N$ externally-amputated gauge-fixing functions:

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

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

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

We can prove the above identity by computing the amplitude on its right-hand
side. For this, we expand $\Theta_{n}$ in terms of $\,\Theta_{n}\\!=h_{n}^{\pm
1}\\!-\underline{\mathbb{F}}_{n}$ for each external state of Eq.(67lg) and
thus deduce the left-hand side of Eq.(67lg) after applying the identity
Eq.(67lf) to eliminate each extenal state of $\underline{\mathbb{F}}_{n}$. The
above Eq.(67lg) just reproduce the GRET identity (67ce) in the main text.

##  Appendix C Feynman Rules for Warped KK Gauge and Gravity Theories

In this Appendix, we present the relevant Feynman rules for the warped KK
gauge and gravity theories under the 5d compactification of
$S^{1}\\!/\mathbb{Z}_{2}\hskip 0.85358pt$, which are needed for explicitly
computing the 3-point and 4-point KK scattering amplitudes in the main text.

###  C.1 Feynman Rules for Warped KK Gauge Theory

In this sub-Appendix, we present the relevant Feynman rules for the warped KK
gauge theory. The propagators for KK gauge boson and the KK Goldstone boson
take the following forms in the $R_{\xi}$ gauge:

$\displaystyle\mathcal{D}_{nm}^{\mu\nu}(p)$
$\displaystyle=-\frac{\text{i}\hskip
0.85358pt\delta_{nm}}{\,p^{2}\\!+\\!\mathbb{M}_{n}^{2}\,}\\!\left[\eta^{\mu\nu}\\!+(\xi_{n}\\!-\\!1)\frac{p^{\mu}p^{\nu}}{\,p^{2}+\xi_{n}\mathbb{M}_{n}^{2}\,}\right]\\!,$
(67lha) $\displaystyle\mathcal{D}_{nm}(p)$
$\displaystyle=-\frac{\text{i}\hskip
0.85358pt\delta_{nm}}{~{}p^{2}+\xi_{n}\mathbb{M}_{n}^{2}~{}}\,.$ (67lhb)

The trilinear vertices and quartic vertices for KK gauge bosons (KK Goldstone
bosons) are derived as follows:

$\displaystyle\begin{aligned} &=-g\hskip 0.85358pta_{nm\ell}\hskip
0.85358ptf^{abc}\big{[}\hskip
0.85358pt\eta^{\mu\nu}(p_{1}\\!-p_{2})^{\alpha}\\!+\eta^{\nu\alpha}(p_{2}\\!-p_{3})^{\mu}\\\
&\hskip 73.97716pt+\eta^{\alpha\mu}(p_{3}\\!-p_{1})^{\nu}\hskip
0.85358pt\big{]}\,,\end{aligned}$ (67lia) $\displaystyle=-g\hskip
0.85358pt\tilde{a}_{nm\ell}\hskip 0.85358ptf^{abc}(p_{1}\\!-p_{2})^{\mu}\,,$
(67lib) $\displaystyle=\text{i}g\hskip 0.85358pta_{nm\ell}\hskip
0.85358ptf^{abc}\eta^{\mu\nu}(M_{n}^{2}\\!-\\!M_{m}^{2})M_{\ell}^{-1}\,,$
(67lic) $\displaystyle\begin{array}[]{ll}&=\text{i}\hskip 0.85358ptg^{2}\hskip
0.85358pta_{nm\ell q}\big{[}\hskip
0.85358ptf^{abe}f^{cde}(\eta^{\mu\alpha}\eta^{\nu\beta}\\!-\eta^{\mu\beta}\eta^{\nu\alpha})\\\
&\hskip
56.9055pt+f^{ace}f^{dbe}(\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\eta^{\mu\nu}\eta^{\alpha\beta})\\\
&\hskip
56.9055pt+f^{ade}f^{bce}(\eta^{\mu\nu}\eta^{\alpha\beta}\\!-\eta^{\mu\alpha}\eta^{\nu\beta})\hskip
0.85358pt\big{]},\end{array}$ (67lig)

where the effective cubic and quartic KK coupling coefficients
$(a_{nm\ell},\,\tilde{a}_{nm\ell},\,a_{nm\ell q},$ $\tilde{a}_{nm\ell q})$ are
defined as follows:

$\displaystyle a_{nm\ell}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\text{d}z\,e^{A(z)}\,\mathsf{f}_{n}(z)\mathsf{f}_{m}(z)\mathsf{f}_{\ell}(z)\,,$
(67lja) $\displaystyle\tilde{a}_{nm\ell}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\text{d}z\,e^{A(z)}\,\tilde{\mathsf{f}}_{n}(z)\tilde{\mathsf{f}}_{m}(z)\mathsf{f}_{\ell}(z)\,,$
(67ljb) $\displaystyle a_{nm\ell q}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\text{d}z\,e^{A(z)}\,\mathsf{f}_{n}(z)\mathsf{f}_{m}(z)\mathsf{f}_{\ell}(z)\mathsf{f}_{q}(z)\,,$
(67ljc) $\displaystyle\tilde{a}_{nm\ell q}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\text{d}z\,e^{A(z)}\,\tilde{\mathsf{f}}_{n}(z)\tilde{\mathsf{f}}_{m}(z)\tilde{\mathsf{f}}_{\ell}(z)\tilde{\mathsf{f}}_{q}(z)\,.$
(67ljd)

These coupling coefficients will be used for the double-copy construction in
Sections 4.1-4.2 and for deriving the gravitational sum rules in Appendix D.
In the above Feynman rules (67li), the coupling $g$ is the 4d gauge coupling
and is connected to the 5d gauge coupling $\hat{g}_{5}$ via
$\,g=\hat{g}_{5}/\sqrt{L\,}$. For the 5d physical coordinate $y$ (instead of
the 5d conformal coordinate $z$), the 5d length is $\bar{L}\\!=\\!\pi r_{c}$
and the corresponding 4d gauge coupling is
$\bar{g}\\!=\\!\hat{g}_{5}/\sqrt{\bar{L}\,}$, which is related to the gauge
coupling $g$ via $\,\bar{g}\\!=\\!{g}\sqrt{\bar{L}/L\,}$. Thus, the cubic KK
gauge coupling coefficient $\bar{a}_{nm\ell}$ (defined in 5d physical
coordinate) is connected to the above cubic KK gauge coupling coefficient
${a}_{nm\ell}$ (defined in 5d conformal coordinate) by
$\bar{a}_{nm\ell}\\!=\\!{a}_{nm\ell}\sqrt{\bar{L}/L\,}$. Similarly, for the
quartic KK gauge coupling coefficient, we have the relation, $\bar{a}_{nm\ell
q}\\!=\\!(\bar{L}/L)\hskip 0.85358pt{a}_{nm\ell q}\hskip 0.85358pt$.

###  C.2 Feynman Rules for Warped KK Gravity Theory

The propagators for KK gravitons and gravitational KK Goldstone bosons in a
general $R_{\xi}$ gauge (43)-(44) are given by the following Hang:2021fmp :

$\displaystyle\mathcal{D}_{nm}^{\mu\nu\alpha\beta}(p)=-\frac{\,\text{i}\hskip
0.85358pt\delta_{nm}\,}{2}\left\\{\frac{\,\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta}}{p^{2}\\!+\mathbb{M}_{n}^{2}}+\frac{1}{3}\\!\left[\frac{1}{\,p^{2}\\!+\mathbb{M}_{n}^{2}\,}-\frac{1}{\,p^{2}\\!+\\!(3\hskip
0.85358pt\xi_{n}\\!-\\!2)\mathbb{M}_{n}^{2}\,}\right]\right.$
$\displaystyle\times\\!\left(\\!\eta^{\mu\nu}\\!\\!-\\!\frac{\,2p^{\mu}p^{\nu}\,}{\mathbb{M}_{n}^{2}}\\!\right)\\!\\!\left(\\!\eta^{\alpha\beta}\\!\\!-\\!\frac{\,2p^{\alpha}p^{\beta}\,}{\mathbb{M}_{n}^{2}}\\!\right)\\!+\\!\frac{1}{M_{n}^{2}}\\!\left[\frac{1}{\,p^{2}\\!+\\!\mathbb{M}_{n}^{2}\,}-\frac{1}{\,p^{2}\\!+\xi_{n}\mathbb{M}_{n}^{2}\,}\right]\\!\\!\left(\eta^{\mu\alpha}p^{\nu}p^{\beta}\\!+\eta^{\mu\beta}p^{\nu}p^{\alpha}\right.$
$\displaystyle\left.\left.+\,\eta^{\nu\alpha}p^{\mu}p^{\beta}\\!+\\!\eta^{\nu\beta}p^{\mu}p^{\alpha}\right)\\!+\\!\frac{~{}4\hskip
0.85358ptp^{\mu}p^{\nu}p^{\alpha}p^{\beta}~{}}{\xi_{n}M_{n}^{4}}\\!\left(\frac{1}{~{}p^{2}\\!+\xi_{n}^{2}\mathbb{M}_{n}^{2}~{}}-\frac{1}{~{}p^{2}\\!+\xi_{n}\mathbb{M}_{n}^{2}~{}}\right)\\!\right\\},$
(67lka) $\displaystyle\mathcal{D}^{\mu\nu}_{nm}(p)=-\frac{\text{i}\hskip
0.85358pt\delta_{nm}}{\,p^{2}\\!+\xi_{n}\mathbb{M}_{n}^{2}\,}\\!\left[\eta^{\mu\nu}\\!\\!-\\!\frac{~{}p^{\mu}p^{\nu}(1\\!-\\!\xi_{n})~{}}{\,p^{2}\\!+\xi_{n}^{2}\mathbb{M}_{n}^{2}~{}}\right]\\!,$
(67lkb) $\displaystyle\mathcal{D}_{nm}(p)=-\frac{\text{i}\hskip
0.85358pt\delta_{nm}}{\,p^{2}+(3\xi_{n}\\!\\!-\\!2)\mathbb{M}_{n}^{2}\,}\,.$
(67lkc)

The relevant trilinear gravitational KK vertices are given by

$\displaystyle\hskip 17.07182pt=\text{i}\hskip 0.85358pt\kappa\hskip
0.85358pt\alpha_{nm\ell}\hskip
0.85358pt\Gamma^{\mu\nu\rho\sigma\alpha\beta}_{nm\ell}(p_{1},p_{2},p_{3})\hskip
0.85358pt,$ (67lla) $\displaystyle\hskip 17.64056pt=\frac{\,\text{i}\hskip
0.85358pt\kappa\,}{2}\tilde{\alpha}_{nm\ell}\hskip
0.85358pt\Gamma^{\alpha\beta\mu\nu}_{nm\ell}(p_{1},p_{2})\hskip 0.85358pt,$
(67llb) $\displaystyle\hskip 17.64056pt\begin{aligned}
&=\frac{\,\text{i}\hskip
0.85358pt\kappa\,}{2}\tilde{\beta}_{nm\ell}\Big{\\{}2(p_{1}^{\mu}p_{1}^{\nu}+p_{2}^{\mu}p_{2}^{\nu})+p_{1}^{\mu}p_{2}^{\nu}+p_{1}^{\nu}p_{2}^{\mu}-[2(p_{1}^{2}+p_{2}^{2})\\\
&\hskip
59.75078pt+3p_{1}\\!\cdot\\!p_{2}+\mathbb{M}_{n}\mathbb{M}_{\ell}+\mathbb{M}_{m}\mathbb{M}_{\ell}]\eta^{\mu\nu}\Big{\\}}\hskip
0.85358pt,\end{aligned}$ (67llc) $\displaystyle\hskip
17.64056pt\begin{aligned} &=\frac{\text{i}\hskip
0.85358pt\kappa}{\sqrt{6\,}\,}\hskip
0.85358pt\tilde{\rho}_{nm\ell}\Big{\\{}\\!\\!-\\!2\hskip
0.85358pt(p_{1}^{\mu}p_{1}^{\nu}\\!+p_{2}^{\mu}p_{2}^{\nu})\\!-\\!3\hskip
0.85358ptp_{2}^{\mu}p_{1}^{\nu}\\!+\\![2(p_{1}^{2}\\!+\\!p_{2}^{2})\\\ &\hskip
59.75078pt+3\hskip 0.85358ptp_{1}\\!\cdot\\!p_{2}]\eta^{\mu\nu}\Big{\\}}\hskip
0.85358pt,\end{aligned}$ (67lld) $\displaystyle\hskip
17.64056pt\begin{aligned} &=\text{i}\hskip
0.85358pt2\sqrt{\frac{2}{3}\,}\hskip 0.85358pt\kappa\hskip
0.85358pt\tilde{\omega}_{nm\ell}\big{(}p_{1}\\!\cdot\\!p_{2}\\!+p_{1}\\!\cdot\\!p_{3}\\!+p_{2}\\!\cdot\\!p_{3}\\!-\mathbb{M}_{n}\mathbb{M}_{m}\\\
&\hskip
79.66771pt-\mathbb{M}_{n}\mathbb{M}_{\ell}\\!-\mathbb{M}_{m}\mathbb{M}_{\ell}\big{)}\hskip
0.85358pt.\end{aligned}$ (67lle)

In the above Feynman rules (67ll), the coupling $\kappa$ is the 4d
gravitational coupling and is connected to the 5d gravitational coupling
$\hat{\kappa}_{5}$ via $\,\kappa\\!=\\!\hat{\kappa}_{5}/\sqrt{L\,}$. For the
5d physical coordinate $y$ (instead of the 5d conformal coordinate $z$), the
5d length is $\bar{L}\\!=\\!\pi r_{c}$ and the corresponding 4d gravitational
coupling is $\bar{\kappa}\\!=\\!\hat{\kappa}_{5}/\sqrt{\bar{L}\,}$, which is
related to the gravitational coupling $\kappa$ via
$\,\bar{\kappa}\\!=\\!{\kappa}\sqrt{\bar{L}/L\,}$. Thus, the cubic KK
gravitational coupling coefficient $\bar{\alpha}_{nm\ell}$ (defined in 5d
physical coordinate) is connected to the above cubic KK gravitational coupling
coefficient ${\alpha}_{nm\ell}$ (defined in 5d conformal coordinate) by
$\bar{\alpha}_{nm\ell}\\!=\\!{\alpha}_{nm\ell}\sqrt{\bar{L}/L\,}$.

In the above Eq.(67lla), the trilinear gravitational vertex function
$\Gamma^{\mu\nu\rho\sigma\alpha\beta}_{nm\ell}$ is defined as follows:

$\Gamma^{\mu\nu\rho\sigma\alpha\beta}_{nm\ell}(p_{1},p_{2},p_{3})=\frac{1}{\,8\,}\big{[}(\mathbb{M}_{n}^{2}\hskip
0.85358ptF_{1}\\!-p_{1}^{2}\hskip
0.85358ptF_{2}+F_{3})\\!+\\!(1\\!\leftrightarrow\\!2)\\!+\\!(1\\!\leftrightarrow\\!3)\big{]}\,,$
(67lm)

with the functions $(F_{1},\,F_{2},\,F_{3})$ given by

$\displaystyle F_{1}=$
$\displaystyle-\eta^{\alpha\beta}\eta^{\mu\sigma}\eta^{\nu\rho}-\eta^{\alpha\beta}\eta^{\mu\rho}\eta^{\nu\sigma}+2\eta^{\alpha\beta}\eta^{\mu\nu}\eta^{\rho\sigma}+\eta^{\alpha\mu}\eta^{\beta\sigma}\eta^{\nu\rho}+\eta^{\alpha\mu}\eta^{\beta\rho}\eta^{\nu\sigma}-\eta^{\alpha\mu}\eta^{\beta\nu}\eta^{\rho\sigma}$
$\displaystyle+\eta^{\alpha\sigma}\eta^{\beta\mu}\eta^{\nu\rho}+\eta^{\alpha\rho}\eta^{\beta\mu}\eta^{\nu\sigma}+\eta^{\alpha\nu}\eta^{\beta\sigma}\eta^{\mu\rho}+\eta^{\alpha\nu}\eta^{\beta\rho}\eta^{\mu\sigma}-\eta^{\alpha\nu}\eta^{\beta\mu}\eta^{\rho\sigma}+\eta^{\alpha\sigma}\eta^{\beta\nu}\eta^{\mu\rho}$
$\displaystyle+\eta^{\alpha\rho}\eta^{\beta\nu}\eta^{\mu\sigma}-3\eta^{\alpha\rho}\eta^{\beta\sigma}\eta^{\mu\nu}-3\eta^{\alpha\sigma}\eta^{\beta\rho}\eta^{\mu\nu}\,,$
$\displaystyle F_{2}=$
$\displaystyle~{}3\eta^{\alpha\beta}\eta^{\mu\sigma}\eta^{\nu\rho}+3\eta^{\alpha\beta}\eta^{\mu\rho}\eta^{\nu\sigma}-4\eta^{\alpha\beta}\eta^{\mu\nu}\eta^{\rho\sigma}-2\eta^{\alpha\mu}\eta^{\beta\sigma}\eta^{\nu\rho}-2\eta^{\alpha\mu}\eta^{\beta\rho}\eta^{\nu\sigma}+3\eta^{\alpha\mu}\eta^{\beta\nu}\eta^{\rho\sigma}$
$\displaystyle+3\eta^{\alpha\nu}\eta^{\beta\mu}\eta^{\rho\sigma}-2\eta^{\alpha\nu}\eta^{\beta\sigma}\eta^{\mu\rho}-2\eta^{\alpha\nu}\eta^{\beta\rho}\eta^{\mu\sigma}+4\eta^{\alpha\sigma}\eta^{\beta\rho}\eta^{\mu\nu}+4\eta^{\alpha\rho}\eta^{\beta\sigma}\eta^{\mu\nu}-2\eta^{\alpha\rho}\eta^{\beta\nu}\eta^{\mu\sigma}$
$\displaystyle-2\eta^{\alpha\rho}\eta^{\beta\mu}\eta^{\nu\sigma}-2\eta^{\alpha\sigma}\eta^{\beta\nu}\eta^{\mu\rho}-2\eta^{\alpha\sigma}\eta^{\beta\mu}\eta^{\nu\rho}\,,$
$\displaystyle F_{3}=$
$\displaystyle~{}2\eta^{\mu\nu}\eta^{\rho\sigma}p_{1}^{\alpha}p_{2}^{\beta}+\eta^{\mu\nu}\eta^{\rho\sigma}p_{1}^{\alpha}p_{3}^{\beta}+\eta^{\nu\rho}\eta^{\beta\sigma}p_{1}^{\alpha}p_{2}^{\mu}+\eta^{\beta\rho}\eta^{\nu\sigma}p_{1}^{\alpha}p_{2}^{\mu}+\eta^{\nu\rho}\eta^{\beta\sigma}p_{1}^{\alpha}p_{3}^{\mu}+\eta^{\beta\rho}\eta^{\nu\sigma}p_{1}^{\alpha}p_{3}^{\mu}$
$\displaystyle+\eta^{\mu\rho}\eta^{\beta\sigma}p_{1}^{\alpha}p_{2}^{\nu}+\eta^{\beta\rho}\eta^{\mu\sigma}p_{1}^{\alpha}p_{2}^{\nu}+\eta^{\mu\rho}\eta^{\beta\sigma}p_{1}^{\alpha}p_{3}^{\nu}+\eta^{\beta\rho}\eta^{\mu\sigma}p_{1}^{\alpha}p_{3}^{\nu}-\eta^{\beta\sigma}\eta^{\mu\nu}p_{1}^{\alpha}p_{2}^{\rho}-\eta^{\beta\sigma}\eta^{\mu\nu}p_{1}^{\alpha}p_{3}^{\rho}$
$\displaystyle+\eta^{\beta\nu}\eta^{\mu\sigma}p_{1}^{\alpha}p_{3}^{\rho}+\eta^{\beta\mu}\eta^{\nu\sigma}p_{1}^{\alpha}p_{3}^{\rho}-\eta^{\mu\sigma}\eta^{\nu\rho}p_{1}^{\alpha}p_{2}^{\beta}-\eta^{\mu\sigma}\eta^{\nu\rho}p_{1}^{\alpha}p_{3}^{\beta}-\eta^{\beta\rho}\eta^{\mu\nu}p_{1}^{\alpha}p_{2}^{\sigma}-\eta^{\beta\rho}\eta^{\mu\nu}p_{1}^{\alpha}p_{3}^{\sigma}$
$\displaystyle+\eta^{\beta\nu}\eta^{\mu\rho}p_{1}^{\alpha}p_{3}^{\sigma}+\eta^{\beta\mu}\eta^{\nu\rho}p_{1}^{\alpha}p_{3}^{\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}p_{1}^{\alpha}p_{2}^{\beta}-\eta^{\mu\rho}\eta^{\nu\sigma}p_{1}^{\alpha}p_{3}^{\beta}-\eta^{\beta\nu}\eta^{\rho\sigma}p_{1}^{\alpha}p_{2}^{\mu}-\eta^{\beta\nu}\eta^{\rho\sigma}p_{1}^{\alpha}p_{3}^{\mu}$
$\displaystyle-\eta^{\beta\mu}\eta^{\rho\sigma}p_{1}^{\alpha}p_{2}^{\nu}-\eta^{\beta\mu}\eta^{\rho\sigma}p_{1}^{\alpha}p_{3}^{\nu}+2\eta^{\mu\nu}\eta^{\rho\sigma}p_{1}^{\beta}p_{2}^{\alpha}+\eta^{\mu\nu}\eta^{\rho\sigma}p_{1}^{\beta}p_{3}^{\alpha}+\eta^{\nu\rho}\eta^{\alpha\sigma}p_{1}^{\beta}p_{2}^{\mu}+\eta^{\alpha\rho}\eta^{\nu\sigma}p_{1}^{\beta}p_{2}^{\mu}$
$\displaystyle+\eta^{\nu\rho}\eta^{\alpha\sigma}p_{1}^{\beta}p_{3}^{\mu}+\eta^{\alpha\rho}\eta^{\nu\sigma}p_{1}^{\beta}p_{3}^{\mu}+\eta^{\mu\rho}\eta^{\alpha\sigma}p_{1}^{\beta}p_{2}^{\nu}+\eta^{\alpha\rho}\eta^{\mu\sigma}p_{1}^{\beta}p_{2}^{\nu}+\eta^{\mu\rho}\eta^{\alpha\sigma}p_{1}^{\beta}p_{3}^{\nu}+\eta^{\alpha\rho}\eta^{\mu\sigma}p_{1}^{\beta}p_{3}^{\nu}$
$\displaystyle-\eta^{\alpha\sigma}\eta^{\mu\nu}p_{1}^{\beta}p_{2}^{\rho}-\eta^{\alpha\sigma}\eta^{\mu\nu}p_{1}^{\beta}p_{3}^{\rho}+\eta^{\alpha\nu}\eta^{\mu\sigma}p_{1}^{\beta}p_{3}^{\rho}+\eta^{\alpha\mu}\eta^{\nu\sigma}p_{1}^{\beta}p_{3}^{\rho}-\eta^{\mu\sigma}\eta^{\nu\rho}p_{1}^{\beta}p_{3}^{\alpha}-\eta^{\alpha\rho}\eta^{\mu\nu}p_{1}^{\beta}p_{2}^{\sigma}$
$\displaystyle-\eta^{\alpha\rho}\eta^{\mu\nu}p_{1}^{\beta}p_{3}^{\sigma}+\eta^{\alpha\nu}\eta^{\mu\rho}p_{1}^{\beta}p_{3}^{\sigma}+\eta^{\alpha\mu}\eta^{\nu\rho}p_{1}^{\beta}p_{3}^{\sigma}-\eta^{\mu\rho}\eta^{\nu\sigma}p_{1}^{\beta}p_{2}^{\alpha}-\eta^{\mu\rho}\eta^{\nu\sigma}p_{1}^{\beta}p_{3}^{\alpha}-\eta^{\alpha\nu}\eta^{\rho\sigma}p_{1}^{\beta}p_{2}^{\mu}$
$\displaystyle-\eta^{\alpha\nu}\eta^{\rho\sigma}p_{1}^{\beta}p_{3}^{\mu}-\eta^{\alpha\mu}\eta^{\rho\sigma}p_{1}^{\beta}p_{2}^{\nu}-\eta^{\alpha\mu}\eta^{\rho\sigma}p_{1}^{\beta}p_{3}^{\nu}-\eta^{\nu\rho}\eta^{\mu\sigma}p_{1}^{\beta}p_{2}^{\alpha}-\eta^{\alpha\sigma}\eta^{\beta\rho}p_{1}^{\mu}p_{2}^{\nu}-\eta^{\alpha\rho}\eta^{\beta\sigma}p_{1}^{\mu}p_{2}^{\nu}$
$\displaystyle+\eta^{\alpha\beta}\eta^{\rho\sigma}p_{1}^{\mu}p_{2}^{\nu}-\eta^{\alpha\sigma}\eta^{\beta\rho}p_{1}^{\mu}p_{3}^{\nu}-\eta^{\alpha\rho}\eta^{\beta\sigma}p_{1}^{\mu}p_{3}^{\nu}+\eta^{\alpha\beta}\eta^{\rho\sigma}p_{1}^{\mu}p_{3}^{\nu}-\eta^{\alpha\beta}\eta^{\nu\sigma}p_{1}^{\mu}p_{3}^{\rho}-\eta^{\alpha\beta}\eta^{\nu\rho}p_{1}^{\mu}p_{3}^{\sigma}$
$\displaystyle-\eta^{\alpha\nu}\eta^{\rho\sigma}p_{1}^{\mu}p_{2}^{\beta}-\eta^{\beta\nu}\eta^{\rho\sigma}p_{1}^{\mu}p_{2}^{\alpha}-\eta^{\alpha\sigma}\eta^{\beta\rho}p_{1}^{\nu}p_{2}^{\mu}-\eta^{\alpha\rho}\eta^{\beta\sigma}p_{1}^{\nu}p_{2}^{\mu}+\eta^{\alpha\beta}\eta^{\rho\sigma}p_{1}^{\nu}p_{2}^{\mu}-\eta^{\alpha\sigma}\eta^{\beta\rho}p_{1}^{\nu}p_{3}^{\mu}$
$\displaystyle-\eta^{\alpha\rho}\eta^{\beta\sigma}p_{1}^{\nu}p_{3}^{\mu}+\eta^{\alpha\beta}\eta^{\rho\sigma}p_{1}^{\nu}p_{3}^{\mu}-\eta^{\alpha\beta}\eta^{\mu\sigma}p_{1}^{\nu}p_{3}^{\rho}-\eta^{\alpha\beta}\eta^{\mu\rho}p_{1}^{\nu}p_{3}^{\sigma}-\eta^{\alpha\mu}\eta^{\rho\sigma}p_{1}^{\nu}p_{2}^{\beta}-\eta^{\beta\mu}\eta^{\rho\sigma}p_{1}^{\nu}p_{2}^{\alpha}$
$\displaystyle+\eta^{\beta\nu}\eta^{\mu\sigma}p_{1}^{\rho}p_{2}^{\alpha}+\eta^{\beta\mu}\eta^{\nu\sigma}p_{1}^{\rho}p_{2}^{\alpha}+\eta^{\alpha\nu}\eta^{\mu\sigma}p_{1}^{\rho}p_{2}^{\beta}+\eta^{\alpha\mu}\eta^{\nu\sigma}p_{1}^{\rho}p_{2}^{\beta}+\eta^{\beta\nu}\eta^{\alpha\sigma}p_{1}^{\rho}p_{2}^{\mu}+\eta^{\alpha\nu}\eta^{\beta\sigma}p_{1}^{\rho}p_{2}^{\mu}$
$\displaystyle+\eta^{\beta\nu}\eta^{\alpha\sigma}p_{1}^{\rho}p_{3}^{\mu}+\eta^{\alpha\nu}\eta^{\beta\sigma}p_{1}^{\rho}p_{3}^{\mu}+\eta^{\beta\mu}\eta^{\alpha\sigma}p_{1}^{\rho}p_{2}^{\nu}+\eta^{\alpha\mu}\eta^{\beta\sigma}p_{1}^{\rho}p_{2}^{\nu}-\eta^{\alpha\beta}\eta^{\mu\sigma}p_{1}^{\rho}p_{2}^{\nu}+\eta^{\beta\mu}\eta^{\alpha\sigma}p_{1}^{\rho}p_{3}^{\nu}$
$\displaystyle+\eta^{\alpha\mu}\eta^{\beta\sigma}p_{1}^{\rho}p_{3}^{\nu}-\eta^{\alpha\beta}\eta^{\mu\sigma}p_{1}^{\rho}p_{3}^{\nu}-\eta^{\alpha\sigma}\eta^{\mu\nu}p_{1}^{\rho}p_{2}^{\beta}-\eta^{\alpha\sigma}\eta^{\mu\nu}p_{1}^{\rho}p_{3}^{\beta}-\eta^{\beta\sigma}\eta^{\mu\nu}p_{1}^{\rho}p_{2}^{\alpha}-\eta^{\beta\sigma}\eta^{\mu\nu}p_{1}^{\rho}p_{3}^{\alpha}$
$\displaystyle-\eta^{\alpha\nu}\eta^{\beta\mu}p_{1}^{\rho}p_{2}^{\sigma}-\eta^{\alpha\mu}\eta^{\beta\nu}p_{1}^{\rho}p_{2}^{\sigma}+\eta^{\alpha\beta}\eta^{\mu\nu}p_{1}^{\rho}p_{2}^{\sigma}-\eta^{\alpha\nu}\eta^{\beta\mu}p_{1}^{\rho}p_{3}^{\sigma}-\eta^{\alpha\mu}\eta^{\beta\nu}p_{1}^{\rho}p_{3}^{\sigma}+2\eta^{\alpha\beta}\eta^{\mu\nu}p_{1}^{\rho}p_{3}^{\sigma}$
$\displaystyle-\eta^{\alpha\beta}\eta^{\nu\sigma}p_{1}^{\rho}p_{2}^{\mu}-\eta^{\alpha\beta}\eta^{\nu\sigma}p_{1}^{\rho}p_{3}^{\mu}+\eta^{\beta\nu}\eta^{\mu\rho}p_{1}^{\sigma}p_{2}^{\alpha}+\eta^{\beta\mu}\eta^{\nu\rho}p_{1}^{\sigma}p_{2}^{\alpha}+\eta^{\alpha\nu}\eta^{\mu\rho}p_{1}^{\sigma}p_{2}^{\beta}+\eta^{\alpha\mu}\eta^{\nu\rho}p_{1}^{\sigma}p_{2}^{\beta}$
$\displaystyle+\eta^{\beta\nu}\eta^{\alpha\rho}p_{1}^{\sigma}p_{2}^{\mu}+\eta^{\alpha\nu}\eta^{\beta\rho}p_{1}^{\sigma}p_{2}^{\mu}+\eta^{\beta\nu}\eta^{\alpha\rho}p_{1}^{\sigma}p_{3}^{\mu}+\eta^{\alpha\nu}\eta^{\beta\rho}p_{1}^{\sigma}p_{3}^{\mu}+\eta^{\beta\mu}\eta^{\alpha\rho}p_{1}^{\sigma}p_{2}^{\nu}+\eta^{\alpha\mu}\eta^{\beta\rho}p_{1}^{\sigma}p_{2}^{\nu}$
$\displaystyle-\eta^{\alpha\beta}\eta^{\mu\rho}p_{1}^{\sigma}p_{2}^{\nu}+\eta^{\beta\mu}\eta^{\alpha\rho}p_{1}^{\sigma}p_{3}^{\nu}+\eta^{\alpha\mu}\eta^{\beta\rho}p_{1}^{\sigma}p_{3}^{\nu}-\eta^{\alpha\beta}\eta^{\mu\rho}p_{1}^{\sigma}p_{3}^{\nu}-\eta^{\alpha\rho}\eta^{\mu\nu}p_{1}^{\sigma}p_{2}^{\beta}-\eta^{\alpha\rho}\eta^{\mu\nu}p_{1}^{\sigma}p_{3}^{\beta}$
$\displaystyle-\eta^{\beta\rho}\eta^{\mu\nu}p_{1}^{\sigma}p_{2}^{\alpha}-\eta^{\beta\rho}\eta^{\mu\nu}p_{1}^{\sigma}p_{3}^{\alpha}-\eta^{\alpha\nu}\eta^{\beta\mu}p_{1}^{\sigma}p_{2}^{\rho}-\eta^{\alpha\mu}\eta^{\beta\nu}p_{1}^{\sigma}p_{2}^{\rho}+\eta^{\alpha\beta}\eta^{\mu\nu}p_{1}^{\sigma}p_{2}^{\rho}-\eta^{\alpha\nu}\eta^{\beta\mu}p_{1}^{\sigma}p_{3}^{\rho}$
$\displaystyle-\eta^{\alpha\mu}\eta^{\beta\nu}p_{1}^{\sigma}p_{3}^{\rho}+2\eta^{\alpha\beta}\eta^{\mu\nu}p_{1}^{\sigma}p_{3}^{\rho}-\eta^{\alpha\beta}\eta^{\nu\rho}p_{1}^{\sigma}p_{2}^{\mu}-\eta^{\alpha\beta}\eta^{\nu\rho}p_{1}^{\sigma}p_{3}^{\mu}\,.$
(67ln)

In Eq.(67llb), the trilinear gravitational vertex function
$\Gamma^{\alpha\beta\mu\nu}_{nm\ell}$ is defined as follows:

$\displaystyle\Gamma^{\alpha\beta\mu\nu}_{nm\ell}(p_{1},p_{2})=$
$\displaystyle-2\eta^{\alpha\beta}p_{1}^{\mu}p_{1}^{\nu}+2\eta^{\alpha\beta}\eta^{\mu\nu}p_{1}^{2}+\eta^{\alpha\mu}p_{1}^{\beta}p_{1}^{\nu}+\eta^{\alpha\nu}p_{1}^{\beta}p_{1}^{\mu}+\eta^{\beta\mu}p_{1}^{\alpha}p_{1}^{\nu}-\eta^{\alpha\nu}\eta^{\beta\mu}p_{1}^{2}$
$\displaystyle+\eta^{\beta\nu}p_{1}^{\alpha}p_{1}^{\mu}-\eta^{\alpha\mu}\eta^{\beta\nu}p_{1}^{2}-2\eta^{\mu\nu}p_{1}^{\alpha}p_{1}^{\beta}-\eta^{\alpha\beta}p_{1}^{\nu}p_{2}^{\mu}-\eta^{\alpha\beta}p_{1}^{\mu}p_{2}^{\nu}+3\eta^{\alpha\beta}\eta^{\mu\nu}(p_{1}\\!\cdot\\!p_{2})$
$\displaystyle+\eta^{\alpha\mu}p_{1}^{\nu}p_{2}^{\beta}\\!+\eta^{\alpha\nu}p_{1}^{\mu}p_{2}^{\beta}+\eta^{\beta\mu}p_{1}^{\alpha}p_{2}^{\nu}-\eta^{\alpha\nu}\eta^{\beta\mu}(p_{1}\\!\cdot\\!p_{2})\\!+\eta^{\beta\nu}p_{1}^{\alpha}p_{2}^{\mu}-\eta^{\alpha\mu}\eta^{\beta\nu}(p_{1}\\!\cdot\\!p_{2})$
$\displaystyle-\eta^{\mu\nu}p_{1}^{\beta}p_{2}^{\alpha}-2\eta^{\mu\nu}p_{1}^{\alpha}p_{2}^{\beta}-2\eta^{\alpha\beta}p_{2}^{\mu}p_{2}^{\nu}+2\eta^{\alpha\beta}\eta^{\mu\nu}p_{2}^{2}+\eta^{\alpha\mu}p_{2}^{\beta}p_{2}^{\nu}+\eta^{\alpha\nu}p_{2}^{\beta}p_{2}^{\mu}$
$\displaystyle+\eta^{\beta\mu}p_{2}^{\alpha}p_{2}^{\nu}-\eta^{\alpha\nu}\eta^{\beta\mu}p_{2}^{2}+\eta^{\beta\nu}p_{2}^{\alpha}p_{2}^{\mu}-\eta^{\alpha\mu}\eta^{\beta\nu}p_{2}^{2}-2\eta^{\mu\nu}p_{2}^{\alpha}p_{2}^{\beta}\,.$
(67lo)

Moreover, the trilinear coupling coefficients $(\alpha_{nm\ell},\hskip
0.85358pt\tilde{\alpha}_{nm\ell},\hskip 0.85358pt\tilde{\beta}_{nm\ell},\hskip
0.85358pt\tilde{\rho}_{nm\ell})$ in Eq.(67ll) are defined as bellow:

$\displaystyle\alpha_{nm\ell}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\mathsf{u}_{n}(z)\mathsf{u}_{m}(z)\mathsf{u}_{\ell}(z)\hskip
0.85358pt,$ (67lpa) $\displaystyle\tilde{\alpha}_{nm\ell}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\mathsf{v}_{n}(z)\mathsf{v}_{m}(z)\mathsf{u}_{\ell}(z)\hskip
0.85358pt,$ (67lpb) $\displaystyle\tilde{\beta}_{nm\ell}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\mathsf{w}_{n}(z)\mathsf{w}_{m}(z)\mathsf{u}_{\ell}(z)\hskip
0.85358pt,$ (67lpc) $\displaystyle\tilde{\rho}_{nm\ell}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\mathsf{v}_{n}(z)\mathsf{v}_{m}(z)\mathsf{w}_{\ell}(z)\hskip
0.85358pt,$ (67lpd) $\displaystyle\tilde{\omega}_{nm\ell}$
$\displaystyle=\frac{1}{\,L\,}\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\mathsf{w}_{n}(z)\mathsf{w}_{m}(z)\mathsf{w}_{\ell}(z)\hskip
0.85358pt.$ (67lpe)

In particular, we can derive the trilinear coupling coefficients containing
zero-modes, $\hskip
0.85358pt\alpha_{000}\\!=\\!\alpha_{nn0}\\!=\\!\tilde{\alpha}_{nn0}\\!=\\!\mathsf{u}_{0}\hskip
0.85358pt$ and $\hskip
0.85358pta_{000}\\!=\\!a_{nn0}\\!=\\!\tilde{a}_{nn0}\\!=\\!\mathsf{f}_{0}\hskip
0.85358pt$, based on the normalization condition Eq.(51). We summarize these
coupling coefficients as follows:

$\displaystyle\alpha_{000}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\hskip
0.85358pt\mathsf{u}_{0}^{2}\hskip 0.85358pt\mathsf{u}_{0}=\mathsf{u}_{0}\,,$
(67lqa) $\displaystyle\alpha_{nn0}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\hskip
0.85358pt\mathsf{u}_{n}^{2}\hskip 0.85358pt\mathsf{u}_{0}=\mathsf{u}_{0}\,,$
(67lqb) $\displaystyle\tilde{\alpha}_{nn0}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\hskip
0.85358pt\mathsf{v}_{n}^{2}\hskip 0.85358pt\mathsf{u}_{0}=\mathsf{u}_{0}\,,$
(67lqc) $\displaystyle a_{000}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{A(z)}\hskip
0.85358pt\mathsf{f}_{0}^{2}\hskip 0.85358pt\mathsf{f}_{0}=\mathsf{f}_{0}\,,$
(67lqd) $\displaystyle a_{nn0}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{A(z)}\hskip
0.85358pt\mathsf{f}_{n}^{2}\hskip 0.85358pt\mathsf{f}_{0}=\mathsf{f}_{0}\,,$
(67lqe) $\displaystyle\tilde{a}_{nn0}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{A(z)}\hskip
0.85358pt\tilde{\mathsf{f}}_{n}^{2}\hskip
0.85358pt\mathsf{f}_{0}=\mathsf{f}_{0}\,.$ (67lqf)

In addition, we further define the following quartic coupling coefficients:

$\displaystyle\alpha_{nm\ell q}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\\!\text{d}z\,e^{3A(z)}\hskip
0.85358pt\mathsf{u}_{n}(z)\mathsf{u}_{m}(z)\mathsf{u}_{\ell}(z)\mathsf{u}_{q}(z)\,,$
(67lra) $\displaystyle\tilde{\beta}_{nm\ell q}$
$\displaystyle=\frac{1}{\,L\,}\\!\int_{0}^{L}\\!\\!\\!\text{d}z\,e^{3A(z)}\hskip
0.85358pt\mathsf{w}_{n}(z)\mathsf{w}_{m}(z)\mathsf{w}_{\ell}(z)\mathsf{w}_{q}(z)\,,$
(67lrb)

which will be used for the double-copy construction of Section 4.2 and for
deriving the gravitational sum rules in Appendix D.

Taking the flat space limit $k\\!\rightarrow\\!0\hskip 0.85358pt$, we see that
all the coupling coefficients will be reduced to the simple trigonometric
functions:

$\displaystyle\mathsf{f}_{0}=\mathsf{u}_{0}=\mathsf{w}_{0}=1\,,$
$\displaystyle\mathsf{f}_{n}(z)=\mathsf{u}_{n}(z)=\mathsf{w}_{n}(z)=\sqrt{2\,}\cos\frac{\,n\pi
z\,}{L}\hskip 0.85358pt,$ (67ls)
$\displaystyle\tilde{\mathsf{f}}_{n}(z)=\mathsf{v}_{n}(z)=\sqrt{2\,}\sin\frac{\,n\pi
z\,}{L},$

where $n\in\mathbb{Z}^{+}$. Thus, together with the definitions (67lq)-(67lr),
we deduce the values of these KK coupling coefficients in the flat 5d limit,
which are summarized in Table 1.

Cubic and Quartic Couplings (flat 5d) | Values
---|---
$\tilde{a}_{nn2n},\,\tilde{a}_{nm|n\pm m|},\,\tilde{\alpha}_{nn2n},\,\tilde{\rho}_{nn2n},\,\tilde{\alpha}_{nm|n\pm m|},\,\tilde{\rho}_{nm|n\pm m|}$ | $-\frac{1}{\sqrt{2\,}\,}$
$a_{nm0},\,\tilde{a}_{nm0},\,\alpha_{nm0},\,\tilde{\alpha}_{nm0},\,\tilde{\beta}_{nm0},\,\tilde{\rho}_{nm0},\,\tilde{\omega}_{nm0}$ | 0
$a_{nm\ell q},\,\tilde{a}_{nm\ell q},\,\alpha_{nm\ell q},\,\tilde{\beta}_{nm\ell q}$ | $\frac{1}{\,2\,}$
$a_{nn2n},\,a_{nm|n\pm m|},\,\alpha_{nn2n},\,\tilde{\beta}_{nn2n},\,\tilde{\omega}_{nn2n},\,\alpha_{nm|n\pm m|},\,\tilde{\beta}_{nm|n\pm m|},\,\tilde{\omega}_{nm|n\pm m|}$ | $\frac{1}{\sqrt{2\,}\,}$
$a_{nn0},\,\tilde{a}_{nn0},\,a_{nnmm},\,\alpha_{nn0},\,\tilde{\alpha}_{nn0},\,\tilde{\beta}_{nn0},\,\tilde{\rho}_{nn0},\,\tilde{\omega}_{nn0},\,\alpha_{nnmm},\,\tilde{\alpha}_{nnmm},\,\tilde{\beta}_{nnmm}$ | 1
$a_{nnnn},\,\alpha_{nnnn},\,\tilde{\alpha}_{nnnn},\,\tilde{\beta}_{nnnn}$ | $\frac{\,3\,}{2}$
Table 1: List of relevant cubic and quartic KK coupling coefficients of the
flat 5d gauge and gravity theories under $S^{1}\\!/\mathbb{Z}_{2}$
compactification. The subscripts are the relevant KK indices with
$n\\!\neq\\!m\\!\neq\\!\ell\\!\neq\\!q$ and $(n,m,\ell,q)\in\mathbb{Z}^{+}$.

Finally, we derive the on-shell 3-point KK gluon scattering amplitudes as
follows:

$\displaystyle\mathcal{T}[\\{\epsilon_{i}\\}]$ $\displaystyle=g\hskip
0.85358ptf^{abc}\mathcal{N}[\\{\epsilon_{j}\\}]\hskip 0.85358pt,$ (67maa)
$\displaystyle\mathcal{N}[\\{\epsilon_{j}\\}]$ $\displaystyle=-\text{i}\hskip
0.85358pt2\hskip 0.85358ptg\hskip 0.85358pta_{nm\ell}\big{[}\hskip
0.85358pt(\epsilon_{1}\epsilon_{2})(\epsilon_{3}p_{1})+(\epsilon_{2}\epsilon_{3})(\epsilon_{1}p_{2})+(\epsilon_{3}\epsilon_{1})(\epsilon_{2}p_{3})\hskip
0.85358pt\big{]}\hskip 0.85358pt,$ (67mab)

which reproduces the formula (67fz) in the main text.

Then, we can express the 3-point KK graviton amplitude into following form:

$\displaystyle\mathcal{M}\big{[}h^{\sigma_{1}}_{n}h^{\sigma_{2}}_{m}h^{\sigma_{3}}_{\ell}\big{]}$
$\displaystyle=\sum_{\lambda_{j},\lambda^{\prime}_{j}}\\!\\!\Big{(}\\!\prod_{j}\\!C_{\lambda_{j}\lambda^{\prime}_{j}}^{\hskip
0.85358pt\sigma_{j}}\Big{)}\mathcal{M}\big{[}\\{e_{i}\\}^{\lambda_{j}}\\!,\\{\epsilon_{i}\\}^{\lambda^{\prime}_{j}}\big{]}$
$\displaystyle\equiv\frac{~{}\kappa\hskip
0.85358pt\alpha_{nm\ell}~{}}{4}\\!\sum_{\lambda_{j},\lambda^{\prime}_{j}}\\!\\!\Big{(}\\!\prod_{j}\\!C_{\lambda_{j}\lambda^{\prime}_{j}}^{\hskip
0.85358pt\sigma_{j}}\Big{)}\hbox to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}\big{[}\\{e_{i}\\}^{\lambda_{j}}\\!,\\{\epsilon_{i}\\}^{\lambda^{\prime}_{j}}\big{]}\hskip
0.85358pt,$ (67mb)

where the helicity index $\,\sigma_{j}\\!=\\!\\{\pm 2,\pm
1,0\\}\\!\equiv\\!\\{\pm 2,\pm 1,L\\}\,$ labels the 5 helicity states of each
external massive KK graviton. The polarization tensors (including the
coefficients $C_{\lambda_{j}\lambda^{\prime}_{j}}^{\hskip
0.85358pt\sigma_{j}}$) of the external KK graviton states are defined as in
Eq.(67gf) of the main text. Then, we explicitly compute the on-shell 3-point
KK graviton scattering amplitudes as follows:

$\displaystyle\hbox to0.0pt{\hskip
2.91666pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{M}$}}[\\{e_{i}\\},\\{\epsilon_{i}\\}]=\kappa\hskip
0.85358pt\alpha_{nm\ell}\left[e_{1,\mu}\epsilon_{1,\nu}e_{2,\rho}\epsilon_{2,\sigma}e_{3,\alpha}\epsilon_{3,\beta}\Gamma^{\mu\nu\rho\sigma\alpha\beta}_{nm\ell}(p_{1},p_{2},p_{3})\right]$
$\displaystyle=$ $\displaystyle\frac{~{}\kappa\hskip
0.85358pt\alpha_{nm\ell}~{}}{4}\big{[}(e_{1}\epsilon_{3})(e_{2}p_{3})(e_{3}p_{2})(\epsilon_{1}\epsilon_{2})+(e_{1}e_{3})(e_{2}p_{3})(p_{2}\epsilon_{3})(\epsilon_{1}\epsilon_{2})+(e_{1}p_{3})(e_{2}p_{3})(e_{3}\epsilon_{2})(\epsilon_{1}\epsilon_{3})$
$\displaystyle+(e_{1}\epsilon_{2})(e_{2}p_{3})(e_{3}p_{2})(\epsilon_{1}\epsilon_{3})+(e_{1}p_{3})(e_{2}p_{3})(e_{3}\epsilon_{1})(\epsilon_{2}\epsilon_{3})+(e_{1}e_{3})(e_{2}p_{3})(p_{3}\epsilon_{1})(\epsilon_{2}\epsilon_{3})$
$\displaystyle+(e_{1}\epsilon_{2})(e_{2}p_{3})(e_{3}\epsilon_{1})(p_{2}\epsilon_{3})-2(e_{1}\epsilon_{2})(e_{2}\epsilon_{1})(e_{3}p_{2})(p_{2}\epsilon_{3})-2(e_{1}e_{2})(e_{3}p_{2})(p_{2}\epsilon_{3})(\epsilon_{1}\epsilon_{2})$
$\displaystyle+(e_{1}e_{3})(e_{2}\epsilon_{1})(p_{3}\epsilon_{2})(p_{2}\epsilon_{3})+(e_{1}e_{2})(e_{3}\epsilon_{1})(p_{3}\epsilon_{2})(p_{2}\epsilon_{3})-(e_{1}p_{3})(e_{2}\epsilon_{3})(e_{3}p_{2})(\epsilon_{1}\epsilon_{2})$
$\displaystyle-(e_{1}p_{3})(e_{2}\epsilon_{1})(e_{3}p_{2})(\epsilon_{2}\epsilon_{3})-(e_{1}p_{3})(e_{2}\epsilon_{1})(e_{3}\epsilon_{2})(p_{2}\epsilon_{3})-(e_{1}p_{3})(e_{2}e_{3})(p_{2}\epsilon_{3})(\epsilon_{1}\epsilon_{2})$
$\displaystyle+(e_{1}\epsilon_{3})(e_{2}p_{3})(e_{3}\epsilon_{2})(p_{3}\epsilon_{1})-(e_{1}\epsilon_{2})(e_{2}\epsilon_{3})(e_{3}p_{2})(p_{3}\epsilon_{1})-(e_{1}e_{2})(e_{3}p_{2})(p_{3}\epsilon_{1})(\epsilon_{2}\epsilon_{3})$
$\displaystyle-(e_{1}\epsilon_{2})(e_{2}e_{3})(p_{3}\epsilon_{1})(p_{2}\epsilon_{3})-(e_{1}e_{2})(e_{3}\epsilon_{2})(p_{3}\epsilon_{1})(p_{2}\epsilon_{3})-2(e_{1}p_{3})(e_{2}\epsilon_{3})(e_{3}\epsilon_{2})(p_{3}\epsilon_{1})$
$\displaystyle-2(e_{1}p_{3})(e_{2}e_{3})(p_{3}\epsilon_{1})(\epsilon_{2}\epsilon_{3})+(e_{1}p_{3})(e_{2}\epsilon_{3})(e_{3}\epsilon_{1})(p_{3}\epsilon_{2})+(e_{1}p_{3})(e_{2}e_{3})(p_{3}\epsilon_{2})(\epsilon_{1}\epsilon_{3})$
$\displaystyle+(e_{1}e_{2})(e_{3}p_{2})(p_{3}\epsilon_{2})(\epsilon_{1}\epsilon_{3})+(e_{1}\epsilon_{3})(e_{2}\epsilon_{1})(e_{3}p_{2})(p_{3}\epsilon_{2})-2(e_{1}\epsilon_{3})(e_{2}p_{3})(e_{3}\epsilon_{1})(p_{3}\epsilon_{2})$
$\displaystyle-\\!2(e_{1}e_{3})(e_{2}p_{3})(p_{3}\epsilon_{2})(\epsilon_{1}\epsilon_{3})\\!+\\!(e_{1}\epsilon_{3})(e_{2}e_{3})(p_{3}\epsilon_{1})(p_{3}\epsilon_{2})\\!+\\!(e_{1}e_{3})(e_{2}\epsilon_{3})(p_{3}\epsilon_{1})(p_{3}\epsilon_{2})\big{]},$
(67mc)

where for simplicity we have introduced a shorthand notation,
$(ab)\\!\equiv\\!(a\cdot b)$.

##  Appendix D Proving 3-Point and 4-Point Warped Identities

In this Appendix, we will prove relevant identities and sum rules for the
cubic and quartic coupling coefficients in the warped KK gauge theory and KK
gravity theory.

By imposing the following completeness conditions,

$\displaystyle\delta(z-z^{\prime})$
$\displaystyle=\sum_{j=0}^{\infty}\\!e^{A(z)}\mathsf{f}_{j}(z)\mathsf{f}_{j}(z^{\prime})=\sum_{j=0}^{\infty}\\!e^{A(z)}\tilde{\mathsf{f}}_{j}(z)\tilde{\mathsf{f}}_{j}(z^{\prime})\hskip
0.85358pt,$ (67mda) $\displaystyle\delta(z-z^{\prime})$
$\displaystyle=\sum_{j=0}^{\infty}\\!e^{3A(z)}\mathsf{u}_{j}(z)\mathsf{u}_{j}(z^{\prime})=\sum_{j=0}^{\infty}\\!e^{3A(z)}\mathsf{v}_{j}(z)\mathsf{v}_{j}(z^{\prime})=\sum_{j=0}^{\infty}\\!e^{3A(z)}\mathsf{w}_{j}(z)\mathsf{w}_{j}(z^{\prime})\hskip
0.85358pt,$ (67mdb)

we can derive the sum rules between the cubic and quartic KK gauge/gravity
coupling coefficients respectively:

$\displaystyle\sum_{j=0}^{\infty}a_{nmj}a_{\ell qj}=\,a_{nm\ell q}\hskip
0.85358pt,\qquad~{}~{}\sum_{j=0}^{\infty}\alpha_{nmj}\alpha_{\ell
qj}=\,\alpha_{nm\ell q}\hskip 0.85358pt,$ (67mea)
$\displaystyle\sum_{j=0}^{\infty}\tilde{a}_{nmj}\tilde{a}_{\ell
qj}=\,\tilde{a}_{nm\ell q}\hskip
0.85358pt,\qquad~{}~{}\sum_{j=0}^{\infty}\tilde{\beta}_{nmj}\tilde{\beta}_{\ell
qj}=\,\tilde{\beta}_{nm\ell q}\hskip 0.85358pt.$ (67meb)

For the special cases with some or all of the KK indices being equal, we can
recast above sum rules in the following simpler forms:

$\displaystyle\sum_{j=0}^{\infty}a_{nnj}^{2}$ $\displaystyle=a_{nnnn}\hskip
0.85358pt,\qquad~{}~{}\sum_{j=0}^{\infty}a_{nnj}a_{mmj}=\sum_{j=0}^{\infty}a_{nmj}^{2}=a_{nnmm}\hskip
0.85358pt,$ (67mfa) $\displaystyle\sum_{j=0}^{\infty}\alpha_{nnj}^{2}$
$\displaystyle=\alpha_{nnnn}\hskip
0.85358pt,\qquad~{}~{}\sum_{j=0}^{\infty}\alpha_{nnj}\alpha_{mmj}=\sum_{j=0}^{\infty}\alpha_{nmj}^{2}=\alpha_{nnmm}\hskip
0.85358pt,$ (67mfb) $\displaystyle\sum_{j=0}^{\infty}\tilde{a}_{nnj}^{2}$
$\displaystyle=\tilde{a}_{nnnn}\hskip
0.85358pt,\qquad~{}~{}\sum_{j=0}^{\infty}\tilde{a}_{nnj}\tilde{a}_{mmj}=\sum_{j=0}^{\infty}\tilde{a}_{nmj}^{2}=\tilde{a}_{nnmm}\hskip
0.85358pt,$ (67mfc) $\displaystyle\sum_{j=0}^{\infty}\tilde{\beta}_{nnj}^{2}$
$\displaystyle=\tilde{\beta}_{nnnn}\hskip
0.85358pt,\qquad~{}~{}\sum_{j=0}^{\infty}\tilde{\beta}_{nnj}\tilde{\beta}_{mmj}=\sum_{j=0}^{\infty}\tilde{\beta}_{nmj}^{2}=\tilde{\beta}_{nnmm}\hskip
0.85358pt.$ (67mfd)

Many of these relations will be used in Sections 3 and 4 of the main text.

Next, by utilizing the equation of motion (14a) and integration by parts, we
derive the following relation:

$\displaystyle\int_{0}^{L}\\!\\!\text{d}z\,e^{A(z)}M_{n}^{2}\,\mathsf{f}_{n}\hskip
0.85358pt\mathsf{f}_{m}\mathsf{f}_{\ell}$ $\displaystyle\,=\hskip
0.85358pt-\\!\int_{0}^{L}\\!\\!\text{d}z\,\partial_{z}(e^{A(z)}\partial_{z}\mathsf{f}_{n})\hskip
0.85358pt\mathsf{f}_{m}\mathsf{f}_{\ell}$
$\displaystyle=\int_{0}^{L}\\!\\!\text{d}z\,e^{A(z)}\big{(}\mathsf{f}_{n}^{\prime}\mathsf{f}_{m}^{\prime}\mathsf{f}_{\ell}+\mathsf{f}_{n}^{\prime}\mathsf{f}_{m}\mathsf{f}_{\ell}^{\prime})\hskip
0.85358pt.$ (67mg)

Thus, using the above relation and the definitions given in
Eqs.(67lja)-(67ljb), we obtain the following identities which are connected to
each other by cycling the three KK indices $(n,m,\ell)\hskip 0.85358pt$:

$\displaystyle M_{n}^{2}\,a_{nm\ell}$
$\displaystyle=M_{n}M_{m}\,\tilde{a}_{nm\ell}+M_{n}M_{\ell}\,\tilde{a}_{n\ell
m}\hskip 0.85358pt,$ $\displaystyle M_{m}^{2}\hskip 0.85358pta_{nm\ell}$
$\displaystyle=M_{n}M_{m}\,\tilde{a}_{nm\ell}+M_{m}M_{\ell}\,\tilde{a}_{m\ell
n}\hskip 0.85358pt,$ (67mh) $\displaystyle M_{\ell}^{2}\hskip
0.85358pta_{nm\ell}$ $\displaystyle=M_{n}M_{\ell}\,\tilde{a}_{n\ell
m}+M_{m}M_{\ell}\,\tilde{a}_{m\ell n}\hskip 0.85358pt.$

From these identities, we can further derive a relation connecting the
coupling coefficient $a_{nm\ell}$ to $\tilde{a}_{nm\ell}\hskip 0.85358pt$:

$\big{(}M_{n}^{2}+M_{m}^{2}\\!-M_{\ell}^{2}\big{)}a_{nm\ell}\,=\,2\hskip
0.85358ptM_{n}M_{m}\hskip 0.85358pt\tilde{a}_{nm\ell}\,,$ (67mi)

which just gives Eq.(67dg) in the main text. Similarly, we can derive the
following relation between the wavefunction couplings $\alpha_{nm\ell}$ and
$\tilde{\alpha}_{nm\ell}\hskip 0.85358pt$:

$\big{(}\mathbb{M}_{n}^{2}+\mathbb{M}_{m}^{2}\\!-\mathbb{M}_{\ell}^{2}\big{)}\alpha_{nm\ell}\,=\,2\hskip
0.85358pt\mathbb{M}_{n}\mathbb{M}_{m}\hskip
0.85358pt\tilde{\alpha}_{nm\ell}\,,$ (67mj)

which further gives Eq.(67dr) in the main text.

Moreover, by utilizing Eq.(14a) and integration by parts, we derive the
following relation containing four wavefunctions of the warped KK gauge
theory:

$\int_{0}^{L}\\!\\!\text{d}z\,e^{A(z)}M_{n}^{2}\hskip
0.85358pt\mathsf{f}_{n}\mathsf{f}_{m}\mathsf{f}_{\ell}\hskip
0.85358pt\mathsf{f}_{q}\,=\int_{0}^{L}\\!\\!\text{d}z\,e^{A(z)}\big{(}\mathsf{f}_{n}^{\prime}\mathsf{f}_{m}^{\prime}\mathsf{f}_{\ell}\hskip
0.85358pt\mathsf{f}_{q}\\!+\mathsf{f}_{n}^{\prime}\mathsf{f}_{m}\mathsf{f}_{\ell}^{\prime}\hskip
0.85358pt\mathsf{f}_{q}\\!+\mathsf{f}_{n}^{\prime}\mathsf{f}_{\ell}\hskip
0.85358pt\mathsf{f}_{m}\mathsf{f}_{q}^{\prime}\big{)}\hskip 0.85358pt.$ (67mk)

With this relation, we further derive the following identities by cycling the
four KK indices $(n,m,\ell,q)\hskip 0.85358pt$:

$\displaystyle M_{n}^{2}\,a_{nm\ell q}$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358ptM_{n}M_{m}\hskip 0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\tilde{\mathsf{f}}_{m}\mathsf{f}_{\ell}\hskip
0.85358pt\mathsf{f}_{q}\hskip 0.85358pt\rrbracket\hskip
0.85358pt\\!+\\!M_{n}M_{\ell}\hskip 0.85358pt\hskip 0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\tilde{\mathsf{f}}_{\ell}\hskip
0.85358pt\mathsf{f}_{m}\mathsf{f}_{q}\hskip 0.85358pt\rrbracket\hskip
0.85358pt\\!+\\!M_{n}M_{q}\hskip 0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\tilde{\mathsf{f}}_{q}\mathsf{f}_{m}\mathsf{f}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt,$ $\displaystyle M_{m}^{2}\,a_{nm\ell q}$
$\displaystyle\hskip 0.85358pt=\hskip 0.85358ptM_{n}M_{m}\hskip
0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\tilde{\mathsf{f}}_{m}\mathsf{f}_{\ell}\hskip
0.85358pt\mathsf{f}_{q}\hskip 0.85358pt\rrbracket\hskip
0.85358pt\\!+\\!M_{m}M_{\ell}\hskip 0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{m}\tilde{\mathsf{f}}_{\ell}\hskip
0.85358pt\mathsf{f}_{n}\mathsf{f}_{q}\hskip 0.85358pt\rrbracket\hskip
0.85358pt\\!+\\!M_{m}M_{q}\hskip 0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{m}\tilde{\mathsf{f}}_{q}\mathsf{f}_{n}\mathsf{f}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt,$ $\displaystyle M_{\ell}^{2}\,a_{nm\ell
q}$ $\displaystyle\hskip 0.85358pt=\hskip 0.85358ptM_{n}M_{\ell}\hskip
0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\tilde{\mathsf{f}}_{\ell}\hskip
0.85358pt\mathsf{f}_{m}\mathsf{f}_{q}\hskip 0.85358pt\rrbracket\hskip
0.85358pt\\!+\\!M_{m}M_{\ell}\hskip 0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{m}\tilde{\mathsf{f}}_{\ell}\hskip
0.85358pt\mathsf{f}_{n}\mathsf{f}_{q}\hskip 0.85358pt\rrbracket\hskip
0.85358pt\\!+\\!M_{\ell}M_{q}\hskip 0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{\ell}\hskip
0.85358pt\tilde{\mathsf{f}}_{q}\mathsf{f}_{n}\mathsf{f}_{m}\hskip
0.85358pt\rrbracket\hskip 0.85358pt,$ $\displaystyle M_{q}^{2}\,a_{nm\ell q}$
$\displaystyle\hskip 0.85358pt=\hskip 0.85358ptM_{n}M_{q}\hskip
0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\tilde{\mathsf{f}}_{q}\mathsf{f}_{m}\mathsf{f}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt\\!+\\!M_{m}M_{q}\hskip
0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{m}\tilde{\mathsf{f}}_{q}\mathsf{f}_{n}\mathsf{f}_{\ell}\hskip
0.85358pt\rrbracket\hskip 0.85358pt\\!+\\!M_{\ell}M_{q}\hskip
0.85358pt\llbracket\hskip 0.85358pt\tilde{\mathsf{f}}_{\ell}\hskip
0.85358pt\tilde{\mathsf{f}}_{q}\mathsf{f}_{n}\mathsf{f}_{m}\hskip
0.85358pt\rrbracket\hskip 0.85358pt.$ (67ml)

In the above Eq.(D), we have used brackets $\hskip 0.85358pt\llbracket\hskip
0.85358pt\cdots\hskip 0.85358pt\rrbracket\hskip 0.85358pt$ to denote the
integration over product of the 5d wavefunctions:

$\hskip 0.85358pt\llbracket\hskip 0.85358pt\mathsf{X}_{n_{1}}\\!\cdots\hskip
0.85358pt\mathsf{X}_{n_{N}}\hskip 0.85358pt\rrbracket\hskip
0.85358pt=\frac{1}{\,L\,}\\!\\!\int_{0}^{L}\\!\\!\text{d}z\,e^{rA(z)}\hskip
0.85358pt\mathsf{X}_{n_{1}}\\!(z)\cdots\mathsf{X}_{n_{N}}\\!(z)\hskip
0.85358pt,$ (67mm)

where $\mathsf{X}_{n}$ represents the 5d wavefunctions and the parameter $r$
is chosen as $r\\!=\\!1\,(r\\!=\\!3)$ for the warped KK gauge (gravity)
theory. Then, summing up the four identities of Eq.(D) and making use of
Eq.(67mi), we derive a new sum rule:

$\sum_{j=0}^{\infty}\\!M_{j}^{2}(a_{nmj}a_{\ell qj}\\!+a_{n\ell
j}a_{mqj}\\!+a_{nqj}a_{m\ell
j})=\big{(}M_{n}^{2}\\!+\\!M_{m}^{2}\\!+\\!M_{\ell}^{2}\\!+\\!M_{q}^{2}\big{)}a_{nm\ell
q}\hskip 0.85358pt.$ (67mn)

For the special case of $n\\!=\\!m\\!=\\!\ell\\!=\\!q$, the above identity
reduces to:

$\sum_{j=0}^{\infty}\\!M_{j}^{2}a_{nnj}^{2}=\frac{\,4\,}{3}M_{n}^{2}a_{nnnn}\hskip
0.85358pt,$ (67mo)

which just reproduces the identity (67eib) in the main text. As for the warped
KK gravity theory, we can derive a new sum rule in similar form:

$\sum_{j=0}^{\infty}\\!\mathbb{M}_{j}^{2}(\alpha_{nmj}\alpha_{\ell
qj}\\!+\\!\alpha_{n\ell j}\alpha_{mqj}\\!+\\!\alpha_{nqj}\alpha_{m\ell
j})=\big{(}\mathbb{M}_{n}^{2}\\!+\mathbb{M}_{m}^{2}\\!+\mathbb{M}_{\ell}^{2}\\!+\mathbb{M}_{q}^{2}\big{)}\alpha_{nm\ell
q}\hskip 0.85358pt.$ (67mp)

For the special case of $n\\!=\\!m\\!=\\!\ell\\!=\\!q$, we simplify the above
identity as follows:

$\sum_{j=0}^{\infty}\mathbb{M}_{j}^{2}\alpha_{nnj}^{2}=\frac{\,4\,}{3}\mathbb{M}_{n}^{2}\alpha_{nnnn}\hskip
0.85358pt,$ (67mq)

which reproduces the identity (67ftb) in the main text.

Next, we prove the identity (67dw) in the main text. For this purpose, we
first derive the following relation,

$\displaystyle
L\mathbb{M}_{n}\mathbb{M}_{m}\tilde{\beta}_{nm\ell}=\int_{0}^{L}\\!\\!\text{d}z\,e^{3A(z)}\,(\mathbb{M}_{n}\mathsf{w}_{n})(\mathbb{M}_{m}\mathsf{w}_{m})\mathsf{u}_{\ell}$
$\displaystyle=\int_{0}^{L}\\!\\!\text{d}z\hskip
0.85358pt\Big{[}\partial_{z}(e^{A(z)}\mathsf{v}_{n})\Big{]}\Big{[}e^{2A(z)}(A^{\prime}\\!+\partial_{z})\mathsf{v}_{m}\Big{]}\mathsf{u}_{\ell}$
$\displaystyle=-\\!\\!\int_{0}^{L}\\!\\!\\!\text{d}z\Big{(}e^{A(z)}\mathsf{v}_{n}\Big{)}\partial_{z}\Big{[}e^{2A(z)}(A^{\prime}\\!+\partial_{z})\mathsf{v}_{m}\Big{]}\mathsf{u}_{\ell}-\\!\\!\int_{0}^{L}\\!\\!\\!\text{d}z\Big{(}e^{A(z)}\mathsf{v}_{n}\Big{)}\\!\Big{[}e^{2A}(A^{\prime}\\!+\partial_{z})\mathsf{v}_{m}\Big{]}\mathsf{u}_{\ell}^{\prime}\hskip
17.07164pt$
$\displaystyle=L\mathbb{M}_{m}^{2}\tilde{\alpha}_{nm\ell}-L\mathbb{M}_{m}\mathbb{M}_{\ell}\hskip
0.85358pt\tilde{\rho}_{n\ell m}\hskip 0.85358pt,$ (67mr)

which leads to the identity:

$\mathbb{M}_{n}\tilde{\beta}_{nm\ell}=\mathbb{M}_{m}\tilde{\alpha}_{nm\ell}-\mathbb{M}_{\ell}\hskip
0.85358pt\tilde{\rho}_{n\ell m}\,.$ (67ms)

For the right-hand side of Eq.(67ms), the second term contains the trilinear
coupling coefficient $\tilde{\rho}_{nm\ell}$ for which we can derive the
following relation:

$\displaystyle L\mathbb{M}_{n}\tilde{\rho}_{m\ell
n}=\int_{0}^{L}\\!\\!\\!\text{d}z\Big{[}\partial_{z}(e^{A(z)}\mathsf{v}_{n})\Big{]}\\!\Big{(}e^{A(z)}\mathsf{v}_{m}\Big{)}\\!\Big{(}e^{A(z)}\mathsf{v}_{\ell}\Big{)},$
$\displaystyle=-\\!\\!\int_{0}^{L}\\!\\!\\!\text{d}z\Big{(}e^{A(z)}\mathsf{v}_{n}\Big{)}\\!\Big{[}\partial_{z}\big{(}e^{A(z)}\mathsf{v}_{m}\big{)}\Big{]}\\!\Big{(}e^{A(z)}\mathsf{v}_{\ell}\Big{)}\\!-\\!\int_{0}^{L}\\!\\!\\!\text{d}z\Big{(}\\!e^{A(z)}\mathsf{v}_{n}\Big{)}\\!\Big{(}\\!e^{A(z)}\mathsf{v}_{m}\Big{)}\\!\Big{[}\partial_{z}\big{(}e^{A(z)}\mathsf{v}_{\ell}\big{)}\Big{]}\hskip
14.22636pt$ $\displaystyle=-L\hskip 0.85358pt\mathbb{M}_{m}\hskip
0.85358pt\tilde{\rho}_{n\ell m}\\!-L\hskip 0.85358pt\mathbb{M}_{\ell}\hskip
0.85358pt\tilde{\rho}_{nm\ell}\,,$ (67mt)

where we have used Eqs.(52) and (53). We can reexpress Eq.(D) as follows:

$\mathbb{M}_{n}\tilde{\rho}_{m\ell n}\\!+\mathbb{M}_{m}\hskip
0.85358pt\tilde{\rho}_{n\ell m}\\!+\mathbb{M}_{\ell}\hskip
0.85358pt\tilde{\rho}_{nm\ell}=0\hskip 0.85358pt.$ (67mu)

Using Eqs.(67ms) and (67mj) as well as cycling their three KK indices, we
further derive the following relations:

$\displaystyle 2\hskip
0.85358pt\mathbb{M}_{n}^{2}\mathbb{M}_{m}^{2}\tilde{\beta}_{nm\ell}$
$\displaystyle=\mathbb{M}_{m}^{2}\big{(}\mathbb{M}_{m}^{2}\\!+\\!\mathbb{M}_{n}^{2}\\!-\\!\mathbb{M}_{\ell}^{2}\big{)}\alpha_{nm\ell}-2\hskip
0.85358pt\mathbb{M}_{n}\mathbb{M}_{m}^{2}\mathbb{M}_{\ell}\hskip
0.85358pt\tilde{\rho}_{n\ell m}\hskip 0.85358pt,$ (67mva) $\displaystyle
2\hskip 0.85358pt\mathbb{M}_{n}^{2}\mathbb{M}_{m}^{2}\tilde{\beta}_{nm\ell}$
$\displaystyle=\mathbb{M}_{n}^{2}\big{(}\mathbb{M}_{m}^{2}\\!+\\!\mathbb{M}_{n}^{2}\\!-\\!\mathbb{M}_{\ell}^{2}\big{)}\alpha_{nm\ell}-2\hskip
0.85358pt\mathbb{M}_{n}^{2}\mathbb{M}_{m}\mathbb{M}_{\ell}\hskip
0.85358pt\tilde{\rho}_{m\ell n}\hskip 0.85358pt,$ (67mvb) $\displaystyle
2\hskip 0.85358pt\mathbb{M}_{m}^{2}\mathbb{M}_{\ell}^{2}\tilde{\beta}_{m\ell
n}$
$\displaystyle=\mathbb{M}_{\ell}^{2}\big{(}\mathbb{M}_{\ell}^{2}\\!+\\!\mathbb{M}_{m}^{2}\\!-\\!\mathbb{M}_{n}^{2}\big{)}\alpha_{nm\ell}-2\hskip
0.85358pt\mathbb{M}_{n}\mathbb{M}_{m}\mathbb{M}_{\ell}^{2}\tilde{\rho}_{nm\ell}\hskip
0.85358pt,$ (67mvc) $\displaystyle 2\hskip
0.85358pt\mathbb{M}_{m}^{2}\mathbb{M}_{\ell}^{2}\tilde{\beta}_{m\ell n}$
$\displaystyle=\mathbb{M}_{m}^{2}\big{(}\mathbb{M}_{\ell}^{2}\\!+\\!\mathbb{M}_{m}^{2}\\!-\\!\mathbb{M}_{n}^{2}\big{)}\alpha_{nm\ell}-2\hskip
0.85358pt\mathbb{M}_{n}\mathbb{M}_{n}^{2}\mathbb{M}_{\ell}\hskip
0.85358pt\tilde{\rho}_{n\ell m}\hskip 0.85358pt,$ (67mvd) $\displaystyle
2\hskip 0.85358pt\mathbb{M}_{\ell}^{2}\mathbb{M}_{n}^{2}\tilde{\beta}_{\ell
nm}$
$\displaystyle=\mathbb{M}_{\ell}^{2}\big{(}\mathbb{M}_{\ell}^{2}\\!+\\!\mathbb{M}_{n}^{2}\\!-\\!\mathbb{M}_{m}^{2}\big{)}\alpha_{nm\ell}-2\hskip
0.85358pt\mathbb{M}_{n}\mathbb{M}_{m}\mathbb{M}_{\ell}^{2}\hskip
0.85358pt\tilde{\rho}_{nm\ell}\hskip 0.85358pt,$ (67mve) $\displaystyle
2\hskip 0.85358pt\mathbb{M}_{\ell}^{2}\mathbb{M}_{n}^{2}\tilde{\beta}_{\ell
nm}$
$\displaystyle=\mathbb{M}_{n}^{2}\big{(}\mathbb{M}_{\ell}^{2}\\!+\\!\mathbb{M}_{n}^{2}-\mathbb{M}_{m}^{2}\big{)}\alpha_{nm\ell}-2\hskip
0.85358pt\mathbb{M}_{n}^{2}\mathbb{M}_{m}\mathbb{M}_{\ell}\hskip
0.85358pt\tilde{\rho}_{m\ell n}\hskip 0.85358pt.$ (67mvf)

With the six relations above, we compute the sum
$\mbox{$\frac{\,{3}\,}{2}$}[(a)\\!+\\!(b)]\\!+\\!\mbox{$\frac{\,{1}\,}{2}$}\big{[}(c)\\!-\\!(d)\\!+\\!(e)\\!-\\!(f)\big{]}$
and impose Eq.(D), with which we derive the final identity:

$\Big{[}\big{(}\mathbb{M}_{n}^{2}\\!+\\!\mathbb{M}_{m}^{2}\\!-\\!\mathbb{M}_{\ell}^{2}\big{)}^{\\!2}\\!+\\!2\hskip
0.85358pt\mathbb{M}_{n}^{2}\mathbb{M}_{m}^{2}\Big{]}\alpha_{nm\ell}\,=\,6\hskip
0.85358pt\mathbb{M}_{n}^{2}\mathbb{M}_{m}^{2}\hskip
0.85358pt\tilde{\beta}_{nm\ell}\hskip 0.85358pt.$ (67mw)

This just reproduces the identity (67dw) in the main text.

For the derivation of Eq.(67eb) in the main text, we can sum up the third and
fifth identities in Eq.(67mv) and obtain the following:

$4\hskip
0.85358pt\mathbb{M}_{n}\mathbb{M}_{m}\mathbb{M}_{\ell}^{2}\tilde{\rho}_{nm\ell}=2\hskip
0.85358pt\mathbb{M}_{\ell}^{4}\alpha_{nm\ell}-2\hskip
0.85358pt\mathbb{M}_{m}^{2}\mathbb{M}_{\ell}^{2}\tilde{\beta}_{m\ell
n}-2\hskip
0.85358pt\mathbb{M}_{n}^{2}\mathbb{M}_{\ell}^{2}\tilde{\beta}_{n\ell m}\,.$
(67mx)

With this and further using the identity (67mw), we arrive at

$\Big{[}2\hskip
0.85358pt\mathbb{M}_{\ell}^{4}\\!-\\!\big{(}\mathbb{M}_{n}^{2}\\!-\mathbb{M}_{m}^{2}\big{)}^{\\!2}\\!-\mathbb{M}_{\ell}^{2}\big{(}\mathbb{M}_{n}^{2}\\!+\mathbb{M}_{m}^{2}\big{)}\\!\Big{]}\alpha_{nm\ell}=6\hskip
0.85358pt\mathbb{M}_{n}\mathbb{M}_{m}\mathbb{M}_{\ell}^{2}\hskip
0.85358pt\tilde{\rho}_{nm\ell}\,.$ (67my)

This just reproduces Eq.(67eb) in the main text.

Then, we derive two sum rules Eqs.(67eub) and (67euc) which are used for
computing the inelastic scattering amplitudes of longitudinal KK gauge bosons.
For the Eq.(67eub), we can set $(\ell,q)\\!=\\!(n,m)$ in Eq.(67mn), and derive
the sum rule:

$\sum_{j=0}^{\infty}\\!M_{j}^{2}a_{nmj}^{2}=\sum_{j=0}^{\infty}\\!\left(M_{n}^{2}\\!+\\!M_{m}^{2}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}M_{j}^{2}\right)\\!a_{nnj}a_{mmj}\,.$
(67mz)

As for the derivation of Eq.(67euc), by taking the difference between the
first two identities in Eq.(67mh), we arrive at

$\big{(}M_{n}^{2}-\\!M_{m}^{2}\big{)}a_{nmj}=M_{j}\big{(}M_{n}\tilde{a}_{njm}\\!-\\!M_{m}\tilde{a}_{mjn}\big{)}.$
(67na)

Then, squaring the above formula and sum over $j$, we derive the following:

$\displaystyle\sum_{j=1}^{\infty}\\!\big{(}M_{n}^{2}-\\!M_{m}^{2}\big{)}^{\\!2}M_{j}^{-2}a_{nmj}^{2}=\sum_{j=1}^{\infty}\big{(}M_{n}\tilde{a}_{njm}\\!-\\!M_{m}\tilde{a}_{mjn}\big{)}^{\\!2}$
$\displaystyle=\sum_{j=1}^{\infty}\\!\big{(}M_{n}^{2}\tilde{a}_{njm}^{2}\\!-\\!2M_{n}M_{m}\tilde{a}_{njm}\tilde{a}_{mjn}\\!+\\!M_{m}^{2}\tilde{a}_{mjn}^{2}\big{)}$
$\displaystyle=\sum_{j=1}^{\infty}(M_{n}^{2}\tilde{a}_{nnj}a_{mmj}\\!-2M_{n}M_{m}\tilde{a}_{nmj}a_{nmj}\\!+\\!M_{m}^{2}\tilde{a}_{mmj}a_{nnj})$
$\displaystyle=\sum_{j=1}^{\infty}\\!\left[\mbox{$\frac{\,{1}\,}{2}$}(2M_{n}^{2}\\!-\\!M_{j}^{2})a_{nnj}a_{mmj}\\!-\\!(M_{n}^{2}\\!+\\!M_{m}^{2}-M_{j}^{2})a_{nmj}^{2}\\!+\\!\mbox{$\frac{\,{1}\,}{2}$}(2M_{m}^{2}-M_{j}^{2})a_{mmj}a_{nnj}\right]$
$\displaystyle=\sum_{j=0}^{\infty}M_{j}^{2}\big{(}a_{nmj}^{2}\\!-a_{nnj}a_{mmj}\big{)}\hskip
0.85358pt,$ (67nb)

where for the third equality sign, we have imposed the following completeness
relation:

$\displaystyle\sum_{j=1}^{\infty}\\!\tilde{a}_{njm}\tilde{a}_{\ell jq}=\hskip
0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\mathsf{f}_{m}\tilde{\mathsf{f}}_{\ell}\mathsf{f}_{q}\hskip
0.85358pt\rrbracket\hskip 0.85358pt=\sum_{j=1}^{\infty}\\!\hskip
0.85358pt\llbracket\hskip
0.85358pt\tilde{\mathsf{f}}_{n}\tilde{\mathsf{f}}_{\ell}\mathsf{f}_{j}\hskip
0.85358pt\rrbracket\hskip 0.85358pt\\!\hskip 0.85358pt\llbracket\hskip
0.85358pt\mathsf{f}_{m}\mathsf{f}_{q}\mathsf{f}_{j}\hskip
0.85358pt\rrbracket\hskip 0.85358pt=\sum_{j=1}^{\infty}\\!\tilde{a}_{n\ell
j}a_{mqj}\hskip 0.85358pt.$ (67nc)

For the fourth equality sign of Eq.(D), we have used Eq.(67mi). Moreover,
substituting Eq.(67mz) into Eq.(D), we derive the sum rule identity:

$\sum_{j=1}^{\infty}\\!\\!\big{(}M_{n}^{2}\\!-\\!M_{m}^{2}\big{)}^{\\!2}M_{j}^{-2}a_{nmj}^{2}=\sum_{j=0}^{\infty}\\!\big{(}M_{n}^{2}\\!+\\!M_{m}^{2}\\!-\\!\mbox{$\frac{\,{3}\,}{2}$}M_{j}^{2}\big{)}a_{nnj}a_{mmj}\hskip
0.85358pt,$ (67nd)

which is Eq.(67euc) in the main text.

Next, we derive the sum rule identities in Eq.(67fg) of Sec. 3.2. We prove the
$s$-channel sum rule (67fgc) as an example and other sum rules in Eq.(67fg) be
readily obtained from Eq.(67fgc) by permuting the KK indices. From Eq.(67na),
we derive the following:

$\displaystyle\big{(}M_{n}^{2}\\!-\\!M_{m}^{2}\big{)}\big{(}M_{\ell}^{2}\\!-\\!M_{q}^{2}\big{)}\\!\sum_{j=1}^{\infty}\\!M_{j}^{-2}a_{nmj}\hskip
0.85358pta_{\ell qj}$
$\displaystyle=\sum_{j=1}^{\infty}\\!\left[\\!\big{(}M_{n}^{2}\\!-\\!M_{m}^{2}\big{)}M_{j}^{-1}a_{nmj}\right]\\!\\!\left[\\!\big{(}M_{\ell}^{2}\\!-\\!M_{q}^{2}\big{)}M_{j}^{-1}a_{\ell
qj}\right]$
$\displaystyle=\sum_{j=1}^{\infty}\\!\big{(}M_{n}\tilde{a}_{njm}\\!-\\!M_{m}\tilde{a}_{mjn}\big{)}\\!\big{(}M_{\ell}\tilde{a}_{\ell
jq}\\!-\\!M_{q}\tilde{a}_{qj\ell}\big{)}$
$\displaystyle=\sum_{j=1}^{\infty}\\!\big{(}M_{n}M_{\ell}\tilde{a}_{njm}\tilde{a}_{\ell
jq}\\!-\\!M_{n}M_{q}\tilde{a}_{njm}\tilde{a}_{qj\ell}\\!-\\!M_{m}M_{\ell}\tilde{a}_{mjn}\tilde{a}_{\ell
jq}\\!+\\!M_{m}M_{q}\tilde{a}_{mjn}\tilde{a}_{qj\ell}\big{)}$
$\displaystyle=\sum_{j=1}^{\infty}\\!\\!\big{(}M_{n}M_{\ell}\tilde{a}_{n\ell
j}a_{mqj}\\!-\\!M_{n}M_{q}\tilde{a}_{nqj}a_{m\ell
j}\\!-\\!M_{m}M_{\ell}\tilde{a}_{m\ell
j}a_{nqj}\\!+\\!M_{m}M_{q}\tilde{a}_{mqj}a_{n\ell j}\big{)},$ (67ne)

where we have applied Eqs.(67mh) and (67nc) for the second and fourth equality
signs, respectively. Then, substituting Eq.(67mi) into Eq.(D), we finally
arrive at

$\displaystyle\big{(}M_{n}^{2}-M_{m}^{2}\big{)}\big{(}M_{\ell}^{2}-M_{q}^{2}\big{)}\sum_{j=1}^{\infty}M_{j}^{-2}a_{nmj}a_{\ell
qj}$
$\displaystyle=\sum_{j=1}^{\infty}\\!\Big{[}\mbox{$\frac{\,{1}\,}{2}$}\big{(}M_{n}^{2}\\!+\\!M_{\ell}^{2}\\!-\\!M_{j}^{2}\big{)}a_{n\ell
j}a_{mqj}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}\big{(}M_{n}^{2}\\!+\\!M_{q}^{2}\\!-\\!M_{j}^{2}\big{)}a_{nqj}a_{m\ell
j}$ $\displaystyle\hskip
28.45274pt-\mbox{$\frac{\,{1}\,}{2}$}\big{(}M_{m}^{2}\\!+\\!M_{\ell}^{2}\\!-\\!M_{j}^{2}\big{)}a_{m\ell
j}a_{nqj}\\!+\\!\mbox{$\frac{\,{1}\,}{2}$}\big{(}M_{m}^{2}\\!+\\!M_{q}^{2}\\!-\\!M_{j}^{2}\big{)}a_{mqj}a_{n\ell
j}\Big{]}$
$\displaystyle=\sum_{j=0}^{\infty}\\!M_{j}^{2}\big{(}a_{nqj}a_{m\ell
j}\\!-a_{n\ell j}a_{mqj}\big{)}.$ (67nf)

This just reproduces the sum rule identity (67fgc) given in the main text.

Finally, we derive the identity (67fv) in the main text. With Eq.(67dt) and
Eq.(67mq), we first derive the following identity:

$\sum_{j=0}^{\infty}\\!\alpha_{nnj}\tilde{\alpha}_{nnj}=\frac{1}{\,3\,}\alpha_{nnnn}\hskip
0.85358pt.$ (67ng)

By using Eq.(67dt) and Eq.(67mq), we compute:

$\displaystyle\sum_{j=0}^{\infty}\\!\hat{r}_{j}^{6}\alpha_{nnj}^{2}$
$\displaystyle=\sum_{j=0}^{\infty}\\!4\hskip
0.85358pt\hat{r}_{j}^{2}\big{(}\alpha_{nnj}\\!-\tilde{\alpha}_{nnj}\big{)}^{\\!2}=\sum_{j=0}^{\infty}\\!4\hskip
0.85358pt(4+\hat{r}^{2}_{j})\hskip 0.85358pt\tilde{\alpha}_{nnj}^{2}\hskip
0.85358pt,$ (67nh)

where we have used the notation $\hskip
0.85358pt\hat{r}_{j}\\!=\\!\mathbb{M}_{j}/\mathbb{M}_{n}\hskip 0.85358pt$.
Then, we compute the following:

$\displaystyle\mathbb{M}_{j}^{2}\tilde{\alpha}_{nnj}$
$\displaystyle=\mathbb{M}_{j}^{2}\int_{0}^{L}\\!\\!\text{d}z\hskip
0.85358pt{e^{3A(z)}}\hskip 0.85358pt\mathsf{v}_{n}^{2}\mathsf{u}_{j}$
$\displaystyle=\int_{0}^{L}\\!\\!\text{d}z\hskip 0.85358pt{e^{3A(z)}}\hskip
0.85358pt\mathsf{u}_{j}\Big{[}\\!-\\!12{A^{\prime\prime}}\hskip
0.85358pt\mathsf{v}_{n}^{2}\\!+\\!12{A^{\prime}}\hskip
0.85358pt\mathsf{u}_{n}\mathsf{v}_{n}\\!+\\!2\hskip
0.85358pt\mathsf{v}_{n}^{2}\\!-\\!2\hskip
0.85358pt\mathbb{M}_{n}^{2}\mathsf{u}_{n}^{2}\Big{]},$ (67nia)
$\displaystyle\sum_{j=0}^{\infty}\mathbb{M}_{j}^{2}\tilde{\alpha}_{nnj}^{2}$
$\displaystyle=-2\hskip
0.85358pt\mathbb{M}_{n}^{2}\alpha_{nnj}\tilde{\alpha}_{nnj}+2\hskip
0.85358pt\mathbb{M}_{n}^{2}\tilde{\alpha}_{nnj}^{2}+I_{1}+I_{2}\hskip
0.85358pt,$ (67nib)

where the $I_{1}$ and $I_{2}$ represent two integrals:

$\displaystyle I_{1}$ $\displaystyle=-12\hskip
0.85358pt\mathbb{M}_{n}\\!\\!\int_{0}^{L}\\!\\!\text{d}z\hskip
0.85358pt{e^{3A(z)}A^{\prime}}\hskip
0.85358pt\mathsf{u}_{n}\mathsf{v}_{n}^{3}=-2\hskip
0.85358pt\mathbb{M}_{n}^{2}\sum_{j=0}^{\infty}\\!\big{(}\tilde{\alpha}_{nnj}^{2}\\!-\\!3\hskip
0.85358pt\alpha_{nnj}\tilde{\alpha}_{nnj}\big{)}\hskip 0.85358pt,$ (67nja)
$\displaystyle I_{2}$ $\displaystyle=-12\\!\int_{0}^{L}\\!\\!\text{d}z\hskip
0.85358pt{e^{3A(z)}A^{\prime\prime}}\hskip
0.85358pt\mathsf{v}_{n}^{4}=-\frac{1}{\,2\,}I_{1}\hskip 0.85358pt.$ (67njb)

In the above we have denoted $A^{\prime}\\!\\!=\\!\partial_{z}A(z)$ and
$A^{\prime\prime}\\!\\!=\\!\partial_{z}^{2}A(z)$. Combining Eqs.(67ni) and
(67nj), we further compute the summation of Eq.(67nh):

$\displaystyle\sum_{j=0}^{\infty}\\!\hat{r}_{j}^{6}\alpha_{nnj}^{2}$
$\displaystyle=\sum_{j=0}^{\infty}\\!4\hskip 0.85358pt\big{(}5\hskip
0.85358pt\tilde{\alpha}_{nnj}^{2}\\!+\alpha_{nnj}\tilde{\alpha}_{nnj}\big{)}=\sum_{j=0}^{\infty}\\!\left(\\!5\hskip
0.85358pt\hat{r}_{j}^{4}-\\!\frac{\,16\,}{3}\\!\right)\\!\alpha_{nnj}^{2}\hskip
0.85358pt,$ (67nk)

where we have made use of Eq.(67dt) and Eq.(67mq). We can re-express Eq.(67nk)
as follows:

$\displaystyle\sum_{j=0}^{\infty}\\!\left(\\!\hat{r}_{j}^{6}-5\hskip
0.85358pt\hat{r}_{j}^{4}+\frac{\,16\,}{3}\\!\right)\\!\alpha_{nnj}^{2}=0\hskip
0.85358pt,$ (67nl)

which just reproduces the identity (67fv) in the main text.

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