Structure of Kaluza-Klein Graviton Scattering Amplitudes
from Gravitational Equivalence Theorem and Double-Copy

Yan-Feng Hang<EMAIL_ADDRESS>and Hong-Jian He<EMAIL_ADDRESS>
a Tsung-Dao Lee Institute $\&$ School of Physics and Astronomy,

Key Laboratory for Particle Astrophysics and Cosmology (MOE),

Shanghai Key Laboratory for Particle Physics and Cosmology,

Shanghai Jiao Tong University, Shanghai, China

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

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

###### Abstract

We study the structure of scattering amplitudes of the Kaluza-Klein (KK)
gravitons and of the gravitational KK Goldstone bosons in the compactified 5d
General Relativity (GR). We analyze the geometric “Higgs” mechanism for mass-
generation of KK gravitons under compactification with a general $R_{\xi}$
gauge-fixing, which we prove to be free from the vDVZ discontinuity. With
these, we formulate the Gravitational Equivalence Theorem (GRET) to connect
the longitudinal KK graviton amplitudes to the corresponding KK Goldstone
amplitudes, which is a manifestation of the geometric “Higgs” mechanism at
$S$-matrix level. We directly compute the gravitational KK Goldstone
amplitudes at tree level and show that they equal the corresponding
longitudinal KK graviton amplitudes in the high energy limit. We further
extend the double-copy method with color-kinematics duality to reconstruct the
massive KK longitudinal graviton (Goldstone) amplitudes from the KK
longitudinal gauge boson (Goldstone) amplitudes in the compactified 5d Yang-
Mills (YM) gauge theory under high energy expansion. From these, we
reconstruct the GRET of the KK longitudinal graviton (Goldstone) amplitudes in
the 5d GR theory from the KK longitudinal gauge boson (Goldstone) amplitudes
in the 5d YM theory. Using either the GRET or the double-copy reconstruction,
we provide a theoretical mechanism showing that the sum of all the energy-
power terms up to $\mathcal{O}(E^{10})$ in the high-energy scattering
amplitudes of four longitudinal KK gravitons must cancel down to
$\mathcal{O}(E^{2})$ as enforced by matching the energy dependence of the
corresponding KK Goldstone amplitudes or by matching that of the double-copy
amplitudes from the KK YM theory. With the double-copy approach, we establish
a new correspondence between the two energy-cancellations in the four-point
longitudinal KK scattering amplitudes: $\,E^{4}\\!\\!\rightarrow\\!E^{0}\,$ in
the 5d KK YM theory and $\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$ in the 5d KK GR
theory. We further analyze the structure of the residual term in the GRET and
uncover a new energy-cancellation mechanism therein.
Phys. Rev. D 105 (2022) 084005, no.8 $[$arXiv:2106.04568 [hep-th]].

###### Contents

  1. 1 Introduction
  2. 2 Gauge-Fixing and Propagators without vDVZ Discontinuity
    1. 2.1 Setup and Weak Field Expansion in 5d
    2. 2.2 Geometric Higgs Mechanism and Gauge Fixing under KK Compactification
    3. 2.3 Massless Limit and Absence of vDVZ Discontinuity in $R_{\xi}$ Gauge
  3. 3 Formulation of Gravitational Equivalence Theorem and the Energy Cancellation Mechanism
    1. 3.1 Formulation of Gravitational Equivalence Theorem
    2. 3.2 Energy Cancellation Mechanism for KK Graviton Scattering Amplitudes
  4. 4 Structure of KK Graviton Scattering Amplitudes from Gravitational Equivalence Theorem
    1. 4.1 GRET for the 5d Gravitational Scalar QED
    2. 4.2 Gravitational KK Goldstone Boson Scattering Amplitudes
      1. 4.2.1 Elastic Gravitational KK Goldstone Boson Scattering Amplitudes
      2. 4.2.2 Inelastic Gravitational KK Goldstone Boson Scattering Amplitudes
  5. 5 Construction of Gravitational KK Amplitudes from Gauge KK Amplitudes with Double-Copy
    1. 5.1 Structure of Amplitudes for KK Gauge Bosons and Goldstone Bosons
    2. 5.2 Constructing KK Scattering Amplitudes and GRET by Double-Copy
    3. 5.3 Constructing Mass-Dependent KK Amplitudes from Double-Copy
      1. 5.3.1 Double-Copy of NLO KK Amplitudes under $1/s_{0}$ Expansion
      2. 5.3.2 Improved Double-Copy of NLO KK Amplitudes under $1/s$ Expansion
      3. 5.3.3 Exact NLO Double-Copy based on KK String Approach
    4. 5.4 GRET Residual Terms: Energy Cancellation from Double-Copy
  6. 6 Conclusions
  7. A Kinematics of KK Particle Scattering
  8. B From $R_{\xi}$ Gauge to Unitary Gauge
  9. C Feynman Rules for KK Graviton Interaction with Matter
  10. D Gravitational KK Goldstone Amplitudes from Goldstone Exchanges and Contact Interactions
  11. E Massless Graviton Scattering and Double-Copy in 4d
    1. E.1 Massless Graviton Scattering from Double-Copy Construction in 4d
    2. E.2 Massless Graviton Scattering from Type-II Superstring Theory
  12. F Full Amplitudes of KK Gravitons and Goldstones in 5d GR
  13. G Extending KLT Construction to KK Amplitudes

##  Introduction

The world is apparently four-dimensional, but it could be only part of a
higher dimensional space-time structure, with all the extra spatial dimensions
compactified at the boundaries and with their sizes much smaller than the
present observational limits. The first of such theories was proposed a
century ago by Kaluza and Klein in an attempt to unify the gravitational and
electromagnetic forces with a compactified fifth dimension (5d) [1]. This
intriguing avenue was subsequently extended and explored in various contexts,
including the (super) string/M theories [2] and extra dimensional field
theories with large or small extra dimensions [3][4][ExdRS].

The Kaluza-Klein (KK) compactification of an extra dimension leads to an
infinite tower of massive KK states in the low energy 4d effective field
theory for each type of particles that propagate into the extra dimension. On
one hand, the low-lying KK states in such extra dimensional KK theories have
intrigued much phenomenological and experimental efforts over the past two
decades [5][6], as they may provide the first signatures for the new physics
beyond the standard model (SM), ranging from the KK states of the SM particles
to the spin-2 KK gravitons and possible dark matter candidate. On the other
hand, the mass generation of these KK states has important implications for
the theory side because it is realized by a geometric “Higgs” mechanism
through compactification itself and without invoking any additional Higgs
boson of the conventional Higgs mechanism [7].

For the compactified 5d KK Yang-Mills (YM) gauge theories, it was realized [8]
that each massive KK gauge boson $A_{n}^{a\mu}$ of KK level-$n$ acquires its
mass by absorbing the fifth component $A_{n}^{a5}$ (Goldstone boson) of the 5d
gauge field. This geometric KK “Higgs” mechanism is quantitatively described
by the KK equivalence theorem for compactified gauge theories (KK GAET) at the
$S$-matrix level [8], stating that the scattering amplitude of the
longitudinally-polarized KK gauge bosons ($A^{an}_{L}$) equals that of the
corresponding KK Goldstone bosons in the high energy limit. This is a direct
consequence of the spontaneous geometric breaking of the 5d gauge symmetry
down to the 4d gauge symmetry via KK compactification [8][9]. It was proven
that the nontrivial cancellation of energy-power terms of
$\mathcal{O}(E^{4})\\!\rightarrow\\!\mathcal{O}(E^{0})$ in the four
longitudinal KK gauge boson scattering amplitude in the high energy limit is
generally guaranteed by the KK GAET under which the corresponding KK Goldstone
boson amplitude is manifestly of $\mathcal{O}(E^{0})$ [8]. The extension of KK
GAET to quantum loop level via BRST quantization was given in Ref. [9]. It was
realized that the KK GAET (which ensures the energy-cancellation of
$\,E^{4}\\!\\!\rightarrow\\!E^{0}\hskip 0.85358pt$) [8][9] originates from the
5d gauge symmetry under compactification and the resulting BRST identity. The
5d KK gauge boson scattering amplitudes were further studied in the context of
the deconstructed 5d YM theories [10][9] and the compactified 5d SM [11].

It was realized even earlier that the compactified 5d General Relativity (GR)
also exhibits a geometric mechanism for the mass generation of KK gravitons.
Refs. [12][13] gave formal discussions of such geometric breaking by
formulating an infinite-parameter Virasoro-Kac-Moody group for the 4d
effective KK theory which is spontaneously broken down to the four-dimensional
translations and the U(1) gauge group by the 5d periodic boundary conditions.
It is expected that the 5d gravitational diffeomorphism invariance of the
Einstein-Hilbert (EH) action is spontaneously broken by the boundary
conditions to that of the 4d KK theory via a geometric breaking mechanism,
where at each KK level-$n$ the helicity $\pm 1$ components (${h}^{\mu 5}_{n}$)
and the helicity-0 component (${h}^{55}_{n}$) of the 5d spin-2 graviton
($\hat{h}^{AB}$) are supposed to be absorbed by the KK graviton
(${h}^{\mu\nu}_{n}$) via geometric “Higgs” mechanism under the 5d
compactification. However, there is no quantitative formulation of this
gravitational KK “Higgs” mechanism at the $S$-matrix level so far. There are
recent works [14][15] which gave direct calculations of the four-point
scattering amplitudes of (helicity-zero) longitudinal 5d KK gravitons at tree
level, and explicitly showed large energy cancellations among the individual
contributions of $\mathcal{O}(E^{10})\\!\rightarrow\mathcal{O}(E^{2})$ for
flat or warped 5d model. Following Ref. [14], the authors of Ref. [16] used
Hodge and eigenfunction decompositions [17] to show that at tree level such
energy cancellations of four-point KK graviton amplitudes occur for
compactification on general closed Ricci-flat manifolds. While showing such
intricate large energy cancellations in the tree-level amplitudes of four KK
gravitons are interesting and valuable, it remains to be understood
quantitatively why such nontrivial cancellations must occur at the tree level
and even loop levels for the $N$-particle KK amplitudes ($N\\!\geqslant\\!4$)
in connection to the compactified diffeomorphism (gauge) symmetry with
geometric breaking in the 5d KK GR or in the 5d KK YM gauge theory.

In this work, we present a general formulation of the geometric “Higgs”
mechanism for the compactified 5d GR in the $R_{\xi}$ gauge, at both the
Lagrangian level and scattering $S$-matrix level. For this geometric “Higgs”
mechanism, we will formulate a KK Gravitational Equivalence Theorem (GRET)
which quantitatively connects each scattering amplitude of longitudinally-
polarized KK gravitons to that of the corresponding gravitational KK Goldstone
bosons. The formulation of the KK GRET is highly nontrivial and differs from
the KK GAET of the 5d KK gauge theories [8], because the gravitational
Goldstone bosons contain both spin-0 and spin-1 components. By inspecting the
spin-0 gravitational KK Goldstone scattering amplitudes and the residual term
of the GRET, we show that they are manifestly of $\mathcal{O}(E^{2})$ in the
high energy regime without invoking any extra energy-power cancellation. Using
the GRET (based on BRST quantization), we provide a theoretical mechanism
showing that the sum of all the energy-power terms [up to
$\mathcal{O}(E^{10})$] in the four longitudinal KK graviton scattering
amplitude must cancel down to $\mathcal{O}(E^{2})$ at tree level as enforced
by matching the energy-power dependence in the corresponding KK Goldstone
amplitude (and residual term). We will also extend this conclusion to the case
of $N$-point longitudinal KK graviton scattering amplitudes and up to any loop
level, where we prove that each $N$-point longitudinal KK graviton amplitude
has large energy cancellations by a power of $\hskip 0.85358ptE^{2N}$
($N\\!\\!\geqslant\\!4$). This is in contrast to the case of the Fierz-Pauli
(FP) gravity and alike [18][19] where the four-point massive longitudinal
graviton scattering amplitudes generally scale as $\,E^{10}$ [20]. By
including additional non-linear polynomial interaction terms in the
literature, the high energy behavior of the massive graviton amplitudes could
be improved to no better than $\,E^{6}$ [21][22], which is still much worse
than the final energy-dependence of $\mathcal{O}(E^{2})$ in the massive KK
graviton scattering amplitudes as mentioned above.

In addition, using our general $R_{\xi}$ gauge formulation of the massive KK
graviton propagator, we further demonstrate that the spontaneous breaking of
the 5d gravitational diffeomorphism invariance of the EH action under
geometric “Higgs” mechanism will ensure the absence of the vDVZ (van Dam-
Veltman and Zakharov) discontinuity [23] under the massless limit, in contrast
to the case of the Fierz-Pauli (FP) gravity and alike [18][19] which are
plagued by the longstanding problem of the vDVZ discontinuity.

Furthermore, we attempt to reconstruct the 5d KK graviton scattering
amplitudes from the corresponding 5d KK gauge boson scattering amplitudes [8]
under high energy expansion to the leading order (LO) and the next-to-leading
order (NLO) contributions, by extending the conventional double-copy method of
the color-kinematics (CK) duality of Bern-Carrasco-Johansson (BCJ) [24][25]
which was proposed for connecting the massless gauge theories to the massless
gravity. The BCJ method was inspired by the Kawai-Lewellen-Tye (KLT) [26]
relation which connects the product of the scattering amplitudes of two open
strings to that of the closed string at tree level. Analyzing the properties
of the heterotic string and open string amplitudes can prove and refine parts
of the BCJ conjecture [27]. The conventional double-copy formulation reveals a
deep connection between the GR theory with massless spin-2 gravitons and the
YM theory with massless spin-1 gauge bosons. This may be schematically
presented as follows [28]:

$\mathrm{GR}=(\mathrm{Gauge\,Theory})^{2}\,.$ (1.1)

We extend the double-copy method to the massive 5d KK gravity and KK gauge
theories, and compute the LO and NLO four-particle scattering amplitudes under
the high energy expansion. This provides an extremely simple and efficient way
to construct the complicated KK graviton amplitudes from the 5d KK gauge boson
amplitudes. Indeed, we find that our LO longitudinal KK graviton amplitudes as
reconstructed from the LO amplitudes of 5d KK gauge bosons [8] are equal to
the KK graviton amplitudes as obtained by the lengthy direct calculations of
[14][15]. Because the 5d KK gauge boson amplitudes [8] are of
$\mathcal{O}(E^{0}M_{n}^{0})$, our double-copy approach shows that the
reconstructed KK graviton amplitudes must be of $\mathcal{O}(E^{2}M_{n}^{0})$,
where $M_{n}$ denotes the relevant KK mass. Moreover, we use the KK Goldstone
amplitudes of the 5d YM theory [which are manifestly of
$\mathcal{O}(E^{0}M_{n}^{0})$] to reconstruct the corresponding gravitational
KK Goldstone amplitudes by the double-copy method, and find that these
gravitational KK Goldstone amplitudes must be of
$\mathcal{O}(E^{2}M_{n}^{0})$. We further compare the reconstructed
gravitational KK Goldstone amplitudes with the reconstructed longitudinal KK
graviton amplitudes under the high energy expansion, and find that they are
equal to each other at the leading order of $\,\mathcal{O}(E^{2}M_{n}^{0})$
and their difference is only $\,\mathcal{O}(E^{0}M_{n}^{2})$ . Hence, for the
four-particle scattering processes, we establish the GRET in the 5d KK GR
theory from the KK GAET in the 5d YM theory [8] by using the extended double-
copy reconstruction method. By doing so, we will demonstrate a nontrivial new
correspondence from the energy-cancellation of
$\,E^{4}\\!\\!\rightarrow\\!E^{0}$ in the four-particle amplitudes for
longitudinal KK gauge bosons of the 5d KK YM theory (YM5) to the energy-
cancellation of $\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$ in the four-particle
amplitudes for longitudinal KK gravitons of the 5d KK GR theory (GR5).
Schematically, we illustrate this correspondence between the two energy-
cancellations as follows:

$E^{4}\\!\rightarrow\\!E^{0}\,\mathrm{(YM5)}\leavevmode\nobreak\
\leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\
E^{10}\\!\rightarrow\\!E^{2}\,\mathrm{(GR5)}\,,$ (1.2)

which will be established later in Eq.(2.3al) of section 5.2. In addition,
using the double-copy approach, we analyze the structure of the residual terms
in the GRET and further uncover a new energy-cancellation mechanism of
$\,E^{2}\\!\\!\rightarrow\\!E^{0}\,$ therein. It is clear that the GRET and
its reconstruction from the 5d KK YM gauge theory via double-copy can provide
a deep quantitative understanding on the structure of the KK graviton
(Goldstone) scattering amplitudes and thus the realization of the geometric
“Higgs” mechanism of KK compactification.

This paper is organized as follows. In section 2, we present the general
$R_{\xi}$ gauge quantization for the 5d KK GR. We derive the propagators for
the KK graviton and KK Goldstone bosons. We will show that the KK graviton
propagator in the $R_{\xi}$ gauge is free from the vDVZ discontinuity, in
contrast to that of the Fierz-Pauli gravity. In section 3, we present the
formulation of the GRET and use it to establish a theoretical mechanism which
ensures the nontrivial energy cancellations in the $N$-point longitudinal KK
graviton scattering amplitudes. This cancellation mechanism holds not only for
the four-particle amplitudes at tree level, but also can be applied to the
general $N$-particle amplitudes ($N\\!\\!\geqslant\\!4$) and up to loop levels
in principle. In section 3.1, we first derive the formulation of the GRET,
which nontrivially differs from the KK GAET of the 5d KK gauge theories [8].
Then, in section 3.2 we present a general method of energy power counting to
determine the leading energy dependence of the high energy scattering
amplitudes in the KK GR theory and in the KK YM theory. In section 4, we
present the explicit analyses of the scattering amplitudes of longitudinal KK
gravitons and of the corresponding gravitational KK Goldstone bosons to
demonstrate how the GRET works. In section 5, we establish the double-copy
constructions of the longitudinal KK graviton scattering amplitudes and the
corresponding KK Goldstone scattering amplitudes. We give in section 5.1 the
full scattering amplitudes of the longitudinal KK gauge boson amplitudes and
the KK Goldstone amplitudes, and derive their LO and NLO contributions under
high energy expansion. Then, in section 5.2, we use the double-copy approach
to reconstruct the LO KK graviton amplitudes and KK Goldstone amplitudes. With
these, we establish the GRET in the 5d KK GR theory from the KK GAET in the 5d
YM gauge theory at the LO. In section 5.3, we study the double-copy
construction of the NLO gravitational KK amplitudes of
$\mathcal{O}(E^{0}M_{n}^{2})$. Based on the recent first principle approach of
the KK string theory, we further propose an improved double-copy construction
to derive the exact NLO KK graviton amplitudes. In section 5.4, we analyze the
structure and size of the residual term in the GRET, and establish the
correspondence from the KK GAET to the KK GRET. We conclude in section 6.
Finally, the Appendices A-G present a number of analyses used for the text
discussions.

##  Gauge-Fixing and Propagators without vDVZ Discontinuity

In this section, we first setup the 5d compactification under the
$S^{1}/\mathbb{Z}_{2}$ orbifold, including the notations and KK expansions.
Then, we present the quadratic Lagrangian terms from the 5d EH action,
construct a general $R_{\xi}$ gauge-fixing, and also derive the relevant KK
graviton and KK Goldstone propagators. Finally, we prove that the massive KK
graviton propagator is naturally free from the longstanding puzzle of the vDVZ
discontinuity which plagues the Pauli-Fierz gravity and alike [18][19].

###  Setup and Weak Field Expansion in 5d

For the current study, we consider the five-dimensional general relativity on
a compactified flat space under orbifold $S^{1}\\!/\mathbb{Z}_{2}$ .111The
extension of our present study to the case of non-flat 5d space (such as
warped 5d [29]) does not cause any conceptual difference regarding all the
major conclusions in this work, which will be addressed elsewhere. Thus,
the compactified fifth dimension is a line segment with
$\,0\\!\leqslant\\!x^{5}\\!\leqslant\\!\pi r_{c}$ , where $r_{c}$ stands for
the compactification radius. Based on this, the 5d Einstein-Hilbert (EH)
action is given by

$S_{\mathrm{EH}}\,=\,\int\\!\text{d}^{5}{x}\,\frac{2}{\,\hat{\kappa}^{2}\,}\sqrt{\\!-\hat{g}\,}\hat{R}\,,$
(2.1)

where $\hat{R}$ is the 5d Ricci scalar curvature, $\,\hat{\kappa}\,$ is the 5d
gravitational coupling with mass-dimension $-\mbox{$\frac{\,{3}\,}{2}$}$ and
it is related to the 5d Newton constant $\,\hat{G}\,$ via
$\,\hat{\kappa}=\\!\sqrt{32\pi\hat{G}\,}\,$. The 5d metric tensor is
$\hat{g}_{AB}$ ($A,B=0,1,2,3,5$) and its determinant is given by
$\,\hat{g}=\det(\hat{g}_{AB})\,$. We also adopt the metric signature
$(-,+,+,+,+)$. In addition, we denote the 4d Lorentz indices by the lowercase
Greek letters (such as $\mu=0,1,2,3$), and the 5d Lorentz indices by the
uppercase Latin letters (such as $A=\mu,5$).

We make the following weak field expansion of the 5d EH action (2.1) around
the flat Minkowski metric $\hat{\eta}_{AB}$ :

$\hat{g}_{AB}=\hat{\eta}_{AB}+\hat{\kappa}\hat{h}_{AB}\,,$ (2.2)

where the graviton field $\hat{h}_{AB}$ has the mass-dimension
$\frac{\,{3}\,}{2}$. Then, it is straightforward to derive

$\displaystyle\hat{g}^{AB}$
$\displaystyle\,=\,\hat{\eta}^{AB}\\!-\hat{\kappa}\hat{h}^{AB}\\!+\hat{\kappa}^{2}\hat{h}^{AC}\tensor{\hat{h}}{{}_{C}^{B}}\\!-\hat{\kappa}^{3}\hat{h}^{AC}\hat{h}_{CD}\hat{h}^{DB}\\!+\mathcal{O}(\hat{h}^{4})\,$
$\displaystyle\sqrt{-\hat{g}\,}$
$\displaystyle\,=\,1\\!+\\!\frac{\hat{\kappa}}{2}\hat{h}\\!+\\!\frac{\hat{\kappa}^{2}}{8}(\hat{h}^{2}\\!-\\!2\tensor{\hat{h}}{{}_{\\!A}{}_{B}}\tensor{\hat{h}}{{}^{\\!A}{}^{B}})\\!+\\!\frac{\,\hat{\kappa}^{3}}{\,48\,}(\hat{h}^{3}\\!-6\hat{h}\tensor{\hat{h}}{{}_{\\!A}{}_{B}}\tensor{\hat{h}}{{}^{\\!A}{}^{B}}\\!+8\tensor{\hat{h}}{{}_{\\!A}{}_{B}}\tensor{\hat{h}}{{}^{\\!B}{}^{C}}\tensor{\hat{h}}{{}_{C}^{A}})+\mathcal{O}(\hat{h}^{4}),$
(2.3a)

where we have defined $\,\hat{h}=\hat{\eta}^{AB}\hat{h}_{AB}$ . Now, the 5d
scalar curvature $\hat{R}$ can be decomposed in terms of the metric tensors
$\hat{g}_{AB}$ and $\hat{g}^{AB}$ as follows:

$\displaystyle\hat{R}\ $
$\displaystyle\,=\,\hat{g}^{AB}\hat{R}_{AB}=\hat{g}^{AB}\tensor{\hat{R}}{{}_{A}{}_{C}{}_{B}^{C}},$
(2.3da) $\displaystyle\tensor{\hat{R}}{{}_{A}{}_{C}{}_{B}^{C}}\\!$
$\displaystyle\,=\,\partial_{C}\tensor{\hat{\Gamma}}{{}^{C}_{A}{}_{B}}\\!-\partial_{A}\tensor{\hat{\Gamma}}{{}^{C}_{C}{}_{B}}\\!+\tensor{\hat{\Gamma}}{{}^{D}_{A}{}_{B}}\tensor{\hat{\Gamma}}{{}^{C}_{D}{}_{C}}\\!-\tensor{\hat{\Gamma}}{{}^{D}_{C}{}_{B}}\tensor{\hat{\Gamma}}{{}^{C}_{D}{}_{A}}\,,$
(2.3db) $\displaystyle\tensor{\hat{\Gamma}}{{}^{C}_{\\!A}{}_{B}}$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}\,{\hat{g}^{CD}(\partial_{B}\hat{g}_{DA}\\!+\partial_{A}\hat{g}_{BD}\\!-\partial_{D}\hat{g}_{AB})}\,.$
(2.3dc)

With the above formulas, we can expand the 5d EH action
$\,S_{\mathrm{EH}}\\!=\\!\int\\!\text{d}^{5}{x}\,\hat{\mathcal{L}}_{\mathrm{EH}}\,$
shown in Eq.(2.1) as

$\hat{\mathcal{L}}_{\mathrm{EH}}\,=\,\hat{\mathcal{L}}_{0}+\hat{\kappa}\,\hat{\mathcal{L}}_{1}+\hat{\kappa}^{2}\hat{\mathcal{L}}_{2}+\hat{\kappa}^{3}\hat{\mathcal{L}}_{3}+\cdots\,,$
(2.3e)

where each expanded Lagrangian term $\,\hat{\mathcal{L}}_{j}$ $(j=0,1,\cdots)$
contains $\,j+2$ graviton fields. The effective 4d Lagrangian is obtained by
integrating over the extra dimension coordinate $x^{5}$ under proper
compactification:

$\mathcal{L}_{\mathrm{eff}}\,=\leavevmode\nobreak\
\sum_{j=0}^{\infty}\int_{0}^{L}\\!\\!\text{d}x^{5}\,\hat{\kappa}^{j}\hat{\mathcal{L}}_{j}\,.$
(2.3f)

The realization of 5d compactification will be given in the next subsection.
Finally, the corresponding effective 4d coupling $\,\kappa=\\!\sqrt{32\pi
G\,}\,$ is connected to the $\hat{\kappa}$ and the reduced Planck mass
$M_{\mathrm{Pl}}$ via

$\kappa=\frac{\hat{\kappa}}{\sqrt{L\,}\,}=\frac{2}{M_{\text{Pl}}}\,,$ (2.3g)

where we have denoted $\,L=\pi r_{c}\,$ as the length of the 5th dimension
under the compactification of $S^{1}\\!/\mathbb{Z}_{2}\,$, and the reduced
Planck mass is represented as $M_{\text{Pl}}\\!=(8\pi G)^{-1/2}\,$.

###  Geometric Higgs Mechanism and Gauge Fixing under KK Compactification

In this subsection, we will make KK compactification of the 5d EH action. This
can be realized for the 5d obifold compactification $S^{1}\\!/\mathbb{Z}_{2}$
with proper boundary conditions, and the resulting 4d effective KK theory
contains the KK tower of massive graviton states. The 5d gravitational
diffeomorphism invariance of the EH action is expected to be spontaneously
broken by the boundary conditions to that of the 4d KK theory via a geometric
breaking mechanism, where at each KK level-$n$ the vector components ($h^{\mu
5}_{n}$) and the scalar component ($h^{55}_{n}$) of the 5d spin-2 graviton
($\hat{h}^{AB}$) are supposed to be absorbed by the KK graviton
($h^{\mu\nu}_{n}$). There are formal discussions of such geometric breaking in
the literature [12][13], by formulating an infinite-parameter Virasoro-Kac-
Moody group for the 4d effective KK theory which is spontaneously broken down
to the four-dimensional translations and the U(1) gauge group. These formal
discussions [12][13] did not provide a practical formulation as needed for our
current study of perturbative KK theory and for the scattering amplitudes at
the $S$-matrix level.

In the following, we present an explicit formulation of this geometric “Higgs”
mechanism at the Lagrangian level, and then at the $S$-matrix level via the
GRET (section 3). The 5d geometric “Higgs” mechanism was previously
established for the compactified 5d Yang-Mills theories in Ref. [8].222The
extension to the deconstructed 5d YM theories was given in Ref. [10] and to
the compactified 5d SM was given in Ref. [11]. In this study, we present an
explicit formulation of the 5d geometric “Higgs” mechanism for the 5d Einstein
gravity, with which we will identify the gravitational Goldstone bosons
$(h_{n}^{\mu 5},\,h_{n}^{55})$ for each massive KK graviton $h_{n}^{\mu\nu}$.
Then, we explicitly construct the $R_{\xi}$ gauge-fixing term and derive the
propagators for KK gravitons and their corresponding Goldstone bosons.

The 5d graviton field $\hat{h}_{AB}$ can be parametrized as

$\hat{h}_{AB}=\begin{pmatrix}\hat{h}_{\mu\nu}\\!+w\eta_{\mu\nu}\hat{\phi}&\leavevmode\nobreak\
\hat{h}_{\mu 5}\\\\[5.69054pt] \hat{h}_{5\nu}&\leavevmode\nobreak\
\hat{\phi}\end{pmatrix}\\!,$ (2.3h)

where the (1,1) block is the 4d component of $\,\hat{h}_{AB}$ and the
additional term $\,w\eta_{\mu\nu}\hat{\phi}\,$ corresponds to a Weyl
transformation333More precisely, under the Weyl transformation the 4d metric
is rescaled as
$\,\hat{g}_{\mu\nu}\\!\rightarrow\\!\hat{g}_{\mu\nu}^{\prime}=e^{w\hat{\kappa}\hat{\phi}}\hat{g}_{\mu\nu}\,$.
with a nonzero coefficient $w$ .444In Ref.[19], $w$ is expressed as
$\,w=2/(d-2)\,$, which gives $w=1$ in 4d and $\,w=2/3\,$ in 5d. We will
determine the value of $\,w\,$ from a consistency requirement in the following
analysis. The (2,2) block of $\hat{h}_{AB}$ is a scalar field known as the
radion field ( $\hat{\phi}\equiv\hat{h}_{55}$ ). The blocks (1,2) and (2,1)
correspond to the vector component of the 5d graviton field $\hat{h}_{AB}$ .

With the 5d metric tensor (2.2) and the 5d graviton field (2.3h), we derive
the squared 5d interval

$\text{d}\hat{s}^{2}=[\eta_{\mu\nu}\\!+\hat{\kappa}(\hat{h}_{\mu\nu}\\!+\\!w\eta_{\mu\nu}\hat{\phi})]\text{d}x^{\mu}\text{d}x^{\nu}\\!+2\hat{\kappa}\,\hat{h}_{\mu
5}\,\text{d}x^{\mu}\text{d}x^{5}\\!+(1\\!+\\!\hat{\kappa}\hat{\phi})\text{d}x^{5}\text{d}x^{5}\,.$
(2.3i)

We compactify the 5d space under $S^{1}/\mathbb{Z}_{2}$ orbifold and require
$\,\text{d}\hat{s}^{2}\,$ to be invariant under a $\mathbb{Z}_{2}$ orbifold
reflection $\,x_{5}\\!\rightarrow\\!-x_{5}$. Hence, this requires that the
graviton’s tensor component $\,\hat{h}_{\mu\nu}\,$ and the scalar component
$\,\hat{\phi}\,$ to be even under $\mathbb{Z}_{2}$ symmetry, while the vector
component $\hat{h}_{\mu 5}$ should be $\mathbb{Z}_{2}$ odd:

$\displaystyle\hat{h}_{\mu\nu}(x_{\rho},x_{5})\,$
$\displaystyle=\,\hat{h}_{\mu\nu}(x_{\rho},-x_{5})\,,$ (2.3ja)
$\displaystyle\hat{h}_{\mu 5}(x_{\rho},x_{5})\,$
$\displaystyle=\,-\hat{h}_{\mu 5}(x_{\rho},-x_{5})\,,$ (2.3jb)
$\displaystyle\hat{\phi}(x_{\rho},x_{5})\,$
$\displaystyle=\,\hat{\phi}(x_{\rho},-x_{5})\,.$ (2.3jc)

This is equivalent to imposing the Neumann boundary conditions on
$\,\hat{h}_{\mu\nu}$ and $\,\hat{\phi}\,$ at the ends of the 5d interval
$[0,\,L]$, and imposing the Dirichlet boundary condition on $\hat{h}_{\mu 5}$
,

$\left.\partial_{5}\hat{h}_{\mu\nu}\right|_{x_{5}\,=\,0,L}\\!=0\,,\hskip
28.45274pt\left.\partial_{5}\hat{\phi}\right|_{x_{5}\,=\,0,L}\\!=0\,,\hskip
28.45274pt\left.\hat{h}_{\mu 5}\right|_{x_{5}\,=\,0,L}\\!=0\,.$ (2.3k)

With these, we can make the following KK expansions for the 5d graviton fields
via Fourier series in terms of their zero-modes and KK states,

$\displaystyle\hat{h}^{\mu\nu}(x^{\rho},x^{5})$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\\!\left[\,h^{\mu\nu}_{0}(x^{\rho})+\sqrt{2}\sum_{n=1}^{\infty}h^{\mu\nu}_{n}(x^{\rho})\cos\\!\frac{n\pi
x^{5}}{L}\\!\,\right]\\!,$ (2.3la) $\displaystyle\hat{h}^{\mu
5}(x^{\rho},x^{5})$
$\displaystyle\,=\,\sqrt{\frac{2}{L}\,}\sum_{n=1}^{\infty}h^{\mu
5}_{n}(x^{\rho})\sin\\!\frac{n\pi x^{5}}{L}\,,$ (2.3lb)
$\displaystyle\hat{\phi}(x^{\rho},x^{5})$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\left[\,\phi_{0}(x^{\rho})+\sqrt{2}\sum_{n=1}^{\infty}\phi_{n}(x^{\rho})\cos\\!\frac{n\pi
x^{5}}{L}\\!\,\right]\\!.$ (2.3lc)

Then, we examine the quadratic Lagrangian $\hat{\mathcal{L}}_{0}$, which takes
the following form:

$\hat{\mathcal{L}}_{0}\,=\,\mbox{$\frac{\,{1}\,}{2}$}(\partial_{A}\hat{h})^{2}-\mbox{$\frac{\,{1}\,}{2}$}(\partial_{C}\hat{h}_{AB})^{2}-\partial_{A}\hat{h}^{AB}\partial_{B}\hat{h}+\partial_{A}\hat{h}^{AC}\partial^{B}\hat{h}_{BC}\,.$
(2.3m)

Substituting Eq.(2.3h) into the quadratic Lagrangian (2.3m), we thus derive

$\displaystyle\hat{\mathcal{L}}_{0}\,=$ $\displaystyle\
\mbox{$\frac{\,{1}\,}{2}$}(\partial_{\mu}\hat{h})^{2}\\!+\\!\mbox{$\frac{\,{1}\,}{2}$}(\partial_{5}\hat{h})^{2}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}(\partial_{\rho}\hat{h}_{\mu\nu})^{2}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}(\partial_{5}\hat{h}_{\mu\nu})^{2}\\!-\\!(\partial_{\mu}\hat{h}_{\nu
5})^{2}\\!+\\!(\partial_{\mu}\hat{h}^{\mu
5})^{2}\\!+\\!3w(w\\!+\\!1)(\partial_{\mu}\hat{\phi})^{2}$
$\displaystyle+6w^{2}(\partial_{5}\hat{\phi})^{2}\\!-\partial_{\mu}\hat{h}^{\mu\nu}\partial_{\nu}\hat{h}+\partial_{\mu}\hat{h}^{\mu\rho}\partial^{\nu}\hat{h}_{\nu\rho}\\!-(2w\\!+\\!1)(\partial_{\mu}\hat{h}^{\mu\nu}\partial_{\nu}\phi-\partial_{\mu}\hat{h}\,\partial^{\mu}\hat{\phi})$
$\displaystyle-2\partial_{\mu}\hat{h}^{\mu
5}\partial_{5}\hat{h}+2\partial^{5}\hat{h}^{\mu\nu}\partial_{\mu}\hat{h}_{\nu_{5}}\\!-6w\,\partial_{\mu}\hat{h}^{\mu
5}\partial_{5}\hat{\phi}+3w\,\partial_{5}\hat{h}\partial^{5}\hat{\phi}\,.$
(2.3n)

In terms of the KK expansions (2.3l) and integrating over $x^{5}$, we can
further expand the Lagrangian (2.2) as follows:

$\displaystyle\mathcal{L}_{0}\,=\,$
$\displaystyle\,\sum_{n=0}^{\infty}\left[\mbox{$\frac{\,{1}\,}{2}$}(\partial_{\mu}h_{n})^{2}\\!+\\!\mbox{$\frac{\,{1}\,}{2}$}M_{n}^{2}(h_{n})^{2}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}(\partial^{\rho}h^{\mu\nu}_{n})^{2}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}M_{n}^{2}(h^{\mu\nu}_{n})^{2}\\!-\\!(\partial^{\mu}\mathcal{A}^{\nu}_{n})^{2}\\!+(\partial_{\mu}\mathcal{A}_{n}^{\mu})^{2}\right.$
$\displaystyle\hskip
26.00009pt+3w(w\\!+\\!1)(\partial_{\mu}\phi_{n})^{2}\\!+6w^{2}M_{n}^{2}\phi_{n}^{2}\\!-\partial_{\mu}h^{\mu\nu}_{n}\partial_{\nu}h_{n}+\partial_{\mu}h^{\mu\rho}_{n}\partial^{\nu}h_{\nu\rho,n}$
$\displaystyle\hskip
26.00009pt-(2w\\!+\\!1)(\partial_{\mu}h^{\mu\nu}_{n}\partial_{\nu}\phi_{n}\\!-\partial_{\mu}h_{n}\partial^{\mu}\phi_{n})\\!+2M_{n}h_{n}\partial_{\mu}\mathcal{A}^{\mu}_{n}\\!-2M_{n}h^{\mu\nu}_{n}\partial_{\mu}\mathcal{A}_{\nu,n}$
$\displaystyle\hskip
26.00009pt\left.+3wM_{n}^{2}\,h_{n}\phi_{n}+6wM_{n}\partial_{\mu}\mathcal{A}^{\mu}_{n}\,\phi_{n}\,\right]\\!,$
(2.3o)

where for convenience we have denoted the vector field as
$\,\mathcal{A}_{n}^{\mu}\\!\equiv h^{\mu 5}_{n}$, and $\,M_{n}\\!=\\!n\pi/L$
stands for the mass of KK states of level-$n$.

Inspecting the Lagrangian (2.2), we set
$\,w\\!=\\!-\mbox{$\frac{\,{1}\,}{2}$}$ to remove the two undesirable mixing
terms in its 3rd line. We can further eliminate the rest of the mixing terms
in the 3rd and 4th lines of Eq.(2.2) by introducing the following
$R_{\xi}$-type gauge-fixing terms,

$\mathcal{L}_{\mathrm{GF}}=-\\!\sum_{n=0}^{\infty}\\!\left\\{\frac{1}{\xi_{n}}\\!\\!\left[\,\\!\partial_{\nu}h_{n}^{\mu\nu}\\!-\\!\left(\\!1\\!-\\!\frac{1}{2\xi_{n}}\right)\\!\partial^{\mu}h_{n}\\!+\xi_{n}M_{n}\mathcal{A}^{\mu}_{n}\\!\,\right]^{\\!2}\\!+\frac{\,M_{n}^{2}\,}{\,4\xi_{n}\,}\\!\\!\left(\\!h_{n}\\!\\!-\\!3\xi_{n}\phi_{n}\\!+\\!\frac{2\partial_{\\!\mu}\mathcal{A}_{n}^{\mu}}{M_{n}}\right)^{\\!\\!2}\right\\}\\!,$
(2.3p)

where $\,\xi_{n}\,$ is the gauge-fixing parameter for the zero-mode gravitons
($n=0$) and KK gravitons ($n\geqslant 1$) . By imposing the gauge-fixing term
(2.3p) to remove the quadratic mixing terms, we explicitly verify that both
the vector component $\,\mathcal{A}_{n}^{\mu}\,$ and scalar component
$\,\phi_{n}\,$ are absorbed (“eaten”) by the KK graviton $h^{\mu\nu}_{n}$, and
identify them as the gravitational KK Goldstone fields, which are the direct
outcome of realizing the 5d geometric KK “Higgs” mechanism.

From the above, we can explicitly integrate over $x^{5}$ and derive the
effective 4d KK action at the quadratic order:

$S_{\mathrm{eff}}=\int\\!\\!\text{d}^{4}x\,\sum_{n=0}^{\infty}\frac{1}{2}\\!\left(h^{\mu\nu}_{n}\mathcal{D}^{-1}_{\mu\nu\alpha\beta,nn}h^{\alpha\beta}_{n}+\mathcal{A}^{\mu}_{n}\mathcal{D}_{\mu\nu,nn}^{-1}\mathcal{A}^{\nu}_{n}+\phi_{n}\mathcal{D}^{-1}_{nn}\phi_{n}\right),$
(2.3q)

where the inverse KK propagators take the following forms:

$\displaystyle\mathcal{D}^{-1}_{\mu\nu\alpha\beta,nn}=$
$\displaystyle-\\!\left[\,\\!1\\!-\\!\frac{2}{\xi_{n}}\\!\left(\\!1-\frac{1}{2\xi_{n}}\right)^{\\!\\!2}\,\right]\\!\eta_{\mu\nu}\eta_{\alpha\beta}\partial^{2}+\left(\\!1\\!-\\!\frac{1}{2\xi_{n}}\right)\eta_{\mu\nu}\eta_{\alpha\beta}M_{n}^{2}$
$\displaystyle+\frac{1}{2}\left(\eta_{\mu\alpha}\eta_{\nu\beta}\\!+\eta_{\mu\beta}\eta_{\nu\alpha}\right)\\!\left(\partial^{2}\\!-\\!M_{n}^{2}\right)+\left(\frac{1}{\xi_{n}}\\!-\\!1\\!\right)^{\\!\\!2}\\!\left(\eta_{\mu\nu}\partial_{\alpha}\partial_{\beta}\\!+\\!\eta_{\alpha\beta}\partial_{\mu}\partial_{\nu}\right)$
$\displaystyle-\frac{1}{2}\\!\left(\\!1\\!-\\!\frac{1}{\xi_{n}}\right)\\!\left(\eta_{\mu\alpha}\partial_{\nu}\partial_{\beta}+\eta_{\mu\beta}\partial_{\nu}\partial_{\alpha}+\eta_{\nu\alpha}\partial_{\mu}\partial_{\beta}+\eta_{\nu\beta}\partial_{\mu}\partial_{\alpha}\right),$
(2.3ra) $\displaystyle\mathcal{D}^{-1}_{\mu\nu,nn}=$ $\displaystyle\
\eta_{\mu\nu}(\partial^{2}\\!-\xi_{n}M_{n}^{2})+\frac{\,1\\!-\xi_{n}\,}{\xi_{n}}\,\partial_{\mu}\partial_{\nu}\,,\hskip
28.45274pt$ (2.3rb) $\displaystyle\mathcal{D}^{-1}_{nn}=$ $\displaystyle\
\partial^{2}\\!-(3\xi_{n}\\!-2)M_{n}^{2}\,,$ (2.3rc)

and we have also rescaled the vector and scalar fields by

$\displaystyle\mathcal{A}^{\mu}_{n}\rightarrow\frac{1}{\sqrt{2\,}\,}\mathcal{A}^{\mu}_{n}\,,\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\phi_{n}\rightarrow\sqrt{\frac{2}{3}\,}\phi_{n}\,,$ (2.3s)

which ensure that their kinematic terms have the correct normalization factor
$\frac{\,{1}\,}{2}$ . Furthermore, the propagators of the KK graviton and KK
Goldstone bosons are the inverse of Eq.(2.3r) and satisfy the following
conditions:

$\displaystyle\int\\!\text{d}^{4}z\,\mathcal{D}_{\mu\nu\alpha\beta,nn}^{-1}(x,z)\mathcal{D}^{\alpha\beta\rho\sigma}_{nn}(z,y)$
$\displaystyle\,=\,\frac{\text{i}}{2}({\delta}{{}_{\mu}^{\rho}}{\delta}{{}_{\nu}^{\sigma}}+{\delta}{{}_{\mu}^{\sigma}}{\delta}{{}_{\nu}^{\rho}})\delta^{(4)}(x-y)\,,$
(2.3ta)
$\displaystyle\int\\!\text{d}^{4}z\,\mathcal{D}_{\mu\nu,nn}^{-1}(x,z)\mathcal{D}^{\nu\rho}_{nn}(z,y)$
$\displaystyle\,=\,\text{i}\,{\delta}{{}_{\mu}^{\rho}}\delta^{(4)}(x-y)\,,$
(2.3tb)
$\displaystyle\int\\!\text{d}^{4}z\,\mathcal{D}_{nn}^{-1}(x,z)\mathcal{D}_{nn}(z,y)$
$\displaystyle\,=\,\text{i}\,\delta^{(4)}(x-y)\,.$ (2.3tc)

Substituting Eq.(2.3r) into Eq.(2.3t), we finally derive the following compact
form of the propagators for the KK gravitons $(h_{n}^{\mu\nu})$ and for the KK
Goldstone bosons ($\mathcal{A}^{\mu}_{n}$ and $\phi_{n}$) in momentum space:

$\displaystyle\mathcal{D}_{nm}^{\mu\nu\alpha\beta}(p)\,=\,$
$\displaystyle-\frac{\,\text{i}\delta_{nm}\,}{2}\left\\{\frac{\,(\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta})\,}{\,p^{2}\\!+\\!M_{n}^{2}\,}\right.$
$\displaystyle+\frac{1}{3}\\!\left[\,\\!\frac{1}{\,p^{2}\\!+\\!M_{n}^{2}\,}-\frac{1}{\,p^{2}\\!+\\!(3\xi_{n}\\!-\\!2)M_{n}^{2}\,}\\!\,\right]\\!\\!\left(\\!\eta^{\mu\nu}\\!-\\!\frac{\,2p^{\mu}p^{\nu}}{M_{n}^{2}}\\!\right)\\!\\!\left(\\!\eta^{\alpha\beta}\\!-\\!\frac{\,2p^{\alpha}p^{\beta}}{M_{n}^{2}}\\!\right)$
$\displaystyle+\frac{1}{\,M_{n}^{2}\,}\\!\left[\,\\!\frac{1}{\,p^{2}\\!+\\!M_{n}^{2}}-\frac{1}{\,p^{2}\\!+\\!\xi_{n}M_{n}^{2}\,}\\!\,\right]\\!(\eta^{\mu\alpha}p^{\nu}p^{\beta}\\!+\eta^{\mu\beta}p^{\nu}p^{\alpha}\\!+\eta^{\nu\alpha}p^{\mu}p^{\beta}\\!+\eta^{\nu\beta}p^{\mu}p^{\alpha})$
$\displaystyle\left.+\frac{\,4p^{\mu}p^{\nu}p^{\alpha}p^{\beta}\,}{\xi_{n}M_{n}^{4}}\\!\left(\\!\frac{1}{\,p^{2}\\!+\\!\xi_{n}^{2}M_{n}^{2}\,}-\frac{1}{\,p^{2}\\!+\\!\xi_{n}M_{n}^{2}\,}\\!\right)\\!\right\\}\\!,$
(2.3ua) $\displaystyle\mathcal{D}^{\mu\nu}_{nm}(p)\,=\,$
$\displaystyle\frac{-\text{i}\delta_{nm}}{\,p^{2}\\!+\\!\xi_{n}M_{n}^{2}\,}\\!\left[\eta^{\mu\nu}\\!\\!-\\!\frac{\,p^{\mu}p^{\nu}(1\\!-\\!\xi_{n})\,}{\,p^{2}\\!+\xi_{n}^{2}M_{n}^{2}}\right]\\!,$
(2.3ub) $\displaystyle\mathcal{D}_{nm}(p)\,=\,$
$\displaystyle\frac{-\text{i}\delta_{nm}}{\,p^{2}\\!+\\!(3\xi_{n}\\!\\!-\\!2)M_{n}^{2}\,}\,.$
(2.3uc)

The Faddeev-Popov ghosts can be further included for the loop analysis
although this is not needed for our present study of KK scattering amplitudes
at tree level. The unphysical states of the massive KK gravitons correspond to
the spin-0 and spin-1 Goldstone bosons, and we see that the above Goldstone
propagators (2.3ub) and (2.3uc) have the same $\xi_{n}$-dependent unphysical
mass poles as those of the KK graviton propagator (2.3ua).

It is instructive to consider the Feynman-’t Hooft gauge with
$\,\xi_{n}\\!=\\!1\,$. In this gauge, the above $R_{\xi}$-gauge propagators
take the following simple forms:

$\displaystyle\mathcal{D}_{nm}^{\mu\nu\alpha\beta}(p)$
$\displaystyle=-\frac{\,\text{i}\delta_{nm}\,}{2}\frac{\,\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta}\,}{\,p^{2}\\!+\\!M_{n}^{2}\,}\,,$
(2.3va) $\displaystyle\mathcal{D}^{\mu\nu}_{nm}(p)$
$\displaystyle=-\frac{\,\text{i}\eta^{\mu\nu}\delta_{nm}\,}{\,p^{2}\\!+\\!M_{n}^{2}\,}\,,$
(2.3vb) $\displaystyle\mathcal{D}_{nm}(p)$
$\displaystyle=-\frac{\,\text{i}\delta_{nm}\,}{\leavevmode\nobreak\
p^{2}\\!+\\!M_{n}^{2}\leavevmode\nobreak\ }\,.$ (2.3vc)

We can find that all the mass poles are identical to $p^{2}\\!=\\!-M_{n}^{2}$
. Then, we take the limit $\,\xi_{n}\\!\rightarrow\\!\infty\,$ and derive the
propagator under unitary gauge:

$\mathcal{D}^{\mu\nu\alpha\beta}_{nm,\mathrm{UG}}(p)\,=\,-\frac{\text{i}\delta_{nm}\,}{\,2\,}\frac{\leavevmode\nobreak\
\bar{\eta}^{\mu\alpha}\bar{\eta}^{\nu\beta}\\!+\\!\bar{\eta}^{\mu\beta}\bar{\eta}^{\nu\alpha}\\!-\\!\frac{2}{3}\bar{\eta}^{\mu\nu}\bar{\eta}^{\alpha\beta}\leavevmode\nobreak\
}{\,p^{2}+M_{n}^{2}\,}\,,$ (2.3w)

where
$\,\bar{\eta}^{\mu\nu}\\!=\eta^{\mu\nu}\\!+p^{\mu}p^{\nu}\\!/\\!M_{n}^{2}\,$.
As we will discuss in section 2.3, this just coincides with the massive
graviton propagator (2.3y) of the 4d Fierz-Pauli Lagrangian. Appendix B gives
more detailed discussions about the graviton propagator under the unitary
gauge.

###  Massless Limit and Absence of vDVZ Discontinuity in $R_{\xi}$ Gauge

In this subsection, we examine the massless limit
$\,M_{n}\\!\rightarrow\\!0\,$ under the $R_{\xi}$ gauge as constructed in
section 2.2. We will demonstrate that our $R_{\xi}$ propagator (2.3ua) of KK
gravitons has a smooth massless limit and is free from the conventional vDVZ
(van Dam-Veltman and Zakharov) discontinuity [23] of the Fierz-Pauli massive
gravity [18][19].

We recall the 4d Fierz-Pauli Lagrangian for massive graviton fields
$h^{\mu\nu}$ with mass $M$ [18][19]

$\mathcal{L}_{\mathrm{FP}}=\mbox{$\frac{\,{1}\,}{2}$}(\partial_{\mu}h)^{2}\\!-\\!\mbox{$\frac{\,{1}\,}{2}$}(\partial_{\alpha}h_{\mu\nu})^{2}-\partial_{\mu}h^{\mu\nu}\partial_{\nu}h+\partial_{\mu}h^{\mu\alpha}\partial^{\nu}h_{\nu\alpha}\\!+\\!\mbox{$\frac{\,{1}\,}{2}$}M^{2}(h^{2}-h_{\mu\nu}^{2})\,,$
(2.3x)

which has the following propagator

$\mathcal{D}^{\mu\nu\alpha\beta}_{\text{FP}}(p)\,=\,-\frac{\text{i}}{\,2\,}\frac{\leavevmode\nobreak\
\bar{\eta}^{\mu\alpha}\bar{\eta}^{\nu\beta}\\!+\\!\bar{\eta}^{\mu\beta}\bar{\eta}^{\nu\alpha}\\!-\\!\frac{2}{3}\bar{\eta}^{\mu\nu}\bar{\eta}^{\alpha\beta}\leavevmode\nobreak\
}{\,p^{2}+M^{2}\,}\,,$ (2.3y)

where
$\,\bar{\eta}^{\mu\nu}\\!=\eta^{\mu\nu}\\!+\\!p^{\mu}p^{\nu}\\!/\\!M^{2}$. In
comparison, for the 4d Einstein gravity under a harmonic gauge-fixing

$\mathcal{L}_{\mathrm{GF}}=\frac{1}{\,\xi\,}\\!\left(\partial_{\nu}h^{\mu\nu}-\mbox{$\frac{\,{1}\,}{2}$}\partial^{\mu}h\right)^{\\!2},$
(2.3z)

the massless graviton propagator is given by

$\displaystyle\mathcal{D}_{00}^{\mu\nu\alpha\beta}(p)$
$\displaystyle=-\frac{\text{i}}{\,2\,}\\!\left[\,\\!\frac{\,\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta}\,}{p^{2}}+(\xi\\!-\\!1)\frac{\,\eta^{\mu\alpha}p^{\nu}p^{\beta}\\!+\\!\eta^{\mu\beta}p^{\nu}p^{\alpha}\\!+\\!\eta^{\nu\alpha}p^{\mu}p^{\beta}\\!+\\!\eta^{\nu\beta}p^{\mu}p^{\alpha}\,}{p^{4}}\\!\,\right]$
$\displaystyle=-\frac{\text{i}}{\,2\,}\frac{\,\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta}\,}{p^{2}}\,,\qquad(\text{for}\leavevmode\nobreak\
\xi\\!=\\!1).$ (2.3aab)

This can also describe the propagator for the zero-mode gravitons in the KK
theory under the harmonic gauge-fixing (2.3z). We inspect the massless limit
$\,M\\!\\!\rightarrow\\!0\,$ of the massive graviton propagator (2.3y) of
Fierz-Pauli. In the massless limit, we note the following features of the
numerator in Eq.(2.3y): (i). the graviton propagator (2.3y) has singularities
from all the mass-dependent terms like $\,p^{\mu}p^{\nu}\\!/M^{2}\,$ inside
those $\bar{\eta}^{\mu\nu}$ ’s; (ii). the coefficient
$-\mbox{$\frac{\,{2}\,}{3}$}$ of the pure metric term
$\,{\eta}^{\mu\nu}{\eta}^{\alpha\beta}\,$ in the numerator does not match the
coefficient $-1$ of the corresponding term in the massless graviton propagator
(2.3aab), which is the so-called vDVZ discontinuity [23]. This discontinuity
is unique for dealing with the spin-2 massive gravitons à la Fierz-Pauli. We
note that the origin for such vDVZ discontinuity is due to the mismatch of
physical degrees of freedom between the massive gravitons in the Fierz-Pauli
gravity and the massless gravitons in GR: the massive graviton has 5 helicity
states $(\lambda=\pm 2,\,\pm 1,\,0)$, while the massless graviton only has two
($\lambda=\pm 2$), namely, $5\neq 2\,$.

For the singularities mentioned above, we note that similar singularity exists
for the spin-1 gauge fields in of the massive Yang-Mills theory (as well as
the Maxwell theory with a massive photon) when considering the massless limit.
To see this, we recall the propagator of the spin-1 massive gauge fields
$A^{a}_{\mu}$ :

$\mathcal{D}^{\mu\nu}(p)\,=\,-\text{i}\frac{\leavevmode\nobreak\
\eta^{\mu\nu}\\!+p^{\mu}p^{\nu}\\!/M^{2}\leavevmode\nobreak\
}{\,p^{2}\\!+\\!M^{2}\,}\,,$ (2.3ab)

where the term $\,p^{\mu}p^{\nu}\\!/M^{2}$ becomes singular in the massless
limit. The appearance of the singularities in the massive graviton propagator
and massive gauge boson propagator is also due to the mismatch of physical
degrees of freedom. In the case of massive spin-1 gauge field $A^{a\mu}$, it
has 3 helicity states $\lambda\\!=\\!\pm 1,0$, whereas the massless gauge
field only has 2 helicity states $\lambda\\!=\\!\pm 1\,$. This mismatch is the
cause of the singular term $\,p^{\mu}p^{\nu}\\!/M^{2}\,$ in the massless
limit. But in the $R_{\xi}$ gauge of the spontaneously broken gauge theories
with the conventional 4d Higgs mechanism [7] or with the geometric “Higgs”
mechanism under compactification [8], the propagator of a massive gauge boson
$A^{a\mu}$ (with mass $M$) can smoothly reduce to the massless gauge boson
propagator under the limit $\,M\\!\\!\rightarrow\\!0\,$ without causing any
singularity or discontinuity. This is because the massive gauge field
$A^{a\mu}$ (with $M\\!\neq\\!0$) has 3 physical degrees of freedom, and in the
massless limit $\,M\\!\rightarrow\\!0\,$ the physical states of $A^{a\mu}$
reduces to two transverse polarization states and its longitudinal component
disappears while the “eaten” would-be Goldstone boson becomes a physical
massless scalar. Hence, the physical degrees of freedom remain conserved,
$\,3=2+1\,$, before and after taking the massless limit.

Then, we examine the massless limit for the propagators of massive KK
gravitons. For this, we take the massless limit $M_{n}\\!\rightarrow\\!0$ for
the $R_{\xi}$ gauge propagator (2.3ua) and expand it up to the zeroth order of
$M_{n}$. We find that under the limit $\,M_{n}\\!\rightarrow\\!0\,$, the sum
of all the negative powers of $M_{n}$ vanishes, and the remaining nonzero part
takes the form:

$\displaystyle\mathcal{D}_{nm}^{\mu\nu\alpha\beta}(p)=$
$\displaystyle\,-\\!\frac{\,\text{i}\delta_{nm}\,}{\,2\,}\\!\left[\,\\!\frac{\,(\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta})\,}{p^{2}}\\!-\\!\frac{\,1\\!-\\!\xi_{n}\,}{p^{4}}\\!\left(\eta^{\mu\alpha}p^{\nu}p^{\beta}\\!+\\!\eta^{\mu\beta}p^{\nu}p^{\alpha}\\!+\\!\eta^{\nu\alpha}p^{\mu}p^{\beta}\right.\right.$
$\displaystyle\left.\left.\hskip
17.07164pt+\,\eta^{\nu\beta}p^{\mu}p^{\alpha}\\!-\\!2\eta^{\mu\nu}p^{\alpha}p^{\beta}\\!-\\!2\eta^{\alpha\beta}p^{\mu}p^{\nu}\right)\\!-4(1\\!-\\!\xi_{n})^{3}\frac{\,p^{\mu}p^{\nu}p^{\alpha}p^{\beta}\,}{p^{6}}\\!\,\right]\\!,$
(2.3aca) $\displaystyle=$
$\displaystyle-\\!\frac{\,\text{i}\delta_{nm}\,}{\,2\,}\frac{\,\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta}\,}{p^{2}}\,,\hskip
14.22636pt(\text{for}\leavevmode\nobreak\ \xi_{n}\\!=\\!1).$ (2.3acb)

From the above, we see that under the massless limit there is no singular
term, and the pure metric terms
$(\eta^{\mu\alpha}\eta^{\nu\beta}\\!+\\!\eta^{\mu\beta}\eta^{\nu\alpha}\\!-\\!\eta^{\mu\nu}\eta^{\alpha\beta})$
in the numerator agree with the massless graviton propagator (2.3aa) (the
$\xi_{n}\\!\\!=\\!1$ part) in the conventional 4d Einstein gravity. Hence, it
is impressive to see that in the massless limit the $R_{\xi}$ gauge propagator
(2.3ua) of massive KK gravitons is free from singularity and the vDVZ
discontinuity. Our $R_{\xi}$ gauge formulation of the KK theory has a well-
defined massless limit because the physical degrees of freedom are conserved
before and after taking the massless limit under the geometric “Higgs”
mechanism. A massive KK graviton $h^{\mu\nu}_{n}$ (having 5 helicity states
$\lambda\\!=\\!\pm 2,\pm 1,0$) acquires its mass via the geometric “Higgs”
mechanism (compactification) by absorbing (“eating”) the corresponding vector-
Goldstone component $\mathcal{A}^{\mu}_{n}$ (having 2 helicity states
$\lambda\\!=\\!\pm 1$) and scalar-Goldstone component $\phi_{n}$ (having
helicity $\lambda\\!=\\!0$) of the 5d graviton field $\,\hat{h}^{AB}$. In the
massless limit, $h^{\mu\nu}_{n}$ becomes massless (having only 2 helicities
$\lambda\\!=\\!\pm 2$), and the vector and scalar Goldstone bosons
$(\mathcal{A}^{\mu}_{n},\,\phi_{n})$ become massless physical states (having
$2\\!+\\!1$ helicities $\lambda\\!=\\!\pm 1,\,0$). Namely, each massive KK
graviton $h^{\mu\nu}_{n}$ has its 3 extra helicity states ($\lambda\\!=\\!\pm
1,0$) originate from those of the vector component $\mathcal{A}^{\mu}_{n}$
($\lambda\\!=\\!\pm 1$) and the scalar component $\phi_{n}$ ($\lambda=0$).
Hence, we see that the total physical degrees of freedom remain conserved
before and after taking the massless limit: $\,5=2+2+1\,$. This shows that the
compactified KK GR theory provides a consistent description of the massive
spin-2 gravitons and is free from the vDVZ discontinuity as well as
singularities under the massless limit, because the KK gravitons acquire their
masses via the geometric “Higgs” mechanism without explicitly breaking the
diffeomorphism invariance in the 5d bulk (except realizing the
compactification at the 5d boundaries).

Finally, we also note that the $\,\xi_{n}\\!\neq 1\,$ part of our KK graviton
propagator (2.3aca) differs from the conventional massless graviton propagator
(2.3aa) under the harmonic gauge-fixing (2.3z). This is because under the
massless limit our $R_{\xi}$ gauge-fixing term (2.3p) reduces to

$\mathcal{L}_{\mathrm{GF}}\leavevmode\nobreak\
\longrightarrow\leavevmode\nobreak\
-\sum_{n=0}^{\infty}\frac{1}{\,\xi_{n}}\left[\,\\!\partial_{\nu}h_{n}^{\mu\nu}-\left(\\!1-\frac{1}{2\xi_{n}}\right)\\!\partial^{\mu}h_{n}\\!\,\right]^{\\!2},$
(2.3ad)

where the coefficient $(1\\!-\\!\mbox{$\frac{\,{1}\,}{2\xi_{n}}$})$ differs
from that of the conventional harmonic gauge-fixing (2.3z) except for the case
$\,\xi_{n}\\!=\\!1\,$.

##  Formulation of Gravitational Equivalence Theorem and the Energy
Cancellation Mechanism

In the previous section, we have presented the $R_{\xi}$ gauge formulation of
the geometric “Higgs” mechanism for massive KK gravitons $h^{\mu\nu}_{n}$ and
the corresponding KK Goldstone bosons $\mathcal{A}_{n}^{\mu}\,(=h^{\mu
5}_{n})$ and $\phi_{n}\,(=h_{n}^{55})$, under which we can derive the
propagators.

In the subsection 3.1, we apply our $R_{\xi}$ gauge formulation in section 2.2
to establish a Gravitational Equivalence Theorem (GRET) for the 5d KK GR
theory, which quantitatively connects the high-energy scattering amplitude of
the (helicity-zero) longitudinal KK gravitons $h^{n}_{L}$ to that of the
corresponding KK Goldstone bosons $\phi_{n}$ . Then, in the subsection 3.2, we
will show that the GRET identity provides a theoretical mechanism which
guarantees the longitudinal KK graviton scattering amplitudes to have
nontrivial energy-cancellations, such as $\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$
for the four-particle amplitudes and $\,E^{2N+2}\\!\\!\rightarrow\\!E^{2}\,$
for the $N$-particle amplitudes. We derive a generalized naive power counting
method (à la Weinberg [30]) on the leading energy-dependence of the scattering
amplitudes, and apply this to analyze the leading energy-dependence of the
relevant amplitudes on both sides of the GRET identity (2.3o). With these, we
can demonstrate the above-mentioned nontrivial energy-cancellations in the
longitudinal KK graviton scattering amplitudes.

###  Formulation of Gravitational Equivalence Theorem

We first express the $R_{\xi}$ gauge-fixing term (2.3p) in the following form:

$\displaystyle\mathcal{L}_{\mathrm{GF}}$
$\displaystyle\,=\,-\sum_{n=0}^{\infty}\frac{1}{\,\xi_{n}}\\!\left(F_{n}^{A}\right)^{2}=-\sum_{n=0}^{\infty}\frac{1}{\,\xi_{n}}\\!\left[\,\\!\left(F_{n}^{\mu}\right)^{2}+\left(F_{n}^{5}\right)^{2}\\!\,\right],$
(2.3aa) $\displaystyle F_{n}^{\mu}$
$\displaystyle\,=\,\partial_{\nu}h^{\mu\nu}_{n}\\!-\\!\left(\\!1\\!-\\!\frac{1}{\,2\xi_{n}}\right)\\!\partial^{\mu}h_{n}+\xi_{n}M_{n}\mathcal{A}_{n}^{\mu}\,,$
(2.3ab) $\displaystyle F_{n}^{5}$
$\displaystyle\,=\,\frac{1}{\,2\,}\\!\left(M_{n}h_{n}\\!-3\xi_{n}M_{n}\phi_{n}\\!+2\partial_{\mu}\mathcal{A}_{n}^{\mu}\right)\\!.$
(2.3ac)

Accordingly, we can write down the Faddeev-Popov ghost term
$\mathcal{L}_{\mathrm{FP}}$ and the BRST (Becchi-Rouet-Stora-Tyutin) [31]
transformations. With these and using the method of Ref. [32] (cf. Appendix A
of the first paper therein), we can derive a Slavnov-Taylor-type identity

$\langle 0|\hat{T}F_{n_{1}}^{\mu_{1}}(x_{1})F_{n_{2}}^{\mu_{2}}(x_{2})\cdots
F_{m_{1}}^{5}(y_{1})F_{m_{2}}^{5}(y_{2})\cdots\Phi|0\rangle\,=\,0\,,$ (2.3b)

where $\Phi$ denotes any other on-shell physical fields after the LSZ
(Lehmann-Symanzik-Zimmermann) amputation. In the momentum space, the identity
(2.3b) takes the form:

$\langle 0|F_{n_{1}}^{\mu_{1}}(k_{1})F_{n_{2}}^{\mu_{2}}(k_{2})\cdots
F_{m_{1}}^{5}(p_{1})F_{m_{2}}^{5}(p_{2})\cdots\Phi|0\rangle\,=\,0\,,$ (2.3c)

where we will set each external momentum be on-shell (according to the mass of
the corresponding physical KK graviton $h^{\mu\nu}_{n}$):
$k_{j}^{2}=\\!-M_{n_{j}}^{2}$ and $p_{j}^{2}=\\!-M_{m_{j}}^{2}$ (with
$j=1,2,\cdots\,$). For the case of just one external line of $F_{n}^{\mu}$ or
$F_{n}^{5}$, we obtain the following identities of the scattering amplitudes:

$\mathcal{M}[F_{n}^{\mu}(k),\Phi]=0\,,\qquad\mathcal{M}[F_{n}^{5}(k),\Phi]=0\,,$
(2.3d)

where we have not yet imposed the LSZ amputation on the external line
$F_{n}^{\mu}$ or $F_{n}^{5}$.

Now, combining Eqs.(2.3ab) with (2.3ac), we can eliminate the vector Goldstone
field $\,\mathcal{A}_{n}^{\mu}\,$ and derive the expression:

$\partial_{\mu}F_{n}^{\mu}\\!-\xi_{n}M_{n}F_{n}^{5}\,=\,\partial_{\mu}\partial_{\nu}h_{n}^{\mu\nu}\\!-\mbox{$\frac{\,{1}\,}{2}$}\\!\left[\,\\!(2\\!-\xi_{n}^{-1})\partial^{2}\\!+\xi_{n}M_{n}^{2}\,\right]\\!h_{n}\\!+\\!\mbox{$\frac{\,{3}\,}{2}$}\xi_{n}^{2}M_{n}^{2}\phi_{n}\,.$
(2.3e)

Then, choosing the Feynman-’t Hooft gauge $\,\xi_{n}\\!=\\!1\,$ for simplicity
and imposing the on-shell condition $\,k^{2}\\!=\\!-M_{n}^{2}$ in momentum
space, we derive the following formula:

$\displaystyle\text{i}k_{\mu}F_{n}^{\mu}\\!+M_{n}F_{n}^{5}\,=\sqrt{\frac{3}{2}\,}M_{n}^{2}\mathcal{F}_{n}\,,$
(2.3fa) $\displaystyle\leavevmode\nobreak\
\mathcal{F}_{n}\,\equiv\,\sqrt{\frac{2}{3}\,}\frac{\,k_{\mu}k_{\nu}\,}{M_{n}^{2}}h_{n}^{\mu\nu}\\!+\sqrt{\frac{2}{3}\,}h_{n}\\!-\phi_{n},$
(2.3fb)

where we have made the rescaling (2.3s) for $\phi_{n}\,$ and defined the
external momentum $k^{\mu}$ to be incoming in Eq.(2.3fa). For the longitudinal
polarization tensor $\,\varepsilon_{L}^{\mu\nu}\,$ of the massive KK graviton,
we make the high-energy expansion under $E=k^{0}\\!\gg\\!M_{n}$ ,

$\displaystyle\varepsilon_{L}^{\mu\nu}=\frac{1}{\sqrt{6\,}\,}\\!\left(\epsilon_{+}^{\mu}\epsilon_{-}^{\nu}\\!+\epsilon_{-}^{\mu}\epsilon_{+}^{\nu}\\!+2\epsilon_{L}^{\mu}\epsilon_{L}^{\nu}\right)\equiv\sqrt{\frac{2}{3}\,}\frac{\,k^{\mu}k^{\nu}}{M_{n}^{2}}+\tilde{v}^{\mu\nu}=\sqrt{\frac{2}{3}\,}\varepsilon_{S}^{\mu\nu}+\tilde{v}^{\mu\nu},$
(2.3g)

where the longitudinal polarization vector
$\epsilon_{L}^{\mu}=(k^{0}/M_{n})(|\vec{k}|/k^{0},\,\vec{k}/|\vec{k}|)=\epsilon_{S}^{\mu}+v^{\mu}$
with $\,\epsilon_{S}^{\mu}=k^{\mu}/M_{n}\,$ and
$\,v^{\mu}\\!=\\!\mathcal{O}(M_{n}/E_{n})\,$. In the above, the scalar-
polarization tensor is defined to be
$\,\varepsilon_{S}^{\mu\nu}\\!=\\!\epsilon_{S}^{\mu}\epsilon_{S}^{\nu}={k^{\mu}k^{\nu}}\\!/{M_{n}^{2}}\,$
and the residual term has the energy scale
$\,\tilde{v}^{\mu\nu}\\!\\!=\\!\mathcal{O}(E^{0})$. Thus, we can further
express Eq.(2.3fb) as

$\displaystyle\mathcal{F}_{n}$
$\displaystyle=\tilde{h}_{n}^{S}-\overline{\Omega}_{n}=h_{n}^{L}-\Omega_{n}\,,$
(2.3ha) $\displaystyle\overline{\Omega}_{n}$
$\displaystyle=\phi_{n}\\!-\\!\tilde{h}_{n},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Omega_{n}=\overline{\Omega}_{n}\\!+\tilde{v}_{n}=\phi_{n}\\!+\widetilde{\Delta}_{n},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\widetilde{\Delta}_{n}\\!=\tilde{v}_{n}\\!-\tilde{h}_{n},\hskip 28.45274pt$
(2.3hb) $\displaystyle h_{n}^{S}$
$\displaystyle=\varepsilon^{S}_{\mu\nu}h^{\mu\nu}_{n}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\displaystyle\tilde{h}_{n}^{S}=\sqrt{\frac{2}{3}\,}h_{n}^{S},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{h}_{n}^{\mu\nu}=\sqrt{\frac{2}{3}\,}h_{n}^{\mu\nu},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{h}_{n}=\eta_{\mu\nu}\tilde{h}_{n}^{\mu\nu},$ (2.3hc) $\displaystyle
h_{n}^{L}$
$\displaystyle=\varepsilon^{L}_{\mu\nu}h^{\mu\nu}_{n}=\tilde{h}_{n}^{S}+\tilde{v}_{n},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{v}_{n}=\tilde{v}_{\mu\nu}h^{\mu\nu}_{n},$ (2.3hd)
$\displaystyle\epsilon_{S}^{\mu}$
$\displaystyle=\frac{\,k^{\mu}}{M_{n}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\varepsilon_{S}^{\mu\nu}=\epsilon_{S}^{\mu}\epsilon_{S}^{\nu}=\frac{\,k^{\mu}k^{\nu}\,}{M_{n}^{2}}\,.\hskip
42.67912pt$ (2.3he)

Then, using Eqs.(2.3d) and (2.3fa), we deduce

$\mathcal{M}[\mathcal{F}_{n}(k),\,\Phi]=0\,,$ (2.3i)

for one external $\mathcal{F}_{n}$ line. In the Feynman-’t Hooft gauge, all
the KK fields of level-$n$ have mass-pole $\,k^{2}\\!=\\!-M_{n}^{2}$ . Also,
due to our $R_{\xi}$ gauge-fixing (2.3aa) or (2.3p), all the KK fields have
diagonal propagators at tree level. So we can amputate the external line
$\mathcal{F}_{n}$ à la LSZ by multiplying the propagator-inverse
$(k^{2}\\!+\\!M_{n}^{2})\\!\rightarrow\\!0$ . Thus, the amplitude in Eq.(2.3i)
will take the same form except that the external line $\mathcal{F}_{n}$ is
amputated. After this, we can rewrite the identity (2.3i) as follows:

$\mathcal{M}[\tilde{h}_{n}^{S}(k),\,\Phi]\,=\,\mathcal{M}[\overline{\Omega}_{n}(k),\Phi]\,,$
(2.3j)

or, equivalently,

$\displaystyle\mathcal{M}[h_{n}^{L}(k),\Phi]$
$\displaystyle\,=\,\mathcal{M}[\overline{\Omega}_{n}(k),\Phi]+\mathcal{M}[\tilde{v}_{n}(k),\Phi]$
(2.3ka)
$\displaystyle\,=\,\mathcal{M}[\phi_{n}(k),\Phi]+\mathcal{M}[\widetilde{\Delta}_{n}(k),\Phi]\,,$
(2.3kb)

where $\,\overline{\Omega}_{n}=\phi_{n}\\!-\\!\tilde{h}_{n}\,$ and
$\,\widetilde{\Delta}_{n}\\!=\tilde{v}_{n}-\tilde{h}_{n}\,$.

For the $N$ external $\mathcal{F}_{n}$ lines, we thus deduce the following
identity with all $\mathcal{F}_{n}$ lines amputated and on-shell

$\mathcal{M}[\mathcal{F}_{n_{1}}\\!(k_{1}),\mathcal{F}_{n_{2}}\\!(k_{2}),\cdots\\!,\mathcal{F}_{n_{N}}\\!(k_{N}),\Phi]\,=\,0\,,$
(2.3l)

where $\,\mathcal{F}_{n}\\!=\\!h_{n}^{L}-\Omega_{n}\,$ and $\,\Phi\,$ denotes
any possible amputated on-shell external physical fields. Then, we derive an
identity for the scattering amplitude of $N$ longitudinally-polarized KK
gravitons:

$\mathcal{M}[h^{L}_{n_{1}}(k_{1}),\cdots\\!,h^{L}_{n_{N}}(k_{N}),\Phi]\,=\,\mathcal{M}[\Omega_{n_{1}}(k_{1}),\cdots\\!,\Omega_{n_{N}}(k_{N}),\Phi]\,.$
(2.3m)

Using the identity (2.3l), we can prove the GRET identity (2.3m) directly by
computing its righ-hand-side (RHS)

$\displaystyle\mathcal{M}[\Omega_{n_{1}}(k_{1}),\cdots\\!,\Omega_{n_{N}}(k_{N}),\Phi]$
$\displaystyle=\mathcal{M}[h_{n_{1}}^{L}\\!(k_{1})\\!-\\!\mathcal{F}_{n_{1}}\\!(k_{1}),\cdots\\!,h_{n_{N}}^{L}\\!(k_{N})\\!-\\!\mathcal{F}_{n_{N}}\\!(k_{N}),\Phi]$
$\displaystyle=\mathcal{M}[h_{n_{1}}^{L}(k_{1}),\cdots\\!,h_{n_{N}}^{L}(k_{N}),\Phi]\,.$
(2.3n)

In the last step of the above derivation, we have used the fact that an
amplitude including one (or more) external $\mathcal{F}_{n}$ line plus any
other external on-shell physical fields must vanish according to the identity
(2.3l).

Expanding the RHS of Eq.(2.3m), we can derive an identity that connects the
longitudinal KK graviton amplitude to the corresponding KK Goldstone boson
amplitude and will be called the GRET identity hereafter:

$\displaystyle\mathcal{M}[h^{L}_{n_{1}}(k_{1}),\cdots\\!,h^{L}_{n_{N}}(k_{N}),\Phi]\,=\,\mathcal{M}[\phi_{n_{1}}(k_{1}),\cdots\\!,\phi_{n_{N}}(k_{N}),\Phi]+\mathcal{M}_{\\!\Delta}\,,$
(2.3oa) $\displaystyle\mathcal{M}_{\\!\Delta}\,\equiv\,\sum_{1\leqslant
j\leqslant
N}\\!\\!\\!\\!\mathcal{M}[\\{\widetilde{\Delta}_{n_{j}},\phi_{n_{\\!j^{\prime}}}\\},\Phi]\,,$
(2.3ob)

where $\,\widetilde{\Delta}_{n}\\!=\tilde{v}_{n}\\!-\tilde{h}_{n}\,$ with the
notations $\,\tilde{v}_{n}\\!=\\!\tilde{v}_{\mu\nu}h_{n}^{\mu\nu}$ and
$\,\tilde{h}_{n}\\!=\\!\eta_{\mu\nu}\tilde{h}^{\mu\nu}_{n}\,$. The last term
$\,\mathcal{M}_{\\!\Delta}\,$ on the RHS of Eq.(2.3o) denotes the residual
term of GRET which is the sum of individual amplitudes where each amplitude
$\,\mathcal{M}[\\{\widetilde{\Delta}_{n_{j}},\phi_{n_{\\!j^{\prime}}}\\},\Phi]\,$
contains $\,j\,$ external states of $\,\widetilde{\Delta}_{n_{j}}$ with
$\,j\\!\in\\!\\{1,2,\cdots\\!,N\\}\,$ and
$\,{j^{\prime}}\,(=\\!N\\!\\!-\\!j)\,$ external states of
$\phi_{n_{\\!j^{\prime}}}$ . We note that an on-shell KK graviton has five
physical helicity states ($\lambda\\!=\\!\pm 2,\,\pm 1,\,0$) and their
polarization tensors, as given by Eq.(A.6) of Appendix A, are all traceless.
Hence, the external KK graviton $\,\tilde{h}_{n}\,$ is an unphysical state.
This means that the amplitudes containing one or more external
$\,\tilde{h}_{n}$ state(s) are unphysical amplitudes. This is why we arrange
all the $\,\tilde{h}_{n}$-related amplitudes on the RHS of the GRET identity
(2.3o) as part of the summed residual term $\,\mathcal{M}_{\\!\Delta}\,$.

Besides, we can further extend the above proof of the GRET identity (2.3o)
beyond tree-level and to be valid for all $R_{\xi}$ gauges by using the
gravitational BRST identities. Then, each external Goldstone boson state
$\phi_{n}$ in the amplitudes on the RHS of Eq.(2.3o) will receive a
multiplicative modification factor
$\,C_{\mathrm{mod}}\\!=\\!1+\mathcal{O}(\mathrm{loop})$, which is energy-
independent and similar to the case of the KK GAET formulation in the
compactified 5d YM theories [9] and in the 4d SM [32][33][34].555Our KK GRET
formulation is based on the quantized BRST symmetry and thus can be readily
extended up to loop levels. This means that our new mechanism of energy-
cancellation based on the KK GRET or KK GAET (cf. sections 4-5) will generally
hold up to loop orders, which differs from the recent literatures for the
explicit verifications of energy-cancellations in the tree-level KK graviton
amplitudes[14][15][16]. So, such energy-independent factor
$\,C_{\mathrm{mod}}$ does not affect the energy-power counting of the
(Goldstone-related) amplitudes of Eq.(2.3o) at loop levels. Since we focus on
the scattering amplitudes and the application of GRET at tree level for the
current study, we will present a generalized loop-level formulation elsewhere
[35].666The 4d ET in the presence of the Higgs-gravity interactions was
established in Refs. [36][37] which can be applied to studying cosmological
models (such as the Higgs inflation [37][38][39]) or to testing self-
interactions of weak gauge bosons and Higgs bosons [36][37][40].

Next, inspecting both sides of the GRET identity (2.3oa), we can readily make
naive power counting on the energy-dependence of the individual Feynman
diagrams for each scattering amplitude. For the four-particle scattering at
tree level, the longitudinal KK graviton amplitude on the LHS of the identity
(2.3oa) contains the contributions by individual diagrams via quartic contact
interactions or via exchanging KK (or zero-mode) gravitons. Since each exteral
longitudinal KK graviton has polarization tensor (2.3g) scales like
$\,\varepsilon_{L}^{\mu\nu}\\!\propto\\!k^{\mu}k^{\nu}/M_{n}^{2}\,$ in the
high energy limit, the contribution by each individual diagram behaves as
$\mathcal{O}(E^{10})$, where the energy-power $\,10=8+2$ contains the energy-
power of $\,8=2\\!\times\\!4\,$ arising from the four external longitudinal KK
gravitons and the energy-power 2 contributed by the internal couplings and
propagators. On the other hand, we can make naive power counting on the
energy-dependence of the individual diagrams in each amplitude of the RHS of
Eq.(2.3oa). Because the external states (either the KK Goldstone boson
$\phi_{n}$, or, the KK gravitons such as
$\,\tilde{v}_{n}\\!=\\!\tilde{v}_{\mu\nu}h_{n}^{\mu\nu}$ or
$\,\tilde{h}_{n}\\!=\\!\eta_{\mu\nu}\tilde{h}^{\mu\nu}_{n}\,$) in all such
amplitudes have no extra enhancement or suppression factor, we can readily
make naive power counting on their energy-dependence and deduce that they all
behave as $\mathcal{O}(E^{2})$ under the high energy expansion. Hence, the
GRET identity (2.3oa) provides a general mechanism for the energy-power
cancellation of $\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$ in the longitudinal KK
graviton scattering amplitudes at tree level.

We note that on the RHS of Eq.(2.3oa) the residual term
$\,\mathcal{M}_{\Delta}\,$ contains individual amplitude
$\,\mathcal{M}[\\{\widetilde{\Delta}_{n_{j}},\phi_{n_{\\!j^{\prime}}}\\},\Phi]\,$
with external states of the type
$\,\widetilde{\Delta}_{n}\\!=\tilde{v}_{n}-\tilde{h}_{n}\,$. The external
state $\,\tilde{v}_{n}\\!\\!=\\!\tilde{v}_{\mu\nu}h^{\mu\nu}_{n}\,$ is not
suppressed under high energy expansion due to
$\,\tilde{v}^{\mu\nu}\\!\\!=\\!\mathcal{O}(E^{0})\,$, and the external state
$\,\tilde{h}_{n}\\!=\\!\eta_{\mu\nu}\tilde{h}_{n}^{\mu\nu}\,$ is unsuppressed
either by any factor of $M_{n}/E$ . Thus, there is no apparent “equivalence”
between the (helicity-zero) longitudinal KK graviton $h_{L}^{n}$-amplitude and
the KK Goldstone $\phi_{n}$-amplitude in Eq.(2.3oa) under the high energy
expansion. This differs essentially from the conventional equivalence theorem
(ET) for the spin-1 massive gauge bosons in the SM and in the compactified KK
gauge theory, where the residual term is suppressed in the high energy limit
because of the corresponding residual factor
$\,v^{\mu}\\!=\\!\epsilon_{L}^{\mu}\\!-\\!\epsilon_{S}^{\mu}\\!=\\!\mathcal{O}(M_{n}/E_{n})\,$.
In fact, we observe that the GRET residual term $\,\mathcal{M}_{\Delta}\,$ in
Eq.(2.3ob) is given by the sum of amplitudes like
$\,\mathcal{M}[\\{\widetilde{\Delta}_{n},\phi_{n}\\},\Phi]\,$ with
$\,\widetilde{\Delta}_{n}\\!=\tilde{v}_{n}\\!-\tilde{h}_{n}\,$ containing both
the external fields $\,\tilde{v}_{n}\,$ and $\,\tilde{h}_{n}\,$, which do not
receive additional suppression under the high energy expansion. As we will
show in sections 4.2 and 5.4 for the four longitudinal KK graviton scattering,
the residual term $\,\mathcal{M}_{\Delta}\,$ as a sum of the
$\widetilde{\Delta}_{n}$-dependent individual amplitudes in Eq.(2.3o) has
$\mathcal{O}(E^{2})$ by the naive power counting and will be further cancelled
down to $\mathcal{O}(E^{0})$ in comparison with the leading Goldstone
$\phi_{n}$-amplitude of $\mathcal{O}(E^{2})$ under the high energy expansion.

With the above observations, we can express the GRET as follows:

$\mathcal{M}[h^{L}_{n_{1}}(k_{1}),\cdots\\!,h^{L}_{n_{N}}(k_{N}),\Phi]\,=\,\mathcal{M}[\phi_{n_{1}}(k_{1}),\cdots\\!,\phi_{n_{N}}(k_{N}),\Phi]\,+\,\mathcal{O}(\widetilde{\Delta}_{n})\,,$
(2.3p)

where the residual term $\,\mathcal{M}_{\Delta}\,$ is denoted by
$\mathcal{O}(\widetilde{\Delta}_{n})$ summing up all the remaining amplitudes
with at least one external state being $\,\widetilde{\Delta}_{n}\,$. We will
demonstrate later in sections 4.2 and 5.4 that the sum of residual terms
$\mathcal{O}(\widetilde{\Delta}_{n})$ is indeed suppressed by $M_{n}/E$
factors relative to the leading Goldstone amplitude on the RHS of the GRET
(2.3p) for the high energy scattering processes (with two or more external
longitudinal KK gravitons).

In principle, the GRET identity (2.3oa) and the GRET (2.3p) hold for any
number of external longitudinal KK graviton states, although in the above we
take the case of four longitudinal KK graviton scattering ($N\\!\\!=\\!4$) at
tree level as an important example for discussing the naive energy-power-
counting and energy cancellations. In the following, we will extend the above
naive power counting analysis on energy-dependence of the longitudinal KK
graviton amplitudes, the KK Goldstone amplitudes and the residual-term
amplitudes in the GRET identity (2.3oa) to the general case of
$\,N\\!\\!\geqslant\\!4$ and up to loop levels.

###  Energy Cancellation Mechanism for KK Graviton Scattering Amplitudes

We recall that Steven Weinberg777On July 23, 2021, we reposted an update
version of this paper to arXiv:2106.04568 in which we systematically presented
this generalized power counting method for the compactified KK gauge theories
and KK GR theories in section 3.2. On the following day we learnt the sad news
that Steven Weinberg passed away on July 23. One of us (HJH) wishes to express
his deep gratitude to Steve for his inspirations over the years (including the
discussion of his original work on the power counting analysis [30]),
especially during his times at UT Austin where his office was only a few doors
away from that of Steve. originally derived a power counting rule of energy
dependence for the ungauged nonlinear $\sigma$-model as a description of low
energy QCD interactions [30]. This power counting rule has two major
ingredients: (i). The total mass-dimension $D_{\mathbb{S}}$ of a scattering
$S$-matrix element $\,\mathbb{S}\,$ is determined by the number of external
states ($\mathcal{E}$) and the spacetime dimension, namely,
$\,D_{\mathbb{S}}=4-\mathcal{E}\,$, for 4d field theories. (ii). Consider that
the typical scattering energy $E$ is much larger than all the relevant mass-
poles in the internal propagators of the scattering amplitude $\,{\mathbb{S}}$
. Then the total mass-dimension $D_{C}$ of the $E$-independent coupling
constants contained in the amplitude $\mathbb{S}$ can be directly counted
according to the type of vertices therein. With these, one can deduce the
total energy-power dependence $D_{E}$ of the amplitude $\,\mathbb{S}\,$ as
$\,D_{E}=D_{\mathbb{S}}-D_{C}\,$. We note that the point (i) is fully general,
and the point (ii) holds for any field theory in which the particle masses are
much smaller than the scattering energy $E$ and the nontrivial energy-
dependence of the polarization tensors (vectors) for the possible
longitudinally polarized KK gravitons (gauge bosons) can be properly taken
into account. Hence, we can generalize Weinberg’s power counting rule to the
compactified 5d theories888Weinberg’s power counting rule was extended
previously [34][41] to the 4d gauge theories including the SM, the SM
effective theory (SMEFT), and the electroweak chiral Lagrangian. including KK
graviton (Goldstone) fields and/or KK gauge (Goldstone) fields, and study the
high energy scattering amplitudes of KK particles whose masses are much
smaller than the scattering energy $E$ .

Consider a scattering $S$-matrix element $\,\mathbb{S}\,$ having
$\,\mathcal{E}\,$ external states and $L$ loops ($L\\!\geqslant\\!0$). Thus,
the amplitude $\,\mathbb{S}\,$ has a mass-dimension:

$D_{\mathbb{S}}\,=\,4-\mathcal{E}\,,$ (2.3q)

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

$D_{C}\,=\,\sum_{j}\mathcal{V}_{j}\\!\left(4-d_{j}\\!-b_{j}\\!-\mbox{$\frac{\,{3}\,}{2}$}f_{j}\right)\\!.$
(2.3r)

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

$L=1+I-\mathcal{V}\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{j}\mathcal{V}_{j}b_{j}=2I_{B}+\mathcal{E}_{B}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \sum_{j}\mathcal{V}_{j}f_{j}=2I_{F}+\mathcal{E}_{F}\,,$
(2.3s)

where $\,\mathcal{V}=\sum_{j}\\!\mathcal{V}_{j}\,$ is the total number of
vertices in a given Feynman diagram. The amplitude $\,\mathbb{S}\,$ may
include $\,\mathcal{E}_{h_{L}}\\!$ external longitudinal KK graviton states.
Then, using Eqs.(2.3q)-(2.3s), we deduce the leading energy-power dependence
$\,D_{E}=D_{\mathbb{S}}-D_{C}\,$ of the high energy scattering amplitude
$\mathbb{S}$ as follows:

$D_{E}\leavevmode\nobreak\ =\leavevmode\nobreak\
2\mathcal{E}_{h_{L}}\\!\\!+(2L\\!+2)+\sum_{j}\mathcal{V}_{j}\\!\left(d_{j}\\!-2+\\!\mbox{$\frac{\,{1}\,}{2}$}f_{j}\right).$
(2.3t)

Then, we consider the pure 5d KK GR theory without involving any matter
fields. Thus, for the pure longitudinal KK graviton scattering amplitude with
$\,N\,$ external states
$\,\mathbb{S}\\!=\\!\mathcal{M}[h^{L}_{n_{1}},\cdots\\!,h^{L}_{n_{N}}]\,$, we
have $\,\mathcal{E}_{h_{L}}\\!\\!\\!=\\!N\,$ and $\,f_{j}\\!=\\!0\,$. Each
pure KK graviton vertex always contains two partial derivatives and thus
$\,d_{j}\\!=2\,$. For the loop level ($L\geqslant 1$), the amplitude may
contain gravitational ghost loop which involves graviton-ghost-antighost
vertex, but the number of partial derivatives $\,d_{j}$ should be no more than
two. This means that the leading energy dependence is always given by the
diagrams containing only the KK gravitons and/or zero-mode gravitons. Hence,
to count the leading energy dependence of the pure longitudinal KK graviton
scattering amplitudes, we can further derive the power counting formula (2.3t)
as

$D_{E}[Nh^{L}_{n}]\,=\,2(N\\!+\\!1)\\!+2L\,,$ (2.3u)

where the notation $\,[Nh^{L}_{n}]\,$ just denotes the $N$ external
longitudinal KK graviton states ($h^{L}_{n}$) whose KK indices can differ from
each other in an inelastic scattering amplitude. Similar notations, such as
$\,[N\phi_{n}]$ for $N$ external KK Goldstone states and so on, will be used
for other amplitudes.

Next, we consider the corresponding gravitational KK Goldstone boson
scattering amplitude $\mathcal{M}[\phi_{n_{1}},\cdots\\!,\phi_{n_{N}}]\,$ with
$N$ external states. Its leading energy dependence is given by the diagrams
containing $\phi_{n}$-$\phi_{m}$-$h^{\mu\nu}_{\ell}$ type of cubic vertices
and the pure (KK) graviton self-interaction vertices, where each of these
vertices includes two derivatives ($d_{j}\\!=2$). Hence, to count the leading
energy dependence, we can further derive the power counting formula (2.3t) as
follows:

$D_{E}[N\phi_{n}]\,=\,2+2L\,.$ (2.3v)

Here we also note that each external Goldstone boson state $\phi_{n}$ in the
amplitudes on the RHS of Eq.(2.3oa) will receive a multiplicative modification
factor $\,C_{\mathrm{mod}}\\!=\\!1\\!+\mathcal{O}(\mathrm{loop})$ at loop
level, which is energy-independent as mentioned earlier. Hence such loop
factor $\,C_{\mathrm{mod}}$ will not affect the energy power counting of the
Goldstone $\phi_{n}$-amplitudes. Comparing the energy power counting formulas
(2.3u) and (2.3v), we note that their difference arises from the leading
energy-dependence of the polarization tensors
$\,\varepsilon_{L}^{\mu\nu}\\!\\!\sim\\!k^{\mu}k^{\nu}/M_{n}^{2}\,$ for the
$N$ external longitudinal KK gravitons in the high energy scattering:

$D_{E}[Nh^{L}_{n}]-D_{E}[N\phi_{n}]\,=\,2N\,.$ (2.3w)

We further examine the leading $E$-power dependence of the individual
amplitudes in the residual term $\,\mathcal{M}_{\Delta}\,$ of the GRET (2.3o).
A typical leading amplitude can be
$\mathcal{M}[\tilde{v}_{n_{1}},\cdots\\!,\tilde{v}_{n_{N}}]\,$, in which all
the external states are KK gravitons contracted with the tensor
$\,\tilde{v}^{\mu\nu}\\!=\varepsilon_{L}^{\mu\nu}\\!\\!-\varepsilon_{S}^{\mu\nu}\\!=\mathcal{O}(E^{0})\,$,
such as $\,\tilde{v}_{n}\\!=\tilde{v}_{\mu\nu}h_{n}^{\mu\nu}\,$. Hence, we can
count the leading energy dependence of this amplitude in the same way as
Eq.(2.3u) for the longitudinal KK graviton amplitude
$\mathcal{M}[h^{L}_{n_{1}},\cdots\\!,h^{L}_{n_{N}}]\,$ except taking out the
energy-enhancement factor $E^{2}$ from each external longitudinal polarization
tensor $\varepsilon_{L}^{\mu\nu}$. Then, we deduce the following energy power
dependence of the leading residual amplitude
$\mathcal{M}[\tilde{v}_{n_{1}},\cdots\\!,\tilde{v}_{n_{N}}]\,$:

$D_{E}[N\tilde{v}_{n}]\,=\,2+2L\,,$ (2.3x)

which gives the same energy power dependence as Eq.(2.3v) for the leading
scattering amplitude of $N$ KK Goldstone bosons. We will establish a further
energy cancellation in the residual term $\mathcal{M}_{\Delta}$ in section 5.4
based upon the double-copy construction.

Applying the leading energy-power counting results (2.3u)-(2.3x) to both sides
of the GRET identity (2.3oa), we thus establish an energy cancellation by
$\,E^{2N}\,$ in a scattering amplitude of $N$ longitudinal KK gravitons
$\mathcal{M}[h^{L}_{n_{1}},\cdots\\!,h^{L}_{n_{N}}]\,$. For the case of four
longitudinal KK graviton scattering amplitudes ($N\\!\\!=\\!4$) at tree level
($L\\!=\\!0$), we can deduce the energy power cancellation
$\,E^{10}\\!\\!\rightarrow\\!E^{2}\,$, which reduces the energy powers by
$\,(10-2)\\!=8\,$, as we mentioned earlier. For another case of four KK
graviton scattering amplitudes containing two external longitudinal KK
gravitons and two external transverse KK gravitons
($\,\mathcal{E}_{h_{L}}\\!\\!=\\!2\,$), we have the $E$-power counting
$\,D_{E}[2h_{L}^{n}\\!+2h_{T}^{n}]=6+2L\,$. For the corresponding KK Goldstone
amplitudes, we have energy counting
$\,D_{E}[2\phi_{n}+\\!2h_{T}^{n}]=2+\\!2L\,$. The leading residual term
contains the amplitudes such as
$\mathcal{M}[\tilde{v}_{n_{1}},\tilde{v}_{n_{2}},h^{T}_{n_{3}},h^{T}_{n_{4}}]\,$,
which has the same energy-power dependence as the residual term amplitude with
all external states being $\tilde{v}_{n}$’s [cf. Eq.(2.3x)]. Namely, we can
deduce $\,D_{E}[2\tilde{v}_{n}\\!+\\!2h_{T}^{n}]=2+2L\,$. Hence, from the GRET
identity (2.3oa), we deduce that the KK graviton amplitude
$\mathcal{M}[h^{L}_{n_{1}},h^{L}_{n_{2}},h^{T}_{n_{3}},h^{T}_{n_{4}}]\,$ has
an energy cancellation down by a factor of $\,E^{4}\,$. This energy mechanism
holds not only for the tree level, but also for the loop levels
($L\\!\geqslant\\!1$) since, as we noted earlier, the loop-induced
multiplicative modification factor
$\,C_{\mathrm{mod}}\\!=\\!1\\!+\mathcal{O}(\mathrm{loop})$ associated with
each external KK Goldstone state is energy-independent and thus does not
affect the naive energy-power counting on the RHS of Eq.(2.3o).

In the rest of this subsection, we consider the energy power counting in the
compactified 5d KK YM theory (YM5) under $S^{1}/\mathbb{Z}_{2}$ [8]. For a
scattering amplitude containing $\,\mathcal{E}_{\\!A_{L}^{n}}\\!$ external
longitudinal KK gauge bosons $A_{L}^{an}$ and $\,\mathcal{E}_{v}\,$ external
KK gauge bosons $\,v_{n}^{a}\\!=\\!v_{\mu}A^{a\mu}\,$ (with
$v^{\mu}\\!=\\!\epsilon_{L}^{\mu}\\!-\epsilon_{S}^{\mu}\,$), we can derive the
following leading energy dependence
$\,D_{E}\\!=\\!D_{\mathbb{S}}\\!-\\!D_{C}\,$ from Eqs.(2.3q)-(2.3s),

$D_{E}\,=\,\mathcal{E}_{\\!A_{L}^{n}}\\!-\mathcal{E}_{v}\\!+(2L\\!+2)+\sum_{j}\mathcal{V}_{j}\\!\left(d_{j}\\!-2+\\!\mbox{$\frac{\,{1}\,}{2}$}f_{j}\right)\\!.$
(2.3y)

Inspecting the interaction Lagrangian of the zero-modes and KK-modes of gauge
bosons, we note that it contains only cubic and quartic vertices. Some of the
cubic vertices contain one partial derivative and others do not (including all
quartic gauge boson vertices). For notational convenience, we denote the gauge
fields $\,V_{0}\\!=\\!A_{0}^{a\mu}$, $V_{n}\\!=\\!A_{n}^{a\mu}$, and
$\,\widetilde{V}_{n}\\!=\\!A_{n}^{a5}$. After the BRST quantization, the ghost
term contains the cubic interactions between KK ghost-antighost
$(c_{n}^{a},\,\bar{c}_{m}^{b})$ and KK gauge bosons with one partial
derivative in each vertex [9]. Thus, the cubic vertices with one partial
derivative have the types of
$\,(V_{0}V_{n}V_{n},\,V_{n}V_{m}V_{\ell},\,c_{n}\bar{c}_{m}V_{\ell})$ and
$(V_{0}\widetilde{V}_{n}\widetilde{V}_{n},\,V_{n}\widetilde{V}_{m}\widetilde{V}_{\ell})$.
Hence, we have

$\displaystyle\sum_{j}\mathcal{V}_{j}d_{j}\,=\,\mathcal{V}_{d}\,,$ (2.3za)
$\displaystyle\mathcal{V}_{d}\,=\,\mathcal{V}_{3}(V_{0}V_{n}V_{n})+\mathcal{V}_{3}(V_{n}V_{m}V_{\ell})+\mathcal{V}_{3}(c_{n}\bar{c}_{m}V_{\ell})+\mathcal{V}_{3}(V_{0}\widetilde{V}_{n}\widetilde{V}_{n})+\mathcal{V}_{3}(V_{n}\widetilde{V}_{m}\widetilde{V}_{\ell})\,,\hskip
14.22636pt$ (2.3zb)

where $\,\mathcal{V}_{d}$ denotes the number of all cubic vertices including
one partial derivative and $\,\mathcal{V}_{3}(XYZ)$ denotes the number of
cubic vertices of type $XYZ$. For the YM5 theory, we further have the
following relations:

$\displaystyle\mathcal{V}\,=\,$ $\displaystyle\leavevmode\nobreak\
\sum_{j}\mathcal{V}_{j}\,=\,\mathcal{V}_{3}+\mathcal{V}_{4}\,,$ (2.3aaa)
$\displaystyle\mathcal{V}_{3}\,=\,$ $\displaystyle\leavevmode\nobreak\
\mathcal{V}_{d}+\mathcal{V}_{F}+\overline{\mathcal{V}}_{3}\,,$ (2.3aab)
$\displaystyle\mathcal{V}_{F}\,=\,$ $\displaystyle\leavevmode\nobreak\
\mathcal{V}_{3}(V_{0}f_{n}\bar{f}_{n})+\mathcal{V}_{3}(V_{n}f_{m}\bar{f}_{\ell})+\mathcal{V}_{3}(\widetilde{V}_{n}f_{m}\bar{f}_{\ell})\,,$
(2.3aac) $\displaystyle\overline{\mathcal{V}}_{3}\,=\,$
$\displaystyle\leavevmode\nobreak\
\mathcal{V}_{3}(V_{0}V_{n}\widetilde{V}_{n})+\mathcal{V}_{3}(V_{n}V_{m}\widetilde{V}_{\ell})+\mathcal{V}_{3}(c_{n}\bar{c}_{m}\widetilde{V}_{\ell})\,,$
(2.3aad) $\displaystyle\mathcal{V}_{4}\,=\,$
$\displaystyle\leavevmode\nobreak\
\mathcal{V}_{4}(V_{0}V_{0}V_{n}V_{n})+\mathcal{V}_{4}(V_{0}V_{n}V_{m}V_{\ell})+\mathcal{V}_{4}(V_{n}V_{m}V_{k}V_{\ell})$
$\displaystyle\leavevmode\nobreak\
+\mathcal{V}_{4}(V_{0}V_{0}\widetilde{V}_{n}\widetilde{V}_{n})+\mathcal{V}_{4}(V_{0}V_{n}\widetilde{V}_{m}\widetilde{V}_{\ell})+\mathcal{V}_{4}(V_{n}V_{m}\widetilde{V}_{k}\widetilde{V}_{\ell})\,,$
(2.3aae)

where the possible fermions and their KK states are included although they are
not needed for analyzing the pure KK gauge theory in the present work. Using
Eqs.(2.3z)-(2.3aa), we further derive the leading energy-power dependence
(2.3y) as follows:

$D_{E}\,=\leavevmode\nobreak\
\mathcal{E}_{\\!A_{L}^{n}}\\!-\mathcal{E}_{v}\\!+(2L\\!+\\!2)-(\mathcal{V}_{d}\\!+\\!\mathcal{V}_{F}\\!+2\overline{\mathcal{V}}_{3}\\!+2\mathcal{V}_{4})\,.$
(2.3ab)

Then, using the general relation $\,L\\!=I\\!+1\\!-\mathcal{V}\,$ given by
Eq.(2.3s) and the following relation of the YM5 theory

$2I+\mathcal{E}\,=\,3\mathcal{V}_{3}+4\mathcal{V}_{4}\,,$ (2.3ac)

we can express the leading energey dependence (2.3ab) as

$D_{E}\,=\,(4-\mathcal{E})+(\mathcal{E}_{\\!A_{L}^{n}}\\!-\mathcal{E}_{v})-\overline{\mathcal{V}}_{3}\,,$
(2.3ad)

where $\,\mathcal{E}$ stands for the total number of the external states and
$\,\overline{\mathcal{V}}_{3}\,$ denotes the number of cubic vertices
containing no partial derivative. In Eq.(2.3ad), $\mathcal{E}_{v}$ denotes of
number of external KK gauge bosons contracted with the vector
$\,v^{\mu}\\!=\epsilon_{L}^{\mu}\\!-\epsilon_{S}^{\mu}=\mathcal{O}(M_{n}/E)\,$.
So each external state $\,v_{n}^{a}=v_{\mu}A_{n}^{a\mu}\,$ contributes an
energy suppression factor $\,E^{-1}$ . The naive power counting formula
(2.3ad) does not depend on the loop number $L$ and takes similar form to that
of the SM case [41], because the structure of each individual vertex of the KK
YM5 theory is similar to that of the SM while the non-renormalizability nature
of the KK YM5 theory is reflected by its infinite tower of KK states.

Inspecting Eq.(2.3ad), we note that for the pure longitudinal KK gauge boson
scattering amplitude with
$\,\mathcal{E}\\!=\mathcal{E}_{\\!A_{L}^{n}}\\!=\\!N(\geqslant\\!4)\,$ and
$\,\mathcal{E}_{v}\\!=\\!0\,$, the leading energy dependence is given by

$D_{E}[N\\!A_{L}^{n}]\,=\,4\,,$ (2.3ae)

which corresponds to $\,\overline{\mathcal{V}}_{3}\\!\\!=\\!0\,$. This means
that the leading energy-power dependence of the pure longitudinal KK gauge
boson scattering is always given by the diagrams containing only cubic
derivative gauge vertices and/or quartic gauge vertices. We stress that the
leading energy dependence $\,D_{E}=4\,$ does not depend on the number of
external longitudinal KK gauge bosons ($\,\mathcal{E}_{\\!A_{L}^{n}}\\!=\\!N$
). The case of $\,N\\!=4\,$ scattering amplitudes was studied before [8].
Then, we consider the scattering amplitudes of pure KK Goldstone bosons
($A_{n}^{a5}$) with $\,\mathcal{E}\\!=\mathcal{E}_{\\!A_{5}^{n}}\\!=\\!N\,$
external $A_{n}^{a5}$ states. This also means
$\,\mathcal{E}_{\\!A_{L}^{n}}\\!\\!=0\,$ and $\,\mathcal{E}_{v}\\!=0\,$. Thus,
using Eq.(2.3ad), we deduce the leading energy dependence of $N$ KK Goldstone
boson scattering amplitude as

$D_{E}[N\\!A_{5}^{n}]\,=\,4-N-\overline{\mathcal{V}}_{3}^{\min}\,,$ (2.3af)

where the number of the external KK Goldstone states $\,N\\!\\!\geqslant\\!4$
and the involved minimal number of non-derivative cubic vertices
$\,\overline{\mathcal{V}}_{3}^{\min}=0\,(1)$ for $\,N\\!=\,$even (odd).

It was established [8][9] that the longitudinal KK gauge boson scattering
amplitude and the corresponding KK Goldstone boson scattering amplitude are
connected by the KK equivalence theorem for gauge theory (KK GAET) under the
high energy expansion:

$\displaystyle\mathcal{T}[A_{L}^{a_{1}n_{1}},\cdots\\!,A_{L}^{a_{N}n_{N}},\Phi]\leavevmode\nobreak\
=\leavevmode\nobreak\
C_{\mathrm{mod}}\mathcal{T}[A_{5}^{a_{1}n_{1}},\cdots\\!,A_{5}^{a_{N}n_{N}},\Phi]\,+\,\mathcal{T}_{v}\,,$
(2.3aga)
$\displaystyle\mathcal{T}_{v}\,=\,\sum_{\ell=1}^{N}\\!C_{\mathrm{mod}}^{\prime}\mathcal{T}[v^{a_{1}n_{1}},\cdots\\!,v^{a_{\ell}n_{\ell}},A_{5}^{a_{\ell+1}n_{\ell+1}},\cdots\\!,A_{5}^{a_{N}n_{N}},\Phi]\,=\,\mathcal{O}(M_{n}/E)\,,$
(2.3agb)

where $\,\Phi\,$ denotes any other external physical state(s). The
modification factors
$\,C_{\mathrm{mod}},C_{\mathrm{mod}}^{\prime}\\!=1\\!+\mathcal{O}(\mathrm{loop})\,$
are energy-independent constants and do not affect the energy power counting,
which are generated at loop level [9][34] and are not needed for the tree-
level analysis in the current study.

Then, we consider the scattering amplitudes of $N$ longitudinal KK gauge
bosons and of the corresponding $N$ KK Goldstone bosons. Their leading energy
powers are given by Eqs.(2.3ae) and (2.3af). Thus, we deduce the following
difference between their leading energy powers:

$D_{E}[N\\!A^{L}_{n}]-D_{E}[N\\!A_{5}^{n}]\,=\,N+\overline{\mathcal{V}}_{3}^{\min}\,,$
(2.3ah)

where $\,\overline{\mathcal{V}}_{3}^{\min}$ denotes the involved minimal
number of non-derivative cubic vertices in the KK Goldstone amplitude and
$\,\overline{\mathcal{V}}_{3}^{\min}=0\,(1)$ for $\,N\\!=\,$even (odd) . Next,
we make naive energy counting on the residual term $\,\mathcal{T}_{v}\,$ of
the KK GAET (2.3ag). To extract the leading energy dependence, we start with
the pure KK Goldstone amplitude
$\,\mathcal{T}[A_{5}^{a_{1}n_{1}},\cdots\\!,A_{5}^{a_{N}n_{N}}]\,$ and replace
one external KK Goldstone state (say, $A_{5}^{a_{1}n_{1}}$) by the KK gauge
boson contracted with the $\,v^{\mu}$ factor
($v^{\mu}\\!A_{\mu}^{a_{1}n_{1}}\\!=v^{a_{1}n_{1}}$). For the case of
$\,N\\!=\mathrm{even}$, this means to replace a derivative vertex by a non-
derivative vertex and add the factor $v^{\mu}$, so the leading energy
dependence $D_{E}$ will be reduced by $E^{-2}$. For the case of
$\,N\\!=\mathrm{odd}$, this means to replace a non-derivative cubic vertex by
a derivative cubic vertex and add a $v^{\mu}$ factor. So the leading energy
dependence $D_{E}$ will not change. Thus, we conclude that the leading energy
dependence of the residual term (2.3agb) is given by

$\displaystyle D_{E}[\mathcal{T}_{v}]\,$
$\displaystyle=\,2-N\,,\qquad(\text{for}\leavevmode\nobreak\
N\\!=\mathrm{even}),$ (2.3aia) $\displaystyle D_{E}[\mathcal{T}_{v}]\,$
$\displaystyle=\,3-N\,,\qquad(\text{for}\leavevmode\nobreak\
N\\!=\mathrm{odd}).$ (2.3aib)

Comparing this with the leading energy-power counting (2.3af) of the $N$ KK
Goldstone boson amplitudes in the high energy scattering, we deduce that for
the case of $\,N\\!=\\!\mathrm{even}$ the residual term (2.3agb) is suppressed
by $M_{n}^{2}/E^{2}\,$ factor relative to the leading KK Goldstone amplitude
on the RHS of the KK GAET (2.3aga) and thus can be ignored, while for the case
of $\,N\\!=\mathrm{odd}$ the residual term (2.3agb) has the same leading
energy dependence as that of the leading KK Goldstone amplitude. In either
case, the KK GAET (2.3ag) guarantees that the leading energy dependence
$\,E^{4}\,$ of the pure longitudinal KK gauge boson amplitudes in Eq.(2.3ae)
has to be cancelled down to the leading energy dependence of the corresponding
KK Goldstone amplitudes in Eq.(2.3af). This energy cancellation shows that
even though the $N$-particle longitudinal KK gauge boson scattering amplitudes
have superficial leading energy dependence $E^{4}$ as contributed by
individual Feynman diagrams, these must be cancelled down by an energy factor
$\,E^{\delta D_{\\!E}}\,$ to match the leading energy dependence of the
corresponding KK Goldstone boson amplitudes, where the energy power factor
changes by

$\delta D_{E}\,=\,N+\frac{\,1\\!-\\!(-1)^{N}\,}{2}\,.$ (2.3aj)

This energy cancellation of $\delta D_{E}$ coincides with the above formula
(2.3ah). For the case of four longitudinal KK gauge boson scattering
amplitudes ($N=4$), it was proven [8] that the leading energy cancellation
$\,E^{4}\\!\rightarrow\\!E^{0}\,$ is guaranteed by the KK GAET to match the
leading energy dependence of the corresponding KK Goldstone boson amplitudes.
This fully agrees with the above general analysis for the $N$-particle
scattering amplitudes. In the following, we will focus on the four-particle KK
amplitudes ($N\\!=4$) for the explicit analysis of the GRET in section 4 and
for the double-copy construction in section 5. We will pursue the analysis of
$N\\!>\\!4\,$ case in future works [35].

##  Structure of KK Graviton Scattering Amplitudes from Gravitational
Equivalence Theorem

The compactified five-dimensional Yang-Mills theory under orbifold
$S^{1}/\mathbb{Z}_{2}$ generates a tower of massive gauge bosons via KK
construction. The KK gauge boson mass-generation can be formulated by the
geometric “Higgs” mechanism in a generic $R_{\xi}$ gauge [8], where each
massive longitudinal KK gauge boson $A^{a\mu}_{n}$ acquires its mass by
absorbing the corresponding KK-state Goldstone $A^{a5}_{n}$ from the fifth
component of the 5d gauge field. Ref. [8] has established the KK GAET which
states that each on-shell scattering amplitude of the longitudinal KK gauge
bosons ($A^{aL}_{n}$) equals the amplitude of the corresponding Goldstone
bosons ($A^{a5}_{n}$) down to $\mathcal{O}(E^{0})$ under the high energy
expansion,

$\mathcal{T}\\!\left[\,\\!A^{aL}_{n_{1}}A^{bL}_{n_{2}}\\!\rightarrow\\!A^{cL}_{n_{3}}A^{dL}_{n_{4}}\,\right]\,=\,\mathcal{T}\\!\left[\,\\!A^{a5}_{n_{1}}A^{b5}_{n_{2}}\\!\rightarrow\\!A^{c5}_{n_{3}}A^{d5}_{n_{4}}\,\right]+\mathcal{O}(M_{n_{i}}^{2}/E^{2})\,.$
(2.3a)

This formulation was extended to gauge theories in deconstructed extra
dimension [10] and to the realistic compactified 5d standard model [11].

In this section, we will systematically compute the $2\\!\rightarrow\\!2$
scattering amplitudes of gravitational KK Goldstone bosons for the first time.
Then, we will explicitly demonstrate the validity of the GRET by comparing our
gravitational KK Goldstone amplitudes with the corresponding helicity-zero KK
graviton amplitudes obtained in [15]. For the case of $2\\!\rightarrow\\!2$
scattering, we first deduce the GRET identity from Eq.(2.3oa):

$\mathcal{M}\\!\left[\,\\!h^{L}_{n_{1}}h^{L}_{n_{2}}\\!\\!\rightarrow\\!h^{L}_{n_{3}}h^{L}_{n_{4}}\\!\,\right]\,=\,\mathcal{M}\\!\left[\,\\!\Omega_{n_{1}}\Omega_{n_{2}}\\!\\!\rightarrow\\!\Omega_{n_{3}}\Omega_{n_{4}}\\!\,\right]\,,$
(2.3b)

where $\,\Omega_{n}\\!=\phi_{n}\\!+\widetilde{\Delta}_{n}$ and
$\,\widetilde{\Delta}_{n}\\!=\tilde{v}_{n}\\!-\tilde{h}_{n}\,$. Furthermore,
according to Eq.(2.3p), we reexpress our four-point GRET identity (2.3b) as

$\mathcal{M}\\!\left[\,\\!h^{L}_{n_{1}}h^{L}_{n_{2}}\\!\\!\rightarrow\\!h^{L}_{n_{3}}h^{L}_{n_{4}}\\!\,\right]\,=\,\mathcal{M}\\!\left[\,\\!\phi_{n_{1}}\phi_{n_{2}}\\!\\!\rightarrow\\!\phi_{n_{3}}\phi_{n_{4}}\\!\,\right]\,+\,\mathcal{O}(\widetilde{\Delta}_{n})\,.$
(2.3c)

As we will show in the following section 4.2, the leading gravitational KK
Goldstone amplitude on the RHS of the GRET (2.3c) is of $\mathcal{O}(E^{2})$
and equals the corresponding leading longitudinal KK graviton amplitude on the
LHS of Eq.(2.3c). However, it is highly nontrivial to demonstrate that the
full residual term
$\,\mathcal{O}(\widetilde{\Delta}_{n})\\!=\\!\mathcal{O}(E^{0})\,$ actually
holds and thus can be neglected relative to the leading gravitational KK
Goldstone amplitude on the RHS of the GRET (2.3c). This is because the naive
power counting shows each individual amplitude in the residual term
$\,\mathcal{O}(\widetilde{\Delta}_{n})\,$ is of $\,\mathcal{O}(E^{2})\,$. This
can be understood by noting that the tensor
$\,\tilde{v}^{\mu\nu}\\!\\!=\\!\mathcal{O}(E^{0})$ and thus the external state
$\,\tilde{v}_{n}=\tilde{v}_{\mu\nu}h_{n}^{\mu\nu}\,$ is unsuppressed under
high energy expansion. The same is true for the external state
$\,\tilde{h}_{n}=\eta_{\mu\nu}\tilde{h}_{n}^{\mu\nu}\,$ which has no extra
suppression factor. Thus, by naive power counting of energy, each individual
residual term
$\,\mathcal{O}(\widetilde{\Delta}_{n})\\!=\\!\mathcal{O}(E^{2})\,$ which has
the same energy-dependence as the leading Goldstone amplitude and is not
superficially suppressed. This is an essential difference from the KK GAET [8]
of the compactified 5d KK gauge theories [8], where the residual term is
suppressed by the vector
$\,v^{\mu}\\!=\epsilon^{\mu}_{L}\\!-\epsilon^{\mu}_{S}=\mathcal{O}(M_{n}/E)\,$
and thus is of $\,\mathcal{O}(M_{n}^{2}/E^{2})$ for the case of four-particle
scattering process as shown in Eq.(2.3a).999The residual term of
$\mathcal{O}(v_{n})$ is defined as the difference between the longitudinal
gauge boson amplitude and the corresponding Goldstone amplitude. In the 5d KK
GAET for spin-1 KK gauge bosons [8], the residual term has the size of
$\mathcal{O}(M_{n}^{2}/E^{2})$, which is similar to that of the conventional
ET of 4d gauge theories [42]. We will demonstrate this additional energy-
cancellation of $\,E^{2}\\!\rightarrow\\!E^{0}\,$ in the residual term
$\,\mathcal{O}(\widetilde{\Delta}_{n})\,$ in section 4.2 by the explicit
calculations and in section 5.4 by the double-copy construction from the KK
GAET of 5d YM theory.

###  GRET for the 5d Gravitational Scalar QED

In this subsection, we first consider the 5d gravitational scalar QED (GSQED5)
compactified under $S^{1}/\mathbb{Z}_{2}$, as an example to explicitly test
the GRET. This will provide important insights for our general formulation of
the GRET and double-copy reconstruction analysis in section 5.

In this GSQED5, both graviton and scalar fields live in the 5d bulk.
Therefore, we can write down the 5d action for the matter part, including a
general gauge-fixing term for the gauge field,

$S_{\mathrm{m}}=\int\\!\text{d}^{5}x\sqrt{\\!-\hat{g}\,}\left\\{-\frac{1}{4}\hat{g}^{MP}\hat{g}^{NQ}\hat{F}_{MN}\hat{F}_{PQ}-\frac{1}{2\zeta}(\partial_{M}\hat{A}^{M})^{2}+|D_{M}\hat{\SS}|^{2}+m_{0}^{2}|\hat{\SS}|^{2}\right\\},$
(2.3d)

where $\,\hat{F}_{MN}=\partial_{M}\hat{A}_{N}-\partial_{N}\hat{A}_{M}$ and
$\,D_{M}=\partial_{M}+\text{i}\hat{e}\hat{A}_{M}$.101010In Eq.(2.3d), we have
imposed a minimal gauge-fixing term for photon field with gauge-fixing
function $\,(\partial_{M}\hat{A}^{M})\,$. One could optionally choose the
usual covariant gauge-fixing function for photon $\,(\nabla_{M}\hat{A}^{M})\,$
[43], which contains additional interaction vertices proportional to $1/\zeta$
and will not affect physics. We have explicitly verified that for the
scattering amplitudes of relevant physical processes, the sum of all
$\zeta$-dependent contributions vanishes at tree level, as expected. From
this, we derive the action of the graviton-matter interactions:

$\displaystyle S_{\mathrm{int}}$
$\displaystyle\,=\,-\frac{\hat{\kappa}}{2}\int\\!\text{d}^{5}x\\!\left(\hat{h}^{MN}\hat{T}_{MN}\right)$
$\displaystyle\,=\,-\frac{\hat{\kappa}}{4}\int\\!\text{d}^{5}x\\!\left[\,2\hat{h}^{\mu\nu}\hat{T}_{\mu\nu}\\!+4\hat{h}^{\mu
5}\hat{T}_{\mu
5}-\hat{h}^{55}\,(\tensor{\hat{T}}{{}^{\mu}_{\mu}}\\!\\!-2\hat{T}_{55})\\!\,\right]\\!,$
(2.3e)

where the 5d energy-momentum tensor is defined as

$\hat{T}_{MN}\,=\,\left.\frac{2}{\sqrt{-\hat{g}\,}\,}\frac{\delta
S_{\mathrm{m}}}{\,\delta\hat{g}^{MN}\,}\right|_{\hat{g}\rightarrow\hat{\eta}}\,.$
(2.3f)

Therefore, we can derive the energy-momentum tensors for both the photon field
and scalar field as follows:

$\displaystyle\hat{T}_{MN}^{A}\,=\,$ $\displaystyle\
\frac{\hat{\eta}_{MN}}{4}\hat{F}_{PQ}^{2}+\hat{F}_{M}^{\ \
P}\hat{F}_{PN}+\frac{\hat{\eta}_{MN}}{2\zeta}(\partial^{P}\\!\hat{A}_{P})^{2}\,,$
(2.3ga) $\displaystyle\hat{T}_{MN}^{\mathcal{S}}\,=\,$ $\displaystyle\
(D_{M}\hat{\mathcal{S}})^{*}D_{N}\hat{\mathcal{S}}+(D_{N}\hat{\mathcal{S}})^{*}D_{M}\hat{\mathcal{S}}-\hat{\eta}_{MN}(|D_{P}\hat{\mathcal{S}}|^{2}+m_{0}^{2}|\hat{\mathcal{S}}|^{2})\,.$
(2.3gb)

Then, we make KK expansions for the 5d photon field and scalar field, under
the boundary conditions of the orbifold $S^{1}\\!/\mathbb{Z}_{2}$

$\displaystyle\hat{A}^{\mu}(x^{\nu},x^{5})$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\\!\left[\,\\!A^{\mu}_{0}(x^{\nu})+\sqrt{2}\sum_{n=1}^{\infty}\\!A^{\mu}_{n}(x^{\nu})\cos\\!\frac{\,n\pi
x^{5}}{L}\,\right]\\!,$ (2.3ha) $\displaystyle\hat{A}^{5}(x^{\nu},x^{5})$
$\displaystyle\,=\,\sqrt{\frac{2}{L}\,}\sum_{n=1}^{\infty}A^{5}_{n}(x^{\nu})\sin\\!\frac{\,n\pi
x^{5}\,}{L}\,,$ (2.3hb) $\displaystyle\hat{\mathcal{S}}(x^{\nu},x^{5})$
$\displaystyle\,=\,\frac{1}{\sqrt{L\,}\,}\\!\left[\,\\!\mathcal{S}_{0}(x^{\nu})+\sqrt{2}\sum_{n=1}^{\infty}\mathcal{S}_{n}(x^{\nu})\cos\\!\frac{\,n\pi
x^{5}\,}{L}\,\right]\\!.$ (2.3hc)

With these, we can derive the effective KK Lagrangian in 4d and obtain the
corresponding Feynman rules, which are presented in Appendix C.

To test the GRET explicitly, we consider the scattering of zero-mode photon
and KK graviton into a pair of scalar bosons,
$h_{n}^{L}(p_{1})A_{0}^{T}(p_{2})\\!\rightarrow\\!\SS^{-}_{0}(p_{3})\SS^{+}_{n}(p_{4})$
and
$\tilde{h}_{n}^{S}(p_{1})A_{0}^{T}(p_{2})\\!\rightarrow\\!\SS^{-}_{0}(p_{3})\SS^{+}_{n}(p_{4})$,
where the initial state KK graviton is either longitudinally-polarized
$\,h_{n}^{L}$ or scalar-polarized $\,\tilde{h}_{n}^{S}$, and the zero-mode
photon $\,A_{0}^{T}=\epsilon_{\mu}^{T}A^{\mu}_{0}$ is massless. The final
state includes the zero-mode scalar boson $\SS^{-}_{0}$ and the KK scalar
boson $\SS^{+}_{n}$. We present the relevant Feynman diagrams in Fig. 1.

Figure 1: Scattering processes of zero-mode photon and longitudinally
(scalar)-polarized KK graviton,
$\,h_{n}^{L}A^{T}_{0}\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}$ and
$\,\tilde{h}_{n}^{S}A^{T}_{0}\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}$, via the
$(s,t,u)$-channels and contact interactions. Here the blue double-waved-line
denotes the KK graviton $h_{n}^{\mu\nu}$, the black waved-line denotes zero-
mode photon $A_{0}^{\mu}$ , and the black dashed line denotes the zero-mode
scalar $\SS_{0}$ or KK scalar $\SS_{n}$.

We first compute the diagrams in Fig. 1 for the initial state with scalar-
polarized KK graviton $\tilde{h}_{n}^{S}$ . Thus, the scattering amplitude is
derived as

$\mathcal{M}[\tilde{h}_{n}^{S}]\,=\,-\sqrt{\frac{3}{2}\,}e\kappa\,(\hskip
0.85358ptp_{3}\\!\cdot\epsilon_{2}^{\pm})\,.$ (2.3i)

Then, we consider the corresponding scattering amplitudes
$\,\phi_{n}A^{T}_{0}\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}$ and
$\,\tilde{h}_{n}A^{T}_{0}\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}$, as shown in
Fig. 2. From Fig. 2, we compute the scattering amplitudes with initial state
KK Goldstone boson $\phi_{n}$ and the unphysical trace-part of the KK graviton
field $\tilde{h}_{n}$, respectively. We further derive their summed scattering
amplitude. Now, these scattering amplitudes are presented as follows:

$\displaystyle\mathcal{M}[\phi_{n}]$
$\displaystyle=\frac{1}{2}\sqrt{\frac{2}{3}}\,e\kappa\,(\hskip
0.85358ptp_{3}\\!\cdot\epsilon_{2}^{\pm})\,,\qquad\mathcal{M}[\tilde{h}_{n}]=2\sqrt{\frac{2}{3}}\,e\kappa\,(\hskip
0.85358ptp_{3}\\!\cdot\epsilon_{2}^{\pm})\,,$ (2.3ja)
$\displaystyle\mathcal{M}[\,\overline{\Omega}_{n}]$
$\displaystyle=\mathcal{M}[\phi_{n}]-\mathcal{M}[\tilde{h}_{n}]\,=\,-\sqrt{\frac{3}{2}}\,e\kappa\,(\hskip
0.85358ptp_{3}\\!\cdot\epsilon_{2}^{\pm})\,,$ (2.3jb)

where the notation $\,\overline{\Omega}_{n}\\!=\phi_{n}\\!-\tilde{h}_{n}$ was
introduced in Eq.(2.3h). Inspecting the scalar-polarized KK graviton amplitude
(2.3i) and the summed amplitude (2.3jb), we deduce an equality,

$\mathcal{M}[\tilde{h}_{n}^{S}]\,=\,\mathcal{M}[\,\overline{\Omega}_{n}]\,,$
(2.3k)

which explicitly verifies the GRET identity (2.3j). We also note that for the
current scattering process, the $\overline{\Omega}_{n}$-amplitude contains
contributions by both the gravitational KK Goldstone boson $\,\phi_{n}\,$ and
the trace-part of the KK graviton $\,\tilde{h}_{n}\,$, which are of the same
order of magnitude. This shows an essential difference from the case of the
pure KK gauge theories (without gravity), where for each longitudinal KK gauge
boson $A_{n}^{L}$, its corresponding KK Goldstone boson is just given by the
scalar component $A_{n}^{5}$ [8].

Figure 2: Scattering processes of the zero-mode photon and the KK
gravitational Goldstone boson or the trace-part of KK graviton,
$\,\phi_{n}A^{T}_{0}\\!\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}$ and
$\,\tilde{h}_{n}A^{T}_{0}\\!\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}$ , via the
$(s,t,u)$-channels, where the red solid-line denotes the KK gravitational
scalar Goldstone $\phi_{n}$, the blue double-line denotes the trace-part of
the KK graviton $\tilde{h}_{n}$.

Then, in order to compute the scattering amplitudes explicitly, we choose the
momenta in the center-of-mass frame and make the initial state particles move
along the $z$-axis. Then, the momenta for the initial state particles and
final state particles are given by

$\displaystyle p_{1}^{\mu}=$ $\displaystyle-\\!E(1,0,0,\beta),$
$\displaystyle\hskip 28.45274ptp_{2}^{\mu}=-E\beta(1,0,0,-1),$ $\displaystyle
p_{3}^{\mu}=$ $\displaystyle\,E\beta(1,s_{\theta},0,c_{\theta}),$
$\displaystyle\hskip 28.45274ptp_{4}^{\mu}=E(1,-\beta s_{\theta},0,-\beta
c_{\theta}),$ (2.3l)

where $\,\beta=\\!\sqrt{1\\!-\\!M_{n}^{2}/E^{2}\,}$ and
$(s_{\theta},\,c_{\theta})\\!=\\!(\sin\theta,\,\cos\theta)$ with $\theta$
being the scattering angle. For simplicity of illustration, we consider the
zero-mode mass $\,m_{0}\\!\ll\\!M_{n}$ and thus $\,m_{0}$ is negligible for
this analysis. The polarization vectors of the KK graviton $h_{n}^{L}(p_{1})$
and zero-mode photon $A_{0}^{T}(p_{2})$ in the initial state take the
following forms:

$\displaystyle\varepsilon_{1L}^{\mu\nu}$
$\displaystyle=\frac{1}{\sqrt{6\,}\,}(\epsilon^{\mu}_{1+}\epsilon^{\nu}_{1-}\\!+\epsilon^{\mu}_{1-}\epsilon^{\nu}_{1+}\\!+2\epsilon^{\mu}_{1L}\epsilon^{\nu}_{1L}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\epsilon^{\mu}_{1\pm}=\frac{1}{\sqrt{2\,}\,}(0,\pm
1,-\text{i},0),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\epsilon_{1L}^{\mu}$
$\displaystyle=-\frac{E}{M_{n}}(\beta,0,0,1),\leavevmode\nobreak\
\leavevmode\nobreak\ \quad\epsilon^{\mu}_{2\pm}=-\frac{1}{\sqrt{2\,}\,}(0,\pm
1,\text{i},0)\,.$ (2.3m)

With the above, we compute explicitly the scattering amplitudes of
$\,h_{n}^{L}A_{0}^{T}\\!\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}$ and
$\,\tilde{h}_{n}^{S}A_{0}^{T}\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}$ under
the high energy expansion:

$\displaystyle\mathcal{M}[h_{n}^{L}]$
$\displaystyle\,=\,-\frac{5\sqrt{3}\,e\kappa}{6}(Es_{\theta})+\frac{\,\sqrt{3}\,e\kappa\,}{6}\frac{\,M_{n}^{2}(4\\!-\\!c_{\theta})\,}{\,E\tan(\theta/2)\,}+\mathcal{O}(E^{-3})\,,$
(2.3na) $\displaystyle\mathcal{M}[\phi_{n}]$
$\displaystyle\,=\,-\frac{\,\sqrt{3}\,e\kappa\,}{6}(Es_{\theta})+\frac{\,\sqrt{3}\,e\kappa\,}{12}\frac{\,M_{n}^{2}\,s_{\theta}\,}{E}+\mathcal{O}(E^{-3})\,,$
(2.3nb)

where we have chosen the transverse polarization $\epsilon^{\mu}_{2+}$ for the
initial state photon $A_{0}^{T}$. For the other transverse polarization
$\epsilon^{\mu}_{2-}$ of $A_{0}^{T}$, all of the corresponding amplitudes will
flip an overall sign.

According to the GRET identities (2.3j) and (2.3kb), we can compute the
residual term:

$\displaystyle\mathcal{M}[\widetilde{\Delta}_{n}]\,=\,\mathcal{M}[\tilde{v}_{n}]-\mathcal{M}[\tilde{h}_{n}]\,=\,\mathcal{M}[h_{n}^{L}]-\mathcal{M}[\phi_{n}]\,,$
(2.3o)

where we have used the abbreviations
$\,\mathcal{M}[\widetilde{\Delta}_{n}]\\!\equiv\\!\mathcal{M}[\widetilde{\Delta}_{n}A_{0}^{T}\\!\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}]$,
$\,\mathcal{M}[\tilde{v}_{n}]\\!\equiv\\!\mathcal{M}[\tilde{v}_{n}A_{0}^{T}\\!\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}]$,
and
$\,\mathcal{M}[\tilde{h}_{n}]\\!\equiv\\!\mathcal{M}[\tilde{h}_{n}A_{0}^{T}\\!\\!\rightarrow\\!\SS^{-}_{0}\SS^{+}_{n}]\,$.
Using the longitudinal KK graviton amplitude (2.3na) and KK Goldstone
amplitude (2.3nb), we derive the residual term (2.3o) as follows:

$\mathcal{M}[\widetilde{\Delta}_{n}]\,=\,-\frac{\,2\sqrt{3}\,e\kappa\,}{3}(Es_{\theta})+\mathcal{O}\\!\left(\\!\frac{M_{n}^{2}}{E}\right)\\!,$
(2.3p)

which has the same energy order as the longitudinal KK graviton amplitude
$\mathcal{M}[h_{n}^{L}]$ . This demonstrates that for the case of one external
KK graviton line, although the GRET identity (2.3k) holds as expected,

$\mathcal{M}[h_{n}^{L}]\,=\,\mathcal{M}[\,\overline{\Omega}_{n}]+\mathcal{M}[\tilde{v}_{n}]\,=\,\mathcal{M}[\phi_{n}]+\mathcal{M}[\widetilde{\Delta}_{n}]\,,$
(2.3q)

the GRET itself no longer holds. This is because the residual term
$\,\mathcal{M}[\widetilde{\Delta}_{n}]\,$ in Eq.(2.3p) has the same order of
magnitude as the longitudinal KK graviton amplitude
$\,\mathcal{M}[h_{n}^{L}]\,$ or the KK Goldstone amplitude
$\,\mathcal{M}[\phi_{n}]\,$ in Eq.(2.3n) under the high energy expansion.

###  Gravitational KK Goldstone Boson Scattering Amplitudes

In this subsection, we explicitly compute the elastic and inelastic scattering
amplitudes of four gravitational KK Goldstone bosons in the compactified 5d
GR, which will be compared quantitatively with the corresponding longitudinal
(helicity-zero) KK graviton scattering amplitudes.

####  Elastic Gravitational KK Goldstone Boson Scattering Amplitudes

To compute the scattering amplitudes of the gravitational KK Goldstone bosons,
we first derive the relevant interaction vertices. We will show that the
leading contributions arise from the Feynman diagrams with zero-mode graviton
and KK graviton exchanges. For the cubic interaction vertices containing one
graviton and two KK scalar-Goldstone bosons, we expand EH Lagrangian up to
$\mathcal{O}(\hat{\kappa}^{3})$, denoted as
$\,\hat{\mathcal{L}}_{1}[\hat{h}\hat{\phi}^{2}]$ . We inspect the structure of
$\,\hat{\mathcal{L}}_{1}[\hat{h}\hat{\phi}^{2}]$ and classify it into 12
Lorentz-invariant terms, as presented in Table 1.

$\hat{\mathcal{L}}_{1}[\hat{h}\hat{\phi}^{2}]$ | $\hat{h}^{\mu\nu}\partial_{\mu}\hat{\phi}\,\partial_{\nu}\hat{\phi}$ | $\hat{h}^{\mu\nu}\hat{\phi}\,\partial_{\mu}\partial_{\nu}\hat{\phi}$ | $\hat{h}\,\partial_{\mu}\hat{\phi}\,\partial^{\mu}\hat{\phi}$ | $\hat{h}\,\hat{\phi}\,\partial_{\mu}^{2}\hat{\phi}$ | $\hat{h}\,\partial_{5}\hat{\phi}\,\partial^{5}\hat{\phi}$ | $\hat{h}\,\hat{\phi}\,\partial_{5}^{2}\hat{\phi}$
---|---|---|---|---|---|---
$\partial_{\mu}\hat{h}^{\mu\nu}\hat{\phi}\,\partial_{\nu}\hat{\phi}$ | $(\partial_{\mu}\partial_{\nu}\hat{h}^{\mu\nu})\hat{\phi}^{2}$ | $\partial_{\mu}\hat{h}\,\hat{\phi}\,\partial^{\mu}\hat{\phi}$ | $(\partial_{\mu}^{2}\hat{h})\hat{\phi}^{2}$ | $\partial_{5}\hat{h}\,\hat{\phi}\,\partial^{5}\hat{\phi}$ | $(\partial_{5}^{2}\hat{h})\hat{\phi}^{2}$
Table 1: Classification of the 12 Lorentz-invariant interaction vertices in
$\hat{\mathcal{L}}_{1}[\hat{h}\hat{\phi}^{2}]$, where the 6 operators in the
second row (black color) can be converted into the combinations of the
operators in the first row (red color) via integration by parts.

We note that in Table 1 all the 6 operators in the second row (black color)
contain partial derivatives acting on the graviton fields, but we can always
shift the partial derivatives on to the scalar fields via integration by
parts, and thus they can be converted into combinations of the 6 operators in
the first row (red color). In this way, we can organize the cubic vertices in
the Lagrangian $\,\hat{\mathcal{L}}_{1}[\hat{h}\hat{\phi}^{2}]$ as follows:

$\displaystyle\hat{\mathcal{L}}_{1}[\hat{h}\hat{\phi}^{2}]\,=\,\leavevmode\nobreak\
$ $\displaystyle
a_{1}\,\hat{h}^{\mu\nu}\partial_{\mu}\hat{\phi}\,\partial_{\nu}\hat{\phi}+a_{2}\,\hat{h}^{\mu\nu}\hat{\phi}\,\partial_{\mu}\partial_{\nu}\hat{\phi}+a_{3}\,\hat{h}(\partial_{\mu}\hat{\phi})^{2}+a_{4}\,\hat{h}\,\hat{\phi}\,\partial_{\mu}^{2}\hat{\phi}$
$\displaystyle+a_{5}\,\hat{h}(\partial_{5}\hat{\phi})^{2}+a_{6}\,\hat{h}\,\hat{\phi}\,\partial_{5}^{2}\hat{\phi}\,,$
(2.3t)

where the coefficients are given by

$(\hskip 0.85358pta_{1},\,a_{2},\,a_{3},\,a_{4},\,a_{5},\,a_{6}\hskip
0.85358pt)\,=\,\left(\\!-\frac{1}{\,2\,},\,-1,\,\frac{3}{\,4\,},\,1,\,-\frac{1}{\,2\,},\,-\frac{1}{\,2\,}\hskip
0.85358pt\right)\\!.$ (2.3u)

Next, by substituting Eqs.(2.3la)-(2.3lc) into the Lagrangian (2.3t) and
integrating over $x^{5}$, we derive the corresponding effective Lagrangian in
4d,

$\displaystyle\mathcal{L}_{1}[h\phi^{2}]=\frac{\kappa}{\sqrt{2}}\sum_{n,m,\ell=1}^{\infty}\\!\biggl{\\{}\\!$
$\displaystyle
a_{1}\\!\left[\\!\sqrt{2}\left(h_{0}^{\mu\nu}\partial_{\mu}\phi_{0}\partial_{\nu}\phi_{0}+h_{0}^{\mu\nu}\partial_{\mu}\phi_{m}\partial_{\nu}\phi_{\ell}\delta_{m\ell}+h_{n}^{\mu\nu}\partial_{\mu}\phi_{m}\partial_{\nu}\phi_{0}\delta_{nm}+h_{n}^{\mu\nu}\partial_{\mu}\phi_{0}\partial_{\nu}\phi_{\ell}\delta_{n\ell}\right)\right.$
$\displaystyle\left.+\,h_{n}^{\mu\nu}\partial_{\mu}\phi_{m}\partial_{\nu}\phi_{\ell}\Delta_{3}(n,m,\ell)\right]+a_{2}\left[\\!\sqrt{2}\left(h_{0}^{\mu\nu}\phi_{0}\partial_{\mu}\partial_{\nu}\phi_{0}+h_{0}^{\mu\nu}\phi_{m}\partial_{\mu}\partial_{\nu}\phi_{\ell}\delta_{m\ell}+h_{n}^{\mu\nu}\phi_{m}\partial_{\mu}\partial_{\nu}\phi_{0}\delta_{nm}\right.\right.$
$\displaystyle\left.\left.+\,h_{n}^{\mu\nu}\phi_{0}\partial_{\mu}\partial_{\nu}\phi_{\ell}\delta_{n\ell}\right)+h_{n}^{\mu\nu}\phi_{m}\partial_{\mu}\partial_{\nu}\phi_{\ell}\Delta_{3}(n,m,\ell)\right]+a_{3}\left[\,\\!\\!\sqrt{2}\left(h_{0}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{0}+h_{0}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{\ell}\delta_{m\ell}\right.\right.$
$\displaystyle\left.\left.+\,h_{n}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{0}\delta_{nm}+h_{n}\partial_{\mu}\phi_{0}\partial^{\mu}\phi_{\ell}\delta_{n\ell}\right)+h_{n}\partial_{\mu}\phi_{m}\partial^{\mu}\phi_{\ell}\Delta_{3}(n,m,\ell)\right]+a_{4}\left[\\!\sqrt{2}\left(h_{0}\phi_{0}\partial^{2}_{\mu}\phi_{0}\right.\right.$
$\displaystyle\left.\left.+\,h_{0}\phi_{m}\partial^{2}_{\mu}\phi_{\ell}\delta_{m\ell}+h_{n}\phi_{m}\partial^{2}_{\mu}\phi_{0}\delta_{nm}+h_{n}\phi_{0}\partial^{2}_{\mu}\phi_{\ell}\delta_{n\ell}\right)+h_{n}\phi_{m}\partial^{2}_{\mu}\phi_{\ell}\Delta_{3}(n,m,\ell)\right]$
$\displaystyle+a_{5}M_{m}M_{\ell}\left[\,\\!\\!\sqrt{2}h_{0}\phi_{m}\phi_{\ell}\delta_{m\ell}h_{n}\phi_{m}\phi_{\ell}\widetilde{\Delta}_{3}(n,m,\ell)\\!\,\right]-a_{6}M_{\ell}^{2}\left[\,\\!\\!\sqrt{2}\left(h_{0}\phi_{m}\phi_{\ell}\delta_{m\ell}+h_{n}\phi_{0}\phi_{\ell}\delta_{n\ell}\right)\right.$
$\displaystyle\left.+\,h_{n}\phi_{m}\phi_{\ell}\Delta_{3}(n,m,\ell)\,\right]\biggr{\\}},$
(2.3v)

where $\Delta_{3}(n,m,\ell)$ and $\widetilde{\Delta}_{3}(n,m,\ell)$ are given
by

$\displaystyle\Delta_{3}(n,m,\ell)$
$\displaystyle\,=\,\delta(n+m-\ell)+\delta(n-m-\ell)+\delta(n-m+\ell)\,,$
(2.3wa) $\displaystyle\widetilde{\Delta}_{3}(n,m,\ell)$
$\displaystyle\,=\,\delta(n+m-\ell)-\delta(n-m-\ell)+\delta(n-m+\ell)\,.$
(2.3wb)

Hence, using Eq.(2.3v), we can derive the Feynman rule for graviton-scalar-
scalar interactions as shown in Fig. 3.

Figure 3: Feynman rule for the cubic interaction vertex between KK graviton
and gravitational KK Goldstone bosons, where we define,
$\,\tilde{a}_{4}=a_{4}+(-1)^{\delta_{2n,m}}a_{5}-a_{6}\hskip 0.85358pt$, with
$m\\!=0,\,2n$ .

With the above, we are ready to analyze the elastic scattering of the
gravitational KK Goldstone bosons,
$\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{n}\phi_{n}$ . Fig. 4 shows the Feynman
diagrams at the tree level, which include the scattering via the zero-mode
graviton exchange and the KK graviton exchange at level-$2n$. By
straightforward power counting, we find that each diagram in Fig. 4 has the
leading contribution of $\,\mathcal{O}(E^{2})\,$ in the high energy limit. We
stress that our gravitational KK Goldstone boson scattering amplitudes in our
study do not invoke any energy cancellation among the individual diagrams and
the leading energy dependence of $\mathcal{O}(E^{2})$ is manifest in each
diagram. This feature is an essential difference from the longitudinal KK
graviton amplitudes which involve a complicated large energy cancellations
from $\mathcal{O}(E^{10})$ to $\mathcal{O}(E^{2})$ as in [14][15]. In fact, as
we will demonstrate, our formulation of the GRET (section 3) together with the
double-copy construction (section 5) can provide a general mechanism for these
large energy cancellations.

Figure 4: Elastic scattering of gravitational KK Goldstone bosons,
$\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}$ , via
$(s,\,t,\,u)$-channels mediated by the zero-mode graviton and by the KK
graviton of level-$2n$.

By using the trilinear interaction vertices Fig. 3 and the KK graviton
propagator (2.3va) as well as the kinematics defined in Appendix A, we can
compute all the Feynman diagrams of Fig. 4 in a straightforward way. Summing
up the individual diagrams, we derive the elastic scattering amplitude of
$\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}$ to the leading order
(LO) of $\mathcal{O}(E^{2})$ under the high energy expansion:

$\displaystyle\mathcal{M}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{\,3\kappa^{2}\,}{\,32}\\!\left[\,\\!\\!\frac{\leavevmode\nobreak\
(3+\cos^{2}\\!\theta)^{2}\,}{\sin^{2}\\!\theta}\\!\\!\,\right]\\!s_{0}\,,$
(2.3xa) $\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{\,3\kappa^{2}\,}{\,128}\\!\left[\,\\!\\!\frac{\leavevmode\nobreak\
(7\\!+\cos 2\theta)^{2}\,}{\sin^{2}\\!\theta}\\!\\!\,\right]\\!s_{0}\,.$
(2.3xb)

Then, the expansion to the next-to-leading order (NLO) gives the subleading
amplitude:

$\delta\mathcal{M}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]\,=\,-\frac{\,\kappa^{2}M_{n}^{2}\,}{128}(-1318+2865c_{2\theta}\\!-522c_{4\theta}\\!-c_{6\theta})\csc^{4}\\!\theta\,,\hskip
28.45274pt$ (2.3y)

which is mass-dependent contribution of $\,\mathcal{O}(E^{0}M_{n}^{2})\,$. We
see that this NLO amplitude (2.3y) is much smaller than the LO amplitude
(2.3x) of $\mathcal{O}(E^{2}M_{n}^{0})$ in the high energy scattering.

In order to explicitly demonstrate our GRET, we will first compare our
gravitational KK Goldstone boson amplitude (2.3xb) with the corresponding
longitudinal KK graviton amplitude
$\mathcal{M}[h^{n}_{L}h^{n}_{L}\\!\rightarrow\\!h^{n}_{L}h^{n}_{L}]$ as given
in Ref. [15] (cf. its Eq.(70)). For this comparison, we note a notational
difference: our 4d gravitational coupling constant $\kappa$ is defined in
Eq.(2.3g) as $\,\kappa\\!=\\!{\hat{\kappa}}/{\sqrt{L\,}}$ and differs from
that of Ref. [15] by a factor $\frac{\,{1}\,}{\sqrt{2}}$ since their
definition leads to $\,\kappa\\!=\\!\hat{\kappa}/\\!\sqrt{2L\,}$. Hence, our
KK Goldstone amplitude (2.3xb) should be rescaled by a factor
$\frac{\,{1}\,}{2}$ for the comparison:

$\displaystyle\mathcal{M}\longrightarrow\mathcal{M}\times\frac{1}{2}=\frac{3\kappa^{2}}{\,256}\\!\left[\,\\!\\!\frac{\,(7\\!+\cos
2\theta)^{2}\,}{\sin^{2}\\!\theta}\\!\\!\,\right]\\!s_{0}\,,$ (2.3z)

which equals the KK graviton amplitude in its Eq.(70) of Ref. [15]. This is
truly impressive because our independent computation of the KK Goldstone
amplitude (2.3xb) fully differs from that of the KK graviton amplitude which
contains much more complicated energy cancellations from $\mathcal{O}(E^{10})$
to $\mathcal{O}(E^{2})$. Naively and intuitively, this equivalence seems quite
expected for us because the scalar component of the KK graviton field
$\,{\phi_{n}}\,(\equiv\\!{h}^{55}_{n})$ should be converted to the degree of
freedom of the helicity-zero longitudinal component of the KK graviton, and
thus we would have

$\mathcal{M}[h^{n}_{L}h^{n}_{L}\\!\rightarrow\\!h^{n}_{L}h^{n}_{L}]\,=\,\mathcal{M}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]+\mathcal{O}(M_{n}^{2}E^{0})\,.$
(2.3aa)

However, in the actual situation it is far more nontrivial to quantitatively
demonstrate the equivalence between the two amplitudes in the high energy
limit. This is because our quantative formulation of the GRET (2.3c) (as
systematically presented in section 3) shows that the second term on the RHS
of the GRET contains a combination of both the KK Goldstone bosons
$\,\phi_{n}\,$ and trace-part of graviton $\tilde{h}_{n}$ due to the structure
of our $R_{\xi}$ gauge-fixing functions in Eqs.(2.3ab)-(2.3ac) and (2.3fa). To
fully demonstrate such an equivalence as in Eq.(2.3aa), we have to further
show that all the $\tilde{h}_{n}$-related Goldstone amplitudes on the RHS of
the GRET (2.3c) together with the $\mathcal{O}(\tilde{v}_{n})$ amplitudes
could be of $\mathcal{O}(M_{n}^{2}E^{0})$ at most. We will present this
nontrivial demonstration in section 5 based on our double-copy construction.

Figure 5: Gravitational KK Goldstone boson scattering
$\,\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}$ from Feynman diagrams
of subleading contributions. The diagrams in the first row arise from
exchanging the KK vector $\mathcal{A}_{2n}^{\mu}$ via $(s,t,u)$-channels,
while the diagrams in the second row arise from exchanging both the zero-mode
and KK scalar ($\phi_{0}\,,\phi_{2n}$) via $(s,t,u)$-channels and a KK scalar-
Goldstone contact interaction.

Next, we compute the subleading contributions to the elastic KK Goldstone
amplitude $\,\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{n}\phi_{n}\,$ as shown in
Fig. 5, where the relevant Feynman rules are presented in Appendix D. These
include the subleading contributions via $(s,t,u)$-channels mediated by a
vector $\mathcal{A}_{2n}^{\mu}$ (the first row), a scalar $\phi_{0}$ or
$\phi_{2n}$ (the second row), and a contact interaction (the second row).
Thus, we derive the following three kinds of subleading contributions
accordingly under the high energy expansion:

$\displaystyle\mathcal{M}_{\mathcal{A}}[\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{n}\phi_{n}]$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{\,3\,}{4}\kappa^{2}M_{n}^{2}\,,$ (2.3aba)
$\displaystyle\mathcal{M}_{\phi}[\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{n}\phi_{n}]$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
-18\kappa^{2}M_{n}^{2}\,,$ (2.3abb)
$\displaystyle\mathcal{M}_{c}[\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{n}\phi_{n}]$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\frac{\,9\,}{2}\kappa^{2}M_{n}^{2}\,.$ (2.3abc)

Their sum is given by

$\mathcal{M}_{\mathcal{A}}+\mathcal{M}_{\phi}+\mathcal{M}_{c}\,=\,-\frac{\,51\,}{4}\kappa^{2}M_{n}^{2}\,.$
(2.3ac)

We see that the above subleading contributions are all of
$\mathcal{O}(E^{0}M_{n}^{2})$. The same feature also holds for the subleading
contributions to the inelastic channels.

Finally, we sum up the contributions of both Fig. 4 and Fig. 5, and derive the
complete elastic scattering amplitude of KK scalar-Goldstone bosons without
energy expansion:

$\mathcal{M}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]\,=\,\frac{\,\kappa^{2}M_{n}^{2}(\widetilde{X}^{0}_{0}\\!+\\!\widetilde{X}^{0}_{2}c_{2\theta}\\!+\\!\widetilde{X}^{0}_{4}c_{4\theta}\\!+\\!\widetilde{X}^{0}_{6}c_{6\theta})}{\leavevmode\nobreak\
512\bar{s}_{0}(\bar{s}_{0}\\!+\\!4)\left[\,2s_{\theta}^{2}\bar{s}_{0}^{2}\\!+\\!32\bar{s}_{0}\\!+\\!128\,\right]\\!s_{\theta}^{2}\leavevmode\nobreak\
}\,,\hskip 28.45274pt$ (2.3ad)

where $\,\bar{s}_{0}=s_{0}/M_{n}^{2}$ ,
$\bar{s}=s/M_{n}^{2}=\bar{s}_{0}\\!+\\!4\,$, and

$\displaystyle\widetilde{X}^{0}_{0}$
$\displaystyle=2(255\bar{s}_{0}^{5}+\\!13348\bar{s}_{0}^{4}+168624\bar{s}_{0}^{3}\\!+\\!984384\bar{s}_{0}^{2}+3514368\bar{s}_{0}+6012928)\hskip
0.85358pt,$ (2.3aea) $\displaystyle\widetilde{X}^{0}_{2}$
$\displaystyle=-429\bar{s}_{0}^{5}-12732\bar{s}_{0}^{4}-156288\bar{s}_{0}^{3}-777728\bar{s}_{0}^{2}-2572288\bar{s}_{0}-5210112\hskip
0.85358pt,$ (2.3aeb) $\displaystyle\widetilde{X}^{0}_{4}$
$\displaystyle=-2(39\bar{s}_{0}^{5}-1212\bar{s}_{0}^{4}-7824\bar{s}_{0}^{3}-10688\bar{s}_{0}^{2})\hskip
0.85358pt,$ (2.3aec) $\displaystyle\widetilde{X}^{0}_{6}$
$\displaystyle\,=\,-\left(3\bar{s}_{0}^{5}+\\!4\bar{s}_{0}^{4}\right)\\!.$
(2.3aed)

From the above, we expand the full amplitude (2.3ad)
$(\mathcal{M}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]\equiv\mathcal{M}[4\phi_{n}])$
down to the subleading order under the high energy expansion
$\,s_{0}\\!\gg\\!M_{n}^{2}\,$ (or $\bar{s}_{0}\\!\gg\\!1$),111111As a
clarification of the notations, in sections 3-4 we do not put an extra “tilde”
symbol above the $\phi_{n}$-amplitude $\mathcal{M}\,$ and
$\delta\mathcal{M}\,$ such as those in Eqs.(2.3ad) and (2.3af), but we will
add a “tilde” on top of the same $\phi_{n}$-amplitude symbols such as
$\,\widetilde{\mathcal{M}}\,$ and $\,\delta\widetilde{\mathcal{M}}\,$ in
section 5 as well as in Appendix F for the convenience of notations.

$\displaystyle\mathcal{M}[4\phi_{n}]$
$\displaystyle\,=\,\mathcal{M}_{0}[4\phi_{n}]+\delta\mathcal{M}[4\phi_{n}]\,,$
(2.3afa) $\displaystyle\mathcal{M}_{0}[4\phi_{n}]$
$\displaystyle\,=\,\frac{\,3\kappa^{2}\,}{\,128\,}\\!\left[\,\\!\\!\frac{\,(7\\!+\cos
2\theta)^{2}\,}{\sin^{2}\\!\theta}\\!\\!\,\right]\\!s_{0}\,,$ (2.3afb)
$\displaystyle\delta\mathcal{M}[4\phi_{n}]$
$\displaystyle\,=\,-\frac{\,\kappa^{2}M_{n}^{2}\,}{128}(-706+2049c_{2\theta}\\!-318c_{4\theta}\\!-c_{6\theta})\csc^{4}\\!\theta\,.$
(2.3afc)

We see that the above leading amplitude
$\,\mathcal{M}_{0}[4\phi_{n}]=\mathcal{O}(E^{2}M_{n}^{0})$ is mass-independent
and agrees with Eq.(2.3x), while the subleading amplitude
$\,\delta\mathcal{M}[4\phi_{n}]=\mathcal{O}(M_{n}^{2}E^{0})\,$ is mass-
dependent. As a consistency check, we also note that the above subleading
amplitude $\,\delta\mathcal{M}[4\phi_{n}]\,$ just equals the sum of the two
NLO amplitudes (2.3y) and (2.3ac) which are computed earlier.

####  Inelastic Gravitational KK Goldstone Boson Scattering Amplitudes

In this subsection, we further analyze the inelastic scattering processes for
the gravitational KK Goldstone bosons. Based on the analysis of the previous
section, we have demonstrated that the longitudinal-Goldstone equivalence
(2.3aa) holds down to $\mathcal{O}(E^{2})$ under the high energy expansion,
which is equivalent to taking the high energy limit
$\,M_{n}/E\\!\rightarrow\\!0$ .

From the trilinear interaction vertex Fig. 3, we can deduce a relation between
the $\,h_{0}^{\mu\nu}$-$\phi_{n}$-$\phi_{n}\,$ coupling ($V^{\mu\nu}_{0}$) and
$\,h_{2n}^{\mu\nu}$-$\phi_{n}$-$\phi_{n}\,$ coupling ($V^{\mu\nu}_{2n}$):

$V^{\mu\nu}_{0}=\sqrt{2}\,V^{\mu\nu}_{2n}\,.$ (2.3ag)

Thus, for each channel of the elastic scattering process, the corresponding
amplitudes with the exchanges of zero-mode graviton $h_{0}^{\mu\nu}$ and KK
graviton $h_{2n}^{\mu\nu}$ are connected by the relation:
$\,\mathcal{M}_{j}^{2n}\\!=\\!\mbox{$\frac{\,{1}\,}{2}$}\mathcal{M}_{j}^{0}\,$.
Hence, for a given channel-$j$ , we have
$\,\mathcal{M}_{j}\\!=\mathcal{M}_{j}^{0}\\!+\\!\mathcal{M}_{j}^{2n}\\!=\mbox{$\frac{\,{3}\,}{2}$}\mathcal{M}_{j}^{0}\,$
in the high energy limit $\,M_{n}/E\\!\rightarrow\\!0$ . With these, we can
reproduce the elastic KK Goldstone scattering amplitude (2.3x) by

$\mathcal{M}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]\,=\,\frac{3}{2}\,\sum_{j}\\!\mathcal{M}_{j}^{0}\,,$
(2.3ah)

where $\,j\in(s,\,t,\,u)$. The above amplitude $\,\mathcal{M}_{j}^{0}\,$
arises from the exchange of zero-mode graviton and is given by

$\mathcal{M}_{j}^{0}\leavevmode\nobreak\
=\,-\text{i}\,V_{\mu\nu}^{\,0}\,\mathcal{D}^{\mu\nu\alpha\beta}_{00}\,V_{\alpha\beta}^{\,0}\,.$
(2.3ai)

Figure 6: General four-point scattering process of the gravitational KK
Goldstone bosons,
$\phi_{n_{1}}\phi_{n_{2}}\\!\\!\rightarrow\\!\phi_{n_{3}}\phi_{n_{4}}$ , via
$(s,\,t,\,u)$-channels mediated by a KK graviton of level $N_{s_{j}}$, where
$N_{s_{j}}\\!\\!\geqslant 0$ and $s_{j}\in(s,t,u)$.

With the above, we can extend our analysis of the elastic scattering amplitude
to a general case as shown in Fig. 6, including all the inelastic scattering
channels. In Fig. 6, the external KK Goldstone bosons have KK-levels of
$(n_{1},\,n_{2},\,n_{3},\,n_{4})$, and we denote the intermediate graviton
with levels $(N_{s},\,N_{t},\,N_{u})\\!\geqslant 0$, respectively.

In the following, we consider two types of the inelasic scattering processes:

  1. (i)

For the inelasic scattering
$\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{m}\phi_{m}$ (with $n\\!\neq\\!m$), we
have

$\displaystyle n_{1}=n_{2}=n\,,\quad n_{3}=n_{4}=m\,,$ $\displaystyle
N_{s}=0\,,\quad N_{t}=N_{u}=|n\pm m|\,,$ (2.3aj)

where only the $s$-channel diagram includes the exchange of zero-mode graviton
because of KK number conservation. With these, we compute the inelastic KK
graviton scattering amplitude in the high energy limit as follows:

$\displaystyle\mathcal{M}[\phi_{n}\phi_{n}\\!\rightarrow\\!\phi_{m}\phi_{m}]$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\mathcal{M}_{s}^{0}+2\times\mbox{$\frac{\,{1}\,}{2}$}\left(\mathcal{M}_{t}^{0}+\mathcal{M}_{u}^{0}\,\right)$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\mbox{$\frac{\,{2}\,}{3}$}\,\mathcal{M}[\phi_{n}\phi_{n}\\!\\!\rightarrow\\!\phi_{n}\phi_{n}]\,,$
(2.3ak)

where $\,\mathcal{M}_{j}^{0}\,$ is defined in Eq.(2.3ai) and equals the
elastic amplitude of $\,h_{0}^{\mu\nu}$ exchange in the channel-$j$ .

  2. (ii)

For the inelasic scattering
$\phi_{n}\phi_{k}\\!\rightarrow\\!\phi_{m}\phi_{\ell}$ (with $n\neq k\neq
m\neq\ell$), we have

$\displaystyle n_{1}=n\,,\quad n_{2}=k\,,\quad n_{3}=m\,,\quad n_{4}=\ell\,,$
$\displaystyle N_{s}=|n\pm k|=|m\pm\ell|\,,\quad N_{t}=|n\pm\ell|=|k\pm
m|\,,\quad N_{u}=|n\pm m|=|k\pm\ell|\,.\quad$ (2.3al)

In this case, the process of exchanging zero-mode graviton is prohibited
because of the KK number conservation, while the process by exchanging the
relevant KK gravitons is allowed via $(s,\,t,\,u)$-channels. Thus, we have

$\displaystyle\mathcal{M}[\phi_{n}\phi_{k}\\!\rightarrow\\!\phi_{m}\phi_{\ell}]$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\
\mbox{$\frac{\,{1}\,}{2}$}\left(\mathcal{M}_{s}^{0}+\mathcal{M}_{t}^{0}+\mathcal{M}_{u}^{0}\right)$
$\displaystyle\leavevmode\nobreak\ =\leavevmode\nobreak\