Scattering Amplitudes of Kaluza-Klein Strings
and Extended Massive Double-Copy

Yao Li a,111Email<EMAIL_ADDRESS>Yan-Feng Hang a,222Email:
<EMAIL_ADDRESS>Hong-Jian He a,b,333Email<EMAIL_ADDRESS>Song He
c,d,444Email<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 200240, China

b Institute of Modern Physics and Department of Physics,

Tsinghua University, Beijing 100084, China;

Center for High Energy Physics, Peking University, Beijing 100871, China

c CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, China

d School of Fundamental Physics and Mathematical Sciences,

Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China

###### Abstract

We study the scattering amplitudes of massive Kaluza-Klein (KK) states of open
and closed bosonic strings under toroidal compactification. We analyze the
structure of vertex operators for the KK strings and derive an extended
massive KLT-like relation which connects the $N$-point KK closed-string
amplitude to the products of two KK open-string amplitudes at tree level.
Taking the low energy field-theory limit of vanishing Regge slope, we derive
double-copy construction formula of the $N$-point massive KK graviton
amplitude from the sum of proper products of the corresponding KK gauge boson
amplitudes. Then, using the string-based massive double-copy formula, we
derive the exact tree-level four-point KK gauge boson amplitudes and KK
graviton amplitudes, which fully agree with those given by the KK field-theory
calculations. With these, we give an explicit prescription on constructing the
exact four-point KK graviton amplitudes from the sum of proper products of the
corresponding color-ordered KK gauge boson amplitudes. We further analyze the
string-based double-copy construction of five-point and six-point scattering
amplitudes of massive KK gauge bosons and KK gravitons.
Keywords: Bosonic Strings, Scattering Amplitudes, Gauge-Gravity
Correspondence, Field Theories in Higher Dimensions
JHEP 02 (2022) 120 [arXiv:2111.12042 [hep-th]].

###### Contents

  1. 1 Introduction
  2. 2 KK String Amplitudes and Extended Massive KLT-Like Relation
    1. 2.1 Compactification of Bosonic Strings
    2. 2.2 Vertex Operators of KK String States
    3. 2.3 Open and Closed String Amplitudes for Massive KK States
    4. 2.4 Low Energy Scattering Amplitudes of KK Gauge Bosons and Gravitons
  3. 3 Massive KK Open String Amplitudes and Field Theory Limit
    1. 3.1 Elastic Amplitudes of KK Gauge Bosons from KK Open Strings
    2. 3.2 Inelastic Amplitudes of KK Gauge Bosons from KK Open Strings
    3. 3.3 Structure of Color-Ordered Massive KK Amplitudes
  4. 4 KK Graviton Amplitudes from Extended Massive Double-Copy
    1. 4.1 Constructing Elastic Scattering Amplitudes of Four KK Gravitons
    2. 4.2 Constructing Inelastic Scattering Amplitudes of Four KK Gravitons
    3. 4.3 Constructing Multi-Point Scattering Amplitudes of KK Gravitons
  5. 5 Conclusions
  6. A Kinematics of Four-Point Scattering Amplitudes of KK States
  7. B Full Scattering Amplitudes of KK Gauge and Goldstone Bosons
    1. B.1 Elastic KK Gauge and Goldstone Boson Scattering Amplitudes
    2. B.2 Inelastic Scattering Amplitudes of KK Gauge and Goldstone Bosons
      1. B.2.1 Inelastic Scattering Amplitudes of $(n,n)\\!\rightarrow\\!(m,m)$
      2. B.2.2 Inelastic Scattering Amplitudes of $(0,0)\\!\rightarrow\\!(n,n)$

##  Introduction

Early attempts of unifying the electromagnetic and gravitational forces
pointed to a truly fundamental possibility of a higher dimensional spacetime
structure of 5d with a single extra spatial dimension compactified on a circle
à la Kaluza-Klein (KK) [1]. This intriguing avenue was seriously pursued and
widely explored in various contexts, including the string/M theories [2] and
extra dimensional field theories with large or small extra dimensions [3]. In
fact, the unification among the conventional gauge forces was first realized
through the electroweak theory [4] of the standard model (SM) and subsequently
extended to the grand unification (GUT) of the electroweak and strong forces
[5].

The big obstacle to further unification between the gauge forces and gravity
force lies in the apparently distinctive natures of Einstein’s generality
relativity (GR) including its intricate nonlinearity and perturbative
nonrenormalizability. However, the conjectured double-copy relation of
$\,\text{GR}=(\text{Gauge\leavevmode\nobreak\ Theory})^{2}\,$ points to
fundamental clues to the deep gauge-gravity connection. The Kawai-Lewellen-Tye
(KLT) relation [6] was constructed to connect the scattering amplitudes of
closed strings to the products of scattering amplitudes of open strings at
tree level. In the low energy field-theory limit, the KLT relation leads to
the connection of scattering amplitudes of massless gravitons to the products
of scattering amplitudes of massless gauge bosons with proper kinematic
factors. This was then extended to the field theory framework through the
double-copy method of color-kinematics duality of Bern-Carrasco-Johansson
(BCJ) [7][8] which links the scattering amplitudes of massless gauge theories
to that of the massless gravity. Analyzing the properties of the heterotic
string and open string amplitudes can prove and refine parts of the BCJ
conjecture [9]. The Cachazo-He-Yuan (CHY) formalism [10][11] shows that the
KLT kernel can be interpreted as the inverse amplitudes of bi-adjoint scalars,
and this can be generalized to double-copy relations for other field theories
[11]. So far substantial efforts have been made to formulate and test the
double-copy constructions between the massless gauge theories and massless GR
[8], and some recent works attempted to extend the double-copy method to the
4d massive Yang-Mills (YM) theory versus Fierz-Pauli-like massive gravity
[12], to the KK-inspired effective gauge theory with extra global U(1) [13],
to compactified 5d KK gauge/gravity theories [14], and to the 3d Chern-Simons
theories with or without supersymmetry [15][16][17].

But the extensions of conventional double-copy method to massive gauge/gravity
theories are generally difficult, because many such theories (including the
massive YM theory and massive Fierz-Pauli gravity) violate gauge symmetry and
diffeomorphism invariance (which are the key for successful double-copy
construction). The two important candidates with promise include the
compactified KK gauge/gravity theories and the topologically massive Chern-
Simons (CS) gauge/gravity theories. The massive KK gauge bosons and KK
gravitons acquire their masses from geometric “Higgs” mechanisms [18][19][14]
of the KK compactifications which spontaneously break the higher dimensional
gauge symmetry and diffeomorphism invariance to that of 4d by boundary
conditions. Such geometric “Higgs” mechanisms can be quantitatively formulated
at the scattering $S$-matrix level by the KK gauge boson equivalence theorem
(KK GAET) [18][20][21] and KK gravitational ET (KK GRET) [14], which generally
ensure much better high energy behaviors of the KK scattering amplitudes than
that of other ill-defined massive theories (with explicitly broken
gauge/gravity symmetries) and thus hold real promise for successful double-
copy construction. The 3d CS gauge/gravity theories [22] naturally realize
topological mass-generation for the gauge bosons (gravitons) in a gauge-
invariant (diffeomorphism-invariant) way, which can also ensure good high
energy behaviors of the scattering amplitudes [17] and realize successful
double-copy constructions [15][16][17].

A recent work [14] systematically studied the extended BCJ-type double-copy
construction between the scattering amplitudes of the massive KK gauge bosons
and of the massive KK gravitons in the KK YM gauge theory and KK gravity
theory under the 5d compactification of $\SS^{1}/\mathbb{Z}_{2}\hskip
0.85358pt$. The double-copy construction for the scattering amplitudes of
massive KK gauge bosons and KK gravitons is highly nontrivial even for the
four-point elastic KK amplitudes due to the presence of double-pole-like
structure with exchanges of both zero-modes and KK-modes. Ref. [14] first
proposed an improved double-copy method for massive KK amplitudes by using the
high energy expansion order by order. The leading-order (LO) KK gauge boson
amplitudes were shown [14] to be mass-independent and their numerators obey
the kinematic Jacobi identity, so the extended BCJ-type double-copy
construction can be universally realized to reconstruct the correct LO KK
graviton amplitudes. But the next-to-leading-order (NLO) KK gauge boson
amplitudes were found [14] to be mass-dependent and the corresponding double-
copied KK graviton amplitudes do not always match the exact KK graviton
amplitudes at the NLO. So the naive extension of the BCJ double-copy method
could not fully work and a modified massive double-copy construction was
proposed [14] for the NLO KK amplitudes 111As we will show in Appendix B of
the present paper, we can construct a new type of numerators by making
generalized gauge transformations and properly choosing the energy expansion
parameter such that the kinematic Jacobi identity is obeyed. But further
improvements for realizing precise BCJ-type double-copy are still needed., but
this is yet to be established for all KK scattering processes and for going
beyond the NLO. Hence, it is truly attractive and important to establish the
double-copy construction, from the first principle of KK string theory
formulation, for the exact tree-level massive KK gauge-boson/graviton
amplitudes and in a universal way.

In this work, we take the simplest KK compactification of the 26d bosonic
string theory [2] as a tool to derive the extended massive KLT-like relations
for KK closed/open-string amplitudes and then achieve the doubel-copy
construction of the realistic 5d KK gauge boson/graviton amplitudes in the low
energy field-theory limit. The essential advantage of the compactified KK
string theory is that the connection between the KK closed-string amplitudes
and the proper products of KK open-string amplitudes can be intrinsically
built in from the start. A recent literature [23] studied the general KLT
factorization of winding string amplitudes in the bosonic string theory and
computed explicitly the four-point tachyon amplitudes, but did not consider
the amplitudes in the low energy field-theory limit. We will study the
scattering amplitudes of massive KK states of open and closed bosonic strings,
and derive the corresponding scattering amplitudes of KK gauge-bosons and of
KK gravitons in the field-theory limit. With these, we derive the extended
KLT-like relations which connect the $N$-point KK closed-string amplitude to
the product of the two corresponding open string amplitudes at tree level.
Taking the field-theory limit of vanishing Regge slope, we derive double-copy
construction which formulates the general $N$-point massive KK graviton
amplitude as the sum of proper products of the corresponding KK gauge boson
amplitudes. Then, using the string-based KLT-like massive KK double-copy
formula, we derive the exact four-point elastic and inelastic scattering
amplitudes of KK gravitons from the sum of the proper products of the relevant
color-ordered amplitudes of KK gauge bosons at tree level. We verify that the
reconstructed four-point elastic KK graviton amplitude fully agrees with that
given by the available Feynman-diagram calculations [24][25] in the 5d KK
field theory of GR. Based on our KK-string formulation, we give an explicit
prescription on constructing the exact four-point KK graviton scattering
amplitudes from the sum of relevant products of the corresponding color-
ordered KK gauge boson scattering amplitudes. We further analyze the string-
based double-copy construction of five-point and six-point scattering
amplitudes of massive KK gauge bosons and KK gravitons.

This paper is organized as follows. In section 2, we analyze the structure of
vertex operators for the KK strings and derive an extended massive KLT-like
relation which connects the $N$-point KK closed-string amplitude to the
product of the two corresponding open-string amplitudes at tree level. Then,
we take the low energy field-theory limit of vanishing Regge slope and present
the double-copy construction of the $N$-point massive KK graviton amplitude by
the sum of relevant products of the color-ordered KK gauge boson amplitudes.
In section 3, we systematically derive the four-point elastic and inelastic
color-ordered scattering amplitudes of KK gauge bosons from the field-theory
limit of the corresponding scattering amplitudes of KK open strings. We study
the structure of these color-ordered massive KK gauge boson amplitudes and
demonstrate that they can be obtained from the scattering amplitudes of
massless zero-mode gauge bosons under proper shifts of the Mandelstam
variables. This gives an elegant and efficient way to compute any color-
ordered massive KK gauge boson amplitudes. In section 4, applying the string-
based massive double-copy formula, we derive the exact tree-level four-point
elastic and inelastic KK graviton amplitudes. Then, we give an explicit
prescription on constructing the exact four-point KK graviton amplitudes from
the sum of relevant products of the corresponding color-ordered KK gauge boson
amplitudes. We also use our general string-based double-copy construction to
obtain the five-point and six-point scattering amplitudes of massive KK gauge
bosons and KK gravitons. Finally, we conclude in section 5. Appendix A
provides the notational setup and kinematics formulas for the elastic and
inelastic scattering processes of four KK states. In Appendix B we present the
exact four-point amplitudes of the elastic and inelastic scattering of KK
gauge bosons at tree level, which are shown to fully agree with those obtained
from the corresponding scattering amplitudes of KK open strings under low
energy field-theory limit as given by section 3. This also serves as a
systematic consistency check of our open-string calculations in the main text.

##  KK String Amplitudes and Extended Massive KLT-Like Relation

In this section, we consider the compactifications of the 26d bosonic string
theory with one relatively large extra spatial dimension compactified to a
circle and with all other extra spatial dimensions decoupled due to their
extremely small radii of $\mathcal{O}(M_{\text{Pl}}^{-1})$. We study the mass
spectra of both KK open and closed strings. Then, we explicitly compute the
$N$-point KK open-string amplitudes under compactification by using the
relevant compact photon vertex operators and construct the amplitudes of KK
closed-strings by products of two KK open-string amplitudes. The scattering
amplitudes of KK closed-strings take the KLT-like form. Finally, taking the
low energy field-theory limit $\hskip
0.85358pt\alpha^{\prime}\\!\\!\rightarrow\\!0\,$, we derive the formulas of
general $N$-point amplitudes in the compactified KK field theories, so the
KLT-like relation of KK string amplitudes will result in the double-copy
formula of the corresponding KK graviton amplitudes.

###  Compactification of Bosonic Strings

For the sake of the present study, we consider the bosonic strings propagating
in a 26-dimensional spacetime background $\,\mathbb{R}^{1,24}\times\SS^{1}\,$.
We can first compactify the extra spatial dimensions of coordinates
$\\{X^{4},\cdots,X^{24}\\}$ with their radii $\hskip
0.85358ptr_{j}\\!=\mathcal{O}(M_{\text{Pl}}^{-1})\\!\ll\\!R\,$
($j\\!=\\!4,\cdots,24$), so they are fully decoupled at energy scales much
below the reduced Planck scale $M_{\text{Pl}}=(8\pi G)^{-1/2}$. Thus, we only
need to study the toroidal compactification of the single extra spatial
dimension of coordinate $X^{25}$ on a circle $\hskip 0.85358pt\SS^{1}$ with
radius $R$ , which is much larger than the Planck length $M_{\text{Pl}}^{-1}$
and does not decouple in our low energy effective theory. (Here by low energy,
we mean the energy scale scales which are lower than the reduced Planck scale
$M_{\text{Pl}}$ by about two orders of magnitude or more.) For the present
study, we take bosonic string as a computational tool for establishing the
massive KLT-like relations of KK string states and for deriving the low energy
KK graviton scattering amplitudes.

Then, we can identify the coordinate $X^{25}\hskip 0.85358pt(\hskip
0.85358pt\equiv\\!X\hskip 0.85358pt)$ as a scalar field on the string
worldsheet and it obeys the periodic boundary condition on the circle
$\SS^{1}$ :

$X\,\cong\,X+2\pi R\,.$ (2.1)

For the closed strings, without loss of generality, the periodic boundary
condition further takes the following form:

$X(\tau,\sigma\\!+\\!2\pi)\,=\,X(\tau,\sigma)+2\pi wR\,,$ (2.2)

where $\,w\\!\in\\!\mathbb{Z}\,$ is the winding number describing how many
times a closed string winds around the circle $\hskip 0.85358pt\SS^{1}$. Thus,
the eigenvalues of generators of Virasoro algebra $(L_{0},\hskip
0.85358pt\overline{L}_{0})$ for the oscillation modes of closed string are
derived as follows:

$\displaystyle L_{0}\,$
$\displaystyle=\,\frac{\alpha^{\prime}}{4}p^{\mu}p_{\mu}+\frac{\,\alpha^{\prime}}{4}\\!\left(\frac{\,\hat{n}}{R}+\frac{\hskip
0.85358ptwR\hskip
0.85358pt}{\,\alpha^{\prime}}\right)^{\\!\\!2}\\!+(N\\!-\\!1)\,,$ (2.3a)
$\displaystyle\bar{L}_{0}\,$
$\displaystyle=\,\frac{\,\alpha^{\prime}}{4}p^{\mu}p_{\mu}+\frac{\,\alpha^{\prime}}{4}\\!\left(\frac{\,\hat{n}\hskip
0.85358pt}{R}-\frac{\hskip 0.85358ptwR\hskip
0.85358pt}{\alpha^{\prime}}\right)^{\\!\\!2}\\!+(\widetilde{N}\\!-\\!1)\,,$
(2.3b)

where $\alpha^{\prime}$ is the Regge slope, $\,\hat{n}\in\mathbb{Z}\,$ denotes
the Kaluza-Klein (KK) level, and $(N,\hskip 0.85358pt\widetilde{N})$ represent
the string level. Hence, the mass spectrum of the KK state of a closed string
can be obtained by imposing the physical conditions $\hskip
0.85358ptL_{0}=\bar{L}_{0}=0\,$:

$\displaystyle
M^{2}_{\text{cl}}\,=\left(\\!p_{\hat{n}}+\\!\frac{wR}{\alpha^{\prime}}\right)^{\\!\\!\\!2}\\!+\\!\frac{4}{\alpha^{\prime}}(N\\!-\\!1)\,=\,\left(\\!p_{\hat{n}}-\\!\frac{wR}{\alpha^{\prime}}\right)^{\\!\\!\\!2}\\!+\\!\frac{4}{\alpha^{\prime}}(\widetilde{N}\\!-\\!1)\,,$
(2.4)

where

$p_{\hat{n}}=\frac{\hat{n}}{\,R\,}=\,\text{sgn}\times\\!M_{n}\hskip
0.85358pt,$ (2.5)

with $\hskip 0.85358pt\hat{n}\\!\in\\!\mathbb{Z}\,$ and
$\,\text{sgn}\\!\equiv\text{sign}(\hat{n})=\pm 1\hskip 0.85358pt$. In the
above, $\,M_{n}\\!=|\hat{n}|/R\,$ is the KK mass-parameter. Note that in
Eq.(2.4), the second equality is realized by imposing the level matching
condition $\,nw=\widetilde{N}-N\hskip 0.85358pt$. We can further decompose the
mass spectrum (2.4) as follows:

$M^{2}_{\text{cl}}\,=\,M_{n}^{2}+\frac{\,w^{2}R^{2}\,}{\alpha^{\prime\hskip
0.85358pt2}}+\frac{2}{\,\alpha^{\prime}\,}(N+\widetilde{N}-2)\,,$ (2.6)

where the right-hand-side (RHS) contains the contributions from the squared
KK-mass $M_{n}^{2}=n^{2}/R^{2}\hskip 0.85358pt$ and a squared topological mass
$(wR/\alpha^{\prime})^{2}$ (related to the string winding number). For
studying the low energy limit of string theory as a field theory, we will set
$\hskip 0.85358ptw\\!=\\!0\hskip 0.85358pt$ and thus $\hskip
0.85358ptN\\!=\\!\widetilde{N}\\!=\\!1\hskip 0.85358pt$ for the (KK)
gravitons.

The open string satisfies the Dirichlet boundary condition for the
compactified dimensions and the Neumann boundary condition for non-
compactified dimensions. Each ending point of the open string is attached to a
D-brane [2]. The boundary condition for the open string takes the following
form:

$X(\tau,\hskip 0.85358pt\sigma+\pi)\,=\,X(\tau,\hskip 0.85358pt\sigma)+2\pi
wR+q\hskip 0.85358ptL\,,$ (2.7)

where $q\\!\in\\!\mathbb{Z}$ labels the D-branes transverse to the
compactified circle $\SS^{1}$ with equal distance $L\hskip 0.85358pt$. We make
the following mode expansion for open string:

$X(\tau,\sigma)\,=\,x_{q}+\frac{\sigma}{\,\pi\,}(q^{\prime}L+2\pi
wR)-\sqrt{2\alpha^{\prime}\,}\sum_{m\neq
0}\frac{\,\alpha_{m}\,}{m}e^{-\text{i}\hskip 0.85358ptm\tau}\sin(m\sigma)\,,$
(2.8)

where we set the two ends of the open string at the $q$-th brane and
$(q+q^{\prime})$-th brane:

$\displaystyle X(\tau,0)=x_{q}\,,\quad\leavevmode\nobreak\
\leavevmode\nobreak\ X(\tau,\pi)=x_{q}+q^{\prime}L\hskip 0.85358pt.$ (2.9)

Thus, the mass spectrum of open strings is derived as follows:

$\displaystyle M_{\text{op}}^{2}\,=\,\left(\\!\\!\frac{\leavevmode\nobreak\
\hat{n}L\\!+\\!2\pi wR\leavevmode\nobreak\
}{2\pi\alpha^{\prime}}\\!\right)^{\\!\\!2}+\frac{1}{\,\alpha^{\prime}\,}(N-1)\,.$
(2.10)

For studying the low energy limit of string theory as a field theory, we only
need to consider the $\,w=0\,$ case and thus $\hskip 0.85358ptN\\!=\\!1\hskip
0.85358pt$ for the (KK) gauge bosons. The mass spectra (2.4) and (2.10) are
not necessarily identical in general. But, for the consistent realization of
double-copy construction, we can impose the following matching condition on
Eq.(2.10):

$R\hskip 0.85358ptL\,=\,2\pi\alpha^{\prime}\,,$ (2.11)

and make the rescaling $\,(\alpha^{\prime},R,\hskip
0.85358ptL)\\!\rightarrow\\!\mbox{$\frac{\,{1}\,}{4}$}\\!\left(\alpha^{\prime},R,\hskip
0.85358ptL\right)\hskip 0.85358pt$, such that the mass spectrum (2.10) of open
strings coincides with the mass spectrum (2.4) of closed strings [23].

###  Vertex Operators of KK String States

In this subsection, we present the vertex operators for the KK string states.
For the closed and open strings under compactification, we can write down the
integrated vertex operators for their KK states:

$\displaystyle V_{\text{op}}(\zeta,k,\hat{n})\,$
$\displaystyle=\leavevmode\nobreak\ \text{i}\hskip
0.85358ptg_{\text{op}}\\!\int\\!\\!\text{d}y\,\hskip
0.85358pt\zeta_{\mu}:\\!{\partial X^{\mu}\hskip 0.85358pte^{\text{i}k\cdot
X}e^{\text{i}p_{\hat{n}}\mathcal{Y}}}\\!:\,,$ (2.12a) $\displaystyle
V_{\text{cl}}(\zeta,k,\hat{n})\,$ $\displaystyle=\leavevmode\nobreak\
\text{i}\hskip
0.85358ptg_{\text{cl}}\\!\int\\!\\!\text{d}^{2}z\,\zeta_{\mu\nu}:\\!{\partial
X^{\mu}\bar{\partial}X^{\nu}\hskip 0.85358pte^{\text{i}k\cdot
X}e^{\text{i}p_{\hat{n}}\mathcal{X}}}\\!:\,,$ (2.12b)

where $k$ denotes the momentum in the noncompactified spacetime and
$\,p_{\hat{n}}\\!={\hat{n}}/{R}\,$ ($\hat{n}\\!\in\mathbb{Z}$) is the
quantized momentum in 26d. The compactified 26d string coordinates
$(\mathcal{X},\mathcal{Y})$ are defined as:

$\displaystyle\mathcal{Y}(y)=X_{L}(y)\\!-\\!X_{R}(y)\,,\quad\mathcal{X}(z,\bar{z})=X_{L}(z)\\!+\\!X_{R}(\bar{z})\,,$
(2.13)

where $X_{L}(z)$ and $X_{R}(\bar{z})$ denote the left-moving and right-moving
string coordinates in the 26d, respectively.

Then, for a noncompactified spatial dimension, we can write down the Green
functions for the open strings under the Neumann boundary condition:

$\displaystyle\big{\langle}X^{\mu}_{L}(y_{1})X^{\nu}_{R}(y_{2})\big{\rangle}\,$
$\displaystyle=\,\big{\langle}X^{\mu}_{R}(y_{1})X^{\nu}_{L}(y_{2})\big{\rangle}\,=\,0\,,$
(2.14a)
$\displaystyle\big{\langle}X^{\mu}_{L}(y_{1})X^{\nu}_{L}(y_{2})\big{\rangle}\,$
$\displaystyle=\,\big{\langle}X^{\mu}_{R}(y_{1})X^{\nu}_{R}(y_{2})\big{\rangle}\,=\,-\alpha^{\prime}\eta^{\mu\nu}\ln|y_{1}-y_{2}|\,,$
(2.14b)

where the Lorentz indices $\mu,\nu=0,1,\ldots,24$ . While for a compactified
spatial dimension, we have the Green functions of open strings under the
Dirichlet boundary condition:

$\displaystyle\big{\langle}X_{L}(y_{1})X_{R}(y_{2})\big{\rangle}\,$
$\displaystyle=\,\langle
X_{R}(y_{1})X_{L}(y_{2})\big{\rangle}\,=\,\alpha^{\prime}\ln|y_{1}-y_{2}|\,,$
(2.15a) $\displaystyle\big{\langle}X_{L}(y_{1})X_{L}(y_{2})\rangle\,$
$\displaystyle=\,\big{\langle}X_{R}(y_{1})X_{R}(y_{2})\big{\rangle}\,=\,-\alpha^{\prime}\ln|y_{1}-y_{2}|\,.$
(2.15b)

And the Green functions for closed string is given by

$\big{\langle}X^{M}(z_{1},\bar{z}_{1})X^{N}(z_{2},\bar{z}_{2})\big{\rangle}\,=\,-\frac{\alpha^{\prime}}{2}\eta^{MN}\ln(|z_{1}-z_{2}|^{2})\,,$
(2.16)

where the 26d Lorentz indices $M,N=(0,1,\ldots,25)$ .

In string theory, imposing the orbifold compactification
$\SS^{1}/\mathbb{Z}_{2}$ will generate an anti-periodic boundary condition,
which lifts the vacuum energy on the worldsheet and modifies the mass spectrum
[2]. In consequence, the mass of each KK open-string state gets a shift
$\,\Delta M_{\text{op}}^{2}=\frac{1}{\,16\alpha^{\prime}\,}\,$, and the mass
of each KK closed-string state receives an increase $\Delta
M_{\text{cl}}^{2}=\frac{1}{\,4\alpha^{\prime}\,}$. Thus they will be decoupled
in the field-theory limit of $\,\alpha^{\prime}\\!\rightarrow\\!0\,$. Hence,
we will first make our analysis by using the periodic boundary condition on
$\SS^{1}$ without imposing the $\mathbb{Z}_{2}\hskip 0.85358pt$. Then, we can
construct the vertex operators for KK open strings and KK closed strings with
specified $\mathbb{Z}_{2}$ parity:

$\displaystyle V_{\text{op}}^{\pm}(\zeta,k,n)\,$
$\displaystyle=\,\frac{1}{\sqrt{2\,}\,}\\!\left[\,V_{\text{op}}(\zeta,k,n)\pm
V_{\text{op}}(\zeta,k,-n)\,\right],$ (2.17a) $\displaystyle
V_{\text{cl}}^{\pm}(\zeta,k,n)\,$
$\displaystyle=\,\frac{1}{\sqrt{2\,}\,}\\!\left[\,V_{\text{cl}}(\zeta,k,n)\pm
V_{\text{cl}}(\zeta,k,-n)\,\right],$ (2.17b)

where $\,n\\!\in\\!\mathbb{Z}^{+}$ and the superscript “$+\hskip
0.85358pt(-)$” stands for the $\mathbb{Z}_{2}$-even ($\mathbb{Z}_{2}$-odd)
state. We note that for the zero-modes ($n\\!=\\!0$), vertex operators for
open and closed strings, $V_{\text{op}}(\zeta,k,0)$ and
$V_{\text{cl}}(\zeta,k,0)$, are always $\mathbb{Z}_{2}$-even. In particular,
we will be interested in the above vertex operators having
$\mathbb{Z}_{2}$-even parity because their scattering amplitudes will give, in
the low energy field-theory limit, the corresponding scattering amplitudes of
the KK gauge bosons and of the KK gravitons in the KK gauge/gravty theories
under the 5d compactification of $\hskip 0.85358pt\SS^{1}/\mathbb{Z}_{2}\hskip
0.85358pt$. This will also be valuable for comparison with the literature
[14][18][25] which computed some of these KK amplitudes by Feynman diagram
approach in the KK gauge/gravity field theories under 5d compactification of
$\hskip 0.85358pt\SS^{1}/\mathbb{Z}_{2}\hskip 0.85358pt$.

###  Open and Closed String Amplitudes for Massive KK States

For a $N$-point KK string amplitude, the conservation of the compactified
momentum (KK-number) is achieved by the neutral condition [28] of the vertex
operators on the string worldsheet, i.e., the sum of KK numbers
($\hat{n}_{j}$) of the external states should vanish:

$\sum_{j=1}^{N}\hat{n}_{j}\,=\,\sum_{j=1}^{N}\text{sgn}_{j}\\!\times
n_{j}\,=\,0\,,$ (2.18)

where $\hskip 0.85358pt\hat{n}_{j}\\!\in\\!\mathbb{Z}\,$ is the KK number of
the $j$-th external state, $\hskip
0.85358ptn_{j}\\!=\\!|\hat{n}_{j}|\\!\in\\!\mathbb{Z}^{+}\hskip 0.85358pt$,
and the sign of $\hskip 0.85358pt\hat{n}_{j}\hskip 0.85358pt$ is denoted as
$\,\text{sgn}_{j}\\!=\text{sign}(\hat{n}_{j})\hskip 0.85358pt$. Thus we can
write $\,p_{\hat{n}_{j}}\\!\\!=\text{sgn}_{j}\\!\times\\!M_{n_{j}}$. Hence,
the tree-level $N$-point open-string amplitude can be written as follows:

$\displaystyle\mathcal{A}^{(N)}_{\text{op}}(\zeta,k,n)=\leavevmode\nobreak\
\frac{e^{-\lambda}}{\,\mathrm{Vol}[\hskip
0.85358pt\mathrm{SL}(2,\mathbb{R})]\,}\prod_{j=1}^{N}\int\\!\text{d}y_{j}\,\big{\langle}V_{\text{op}}^{\pm}(\zeta_{1},k_{1},n_{1})\cdots
V_{\text{op}}^{\pm}(\zeta_{N},k_{N},n_{N})\big{\rangle}$
$\displaystyle=\leavevmode\nobreak\ \frac{\,(-)^{\omega}\hskip
0.85358pt2^{-\widehat{N}/2}\hskip
0.85358pte^{-\lambda}\,}{\,\mathrm{Vol}[\hskip
0.85358pt\mathrm{SL}(2,\mathbb{R})]\,}\prod_{j=1}^{N}\sum_{\\{\text{sgn}_{j}\\}}\\!\int\\!\\!\text{d}y_{j}\,\big{\langle}V_{\text{op}}(\zeta_{1},k_{1},\text{sgn}_{1}\hskip
0.85358ptn_{1})\cdots V_{\text{op}}(\zeta_{N},k_{N},\text{sgn}_{N}\hskip
0.85358ptn_{N})\big{\rangle}\hskip 0.85358pt,\leavevmode\nobreak\
\leavevmode\nobreak\ $ (2.19)

where $\,\lambda\,$ denotes the vacuum expectation value of the dilaton and in
the second line we have further expressed the amplitude in terms of open-
string vertex operators from the RHS of Eq.(2.17a). Here we choose each
external state to be $\mathbb{Z}_{2}$ even (odd), corresponding to its vertex
operator $V_{\text{op}}^{\pm}(\zeta_{j},k_{j},n_{j})$ being $\mathbb{Z}_{2}$
even (odd) as indicated by its superscript $+\,(-)\hskip 0.85358pt$. [An
external state of the amplitude $\mathcal{A}^{(N)}_{\text{op}}(\zeta,k,n)$ can
also be chosen as non-eigenstate of $\mathbb{Z}_{2}$ parity and thus its
corresponding vertex operator is $V_{\text{op}}(\zeta_{j},k_{j},\pm n_{j})$.
Such cases can be studied by our formulation as well, although our present
study will focus on the $N$-point amplitude like Eq.(2.19).] In the second
line of Eq.(2.19), the overall coefficient contains the sign factor
$(-)^{\omega}$, where $\hskip 0.85358pt\omega\\!=\\!0\hskip 0.85358pt$ for all
vertex operators being $\mathbb{Z}_{2}$-even and $\hskip 0.85358pt\omega\neq
0\hskip 0.85358pt$ for $N_{-}$ number of vertex operators being
$\mathbb{Z}_{2}$-odd. These $N_{-}$ number of $\mathbb{Z}_{2}$-odd vertex
operators will contribute a nontrivial sign factor $(-)^{\omega}$, where
$\hskip 0.85358pt\omega\\!=\\!\sum_{j=1}^{N_{-}}[1+H(\text{sgn}_{j})]\hskip
0.85358pt$ and the Heaviside step function $H(+)\\!=\\!1$ and
$H(-)\\!=\\!0\hskip 0.85358pt$. In Eq.(2.19), we use $\widehat{N}$ to denote
the number of the external KK excitation states (with $n_{j}\\!\\!>\\!0$) and
the factor $\,2^{-\widehat{N}/2}$ arises from the overall coefficient
$1/\\!\sqrt{2}$ of Eq.(2.17a). So the number of possible external zero-mode
states equals $\hskip 0.85358pt(N\\!-\\!\widehat{N})\hskip 0.85358pt$. In the
second line of Eq.(2.19), the summation over $\hskip
0.85358pt\\{\text{sgn}_{j}\\}\hskip 0.85358pt$ means to sum up all allowed
sign-combinations of KK numbers of external states which obey the condition
(2.18). Integrating over $\hskip 0.85358pty_{j}\hskip 0.85358pt$, we can
reexpress the $N$-point KK open string amplitude as follows:

$\mathcal{A}_{\text{op}}^{(N)}(\zeta,k)\,=\,\left(\frac{1}{2}\right)^{\\!\\!\\!\frac{\widehat{N}}{2}}\\!\\!\sum_{\\{\text{sgn}_{j}\\!\\}}\sum_{\alpha\in
S_{\\!N-1}}\\!\\!\mathcal{A}^{(N)}_{\text{op}}\big{[}\zeta_{j},\hskip
0.85358pt\hat{k}_{j}\hskip 0.85358pt\big{|}\\{1,\alpha(2\cdots
N)\\}\big{]}\\!\times\\!T\\!\left[\,\\!1,\alpha(2\cdots
N)\\!\,\right]\\!,\leavevmode\nobreak\ $ (2.20)

where $\,\hat{k}_{j}\\!=\\!(k_{j}^{\mu},\,\text{sgn}_{j}\times
M_{n_{j}})\hskip 0.85358pt$ is the 26d momentum and the notation
$\,T[1,\alpha(2\cdots N)]\\!=\\!\text{Tr}(T^{1}T^{\alpha(2)}$ $\cdots
T^{\alpha(N)})\,$ denotes the Chan-Paton factor. The partial amplitudes on the
RHS of Eq.(2.20) are not fully independent, among which only $(N\\!-\\!3)!$
partial amplitudes are independent [7][8]. We further express the color-
ordered partial amplitude on the RHS of Eq.(2.20) as follows:

$\mathcal{A}_{\text{op}}^{(N)}\big{[}\zeta_{j},\,\hat{k}_{j}\hskip
0.85358pt\big{|}\\{1,\alpha(2\cdots N)\\}\big{]}=\text{i}\hskip
0.85358pt\hskip 0.85358ptg_{\text{op}}^{N}\hskip
0.85358ptC_{D_{2}}(2\pi)^{26}\hskip
0.85358pt\delta^{(26)}\\!\Big{(}\\!\sum\nolimits_{j}\\!\hat{k}_{j}\Big{)}\hskip
1.42262pt\hbox to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}^{\hskip
0.85358pt(N)}_{\text{op}}\big{[}\zeta_{j},\hskip 0.85358pt\hat{k}_{j}\hskip
0.85358pt\big{|}\\{1,\alpha(2\cdots N)\\}\big{]},\leavevmode\nobreak\ $ (2.21)

where the coefficient $C_{\\!D_{2}}$ is given by
$\,C_{D_{2}}\\!=\\!e^{-\lambda}(C_{D_{2}}^{g}C_{D_{2}}^{X})\hskip 0.85358pt$.
The constants $C_{D_{2}}^{g}$ and $C_{D_{2}}^{X}$ are given by the path
integral of the zero mode of the $bc$ ghost and $X$ scalar field on the string
worldsheet, respectively. We will further compute the reduced amplitude $\hbox
to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}^{\hskip
0.85358pt(N)}_{\text{op}}$ explicitly for the 4-point scattering in section 3.

The closed-string amplitude can be derived from the product of two open-string
amplitudes [6]. Using the open-string amplitude (2.20), we construct the
$N$-point massive KK closed-string amplitude at tree level:

$\displaystyle{\mathcal{A}}_{\text{cl}}^{(N)}(\zeta,k)\,=$
$\displaystyle\leavevmode\nobreak\ \text{i}\hskip
0.85358pt(2\pi)^{26}\delta^{(26)}\\!\Big{(}\\!\sum\nolimits_{j}\\!\hat{k}_{j}\Big{)}\hbox
to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}_{\text{cl}}^{(N)}(\zeta,k)\,,$
(2.22a) $\displaystyle\hbox to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}_{\text{cl}}^{(N)}(\zeta,k)\,=$
$\displaystyle\leavevmode\nobreak\ g_{\text{cl}}^{N}\hskip
0.85358ptC_{S_{2}}\\!\left(\\!\frac{1}{\,2\,}\right)^{\\!\\!\widehat{N}/2}\\!\\!\left(\\!-\frac{\pi\alpha^{\prime}}{2}\right)^{\\!\\!N-3}\\!\\!\sum_{\\{a_{j},\hskip
0.85358ptb_{j}\\}}\sum_{\\{\text{sgn}_{j}\\}}\sum_{\\{\alpha,\beta\\}\in
S_{\\!N-3}}\big{\\{}\widehat{\varrho}_{ab}\,{\mathcal{S}_{\mathrm{ST}}^{\alpha^{\prime}}[\alpha|\beta]_{\hat{k}_{1}}}$
$\displaystyle\hskip 22.76219pt\times\hbox to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}^{\hskip
0.85358pt(N)}_{\text{op}}\big{[}\zeta_{j}^{a_{j}},\hat{k}_{j}\big{|}\\{1,\alpha(2\cdots
N\\!-\\!2),N\\!-\\!1,N\\}\big{]}$ $\displaystyle\hskip 22.76219pt\times\hbox
to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}^{\hskip
0.85358pt(N)}_{\text{op}}\big{[}\zeta_{j}^{b_{j}},\hat{k}_{j}\big{|}\\{N\\!-\\!1,N,\beta(2\cdots
N\\!-\\!2),1\\}\big{]}\big{\\}}\big{|}_{\alpha^{\prime}\rightarrow\hskip
0.85358pt\alpha^{\prime}/4}\,,$ (2.22b)

where
$\,C_{S_{2}}\\!\\!=\\!e^{-2\lambda}(C_{S_{2}}^{g}C_{S_{2}}^{X})\\!\\!=\\!{32\pi}/(\alpha^{\prime\hskip
0.85358pt3}g_{\text{cl}}^{2})\hskip 0.85358pt$. In the above Eq.(2.22b), the
polarization tensor $\,\zeta_{\mu\nu}\,$ of closed strings is expressed as a
sum of the products of polarization vectors of two open strings:

$\zeta_{\mu\nu}\,=\,\varrho_{ab}\,\zeta_{\mu}^{a}\hskip
0.85358pt\zeta_{\nu}^{b}\,,$ (2.23)

where the coefficient $\,\varrho_{ab}\in\mathbb{R}\,$. In Eq.(2.22b), the
coefficient $\hskip 0.85358pt\widehat{\varrho}_{a_{j}b_{j}}\hskip 0.85358pt$
is defined as the product of $\,\varrho_{ab}\,$ for all external graviton
states:

$\widehat{\varrho}_{ab}=\,\prod_{j=1}^{N}\varrho_{a_{j}b_{j}}\,.$ (2.24)

The string momentum kernel
$\,\mathcal{S}_{\mathrm{ST}}^{\alpha^{\prime}}[\alpha|\beta]_{\hat{k}_{1}}$
connects the two open-string amplitudes and takes the following explicit form
[27]:

$\mathcal{S}^{\alpha^{\prime}}_{\mathrm{ST}}[\alpha_{1}\cdots\alpha_{j}|\beta_{1}\cdots\beta_{j}]_{\hat{k}_{1}}=\hskip
0.85358pt\left(\\!\frac{\,\pi\alpha^{\prime}\,}{2}\right)^{\\!\\!-j}\\!\prod_{i=1}^{j}\sin\\!\bigg{[}\pi\alpha^{\prime}\bigg{(}\\!\hat{k}_{1}\\!\cdot\hat{k}_{\alpha_{i}}\\!+\\!\sum_{\ell>i}^{j}\Theta(\alpha_{i},\alpha_{\ell})\hskip
0.85358pt\hat{k}_{\alpha_{i}}\\!\\!\cdot\hat{k}_{\alpha_{\ell}}\bigg{)}\bigg{]},\leavevmode\nobreak\
\leavevmode\nobreak\ $ (2.25)

where the step function $\,\Theta(\alpha_{i},\alpha_{\ell})\\!=\\!1\,$ when
the order of $\,(\alpha_{i},\alpha_{\ell})\,$ is opposite in
$\left(\\{\alpha_{i}\\},\\{\beta_{i}\\}\right)$, while
$\,\Theta(\alpha_{i},\alpha_{\ell})\\!=0\,$ when the order of
$(\alpha_{i},\hskip 0.85358pt\alpha_{\ell})$ is the same as in
$\left(\\{\alpha_{i}\\},\\{\beta_{i}\\}\right)$ .

###  Low Energy Scattering Amplitudes of KK Gauge Bosons and Gravitons

In this subsection, we will derive the extended massive KLT-like relations of
KK states for the low energy field theory. For this purpose, we take the limit
of zero Regge slop $\,\alpha^{\prime}\\!\\!\rightarrow\\!0\,$ for the closed-
string amplitude in Eqs.(2.22) and (2.25). Then, the open/closed-string
amplitudes and the string momentum kernel will reduce to their corresponding
field-theory expressions:

$\mathcal{A}_{\text{op}}\\!\rightarrow\mathcal{T}\,,\quad\mathcal{A}_{\text{cl}}\\!\rightarrow\mathcal{M}\,,\quad\mathcal{S}_{\mathrm{ST}}^{\alpha^{\prime}}\\!\rightarrow\mathcal{S}_{\mathrm{FT}}\,.$
(2.26)

With these, we can derive the following low energy $N$-point graviton
scattering amplitude:

$\displaystyle\mathcal{M}^{(N)}(\zeta,k)\,=\,$ $\displaystyle\hskip
4.2679pt\left(\\!\frac{\,\kappa\,}{\,4\,}\\!\right)^{\\!\\!N-2}\\!2^{-{\widehat{N}}/{2}}\hskip
0.85358pt(-1)^{N+1}\\!\sum_{\\{a_{j},\hskip
0.85358ptb_{j}\\}}\sum_{\\{\text{sgn}_{j}\\}}\\!\sum_{\\{\alpha,\beta\\}\in
S_{\\!N-3}}\\!\\!\big{\\{}\widehat{\varrho}_{ab}\hskip
1.42262pt\mathcal{S}_{\mathrm{FT}}[\alpha|\beta]_{\hat{k}_{1}}$
$\displaystyle\hskip
17.07164pt\times\mathcal{T}^{(N)}\\!\big{[}\zeta^{a_{j}}_{j},\,\hat{k}_{j}\big{|}\\{1,\alpha(2\cdots
N\\!-\\!2),N\\!-\\!1,N\\}\big{]}$ $\displaystyle\hskip
17.07164pt\times\mathcal{T}^{(N)}\\!\big{[}\zeta^{b_{j}}_{j},\,\hat{k}_{j}\big{|}\\{N\\!-\\!1,N,\beta(2\cdots
N\\!-\\!2),1\\}\big{]}\hskip 0.85358pt\big{\\}}\hskip 0.85358pt,$ (2.27)

where the gravitational coupling $\,\kappa\,$ and the closed-string coupling
$\,g_{\text{cl}}\,$ are connected by the relation
$\,\kappa=2\pi\alpha^{\prime}g_{\text{cl}}\hskip 0.85358pt.\hskip 0.85358pt$
In the above, $\hskip
0.85358pt\mathcal{S}_{\mathrm{FT}}[\alpha|\beta]_{\hat{k}_{1}}$ is the
momentum kernel in the field theory limit and takes the following form
[26][27]:

$\mathcal{S}_{\mathrm{FT}}[\alpha_{1}\cdots\alpha_{j}|\beta_{1}\cdots\beta_{j}]_{\hat{k}_{1}}=\leavevmode\nobreak\
\prod_{i=1}^{j}\\!\bigg{(}\\!2\hskip
0.85358pt\hat{k}_{1}\\!\cdot\\!\hat{k}_{\alpha_{i}}\\!+\hskip
0.85358pt2\sum_{\ell>i}^{j}\Theta(\alpha_{i},\alpha_{\ell})\hskip
0.85358pt\hat{k}_{\alpha_{i}}\\!\\!\cdot\hat{k}_{\alpha_{\ell}}\\!\bigg{)}.$
(2.28)

##  Massive KK Open String Amplitudes and Field Theory Limit

In this section, we compute explicitly the four-point color-ordered elastic
and inelastic scattering amplitudes of KK open strings and derive the
corresponding KK gauge boson scattering amplitudes in the low energy field-
theory limit.

Thus, we compute the four-point color-ordered partial amplitudes of KK open-
string scattering with three fixed points $(y_{1},\hskip 0.85358pty_{2},\hskip
0.85358pty_{3})=(0,1,\infty)\hskip 0.85358pt$:

$\displaystyle g_{\text{op}}^{4}\hskip 0.85358ptC_{D_{2}}\hskip 0.85358pt\hbox
to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}_{\text{op}}\hskip
0.85358pt[1243]\,$
$\displaystyle=\,\frac{g_{\text{op}}^{2}}{\,2\alpha^{\prime\hskip
0.85358pt2}\,}\int_{1}^{\infty}\\!\\!F(y_{4})\,\text{d}y_{4}\,,$ (3.1a)
$\displaystyle g_{\text{op}}^{4}\hskip 0.85358ptC_{D_{2}}\hskip 0.85358pt\hbox
to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}_{\text{op}}\hskip
0.85358pt[1423]\,$
$\displaystyle=\,\frac{g_{\text{op}}^{2}}{\,2\alpha^{\prime\hskip
0.85358pt2}\,}\int_{0}^{1}\\!F(y_{4})\,\text{d}y_{4}\,,$ (3.1b) $\displaystyle
g_{\text{op}}^{4}\hskip 0.85358ptC_{D_{2}}\hskip 0.85358pt\hbox to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}_{\text{op}}[4123]\,$
$\displaystyle=\,\frac{g_{\text{op}}^{2}}{\,2\alpha^{\prime\hskip
0.85358pt2}\,}\int_{-\infty}^{0}\\!\\!F(y_{4})\,\text{d}y_{4}\,,$ (3.1c)

where the superscript “$(4)$” in each amplitude $\,\hbox to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}_{\text{op}}\,$
is not displayed for simplicity and the relation
$\,g_{\text{op}}^{2}C_{D_{2}}\\!=\\!1/(2\alpha^{\prime\hskip 0.85358pt2})\,$
is imposed. In the above, the function $F(y_{4})$ is defined as:

$\displaystyle F(y_{4})\,$ $\displaystyle=\,f(y_{4})\hskip
0.85358pt|y_{4}|^{-2\alpha^{\prime}(\hat{k}_{1}\cdot\hskip
1.42262pt\hat{k}_{4})}\hskip
0.85358pt|1\\!-\\!y_{4}|^{-2\alpha^{\prime}(\hat{k}_{1}\cdot\hskip
1.42262pt\hat{k}_{3})}\,,$ (3.2a) $\displaystyle f(y_{4})\,$
$\displaystyle=\,\lim_{y_{1}\rightarrow 0}\lim_{y_{2}\rightarrow
1}\lim_{y_{3}\rightarrow\infty}y_{3}^{2}\hskip 0.85358pt\prod_{i\neq
j}\exp\\!\left[\,\\!\\!\frac{\,2\alpha^{\prime}(\zeta_{i}\\!\cdot\zeta_{j})\,}{(y_{i}\\!-y_{j})^{2}}-\frac{\,2\alpha^{\prime}(\zeta_{i}\\!\cdot\hat{k}_{j})\,}{y_{i}\\!-y_{j}}\\!\\!\,\right]\\!,$
(3.2b)

where we only need to expand up to the linear term of each external
polarization vector. The other three color-ordered partial amplitudes ([1342],
[1432], [4132]) can be obtained by exchanging $(2,3)$. In this section, we
focus on the two partial amplitudes with the color-ordering [1234] and [1243]
for the sake of our later double-copy construction closed-string amplitudes.

###  Elastic Amplitudes of KK Gauge Bosons from KK Open Strings

In this subsection, we study the four-point elastic KK scattering process
$(n,n)\\!\rightarrow\\!(n,n)$ with all external states being
$\mathbb{Z}_{2}$-even. We observe that Eq.(2.18) allows six different
combinations of the KK numbers for the external states of the four-point sub-
amplitudes (which are originally defined under the $\SS^{1}$
compactification):

$\displaystyle\\{+n,\hskip 0.85358pt+n,\hskip 0.85358pt-n,\hskip
0.85358pt-n\\}\hskip 0.85358pt,\hskip 8.53581pt\\{+n,\hskip 0.85358pt-n,\hskip
0.85358pt+n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,\hskip 8.53581pt\\{+n,\hskip
0.85358pt-n,\hskip 0.85358pt-n,\hskip 0.85358pt+n\\}\hskip 0.85358pt,$ (3.3)
$\displaystyle\\{-n,\hskip 0.85358pt-n,\hskip 0.85358pt+n,\hskip
0.85358pt+n\\}\hskip 0.85358pt,\hskip 8.53581pt\\{-n,\hskip 0.85358pt+n,\hskip
0.85358pt-n,\hskip 0.85358pt+n\\}\hskip 0.85358pt,\hskip 8.53581pt\\{-n,\hskip
0.85358pt+n,\hskip 0.85358pt+n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,$

with $n\in\mathbb{Z}^{+}$. This means that the four-point elastic KK open-
string amplitude $\mathcal{A}_{\text{op}}^{(4)}$ can be decomposed into a sum
of the six sub-amplitudes, as presented in Fig. 1. We note that the
compactification under $\SS^{1}$ respects the $\mathbb{Z}_{2}$ parity, so
among the above six combinations of KK numbers only three are independent,
where the three combinations in the first row of Eq.(LABEL:eq:KK-num-4-point)
are connected to the other three corresponding combinations in the second row
by $\mathbb{Z}_{2}$ parity transformation. From the above, our key insight is
that even though the external states on the LHS of Eq.(LABEL:eq:KK-
num-4-point) are all $\mathbb{Z}_{2}$-even, the external states on the RHS
contain two $\mathbb{Z}_{2}$-even states and two $\mathbb{Z}_{2}$-odd states
such that the condition (2.18) is obeyed. This is because our string
compactification of 26d is under $\SS^{1}$ with the periodic boundary
condition (2.1) (without having $\mathbb{Z}_{2}$ orbifold). Hence, even for a
scattering amplitude with $\mathbb{Z}_{2}$-even external states (2.17a), it
contains the combination of individual amplitudes whose external states
include both positive and negative KK numbers, as shown in Eq.(LABEL:eq:KK-
num-4-point) for the case of four-point amplitudes.

Figure 1: Elastic scattering amplitude of $(n,n)\\!\rightarrow\\!(n,n)$ for
massive KK open strings with $\mathbb{Z}_{2}$-even parity (blue color) can be
decomposed to a sum of six sub-amplitudes of massive KK open strings (red
color) under the $\SS^{1}$ compactification of 26d for bosonic strings.

Then, we compute the color-ordered partial amplitudes from Eq.(3.1) under the
field theory limit $\hskip 0.85358pt\alpha^{\prime}\\!\rightarrow\\!0\hskip
0.85358pt$. We present their explicit expressions in Appendix B, where the
open-string coupling $\hskip 0.85358ptg_{\text{op}}\hskip 0.85358pt$ is
replaced by the gauge coupling $\hskip 0.85358ptg\hskip 0.85358pt$ of the YM
theory. Thus, substituting the momenta in Eq.(A.1) and the longitudinal
polarization vectors $\hskip 0.85358pt\zeta^{\mu}_{j,L}\hskip 0.85358pt$ in
Eq.(A.5) into Eq.(3.26), we derive the sub-amplitudes with color ordering
$[1234]$ :

$\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip 0.85358pt2^{\pm n}_{L}\hskip
0.85358pt3^{\mp n}_{L}\hskip 0.85358pt4^{\mp n}_{L}]\,$
$\displaystyle=\,g^{2}\frac{7\\!+c_{2\theta}}{\,(1\\!+\\!c_{\theta})\,}\,,$
(3.4a) $\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip 0.85358pt2^{\mp
n}_{L}\hskip 0.85358pt3^{\pm n}_{L}\hskip 0.85358pt4^{\mp n}_{L}]\,$
$\displaystyle=\,g^{2}\frac{\leavevmode\nobreak\ [\hskip
0.85358pt7\bar{s}^{2}\\!-\\!24\bar{s}\\!+\\!48\\!-\\!16(\bar{s}\\!-\\!4)c_{\theta}\\!+\\!(\bar{s}\\!+\\!4)^{2}c_{2\theta}\hskip
0.85358pt]\sec^{2}\\!\frac{\theta}{2}\leavevmode\nobreak\
}{2\bar{s}(\bar{s}\\!-\\!4)}\,,$ (3.4b) $\displaystyle\mathcal{T}[1^{\pm
n}_{L}\hskip 0.85358pt2^{\mp n}_{L}\hskip 0.85358pt3^{\mp n}_{L}\hskip
0.85358pt4^{\pm n}_{L}]\,$ $\displaystyle=\,g^{2}\frac{\leavevmode\nobreak\
7\bar{s}^{2}\\!-\\!24\bar{s}\\!+\\!48\\!+\\!16(\bar{s}\\!-\\!4)c_{\theta}\\!+\\!(\bar{s}\\!+\\!4)^{2}c_{2\theta}\leavevmode\nobreak\
}{\bar{s}\hskip 0.85358pt[\hskip
0.85358pt\bar{s}\\!+\\!4\\!+\\!(\bar{s}\\!-\\!4)c_{\theta}\hskip
0.85358pt]}\,,$ (3.4c)

and the sub-amplitudes with color ordering $[1243]$ :

$\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip 0.85358pt2^{\pm n}_{L}\hskip
0.85358pt4^{\mp n}_{L}\hskip 0.85358pt3^{\mp n}_{L}]\,$
$\displaystyle=\,g^{2}\frac{\,(7\\!+\\!c_{2\theta})\csc^{2}\\!\frac{\theta}{2}\leavevmode\nobreak\
}{2}\,,$ (3.5a) $\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip 0.85358pt2^{\mp
n}_{L}\hskip 0.85358pt4^{\pm n}_{L}\hskip 0.85358pt3^{\mp n}_{L}]\,$
$\displaystyle=\,g^{2}\frac{\leavevmode\nobreak\
7\bar{s}^{2}\\!-\\!24\bar{s}\\!+\\!48\\!-\\!16(\bar{s}\\!-\\!4)c_{\theta}\\!+\\!(\bar{s}\\!+\\!4)^{2}c_{2\theta}\leavevmode\nobreak\
}{\bar{s}\hskip 0.85358pt[\hskip
0.85358pt\bar{s}\\!+\\!4\\!-\\!(\bar{s}\\!-\\!4)c_{\theta}\hskip
0.85358pt]}\,,$ (3.5b) $\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip
0.85358pt2^{\mp n}_{L}\hskip 0.85358pt4^{\mp n}_{L}\hskip 0.85358pt3^{\pm
n}_{L}]\,$ $\displaystyle=\,g^{2}\frac{\leavevmode\nobreak\ [\hskip
0.85358pt7\bar{s}^{2}\\!-\\!24\bar{s}\\!+\\!48\\!+\\!16(\bar{s}\\!-\\!4)c_{\theta}\\!+\\!(\bar{s}\\!+\\!4)^{2}c_{2\theta}\hskip
0.85358pt]\csc^{2}\frac{\theta}{2}\leavevmode\nobreak\
}{2\bar{s}(\bar{s}\\!-\\!4)}\,,$ (3.5c)

where we have defined $\,\bar{s}=s/M_{n}^{2}$ . With the above, we sum up the
four-point amplitudes in Eq.(3.4) and Eq.(3.5), and derive the following
color-ordered full elastic amplitudes with all external states being
$\mathbb{Z}_{2}$ even:

$\displaystyle\mathcal{T}[1^{n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt3^{n}_{L}\hskip 0.85358pt4^{n}_{L}]$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}(\mathcal{T}[1^{+n}_{L}\hskip
0.85358pt2^{+n}_{L}\hskip 0.85358pt3^{-n}_{L}\hskip
0.85358pt4^{-n}_{L}]+\mathcal{T}[1^{+n}_{L}\hskip 0.85358pt2^{-n}_{L}\hskip
0.85358pt3^{+n}_{L}\hskip 0.85358pt4^{-n}_{L}]+\mathcal{T}[1^{+n}_{L}\hskip
0.85358pt2^{-n}_{L}\hskip 0.85358pt3^{-n}_{L}\hskip 0.85358pt4^{+n}_{L}])$
$\displaystyle\,=\,g^{2}\frac{\leavevmode\nobreak\
(P_{0}\\!+\\!P_{1}c_{\theta}\\!+\\!P_{2}c_{2\theta}\\!+\\!P_{3}c_{3\theta})\sec^{2}\\!\mbox{$\frac{\,{\theta}\,}{2}$}\leavevmode\nobreak\
}{16\bar{s}(\bar{s}\\!-\\!4)\hskip 0.85358pt[\hskip
0.85358pt\bar{s}\\!+\\!4\\!+\\!(\bar{s}\\!-\\!4)c_{\theta}\hskip
0.85358pt]}\,,$ (3.6a) $\displaystyle\mathcal{T}[1^{n}_{L}\hskip
0.85358pt2^{n}_{L}\hskip 0.85358pt4^{n}_{L}\hskip 0.85358pt3^{n}_{L}]$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}(\mathcal{T}[1^{+n}_{L}\hskip
0.85358pt2^{+n}_{L}\hskip 0.85358pt4^{-n}_{L}\hskip
0.85358pt3^{-n}_{L}]+\mathcal{T}[1^{+n}_{L}\hskip 0.85358pt2^{-n}_{L}\hskip
0.85358pt4^{+n}_{L}\hskip 0.85358pt3^{-n}_{L}]+\mathcal{T}[1^{+n}_{L}\hskip
0.85358pt2^{-n}_{L}\hskip 0.85358pt4^{-n}_{L}\hskip 0.85358pt3^{+n}_{L}])$
$\displaystyle\,=\,g^{2}\frac{\leavevmode\nobreak\
(P_{0}\\!-\\!P_{1}c_{\theta}\\!+\\!P_{2}c_{2\theta}\\!-\\!P_{3}c_{3\theta})\csc^{2}\\!\mbox{$\frac{\,{\theta}\,}{2}$}\leavevmode\nobreak\
}{16\bar{s}(\bar{s}\\!-\\!4)\hskip 0.85358pt[\hskip
0.85358pt\bar{s}\\!+\\!4\\!-\\!(\bar{s}\\!-\\!4)c_{\theta}\hskip
0.85358pt]}\,,$ (3.6b)

where the polynomials $\\{P_{j}\\}$ are given by

$\displaystyle P_{0}$ $\displaystyle=42\bar{s}^{3}\\!-96\hskip
0.85358pt\bar{s}^{2}\\!-32\hskip 0.85358pt\bar{s}\hskip
0.85358pt,\quad\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle
P_{1}=45\hskip 0.85358pt\bar{s}^{3}\\!-320\hskip
0.85358pt\bar{s}^{2}\\!+528\hskip 0.85358pt\bar{s}+128\hskip 0.85358pt,$ (3.7)
$\displaystyle P_{2}$ $\displaystyle=2\hskip 0.85358pt\bar{s}\hskip
0.85358pt(3\hskip 0.85358pt\bar{s}^{2}\\!+16\hskip 0.85358pt\bar{s}+16)\hskip
0.85358pt,\quad\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle
P_{3}=3\hskip 0.85358pt\bar{s}^{3}\\!-16\hskip 0.85358pt\bar{s}\\!-128\hskip
0.85358pt.$

The above elastic KK gauge boson amplitudes (3.6)-(3.6) are derived from the
KK open-string amplitudes (3.1). We inspect these color-ordered KK gauge boson
amplitudes (3.6) based on the KK open-string calculation and find that they
can be expressed in the following forms:

$\displaystyle\mathcal{T}[1^{n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt3^{n}_{L}\hskip
0.85358pt4^{n}_{L}]\,=\,g^{2}(-\mathcal{K}_{s}^{\mathrm{el}}+\mathcal{K}_{t}^{\mathrm{el}})\,,\qquad\mathcal{T}[1^{n}_{L}\hskip
0.85358pt2^{n}_{L}\hskip 0.85358pt4^{n}_{L}\hskip
0.85358pt3^{n}_{L}]\,=\,g^{2}(\mathcal{K}_{s}^{\mathrm{el}}-\mathcal{K}_{u}^{\mathrm{el}})\,,$
(3.8)

where the kinematic factors $\\{\mathcal{K}_{j}^{\mathrm{el}}\\}$ are
summarized in Eq.(B.2) of Appendix B. Impressively, we find that the
expressions in Eq.(3.8) fully agree with the corresponding KK gauge boson
amplitudes of Ref. [14] which were computed independently within the
compactified 5d KK YM gauge field theory.

###  Inelastic Amplitudes of KK Gauge Bosons from KK Open Strings

Figure 2: Inelastic scattering amplitude of $\hskip
0.85358pt(n,n)\\!\rightarrow\\!(m,m)\hskip 0.85358pt$ for massive KK open
strings with $\mathbb{Z}_{2}$-even parity (blue color) can be decomposed to a
sum of four sub-amplitudes of massive KK open strings (red color) under the
$\SS^{1}$ compactification of 26d for bosonic strings.

Next, we analyze the four-point color-ordered partial amplitude for the
inelastic channel $(n,\hskip 0.85358ptn)\\!\rightarrow\\!(m,\hskip
0.85358ptm)$ with all external states being $\mathbb{Z}_{2}$-even. We inspect
Eq.(2.18) and find that different from the elastic channel, there are only
four allowed combinations of the external KK states (as originally defined
under the $\SS^{1}$ compactification):

$\displaystyle\hskip 8.53581pt\\{+n,\hskip 0.85358pt-n,\hskip
0.85358pt+m,\hskip 0.85358pt-m\\}\hskip 0.85358pt,\hskip 5.69054pt\\{+n,\hskip
0.85358pt-n,\hskip 0.85358pt-m,\hskip 0.85358pt+m\\}\hskip 0.85358pt,$ (3.9)
$\displaystyle\hskip 8.53581pt\\{-n,\hskip 0.85358pt+n,\hskip
0.85358pt-m,\hskip 0.85358pt+m\\}\hskip 0.85358pt,\hskip 5.69054pt\\{-n,\hskip
0.85358pt+n,\hskip 0.85358pt+m,\hskip 0.85358pt-m\\}\hskip 0.85358pt.$

Thus, the four-point inelastic KK open-string amplitude is given by a sum of
the four sub-amplitudes, as illustrated in Fig. 2.

Then, we derive the four-point inelastic scattering amplitudes of longitudinal
gauge bosons with color-ordering $[1234]$ :

$\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip 0.85358pt2^{\mp n}_{L}\hskip
0.85358pt3^{\pm m}_{L}\hskip 0.85358pt4^{\mp m}_{L}]$
$\displaystyle=g^{2}\hskip 0.85358pt\frac{\leavevmode\nobreak\
(7\bar{s}^{2}\\!-\\!12\hskip 0.85358ptr_{\\!+}^{2}\bar{s}+48\hskip
1.42262ptr^{2})-64\hskip 0.85358ptr\hskip
0.85358pt\bar{q}\bar{q}^{\prime}c_{\theta}\\!+\\!16\hskip
0.85358pt\bar{q}^{2}\bar{q}^{\prime 2}c_{2\theta}\leavevmode\nobreak\
}{2\hskip
0.85358pt\bar{s}\left[(\bar{s}-4r)+4\bar{q}\bar{q}^{\prime}c_{\theta}\right]},$
(3.10a) $\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip 0.85358pt2^{\mp
n}_{L}\hskip 0.85358pt3^{\mp m}_{L}\hskip 0.85358pt4^{\pm m}_{L}]$
$\displaystyle=g^{2}\hskip 0.85358pt\frac{\leavevmode\nobreak\
(7\bar{s}^{2}\\!-\\!12\hskip 0.85358ptr_{\\!+}^{2}\bar{s}+48\hskip
1.42262ptr^{2})+64\hskip 0.85358ptr\hskip
0.85358pt\bar{q}\bar{q}^{\prime}c_{\theta}\\!+\\!16\hskip
0.85358pt\bar{q}^{2}\bar{q}^{\prime 2}c_{2\theta}\leavevmode\nobreak\
}{2\hskip
0.85358pt\bar{s}\left[(\bar{s}+4r)+4\bar{q}\bar{q}^{\prime}c_{\theta}\right]},$
(3.10b)

where we have used the notations,

$\displaystyle q=(E^{2}\\!-\\!M_{n}^{2})^{\frac{1}{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
q^{\prime}\\!=(E^{2}\\!-\\!M_{m}^{2})^{\frac{1}{2}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ r={M_{m}}/{M_{n}}\hskip
0.85358pt,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
r_{\\!+}^{2}=1\\!+r^{2}\hskip 0.85358pt,$ $\displaystyle
s=4E^{2},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
s_{0}\\!=4q^{2},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\bar{s}=s/M_{n}^{2}=\bar{s}_{0}+4\,,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \bar{s}_{0}=s_{0}/M_{n}^{2}\hskip 0.85358pt,$ (3.11)
$\displaystyle\bar{q}^{2}\\!=q^{2}/M_{n}^{2}=\mbox{$\frac{\,{1}\,}{4}$}\bar{s}-\\!1\hskip
0.85358pt,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\bar{q}^{\prime 2}\\!=q^{\prime
2}\\!/M_{n}^{2}\\!=\\!\mbox{$\frac{\,{1}\,}{4}$}\bar{s}-r^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \bar{q}^{2}\bar{q}^{\prime
2}=\mbox{$\frac{\,{1}\,}{\,16\,}$}(\bar{s}\\!-\\!4)(\bar{s}-4\hskip
0.85358ptr^{2})\hskip 0.85358pt.$

For the color-ordering $[1243]$ , we derive the four-point inelastic
scattering amplitudes of longitudinal gauge bosons as follows:

$\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip 0.85358pt2^{\mp n}_{L}\hskip
0.85358pt4^{\pm m}_{L}\hskip 0.85358pt3^{\mp m}_{L}]$
$\displaystyle=g^{2}\hskip 0.85358pt\frac{\leavevmode\nobreak\
(7\bar{s}^{2}\\!-\\!12\hskip 0.85358ptr_{\\!+}^{2}\bar{s}+48\hskip
1.42262ptr^{2})-64\hskip 0.85358ptr\hskip
0.85358pt\bar{q}\bar{q}^{\prime}c_{\theta}\\!+\\!16\hskip
0.85358pt\bar{q}^{2}\bar{q}^{\prime 2}c_{2\theta}\leavevmode\nobreak\
}{2\hskip
0.85358pt\bar{s}\left[(\bar{s}+4r)-4\bar{q}\bar{q}^{\prime}c_{\theta}\right]},$
(3.12a) $\displaystyle\mathcal{T}[1^{\pm n}_{L}\hskip 0.85358pt2^{\mp
n}_{L}\hskip 0.85358pt4^{\mp m}_{L}\hskip 0.85358pt3^{\pm m}_{L}]$
$\displaystyle=g^{2}\hskip 0.85358pt\frac{\leavevmode\nobreak\
(7\bar{s}^{2}\\!-\\!12\hskip 0.85358ptr_{\\!+}^{2}\bar{s}+48\hskip
1.42262ptr^{2})+64\hskip 0.85358ptr\hskip
0.85358pt\bar{q}\bar{q}^{\prime}c_{\theta}\\!+\\!16\hskip
0.85358pt\bar{q}^{2}\bar{q}^{\prime 2}c_{2\theta}\leavevmode\nobreak\
}{2\hskip
0.85358pt\bar{s}\left[(\bar{s}-4r)-4\bar{q}\bar{q}^{\prime}c_{\theta}\right]}.$
(3.12b)

Figure 3: Inelastic scattering amplitudes of $\hskip
0.85358pt(0,0)\\!\rightarrow\\!(n,n)\hskip 0.85358pt$ for massive KK open
string with $\mathbb{Z}_{2}$-even parity (blue color) can be decomposed into a
sum of two sub-amplitudes of massive KK open strings (red color) under the
$\SS^{1}$ compactification of 26d for bosonic strings.

Summing up the four-point inelastic amplitudes (3.10) and (3.12), we derive
the following color-ordered full amplitudes with all external states being
$\mathbb{Z}_{2}$ even:

$\displaystyle\mathcal{T}[1^{n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt3^{m}_{L}\hskip 0.85358pt4^{m}_{L}]$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}(\mathcal{T}[1^{+n}_{L}\hskip
0.85358pt2^{-n}_{L}\hskip 0.85358pt3^{+m}_{L}\hskip
0.85358pt4^{-m}_{L}]+\mathcal{T}[1^{+n}_{L}\hskip 0.85358pt2^{-n}_{L}\hskip
0.85358pt3^{-m}_{L}\hskip 0.85358pt4^{+m}_{L}])$
$\displaystyle\,=\,g^{2}\frac{\leavevmode\nobreak\
P_{0}\\!+\\!P_{1}c_{\theta}\\!+\\!P_{2}c_{2\theta}\\!+\\!P_{3}c_{3\theta}\leavevmode\nobreak\
}{\leavevmode\nobreak\ \bar{s}\hskip 0.85358pt(\hskip 0.85358pt3\hskip
0.85358pt\bar{s}^{2}\\!-\\!4\hskip 0.85358pt\bar{s}\hskip
0.85358ptr_{+}^{2}\\!-\\!16\hskip 0.85358ptr^{2}\\!+\\!16\hskip
0.85358pt\bar{s}\hskip 0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}c_{\theta}+\\!16\hskip 0.85358pt\bar{q}^{2}\hskip
0.85358pt\bar{q}^{\prime 2}c_{2\theta}\hskip 0.85358pt)\leavevmode\nobreak\
}\,,$ (3.13a) $\displaystyle\mathcal{T}[1^{n}_{L}\hskip
0.85358pt2^{n}_{L}\hskip 0.85358pt4^{m}_{L}\hskip 0.85358pt3^{m}_{L}]$
$\displaystyle\,=\,\mbox{$\frac{\,{1}\,}{2}$}(\mathcal{T}[1^{+n}_{L}\hskip
0.85358pt2^{-n}_{L}\hskip 0.85358pt4^{+m}_{L}\hskip
0.85358pt3^{-m}_{L}]+\mathcal{T}[1^{+n}_{L}\hskip 0.85358pt2^{-n}_{L}\hskip
0.85358pt4^{-m}_{L}\hskip 0.85358pt3^{+m}_{L}])$
$\displaystyle\,=\,g^{2}\frac{\leavevmode\nobreak\
P_{0}\\!-\\!P_{1}c_{\theta}\\!+\\!P_{2}c_{2\theta}\\!-\\!P_{3}c_{3\theta}\leavevmode\nobreak\
}{\leavevmode\nobreak\ \bar{s}\hskip 0.85358pt(\hskip 0.85358pt3\hskip
0.85358pt\bar{s}^{2}\\!-\\!4\hskip 0.85358pt\bar{s}\hskip
0.85358ptr_{+}^{2}\\!-\\!16\hskip 0.85358ptr^{2}\\!-\\!16\hskip
0.85358pt\bar{s}\hskip 0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}c_{\theta}+\\!16\hskip
0.85358pt\bar{q}^{2}\bar{q}^{\prime 2}c_{2\theta}\hskip 0.85358pt)}\,,$
(3.13b)

where the polynomials $\\{P_{j}\\}$ take the forms:

$\displaystyle P_{0}$ $\displaystyle=\bar{s}\hskip
0.85358pt(7\bar{s}^{2}\\!-12\hskip 0.85358pt\bar{s}\hskip
0.85358ptr_{+}^{2}+48\hskip 0.85358ptr^{2})\hskip
0.85358pt,\quad\leavevmode\nobreak\ \leavevmode\nobreak\ $ $\displaystyle
P_{1}=2\hskip 0.85358pt\bar{q}\hskip 0.85358pt\bar{q}^{\prime}(15\hskip
0.85358pt\bar{s}^{2}\\!-20\hskip 0.85358pt\bar{s}\hskip
0.85358ptr_{+}^{2}-16\hskip 0.85358ptr^{2})\hskip 0.85358pt,$ (3.14)
$\displaystyle P_{2}$ $\displaystyle=\bar{s}\hskip
0.85358pt(\bar{s}^{2}\\!+4\hskip 0.85358pt\bar{s}\hskip
0.85358ptr_{+}^{2}\\!+16r^{2})\hskip 0.85358pt,\quad\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle P_{3}=2\hskip 0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}(\bar{s}^{2}\\!+4\hskip 0.85358pt\bar{s}\hskip
0.85358ptr_{+}^{2}+16\hskip 0.85358ptr^{2})\hskip 0.85358pt.$

Inspecting the string-based KK gauge boson amplitudes (3.13), we can re-
express them in the following forms:

$\displaystyle\mathcal{T}[1^{n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt3^{m}_{L}\hskip 0.85358pt4^{m}_{L}]\hskip 0.85358pt=\hskip
0.85358ptg^{2}(-\mathcal{K}_{s}^{\mathrm{in}}+\mathcal{K}_{t}^{\mathrm{in}})\hskip
0.85358pt,\qquad\mathcal{T}[1^{n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt4^{m}_{L}\hskip 0.85358pt3^{m}_{L}]\hskip 0.85358pt=\hskip
0.85358ptg^{2}(\mathcal{K}_{s}^{\mathrm{in}}-\mathcal{K}_{u}^{\mathrm{in}})\hskip
0.85358pt,$ (3.15)

where the kinematic functions $\\{\mathcal{K}_{j}^{\mathrm{in}}\\}$ are
summarized in Eq.(B.15) of Appendix B. We find that these $\hskip
0.85358pt\\{\mathcal{K}_{j}^{\mathrm{in}}\\}\hskip 0.85358pt$ functions fully
agree with what we derived independently from the compactified 5d KK YM gauge
field theory.

Next, we study the mixed inelastic channel of gauge boson scattering $\hskip
0.85358pt(0,\hskip 0.85358pt0)\\!\rightarrow\\!(n,\hskip 0.85358ptn)\hskip
0.85358pt$. We find that the condition (2.18) allows only two combinations of
the external KK states (as originally defined under the $\SS^{1}$
compactification of 26d bosonic strings),

$\\{n_{1},\hskip 0.85358ptn_{2},\hskip 0.85358ptn_{3},\hskip
0.85358ptn_{4}\\}\,=\leavevmode\nobreak\ \\{0,\hskip 0.85358pt0,\hskip
0.85358pt+n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,\hskip 8.53581pt\\{0,\hskip
0.85358pt0,\hskip 0.85358pt-n,\hskip 0.85358pt+n\\}\,.$ (3.16)

Hence, this inelastic KK open-string amplitude equals a sum of six sub-
amplitudes, as shown in Fig. 3. Then, we compute the color-ordered gauge boson
amplitudes as follows:

$\displaystyle\mathcal{T}[1^{0}_{+1}\hskip 0.85358pt2^{0}_{-1}\hskip
0.85358pt3^{\pm n}_{L}\hskip 0.85358pt4^{\mp n}_{L}]$
$\displaystyle\,=\,-\frac{g^{2}\left(\bar{s}+4\right)s^{2}_{\theta}}{\leavevmode\nobreak\
\bar{s}+2\hskip 0.85358pt\bar{s}^{{1}/{2}}\hskip 0.85358pt\bar{q}\hskip
0.85358ptc_{\theta}\leavevmode\nobreak\ }\,,$ (3.17a)
$\displaystyle\mathcal{T}[1^{0}_{+1}\hskip 0.85358pt2^{0}_{-1}\hskip
0.85358pt4^{\pm n}_{L}\hskip 0.85358pt3^{\mp n}_{L}]$
$\displaystyle\,=\,-\frac{g^{2}\hskip 0.85358pt(\bar{s}+4)\hskip
0.85358pts^{2}_{\theta}}{\leavevmode\nobreak\ \bar{s}-2\hskip
0.85358pt\bar{s}^{{1}/{2}}\hskip 0.85358pt\bar{q}\hskip
0.85358ptc_{\theta}\leavevmode\nobreak\ }\,,$ (3.17b)

where $\hskip 0.85358pt\bar{s}\hskip 0.85358pt$ and $\hskip
0.85358pt\bar{q}\hskip 0.85358pt$ are defined in Eq.(3.11).

Then, from the four-point inelastic amplitudes in Eq.(3.17), we can obtain the
following color-ordered amplitudes with all external states being
$\mathbb{Z}_{2}$ even:

$\displaystyle\mathcal{T}[1^{0}_{+1}\hskip 0.85358pt2^{0}_{-1}\hskip
0.85358pt3^{n}_{L}\hskip 0.85358pt4^{n}_{L}]$
$\displaystyle\,=\,\mathcal{T}[1^{0}_{+1}\hskip 0.85358pt2^{0}_{-1}\hskip
0.85358pt3^{+n}_{L}\hskip 0.85358pt4^{-n}_{L}]\,=\,-\frac{g^{2}\hskip
0.85358pt(\bar{s}+4)\hskip 0.85358pts^{2}_{\theta}}{\leavevmode\nobreak\
\bar{s}+2\hskip 0.85358pt\bar{s}^{{1}/{2}}\hskip 0.85358pt\bar{q}\hskip
0.85358ptc_{\theta}\leavevmode\nobreak\ }\,,$ (3.18a)
$\displaystyle\mathcal{T}[1^{0}_{+1}\hskip 0.85358pt2^{0}_{-1}\hskip
0.85358pt4^{n}_{L}\hskip 0.85358pt3^{n}_{L}]$
$\displaystyle\,=\,\mathcal{T}[1^{0}_{+1}\hskip 0.85358pt2^{0}_{-1}\hskip
0.85358pt4^{+n}_{L}\hskip 0.85358pt3^{-n}_{L}]\,=\,-\frac{g^{2}\hskip
0.85358pt(\bar{s}+4)\hskip 0.85358pts^{2}_{\theta}}{\leavevmode\nobreak\
\bar{s}-2\hskip 0.85358pt\bar{s}^{{1}/{2}}\hskip 0.85358pt\bar{q}\hskip
0.85358ptc_{\theta}\leavevmode\nobreak\ }\,.$ (3.18b)

We can re-express the above string-based inelastic amplitudes in the following
forms:

$\displaystyle\mathcal{T}[1^{0}_{+1}\hskip 0.85358pt2^{0}_{-1}\hskip
0.85358pt3^{n}_{L}\hskip
0.85358pt4^{n}_{L}]\,=\,g^{2}(-\mathcal{K}_{s}^{\mathrm{in}}+\mathcal{K}_{t}^{\mathrm{in}})\hskip
0.85358pt,\qquad\mathcal{T}[1^{0}_{+1}\hskip 0.85358pt2^{0}_{-1}\hskip
0.85358pt4^{n}_{L}\hskip
0.85358pt3^{n}_{L}]\,=\,g^{2}(\mathcal{K}_{s}^{\mathrm{in}}-\mathcal{K}_{u}^{\mathrm{in}})\hskip
0.85358pt,$ (3.19)

where the kinematic functions $\\{\mathcal{K}_{j}^{\mathrm{in}}\\}$ are
summarized in Eq.(B.29) of Appendix B and fully agree with what we derived
independently from the compactified 5d KK YM gauge field theory.

###  Structure of Color-Ordered Massive KK Amplitudes

In this subsection, we study the structure of the color-ordered scattering
amplitudes of massive KK gauge bosons. We demonstrate that the tree-level
massive KK gauge boson amplitudes can be obtained from the corresponding
color-ordered amplitudes of the massless zero-mode gauge bosons by making
proper shifts of the Mandelstam variables.

From the formulation of the open-string amplitudes in section 2, we observe
that color-ordered massive KK sub-amplitudes in $d$-dimensions, such as
$\hskip 0.85358pt\mathcal{T}[1^{+n}\hskip 0.85358pt2^{+n}\hskip
0.85358pt3^{-n}\hskip 0.85358pt4^{-n}]\hskip 0.85358pt$, can be viewed as the
massless amplitudes in $(d+\\!1)$-dimensions with the $(d+\\!1)$-th component
of each momentum being discretized since the $(d+\\!1)$-th spatial dimension
is compactified on $\SS^{1}$. Namely, we can express the
$(d\\!+\\!1)$-dimensional momentum $\hat{k}^{\hat{\mu}}$ in terms of the
$d$-dimensional momentum $k^{\mu}$ plus an extra discretized $(d\\!+\\!1)$-th
component: $\hat{k}^{\hat{\mu}}\\!=\\!(k^{\mu},\,\hat{n}/R)\hskip 0.85358pt$.
For a given polarization vector $\zeta^{\mu}$ of the on-shell gauge boson in
$d$-dimensions, we can symbolically express it as a $(d+\\!1)$-dimensional
polarization vector $\hskip
0.85358pt\hat{\zeta}^{\hat{\mu}}\\!=(\zeta^{\mu},\hskip 0.85358pt0)\hskip
0.85358pt$. Thus, we have $\hskip
0.85358pt\hat{\zeta}_{i}\cdot\hat{\zeta}_{j}=\zeta_{i}\cdot\zeta_{j}\hskip
0.85358pt$ and $\hskip 0.85358pt\hat{\zeta}_{i}\cdot\hat{k}_{j}=\zeta_{i}\cdot
k_{j}\hskip 0.85358pt$, where the subscripts $(i,j)$ denote the particle
numbers of the external states.222Our later explicit calculations of the KK
scattering amplitudes will be always performed in the effective
(3+1)-dimensional spacetime with $d\\!=4\hskip 0.85358pt$, and with a single
compactified extra spatial dimension of coordinate $X^{25}$. The other extra
spatial dimensions of coordinates $\\{X^{4},\cdots,X^{24}\\}$ have much
smaller radii $\hskip 0.85358ptr_{j}\\!=\mathcal{O}(M_{\text{Pl}}^{-1})\,$
($j\\!=\\!4,\cdots,24$), so they are fully decoupled at energy scales much
below the reduced Planck scale $M_{\text{Pl}}\hskip 0.85358pt$, as we
discussed at the beginning of Sec. 2.1. Thus, the bosonic strings effectively
propagate in (4+1)d spacetime with the single extra spatial dimension of
$X^{25}$ compactified on $\SS^{1}$. Keeping these in mind, we can first
compute a $(d\\!+\\!1)$-dimensional massless scattering amplitude and then we
deduce the corresponding $d$-dimensional massive KK amplitude by using
relations

$\hat{\zeta}_{i}\cdot\hat{\zeta}_{j}=\zeta_{i}\cdot\zeta_{j}\hskip
0.85358pt,\hskip 28.45274pt\hskip
0.85358pt\hat{\zeta}_{i}\cdot\hat{k}_{j}=\zeta_{i}\cdot k_{j}\hskip
0.85358pt,$ (3.20)

and the relation between the two sets of Mandelstam variables

$\hat{s}_{ij}=\,-2\hskip
1.42262pt\hat{k}_{i}\cdot\hat{k}_{j}=\,s_{ij}-\left(\\!\frac{\leavevmode\nobreak\
\hat{n}_{i}\\!+\hat{n}_{j}\leavevmode\nobreak\ }{R}\\!\right)^{\\!\\!2},$
(3.21)

where the $(d\\!+\\!1)$-dimensional momenta obey the on-shell conditions
$\hskip 0.85358pt\hat{k}^{2}_{i}\\!=\hat{k}^{2}_{j}\\!=0\,$. Optionally, we
can first write a $d$-dimensional massless (zero-mode) scattering amplitude
$\,\mathcal{T}_{(0)}(s_{ij})\,$ with all polarization vectors and momenta of
the external states in symbolic format, and then we deduce the corresponding
KK sub-amplitude $\,\mathcal{T}_{\text{KK}}^{\hskip
0.85358pt\text{sub}}({s}_{ij})\,$ as follows:

$\mathcal{T}_{\text{KK}}^{\hskip
0.85358pt\text{sub}}(s_{ij})\,=\,\mathcal{T}_{(0)}(\hat{s}_{ij})\hskip
0.85358pt,$ (3.22)

with each $d$-dimensional external state having its momentum obey the on-shell
condition of the massive KK gauge boson ($k_{j}^{2}=\\!-M_{j}^{2}$) and its
polarization vector replaced by the polarization vector $\zeta_{j}$ of the KK
gauge boson. In this way, we can derive all the $d$-dimensional massive KK
gauge boson amplitudes according to the structure of the corresponding
$d$-dimensional massless (zero-mode) gauge boson amplitudes.

Next, we compute a four-point massless (zero-mode) gauge boson scattering
amplitude with color ordering [1234]. This can be done either in the massless
YM field theory, or, we can deduce it by taking the field theory limit
$\alpha^{\prime}\\!\\!\rightarrow\\!0\hskip 0.85358pt$ of the open-string
scattering amplitude (3.1):

$\displaystyle\mathcal{T}[1^{0}\hskip 0.85358pt2^{0}\hskip
0.85358pt3^{0}\hskip 0.85358pt4^{0}]=$ $\displaystyle\hskip
4.2679ptg^{2}\Big{\\{}2\big{[}(\zeta_{1}\\!\cdot\\!\zeta_{3})(\zeta_{2}\\!\cdot\\!\zeta_{4})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{4})(\zeta_{2}\\!\cdot\\!\zeta_{3})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{2})(\zeta_{3}\\!\cdot\\!\zeta_{4})\big{]}$
$\displaystyle\hskip
14.22636pt+\\!\frac{\,4\,}{\,t\,}\big{[}(\zeta_{2}\\!\cdot\\!\zeta_{3})(\zeta_{1}\\!\cdot\\!k_{2})(\zeta_{4}\\!\cdot\\!k_{1})\\!-\\!(\zeta_{2}\\!\cdot\\!\zeta_{3})(\zeta_{1}\\!\cdot\\!k_{4})(\zeta_{4}\\!\cdot\\!k_{2})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{4})(\zeta_{2}\\!\cdot\\!k_{4})(\zeta_{3}\\!\cdot\\!k_{2})$
$\displaystyle\hskip
14.22636pt+\\!(\zeta_{2}\\!\cdot\\!\zeta_{4})(\zeta_{1}\\!\cdot\\!k_{4})(\zeta_{3}\\!\cdot\\!k_{2})-(\zeta_{1}\\!\cdot\\!\zeta_{4})(\zeta_{2}\\!\cdot\\!k_{1})(\zeta_{3}\\!\cdot\\!k_{4})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{4})(\zeta_{2}\\!\cdot\\!k_{4})(\zeta_{3}\\!\cdot\\!k_{4})$
$\displaystyle\hskip
14.22636pt+\\!(\zeta_{3}\\!\cdot\\!\zeta_{4})(\zeta_{1}\\!\cdot\\!k_{4})(\zeta_{2}\\!\cdot\\!k_{1})\\!+\\!(\zeta_{3}\\!\cdot\\!\zeta_{4})(\zeta_{1}\\!\cdot\\!k_{4})(\zeta_{2}\\!\cdot\\!k_{4})-(\zeta_{1}\\!\cdot\\!\zeta_{3})(\zeta_{2}\\!\cdot\\!k_{1})(\zeta_{4}\\!\cdot\\!k_{1})$
$\displaystyle\hskip
14.22636pt-\\!(\zeta_{1}\\!\cdot\\!\zeta_{3})(\zeta_{2}\\!\cdot\\!k_{4})(\zeta_{4}\\!\cdot\\!k_{1})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{2})(\zeta_{3}\\!\cdot\\!k_{2})(\zeta_{4}\\!\cdot\\!k_{1})\\!-\\!\frac{\,s\,}{\,2\,}(\zeta_{1}\\!\cdot\\!\zeta_{4})(\zeta_{2}\\!\cdot\\!\zeta_{3})]$
(3.23) $\displaystyle\hskip 14.22636pt+\\!\frac{\,4\,}{\,s\,}[\hskip
0.85358pt(\zeta_{2}\\!\cdot\\!\zeta_{4})(\zeta_{1}\\!\cdot\\!k_{2})(\zeta_{3}\\!\cdot\\!k_{4})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{4})(\zeta_{2}\\!\cdot\\!k_{1})(\zeta_{3}\\!\cdot\\!k_{4})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{2})(\zeta_{4}\\!\cdot\\!k_{2})(\zeta_{3}\\!\cdot\\!k_{4})$
$\displaystyle\hskip
14.22636pt+\\!(\zeta_{3}\\!\cdot\\!\zeta_{4})(\zeta_{1}\\!\cdot\\!k_{4})(\zeta_{2}\\!\cdot\\!k_{1})\\!-\\!(\zeta_{3}\\!\cdot\\!\zeta_{4})(\zeta_{1}\\!\cdot\\!k_{2})(\zeta_{2}\\!\cdot\\!k_{4})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{3})(\zeta_{2}\\!\cdot\\!k_{1})(\zeta_{4}\\!\cdot\\!k_{1})$
$\displaystyle\hskip
14.22636pt+\\!(\zeta_{2}\\!\cdot\\!\zeta_{3})(\zeta_{1}\\!\cdot\\!k_{2})(\zeta_{4}\\!\cdot\\!k_{1})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{2})(\zeta_{3}\\!\cdot\\!k_{2})(\zeta_{4}\\!\cdot\\!k_{1})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{3})(\zeta_{2}\\!\cdot\\!k_{1})(\zeta_{4}\\!\cdot\\!k_{2})$
$\displaystyle\hskip
14.22636pt+\\!(\zeta_{2}\\!\cdot\\!\zeta_{3})(\zeta_{1}\\!\cdot\\!k_{2})(\zeta_{4}\\!\cdot\\!k_{2})\\!-\\!(\zeta_{1}\\!\cdot\\!\zeta_{2})(\zeta_{3}\\!\cdot\\!k_{2})(\zeta_{4}\\!\cdot\\!k_{2})\\!-\\!\frac{\,t\,}{\,2\,}(\zeta_{1}\\!\cdot\\!\zeta_{2})(\zeta_{3}\\!\cdot\\!\zeta_{4})\hskip
0.85358pt]\Big{\\}}\,.$

For the other massless color-ordered amplitudes, such as the one with color
ordering [1243], they can be obtained through the relation:

$u\hskip 0.85358pt\mathcal{T}[1^{0}\hskip 0.85358pt2^{0}\hskip
0.85358pt4^{0}\hskip 0.85358pt3^{0}]\,=\,t\hskip
0.85358pt\mathcal{T}[1^{0}\hskip 0.85358pt2^{0}\hskip 0.85358pt3^{0}\hskip
0.85358pt4^{0}]\,.$ (3.24)

For instance, we consider the elastic KK scattering $(n,\hskip
0.85358ptn)\\!\rightarrow\\!(n,\hskip 0.85358ptn)$ as discussed in section
3.1. We can obtain the color-ordered sub-amplitude $\hskip
0.85358pt\mathcal{T}[1^{+n}\hskip 0.85358pt2^{+n}\hskip 0.85358pt3^{-n}\hskip
0.85358pt4^{-n}]\hskip 0.85358pt$ by the replacement
$\,s\\!\rightarrow\\!(s-4M_{n}^{2})\,$ in $\hskip
0.85358pt\mathcal{T}[1^{0}2^{0}3^{0}4^{0}]\hskip 0.85358pt$. In general, the
color-ordered sub-amplitude
$\mathcal{T}[1^{\hat{n}_{1}}2^{\hat{n}_{2}}3^{\hat{n}_{3}}4^{\hat{n}_{4}}]$
can be obtained by the following replacements:

${s}_{ij}\,\longrightarrow\leavevmode\nobreak\
s_{ij}-\left(\\!\frac{\leavevmode\nobreak\
\hat{n}_{i}\\!+\hat{n}_{j}\leavevmode\nobreak\ }{R}\\!\right)^{\\!\\!2}.$
(3.25)

This procedure can be applied to deriving the general $N$-point scattering
amplitudes of massive KK gauge bosons and be extended to the case of KK
gravitons which we will present elsewhere.

For the color-ordered combinations [1234] and [1243], we derive the following
massive KK gauge boson amplitudes according to Eq.(LABEL:eq:KK-num-4-point):

$\displaystyle\mathcal{T}[1^{n}2^{n}3^{n}4^{n}]$
$\displaystyle\,=\,\frac{1}{\,2\,}\left(\mathcal{T}_{1}^{\mathrm{el}}+\mathcal{T}_{2}^{\mathrm{el}}+\mathcal{T}_{3}^{\mathrm{el}}\right)\\!,$
(3.26a) $\displaystyle\mathcal{T}[1^{n}2^{n}4^{n}3^{n}]$
$\displaystyle\,=\,\frac{1}{\,2\,}\\!\left(\\!\frac{\,t\,}{\,u\,}\hskip
0.85358pt\mathcal{T}_{1}^{\mathrm{el}}+\frac{t}{\,u-4M_{n}^{2}\,}\hskip
0.85358pt\mathcal{T}_{2}^{\mathrm{el}}+\frac{\,t-4M_{n}^{2}\,}{u}\hskip
0.85358pt\mathcal{T}_{3}^{\mathrm{el}}\right)\\!,$ (3.26b)

where the basis amplitudes $\\{\mathcal{T}_{1}^{\mathrm{el}},\hskip
0.85358pt\mathcal{T}_{2}^{\mathrm{el}},\hskip
0.85358pt\mathcal{T}_{3}^{\mathrm{el}}\\}$ are given by the following sub-
amplitudes:

$\displaystyle\mathcal{T}_{1}^{\mathrm{el}}$ $\displaystyle=\mathcal{T}[1^{\pm
n}\hskip 0.85358pt2^{\pm n}\hskip 0.85358pt3^{\mp n}\hskip 0.85358pt4^{\mp
n}]=\mathcal{T}[1^{0}\hskip 0.85358pt2^{0}\hskip 0.85358pt3^{0}\hskip
0.85358pt4^{0}]\Big{|}_{s\hskip 1.42262pt\rightarrow\hskip 0.85358pt(s\hskip
0.85358pt-\hskip 0.85358pt4M_{n}^{2})}\,,$ (3.27a)
$\displaystyle\mathcal{T}_{2}^{\mathrm{el}}$ $\displaystyle=\mathcal{T}[1^{\pm
n}\hskip 0.85358pt2^{\mp n}\hskip 0.85358pt3^{\pm n}\hskip 0.85358pt4^{\mp
n}]=\mathcal{T}[1^{0}\hskip 0.85358pt2^{0}\hskip 0.85358pt3^{0}\hskip
0.85358pt4^{0}]\Big{|}_{u\hskip 1.42262pt\rightarrow\hskip 0.85358pt(u\hskip
0.85358pt-\hskip 0.85358pt4M_{n}^{2})}\,,$ (3.27b)
$\displaystyle\mathcal{T}_{3}^{\mathrm{el}}$ $\displaystyle=\mathcal{T}[1^{\pm
n}\hskip 0.85358pt2^{\mp n}\hskip 0.85358pt3^{\mp n}\hskip 0.85358pt4^{\pm
n}]=\mathcal{T}[1^{0}\hskip 0.85358pt2^{0}\hskip 0.85358pt3^{0}\hskip
0.85358pt4^{0}]\Big{|}_{t\hskip 1.42262pt\rightarrow\hskip 0.85358pt(t\hskip
0.85358pt-\hskip 0.85358pt4M_{n}^{2})}\,.$ (3.27c)

We note that the KK numbers of the sub-amplitude $\mathcal{T}[1^{\pm n}\hskip
0.85358pt2^{\mp n}\hskip 0.85358pt3^{\pm n}\hskip 0.85358pt4^{\mp n}]$ in
Eq.(3.27b) makes the shifted mass-term in Eq.(3.25) vanish, so there is
practically no replacement needed.

Following the same procedure, we derive the following color-ordered amplitudes
for the inelastic scattering channel $(n,n)\\!\rightarrow\\!(m,m)\hskip
0.85358pt$:

$\displaystyle\mathcal{T}[1^{n}2^{n}3^{m}4^{m}]\,$
$\displaystyle=\,\frac{1}{2}\\!\left(\mathcal{T}_{1}^{\mathrm{in}}+\mathcal{T}_{2}^{\mathrm{in}}\right),$
(3.28a) $\displaystyle\mathcal{T}[1^{n}2^{n}4^{m}3^{m}]\,$
$\displaystyle=\,\frac{1}{2}\\!\left(\\!\frac{t\\!-\\!M_{n-m}^{2}}{\leavevmode\nobreak\
u\\!-\\!M_{n+m}^{2}\leavevmode\nobreak\ }\hskip
0.85358pt\mathcal{T}_{1}^{\mathrm{in}}+\frac{\,t\\!-\\!M_{n+m}^{2}\,}{\leavevmode\nobreak\
u\\!-\\!M_{n-m}^{2}\leavevmode\nobreak\ }\hskip
0.85358pt\mathcal{T}_{2}^{\mathrm{in}}\right),$ (3.28b)

where the basis amplitudes $\\{\mathcal{T}_{1}^{\mathrm{in}},\hskip
0.85358pt\mathcal{T}_{2}^{\mathrm{in}}\\}$ are obtained by the relations:

$\displaystyle\mathcal{T}_{1}^{\mathrm{in}}\,$
$\displaystyle=\,\mathcal{T}[1^{\pm n}\hskip 0.85358pt2^{\mp n}\hskip
0.85358pt3^{\pm m}\hskip 0.85358pt4^{\mp m}]\,=\,\mathcal{T}[1^{0}\hskip
0.85358pt2^{0}\hskip 0.85358pt3^{0}\hskip 0.85358pt4^{0}]\Big{|}_{t\hskip
0.85358pt\rightarrow\hskip 0.85358pt(t\hskip 0.85358pt-\hskip
0.85358ptM_{n-m}^{2})}\hskip 0.85358pt,$ (3.29a)
$\displaystyle\mathcal{T}_{2}^{\mathrm{in}}\,$
$\displaystyle=\,\mathcal{T}[1^{\pm n}\hskip 0.85358pt2^{\mp n}\hskip
0.85358pt3^{\mp m}\hskip 0.85358pt4^{\pm m}]\,=\,\mathcal{T}[1^{0}\hskip
0.85358pt2^{0}\hskip 0.85358pt3^{0}\hskip 0.85358pt4^{0}]\Big{|}_{t\hskip
0.85358pt\rightarrow\hskip 0.85358pt(t\hskip 0.85358pt-\hskip
0.85358ptM_{n+m}^{2})}\hskip 0.85358pt.$ (3.29b)

For the inelastic scattering channel $(0,0)\rightarrow(n,n)$, we can deduce
its amplitude from that of $(n,n)\rightarrow(m,m)$ by the following
replacements:

$\displaystyle\mathcal{T}[1^{0}\hskip 0.85358pt2^{0}\hskip
0.85358pt3^{n}\hskip 0.85358pt4^{n}]\,$
$\displaystyle=\,\mathcal{T}[1^{n}\hskip 0.85358pt2^{n}\hskip
0.85358pt3^{m}\hskip 0.85358pt4^{m}]\Big{|}_{\\{n\rightarrow 0,\hskip
0.85358ptm\rightarrow n\\}}\hskip 0.85358pt,$ (3.30a)
$\displaystyle\mathcal{T}[1^{0}\hskip 0.85358pt2^{0}\hskip
0.85358pt4^{n}\hskip 0.85358pt3^{n}]\,$
$\displaystyle=\,\mathcal{T}[1^{n}\hskip 0.85358pt2^{n}\hskip
0.85358pt4^{m}\hskip 0.85358pt3^{m}]\Big{|}_{\\{n\rightarrow 0,\hskip
0.85358ptm\rightarrow n\\}}\hskip 0.85358pt.$ (3.30b)

##  KK Graviton Amplitudes from Extended Massive Double-Copy

According to the extended massive KLT-like relation (2.22a), we can construct
explicitly the four-point massive KK closed-string amplitude from the product
of the corresponding massive KK open-string amplitudes as follows:

$\mathcal{A}_{\text{cl}}(\zeta,k)=\frac{\leavevmode\nobreak\
8\kappa^{2}\,}{\leavevmode\nobreak\ \pi\alpha^{\prime\hskip
0.85358pt5}\leavevmode\nobreak\ }\\!\\!\sum_{\\{a_{j},\hskip
1.42262ptb_{j}\\}}\\!\sum_{\\{\text{sgn}_{j}\\}}\\!\widehat{\varrho}_{ab}\sin(\pi\alpha^{\prime}\hat{k}_{1}\\!\cdot\hat{k}_{2})\Big{\\{}\hbox
to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}_{\text{op}}[\zeta_{j}^{a_{j}}\\!,\hskip
0.85358pt\hat{k}_{j}\big{|}1234]\\!\times\\!\hbox to0.0pt{\hskip
2.99998pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\mathcal{A}$}}_{\text{op}}\big{[}\zeta_{j}^{b_{j}}\\!,\hskip
0.85358pt\hat{k}_{j}\big{|}1243\big{]}\Big{\\}},$ (4.1)

where we have replaced the closed-string coupling by the relation $\hskip
0.85358ptg_{\text{cl}}\\!=\\!\kappa/(2\pi\alpha^{\prime})\hskip 0.85358pt$.
For the two massive KK open-string amplitudes inside $\\{\cdots\\}$, the Regge
slope should be rescaled as
$\,\alpha^{\prime}\\!\rightarrow\\!\alpha^{\prime}/4\,$. This is equivalent to
considering the $N$-point low energy field theory formula (2.4) and derive the
four-longitudinal KK graviton amplitude for the case of $\hskip
0.85358ptN\\!=4\hskip 1.42262pt$. We illustrate in Fig. 4 the extended massive
KLT-like relation (4.1) between the four-point scattering amplitude of KK
closed-strings and the products of two color-ordered scattering amplitudes of
KK open-strings.

Figure 4: Four-point scattering amplitude of massive KK closed-strings with
$\mathbb{Z}_{2}$-even parity (marked in blue) can be decomposed into a sum of
the products of two color-ordered amplitudes of massive KK open-strings under
the $\SS^{1}$ compactification of 26d for bosonic strings.

###  Constructing Elastic Scattering Amplitudes of Four KK Gravitons

In this subsection, we construct the four-point elastic scattering amplitudes
of KK gravitons by using the extended massive KLT-like relation (4.1) under
the low energy field theory limit.

Taking the zero-slope limit $\,\alpha^{\prime}\\!\rightarrow\\!0\,$ for the
closed-string amplitude (4.1), we derive the four-longitudinal KK graviton
scattering amplitude for the elastic channel $(n,\hskip
0.85358ptn)\\!\rightarrow\\!(n,\hskip 0.85358ptn)\hskip 0.85358pt$. Thus, we
can express the elastic amplitude of longitudinal KK gravitons as the sum of
products of two color-ordered massive KK gauge boson amplitudes:

$\displaystyle\mathcal{M}[1^{n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt3^{n}_{L}\hskip 0.85358pt4^{n}_{L}]$
$\displaystyle\,=\,\frac{\,\kappa^{2}\,}{\,32\,}\sum_{\\{a_{j},\hskip
1.42262ptb_{j}\\}}\sum_{\\{\text{sgn}_{j}\\}}\\!\widehat{\varrho}_{ab}\hskip
0.85358pt(-\hat{k}_{1}\\!\cdot\hat{k}_{2})\hskip
0.85358pt{\mathcal{T}}\big{[}\zeta^{a_{j}}_{j},\hat{k}_{j}\big{|}1234\hskip
0.85358pt\big{]}\\!\times\\!{\mathcal{T}}\big{[}\zeta^{b_{j}}_{j},\hat{k}_{j}\big{|}1243\hskip
0.85358pt\big{]}$
$\displaystyle\,=\,\frac{\,\kappa^{2}\,}{32}\sum_{\\{a_{j},\hskip
1.42262ptb_{j}\\}}\\!\widehat{\varrho}_{ab}\\!\left\\{\left(s\\!-\\!4M_{n}^{2}\right)\mathcal{T}_{a_{j}}[1^{+n}\hskip
0.85358pt2^{+n}\hskip 0.85358pt3^{-n}\hskip
0.85358pt4^{-n}]\,\mathcal{T}_{b_{j}}[1^{+n}\hskip 0.85358pt2^{+n}\hskip
0.85358pt4^{-n}\hskip 0.85358pt3^{-n}]\right.$ $\displaystyle\hskip
119.78612pt+s\,\mathcal{T}_{a_{j}}[1^{+n}\hskip 0.85358pt2^{-n}\hskip
0.85358pt3^{+n}\hskip 0.85358pt4^{-n}]\,\mathcal{T}_{b_{j}}[1^{+n}\hskip
0.85358pt2^{-n}\hskip 0.85358pt4^{-n}\hskip 0.85358pt3^{+n}]$
$\displaystyle\hskip 120.92421pt\left.+\hskip
0.85358pts\,\mathcal{T}_{a_{j}}[1^{+n}\hskip 0.85358pt2^{-n}\hskip
0.85358pt3^{-n}\hskip 0.85358pt4^{+n}]\,\mathcal{T}_{b_{j}}[1^{+n}\hskip
0.85358pt2^{-n}\hskip 0.85358pt4^{+n}\hskip 0.85358pt3^{-n}]\right\\},$ (4.2)

where in the above second equality we have defined the short-hand notation for
the partial amplitudes as $\hskip 0.85358pt\mathcal{T}_{a_{j}}[1^{+n}\hskip
0.85358pt2^{-n}\hskip 0.85358pt3^{+n}\hskip
0.85358pt4^{-n}]\\!=\\!\mathcal{T}[1^{+n}_{a_{1}}\hskip
0.85358pt2^{-n}_{a_{2}}\hskip 0.85358pt3^{+n}_{a_{3}}\hskip
0.85358pt4^{-n}_{a_{4}}]\hskip 0.85358pt$, and so on. Note that a massive KK
graviton in 4d has 5 physical helicity states and their polarization tensors
are given by

$\displaystyle\zeta^{\mu\nu}_{\pm 2}$ $\displaystyle=\zeta_{\pm
1}^{\mu}\zeta_{\pm 1}^{\nu}\,,\qquad\zeta_{\pm
1}^{\mu\nu}=\mbox{$\frac{\,{1}\,}{\sqrt{2\,}\,}$}\\!\left(\zeta_{\pm
1}^{\mu}\zeta_{L}^{\nu}\\!+\zeta_{L}^{\mu}\zeta_{\pm 1}^{\nu}\right)\\!,$
(4.3a) $\displaystyle\zeta^{\mu\nu}_{L}$
$\displaystyle=\mbox{$\frac{\,{1}\,}{\sqrt{6\,}\,}$}\\!\left(\zeta^{\mu}_{+1}\zeta^{\nu}_{-1}+\zeta^{\mu}_{-1}\zeta^{\nu}_{+1}+2\hskip
0.85358pt\zeta^{\mu}_{L}\zeta^{\nu}_{L}\right)\\!.$ (4.3b)

Thus, in Eq.(4.1) the helicity indices of each partial amplitude are
$\\{a_{j},b_{j}\\}=\\{\pm 1,L\\}$. For the graviton polarization tensor
$\zeta^{\mu\nu}_{\pm 2}$, its coefficient
$\varrho_{a_{j}b_{j}}\\!\\!=\\!\varrho_{\pm 1,\pm 1}\\!=\\!1$; for the
polarization tensor $\zeta^{\mu\nu}_{\pm 1}$, its coefficient
$\varrho_{a_{j}b_{j}}\\!\\!=\\!\varrho_{\pm 1,L}\\!\\!=\\!\varrho_{L,\pm
1}\\!\\!=\\!\mbox{$\frac{\,{1}\,}{\sqrt{2\,}\,}$}$; and for the polarization
tensor $\zeta^{\mu\nu}_{L}$, its coefficient $\varrho_{a_{j}b_{j}}\\!$ takes
the following values:

$\varrho_{a_{j}b_{j}}=\left\\{\begin{array}[]{rcl}\sqrt{\frac{1}{6}}\,,\nobreak\leavevmode\nobreak\leavevmode&\mathrm{for}&\\{a_{j},b_{j}\\}=\\{\pm
1,\mp 1\\}\hskip 0.85358pt,\\\\[4.2679pt]
\sqrt{\frac{2}{3}}\,,\nobreak\leavevmode\nobreak\leavevmode&\mathrm{for}&\\{a_{j},b_{j}\\}=L\hskip
0.85358pt,\\\\[5.69054pt]
0\,,\nobreak\leavevmode\nobreak\leavevmode&\mathrm{for}&\mathrm{others}\hskip
0.85358pt.\end{array}\right.$ (4.4)

We note that the longitudinal graviton’s polarization tensor (4.3b) includes 3
products of two massive spin-1 polarization vectors, among which the two
products $\zeta_{\mu}^{+1}\zeta_{\nu}^{-1}$ and
$\zeta_{\mu}^{-1}\zeta_{\nu}^{+1}$ arise from the transverse polarization
vectors and are important for the successful construction of full massive
graviton amplitudes.

Then, we can compute all the color-ordered four-point gauge boson amplitudes
from Eqs.(3.24) and (3.27) with the relevant external polarization vectors
(A.5) and momenta (A.1). Using these amplitudes, we derive the four-point
elastic longitudinal KK graviton scattering amplitude from the extended
massive double-copy formula (4.1):

$\displaystyle\mathcal{M}[1^{n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt3^{n}_{L}\hskip 0.85358pt4^{n}_{L}]\,=\,-\frac{\leavevmode\nobreak\
\kappa^{2}M_{n}^{2}\hskip
0.85358pt(X_{0}+\\!X_{2}c_{2\theta}+\\!X_{4}c_{4\theta}+\\!X_{6}c_{6\theta})\csc^{2}\\!\theta\leavevmode\nobreak\
}{\leavevmode\nobreak\ 512\hskip 0.85358pt\bar{s}\hskip
0.85358pt(\bar{s}-4)[\hskip
0.85358pt\bar{s}^{2}\\!-(\bar{s}-4)^{2}c_{2\theta}\\!+24\hskip
0.85358pt\bar{s}+\\!16]\leavevmode\nobreak\ }\,,$ (4.5)

where the coefficients $\\{X_{j}\\}$ in the numerator are given by

$\displaystyle X_{0}$ $\displaystyle=-2\hskip 0.85358pt(255\hskip
0.85358pt\bar{s}^{5}\\!+2824\hskip 0.85358pt\bar{s}^{4}\\!-\\!19936\hskip
0.85358pt\bar{s}^{3}\\!+39936\hskip 0.85358pt\bar{s}^{2}\\!-256\hskip
0.85358pt\bar{s}+\\!14336)\hskip 0.85358pt,$ (4.6) $\displaystyle X_{2}$
$\displaystyle=429\hskip 0.85358pt\bar{s}^{5}\\!-10152\hskip
0.85358pt\bar{s}^{4}\\!+30816\hskip 0.85358pt\bar{s}^{3}\\!-27136\hskip
0.85358pt\bar{s}^{2}\\!-49920\bar{s}+34816\hskip 0.85358pt,$ $\displaystyle
X_{4}$ $\displaystyle=2\hskip 0.85358pt(39\hskip
0.85358pt\bar{s}^{5}\\!-312\hskip 0.85358pt\bar{s}^{4}\\!-2784\hskip
0.85358pt\bar{s}^{3}\\!-11264\hskip
0.85358pt\bar{s}^{2}\\!+26368\bar{s}-\\!2048)\hskip 0.85358pt,$ $\displaystyle
X_{6}$ $\displaystyle=3\bar{s}^{5}\\!+40\hskip
0.85358pt\bar{s}^{4}\\!+416\hskip 0.85358pt\bar{s}^{3}\\!-1536\hskip
0.85358pt\bar{s}^{2}\\!-3328\hskip 0.85358pt\bar{s}-2048\hskip 0.85358pt.$

The above formulas take exactly the same form as Eq.(F.3a) and
Eqs.(F.4a)-(F.4d) of Ref. [14]. We can compare our above elastic longitudinal
KK graviton amplitude with that obtained by the previous explicit lengthy
Feynman diagram calculations [25]. It is truly impressive that we find full
agreement between our Eqs.(4.5)-(4.6) and the Eq.(71) of Ref. [25] after
taking into account the notational difference. Hence, our string-based massive
double-copy construction does successfully predict the exact four-point
elastic scattering amplitude of longitudinal KK gravitons at tree level. In
the next subsection, we will further present our string-based massive double-
copy constructions of the inelastic scattering amplitudes of massive KK
gravitons.

We can further derive the following LO and NLO scattering amplitudes of
Eq.(4.5) under high energy expansion:

$\displaystyle\mathcal{M}_{0}[1^{n}_{L}\hskip 1.42262pt2^{n}_{L}\hskip
1.42262pt3^{n}_{L}\hskip 1.42262pt4^{n}_{L}]\,$
$\displaystyle=\,\frac{\leavevmode\nobreak\ 3\hskip
0.85358pt\kappa^{2}\,}{\,128\,}\hskip 0.85358pts\hskip
0.85358pt(7+c_{2\theta})^{2}\csc^{2}\\!\theta\,,$ (4.7a)
$\displaystyle\delta\mathcal{M}[1^{n}_{L}\hskip 1.42262pt2^{n}_{L}\hskip
1.42262pt3^{n}_{L}\hskip 1.42262pt4^{n}_{L}]\,$
$\displaystyle=\,-\frac{\,\kappa^{2}M_{n}^{2}\leavevmode\nobreak\
}{256}(1810+93\hskip 0.85358ptc_{2\theta}+126\hskip
0.85358ptc_{4\theta}+19\hskip 0.85358ptc_{6\theta})\csc^{4}\\!\theta\,.$
(4.7b)

We stress that the string-based double-copy formula (4.1) for 4-point
amplitudes or (2.4) for general $N$-point amplitudes gives an explicit
prescription on how to practically construct the exact tree-level KK graviton
scattering amplitudes from the sum of the products of the corresponding KK
gauge boson amplitudes in the compactified KK field theories. Hence, given our
string-based double-copy formula (2.4) or (4.1), one can practically follow
this explicit prescription to derive the full KK graviton amplitudes without
relying on computing the original KK string amplitudes.

Some further remarks are in order. It is instructive to compare our above
string-based massive double-copy construction (à la extended KLT-like
relations) with the extended BCJ-type double-copy construction under high
energy expansion as given by Sec. 5 of Ref. [14]. At the leading order (LO) of
the high energy expansion, both the 4-point longitudinal KK gauge boson
amplitude and KK graviton amplitude are mass-independent, which are of
$\mathcal{O}(E^{0})$ and $\mathcal{O}(E^{2})$, respectively. We found [14]
that the BCJ-type numerators in the LO gauge boson amplitude satisfy the
kinematic Jacobi identity at the $O(E^{2}M_{n}^{0})$, so the extension from
the conventional massless BCJ method can be realized directly. At the next-to-
leading order (NLO) of the high energy expansion, the longitudinal KK gauge
boson amplitude and KK graviton amplitude become mass-dependent, which are of
$\mathcal{O}(M_{n}^{2}/E^{2})$ and $\mathcal{O}(M_{n}^{2}E^{0})$,
respectively. We further found [14] that the BCJ-type NLO numerators of the KK
gauge boson amplitude can obey the Jacobi identity (after the generalized
gauge transformations) and the double-copied KK graviton amplitude at NLO can
give the correct structure of the exact NLO KK graviton amplitude, but the
numeric coefficients of the double-copied NLO amplitude still differ somewhat
from that of the exact NLO amplitude. So the NLO double-copy construction
needs to be modified [14]. We note that an important reason of this problem is
because the 4-point elastic amplitude of longitudinal KK gauge bosons contains
double-poles with one type of poles from exchanging the massless zero-mode and
another type of mass-poles from exchanging the level-$(2n)$ KK-mode which
contributes to the mass-dependent NLO amplitude. This is beyond the
conventional BCJ double-copy method [7][8], so the deviation from it is
expected at the NLO and a modified BCJ-type double-copy construction was
presented for the NLO KK amplitudes [14]. As another reason, we note that the
polarization tensor $\zeta_{L}^{\mu\nu}$ of the (helicity-zero) longitudinal
KK graviton in Eq.(4.3b) contains the sum of three products of two gauge boson
polarization vectors
$(\zeta^{\mu}_{+1}\zeta^{\nu}_{-1},\,\zeta^{\mu}_{-1}\zeta^{\nu}_{+1},\,\zeta^{\mu}_{L}\zeta^{\nu}_{L})$,
while the simple double-copy by using the longitudinal KK gauge boson
amplitude alone could only provide the polarization-vector product of $\hskip
0.85358pt\zeta^{\mu}_{L}\zeta^{\nu}_{L}$, which does not include the other two
products
$(\zeta^{\mu}_{+1}\zeta^{\nu}_{-1},\,\zeta^{\mu}_{-1}\zeta^{\nu}_{+1})$ of
$\,\zeta_{L}^{\mu\nu}$ as given by the spin-1 helicity combinations $(+1,-1)$
and $(-1,+1)$. However, the sum of all three helicity combinations
$(+1\\!-\\!1,\,-1\\!+\\!1,\,LL)$ for each external longitudinal KK graviton
state in our present string-based massive double-copy formula (4.1) is
automatically built in from the beginning. Another key feature of our string-
based construction (4.1) is that it intrinsically includes a set of KK gauge
boson sub-amplitudes with different KK-number combinations as allowed by the
condition (LABEL:eq:KK-num-4-point) due to the original string
compactification under $\SS^{1}$ (without the orbifold $\mathbb{Z}_{2}$) in
Sec. 2.1. This feature is intrinsically built in for our string-based double-
copy formulation. Using this string-based formulation, we will derive a
precise BCJ-type double-copy formulation in the future work.

We further note that extended BCJ-type double-copy construction of Ref. [14]
does work elegantly for the LO amplitudes under the high energy expansion.
This is highly nontrivial because the longitudinal KK graviton polarization
tensor $\zeta_{L}^{\mu\nu}$ also contains additional products of transverse
polarization vectors
$(\zeta^{\mu}_{+1}\zeta^{\nu}_{-1},\,\zeta^{\mu}_{-1}\zeta^{\nu}_{+1})$ which
can contribute to the LO KK graviton amplitudes of
$\mathcal{O}(E^{2}M_{n}^{0})$. Note that our extended massive double-copy of
LO amplitudes is built upon the KK gauge boson equivalence theorem (KK GAET)
[18][21] which connects the LO longitudinal KK gauge boson amplitude
$\mathcal{T}_{0}[4A_{L}^{an}]$ to its corresponding LO KK Goldstone amplitude
$\widetilde{\mathcal{T}}_{0}[4A_{5}^{an}]$:

$\mathcal{T}_{0}[4A_{L}^{an}]\,=\,\widetilde{\mathcal{T}}_{0}[4A_{5}^{an}]=\mathcal{O}(E^{0}M_{n}^{0})\hskip
0.85358pt,$ (4.8)

where we have introduced the short-hand notations
$\mathcal{T}_{0}[4A_{L}^{an}]\\!=\\!\mathcal{T}_{0}[A_{L}^{an}A_{L}^{bn}\\!\\!\rightarrow\\!A_{L}^{cn}A_{L}^{dn}]$
and
$\widetilde{\mathcal{T}}_{0}[4A_{5}^{an}]\\!=\\!\widetilde{\mathcal{T}}_{0}[A_{5}^{an}A_{5}^{bn}\\!\\!\rightarrow\\!A_{5}^{cn}A_{5}^{dn}]\hskip
0.85358pt$. We note that the KK Goldstone boson $A_{5}^{an}$ is just a scalar
field without any polarization vector, and $A_{5}^{an}$ becomes a massless
physical scalar degree of freedom in the high energy limit. Hence, the double-
copy construction of the gravitational KK Goldstone boson
$h^{55}_{n}$-amplitude from the KK Goldstone $A_{n}^{a5}$-amplitude is
uniquely defined via the correspondence $\,A_{n}^{a5}\otimes
A_{n}^{a5}\rightarrow h^{55}_{n}\,$. Hence, the LO gravitational KK Goldstone
amplitude $\widetilde{\mathcal{M}}_{0}[4h_{n}^{55}]$ as given by the double-
copy of the LO KK Goldstone amplitude
$\widetilde{\mathcal{T}}_{0}[4A_{n}^{a5}]$ is well defined under one-to-one
correspondence. On the other hand, we established the Gravitational
Equivalence Theorem (GRET) [14] which connects the LO longitudinal KK graviton
amplitude $\mathcal{M}_{0}[4h_{L}^{n}]$ to its corresponding LO gravitational
KK Goldstone amplitude $\widetilde{\mathcal{M}}_{0}[4h^{n}_{55}]\hskip
0.85358pt$:

$\mathcal{M}_{0}[4h_{L}^{n}]\,=\,\widetilde{\mathcal{M}}_{0}[4h^{n}_{55}]=\mathcal{O}(E^{2}M_{n}^{0})\hskip
0.85358pt,$ (4.9)

where we have introduced the shorthand notations
$\mathcal{M}_{0}[4h_{L}^{n}]\\!=\\!\mathcal{M}_{0}[h_{L}^{n}h_{L}^{n}\\!\\!\rightarrow\\!h_{L}^{n}h_{L}^{n}]$
and
$\widetilde{\mathcal{M}}_{0}[4h^{n}_{55}]\\!=\\!\mathcal{M}_{0}[h^{n}_{55}h^{n}_{55}\\!\rightarrow\\!h^{n}_{55}h^{n}_{55}]\hskip
0.85358pt$. Thus, the LO equality of the KK GAET (4.8) leads to the LO
equality of the KK GRET (4.9) by the double-copy construction. Hence, the
success of our extended double-copy construction of the LO longitudinal KK
graviton amplitude $\mathcal{M}_{0}[4h_{L}^{n}]$ from the LO longitudinal KK
gauge boson amplitude $\mathcal{T}_{0}[4A_{L}^{an}]$ is fully ensured by the
correct double-copy construction of its corresponding LO gravitational KK
Goldstone amplitude $\widetilde{\mathcal{M}}_{0}[4h_{n}^{55}]$ from the LO
gauge KK Goldstone amplitude $\widetilde{\mathcal{T}}_{0}[4A_{n}^{a5}]$
because we can derive the correspondence GAET $\Longrightarrow$ GRET by using
the LO double-copy construction. This means that we can correctly compute the
LO longitudinal KK graviton amplitude $\mathcal{M}_{0}[4h_{L}^{n}]$ by
defining an effective LO polarization tensor for each longitudinal KK
graviton:

$\zeta_{L(0)}^{\mu\nu}\,=\,\zeta_{L}^{\mu}\hskip 0.85358pt\zeta_{L}^{\nu}\,.$
(4.10)

We have verified this insight by explicitly computing the four-point elastic
scattering amplitude $\mathcal{M}_{0}[4h_{L}^{n}]$ of longitudinal KK
gravitons with the LO polarization tensor (4.10). Namely, using the LO
longitudinal polarization tensor (4.10) can give the same LO longitudinal
graviton amplitude as that of the full longitudinal polarization tensor
(4.3b). The difference between the longitudinal KK graviton amplitudes as
computed by using the two types of polarization tensors (4.3b) and (4.10)
belongs to the mass-dependent residual term of the GRET [14],

$\mathcal{M}_{\Delta}=\,\delta\mathcal{M}[4h_{L}^{n}]-\delta\widetilde{\mathcal{M}}[4h^{n}_{55}]=\,\mathcal{O}(E^{0}M_{n}^{2})\,,$
(4.11)

where $\mathcal{M}\\!=\mathcal{M}_{0}+\delta\mathcal{M}\hskip 0.85358pt$ and
$\widetilde{\mathcal{M}}=\widetilde{\mathcal{M}}_{0}+\delta\widetilde{\mathcal{M}}\hskip
0.85358pt$ under the high energy expansion. The residual term (4.11) is
derived from the GRET identity [14],
$\,\mathcal{M}[4h_{L}^{n}]=\widetilde{\mathcal{M}}[4h^{n}_{55}]+\mathcal{M}_{\Delta}$,
together with the LO GRET equality (4.9).

###  Constructing Inelastic Scattering Amplitudes of Four KK Gravitons

In this subsection, we study the four-point inelastic scattering channels
$(n,\hskip 0.85358ptn)\\!\rightarrow\\!(m,\hskip 0.85358ptm)\hskip 0.85358pt$
and $(0,\hskip 0.85358pt0)\\!\rightarrow\\!(n,\hskip 0.85358ptn)\hskip
0.85358pt$ for the massive KK closed-string amplitudes (4.1). We will take the
low energy field theory limit $\,\alpha^{\prime}\\!\rightarrow\\!\hskip
0.85358pt0\,$ and derive the inelastic scattering amplitudes of four-
longitudinal KK gravitons.

For inelastic scattering channels $(n,\hskip
0.85358ptn)\\!\rightarrow\\!(m,\hskip 0.85358ptm)\hskip 0.85358pt$, we obtain
the following extended massive KLT-like four-point formula, which expresses
the inelastic scattering amplitude of longitudinal KK gravitons as the sum of
products of the corresponding color-ordered KK gauge boson scattering
amplitudes:

$\displaystyle\mathcal{M}[1^{n}_{L}\hskip 1.42262pt2^{n}_{L}\hskip
1.42262pt3^{m}_{L}\hskip 1.42262pt4^{m}_{L}]$
$\displaystyle\,=\,\frac{\,\kappa^{2}\,}{\,32\,}\\!\sum_{\\{a_{j},\hskip
1.42262ptb_{j}\\}}\\!\widehat{\varrho}_{ab}\left\\{s\hskip
1.42262pt\mathcal{T}_{a_{j}}[1^{+n}\hskip 0.85358pt2^{-n}\hskip
0.85358pt3^{+m}\hskip 0.85358pt4^{-m}]\,\mathcal{T}_{b_{j}}[1^{+n}\hskip
0.85358pt2^{-n}\hskip 0.85358pt4^{-m}\hskip 0.85358pt3^{+m}]\right.$
$\displaystyle\hskip 75.3998pt\left.+\hskip 1.42262pts\hskip
1.42262pt\mathcal{T}_{a_{j}}[1^{+n}\hskip 0.85358pt2^{-n}\hskip
0.85358pt3^{-m}\hskip 0.85358pt4^{+m}]\,\mathcal{T}_{b_{j}}[1^{+n}\hskip
0.85358pt2^{-n}\hskip 0.85358pt4^{+m}\hskip 0.85358pt3^{-m}]\right\\}.$ (4.12)

Then, using the color-ordered inelastic KK gauge boson scattering amplitudes
in section 3.1 and our extended massive double-copy formula (4.2), we
construct the four-point inelastic longitudinal KK graviton scattering
amplitude of $\hskip 0.85358pt(n,\hskip
0.85358ptn)\\!\\!\rightarrow\\!(m,\hskip 0.85358ptm)\hskip 0.85358pt$, which
takes the following compact form:

$\mathcal{M}[1^{n}_{L}\hskip 1.42262pt2^{n}_{L}\hskip 1.42262pt3^{m}_{L}\hskip
1.42262pt4^{m}_{L}]\,=\,-\frac{\leavevmode\nobreak\
\kappa^{2}M_{n}^{2}(X_{0}+\\!X_{1}c_{\theta}+\\!X_{2}c_{2\theta})(\hskip
0.85358ptY_{0}+Y_{2}\hskip 0.85358ptc_{2\theta}+Y_{4}\hskip
0.85358ptc_{4\theta}+Y_{6}\hskip 0.85358ptc_{6\theta})\leavevmode\nobreak\
}{\leavevmode\nobreak\ 1024\hskip 0.85358pt\bar{s}\left[\,(\bar{s}-4\hskip
0.85358pt\bar{q}\hskip 0.85358pt\bar{q}^{\prime}c_{\theta})^{2}\\!+\\!16\hskip
0.85358ptr^{2}\,\right]^{2}\left[\,(\bar{s}+4\hskip 0.85358pt\bar{q}\hskip
0.85358pt\bar{q}^{\prime}c_{\theta})^{2}\\!-\\!16\hskip
0.85358ptr^{2}\,\right]\leavevmode\nobreak\ }\,,$ (4.13)

where the coefficients $(X_{j},\,Y_{j})$ in the numerator are polynomial
functions of the scattering energy and momenta,

$\displaystyle X_{0}=$ $\displaystyle\ 3\hskip 0.85358pt\bar{s}^{2}\\!-4\hskip
0.85358ptr_{+}^{2}\bar{s}-\\!16\hskip 0.85358ptr^{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ X_{1}=-16\hskip
0.85358pt\bar{q}\hskip 0.85358pt\bar{q}^{\prime}\bar{s}\hskip
0.85358pt,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
X_{2}=16\hskip 0.85358pt\bar{q}^{2}\bar{q}^{\prime 2},$ (4.14) $\displaystyle
Y_{0}=$ $\displaystyle\hskip 0.85358pt-170\hskip
0.85358pt\bar{s}^{6}\\!-84\hskip 0.85358ptr_{+}^{2}\hskip
0.85358pt\bar{s}^{5}\\!+32\hskip 0.85358pt(39\hskip
0.85358ptr^{4}\\!+\\!135\hskip 0.85358ptr^{2}\\!+39)\hskip
0.85358pt\bar{s}-128\hskip 0.85358pt(17\hskip 0.85358ptr^{6}\\!+\\!69\hskip
0.85358ptr^{4}\\!+\\!69\hskip 0.85358ptr^{2}\\!+17)\hskip
0.85358pt\bar{s}^{3}$ $\displaystyle+512\hskip 0.85358pt(39\hskip
0.85358ptr^{4}\\!+\\!51\hskip 0.85358ptr^{2}\\!+\\!39)\hskip
0.85358ptr^{2}\hskip 0.85358pt\bar{s}^{2}-51200\hskip
0.85358pt(r^{2}\\!+1)\hskip 0.85358pt\bar{s}+57344\hskip 0.85358ptr^{6}\hskip
0.85358pt,$ $\displaystyle Y_{2}=$ $\displaystyle\ 143\hskip
0.85358pt\bar{s}^{6}\\!-1804\hskip
0.85358ptr_{+}^{2}\bar{s}^{5}\\!+\\!16\hskip 0.85358pt(227\hskip
0.85358ptr^{4}\\!+\\!391\hskip 0.85358ptr^{2}\\!+227)\hskip
0.85358pt\bar{s}^{4}\\!-64\hskip 0.85358pt(15\hskip
0.85358ptr^{6}\\!-\\!121\hskip 0.85358ptr^{4}\\!-\\!121\hskip
0.85358ptr^{2}\\!+\\!15)\hskip 0.85358pt\bar{s}^{3}$ $\displaystyle-256\hskip
0.85358pt(61\hskip 0.85358ptr^{4}\\!+345\hskip 0.85358ptr^{2}\\!+61)\hskip
0.85358ptr^{2}\hskip 0.85358pt\bar{s}^{2}\\!+95232\hskip
0.85358ptr_{+}^{2}\hskip 0.85358ptr^{4}\hskip 0.85358pt\bar{s}-69632\hskip
0.85358ptr^{6}\hskip 0.85358pt,$ $\displaystyle Y_{4}=$ $\displaystyle\
2\left[13\hskip 0.85358pt\bar{s}^{6}\\!-\\!44\hskip 0.85358ptr_{+}^{2}\hskip
0.85358pt\bar{s}^{5}\\!-\\!16\hskip 0.85358pt(23\hskip
0.85358ptr^{4}\\!+\\!79\hskip 0.85358ptr^{2}\\!+23)\hskip
0.85358pt\bar{s}^{4}\\!+64\hskip 0.85358pt(9\hskip 0.85358ptr^{6}\\!+29\hskip
0.85358ptr^{4}\\!+29\hskip 0.85358ptr^{2}\\!+9)\hskip
0.85358pt\bar{s}^{3}\right.$ $\displaystyle\left.+\hskip 0.85358pt256\hskip
0.85358pt(25\hskip 0.85358ptr^{4}\\!+\\!53r^{2}\\!+\\!25)\hskip
0.85358ptr^{2}\hskip 0.85358pt\bar{s}^{2}\\!-23552\hskip
0.85358ptr_{+}^{2}\hskip 0.85358ptr^{4}\hskip 0.85358pt\bar{s}+4096\hskip
0.85358ptr^{6}\hskip 0.85358pt\right]\\!,$ $\displaystyle Y_{6}=$
$\displaystyle\ (\bar{s}^{3}\\!+\\!12\hskip 0.85358ptr^{2}\hskip
0.85358pt\bar{s}^{2}\\!-\\!48\hskip 0.85358ptr^{4}\hskip
0.85358pt\bar{s}\\!-64r^{6})(\bar{s}^{3}\\!+\\!12\hskip
0.85358pt\bar{s}^{2}\\!-\\!48\hskip 0.85358pt\bar{s}\\!-64)\hskip 0.85358pt,$

where $\,\bar{s}=s/M_{n}^{2}\,$, $\,r\\!=\\!M_{m}/M_{n}$, and
$\,r_{+}^{2}\\!=\\!1+r^{2}$. From the above Eq.(4.13), we can further derive
the LO and NLO scattering amplitudes of the KK gravitons under high energy
expansion:

$\displaystyle\mathcal{M}_{0}[1^{n}_{L}\hskip 1.42262pt2^{n}_{L}\hskip
1.42262pt3^{m}_{L}\hskip 1.42262pt4^{m}_{L}]\,$
$\displaystyle=\,\frac{\,\kappa^{2}\,}{\,64\,}\hskip 0.85358pts\hskip
0.85358pt(7+c_{2\theta})\csc^{2}\theta\,=\,\frac{2}{\,3\,}\mathcal{M}_{0}[1^{n}_{L}\hskip
1.42262pt2^{n}_{L}\hskip 1.42262pt3^{n}_{L}\hskip 1.42262pt4^{n}_{L}]\,,$
(4.15a) $\displaystyle\delta\mathcal{M}[1^{n}_{L}\hskip
1.42262pt2^{n}_{L}\hskip 1.42262pt3^{m}_{L}\hskip 1.42262pt4^{m}_{L}]\,$
$\displaystyle=\,-\frac{\,\kappa^{2}M_{n}^{2}\,}{128}(1\\!+r^{2})(410+59\hskip
0.85358ptc_{2\theta}+38\hskip 0.85358ptc_{4\theta}+5\hskip
0.85358ptc_{6\theta})\csc^{4}\\!\theta\,.$ (4.15b)

We note that the Eq.(4.15a) just equals $\frac{\,{2}\,}{3}$ times the LO
elastic KK graviton amplitude (4.7a). This factor $\frac{\,{2}\,}{3}$ can be
understood in a transparent way from our string-based double-copy formulation.
The reason is that the elastic scattering $\hskip
0.85358pt(n,n)\\!\rightarrow\\!(n,n)\hskip 0.85358pt$ contains 6 sub-
amplitudes (Fig. 1) and the inelastic scattering $\hskip
0.85358pt(n,n)\\!\rightarrow(m,m)$ includes only 4 sub-amplitudes (Fig. 2).
These sub-amplitude take the same LO form under the high energy limit which is
mass-independent. Hence, we deduce the connection factor $\hskip
0.85358pt\mbox{$\frac{\,{4}\,}{6}$}=\mbox{$\frac{\,{2}\,}{3}$}\hskip
0.85358pt$ between the two LO amplitudes.

Next, we analyze the mixed inelastic channel of KK gauge boson scattering
$\hskip 0.85358pt(0,\hskip 0.85358pt0)\\!\\!\rightarrow\\!\\!(n,\hskip
0.85358ptn)\hskip 0.85358pt$. We find that the condition (2.18) allows only
two combinations of the external KK states (as originally defined under the
$\SS^{1}$ compactification of 26d), $\\{0,\hskip 0.85358pt0,+n,-n\\}$ and
$\\{0,\hskip 0.85358pt0,-n,+n\\}$. Thus, from Eq.(4.1), we derive the
following extended four-point massive KK double-copy formula which expresses
the inelastic amplitude of longitudinal KK gravitons as the sum of products of
the corresponding color-ordered KK gauge boson amplitudes:

$\displaystyle\mathcal{M}[1^{0}_{T}\hskip 1.42262pt2^{0}_{T}\hskip
1.42262pt3^{n}_{L}\hskip
1.42262pt4^{n}_{L}]\,=\,\frac{\,\kappa^{2}\,}{\,32\,}\sum_{\\{a_{j},\hskip
1.42262ptb_{j}\\}}\\!\widehat{\varrho}_{ab}\hskip 1.42262pts\hskip
1.42262pt\mathcal{T}_{a_{j}}[1^{0}\hskip 0.85358pt2^{0}\hskip
0.85358pt3^{+n}\hskip 0.85358pt4^{-n}]\,\mathcal{T}_{b_{j}}[1^{0}\hskip
0.85358pt2^{0}\hskip 0.85358pt4^{-n}\hskip 0.85358pt3^{+n}]\,,$ (4.16)

where in the amplitude $\mathcal{M}[1^{0}_{T}\hskip 0.85358pt2^{0}_{T}\hskip
0.85358pt3^{n}_{L}\hskip 0.85358pt4^{n}_{L}]$ the subscript $\,T\\!=\\!\pm
2\,$ denotes the helicities of the massless zero-mode gravitons in the initial
state.

With the above and using our extended massive double-copy formula (4.16), we
construct the mixed four-point inelastic graviton amplitudes of $\hskip
0.85358pt(0,\hskip 0.85358pt0)\\!\rightarrow\\!(n,\hskip 0.85358ptn)\hskip
0.85358pt$ and derive following form:

$\mathcal{M}[1^{0}_{\pm 2}\hskip 1.42262pt2^{0}_{\mp 2}\hskip
1.42262pt3^{n}_{L}\hskip 1.42262pt4^{n}_{L}]\,=\,\frac{\leavevmode\nobreak\
\kappa^{2}M_{n}^{2}\hskip 0.85358pt(\bar{s}^{2}\\!+\\!16\hskip
0.85358pt\bar{s}+\\!16)\hskip 0.85358pts^{4}_{\theta}\leavevmode\nobreak\
}{16\hskip 0.85358pt[(\bar{s}+4)-(\bar{s}-4)c_{2\theta}]}\,.$ (4.17)

We can further derive the following LO and NLO amplitudes from Eq.(4.17) under
high energy expansion:

$\displaystyle\mathcal{M}_{0}[1^{0}_{\pm 2}\hskip 1.42262pt2^{0}_{\mp 2}\hskip
1.42262pt3^{n}_{L}\hskip 1.42262pt4^{n}_{L}]\,$
$\displaystyle=\,\frac{\,\kappa^{2}}{\leavevmode\nobreak\
32\leavevmode\nobreak\ }s\hskip 0.85358pts_{\theta}^{2}\,,$ (4.18a)
$\displaystyle\delta\mathcal{M}[1^{0}_{\pm 2}\hskip 1.42262pt2^{0}_{\mp
2}\hskip 1.42262pt3^{n}_{L}\hskip 1.42262pt4^{n}_{L}]$
$\displaystyle=\,\frac{\leavevmode\nobreak\
\kappa^{2}M_{n}^{2}\,}{16}(3-5\hskip 0.85358ptc_{2\theta})\,.$ (4.18b)

###  Constructing Multi-Point Scattering Amplitudes of KK Gravitons

Our above analyses of the four-point scattering amplitudes can be further
extended to the $N$-point amplitudes with $\hskip
0.85358ptN\\!\geqslant\\!5\hskip 1.42262pt$. We first consider a five-point
inelastic scattering process $(2n,n)\\!\rightarrow\\!(n,n,n)$, where all the
external KK states are set to be $\mathbb{Z}_{2}$ even. We find that the
condition (2.18) of KK number conservation allows the following combinations
of the KK numbers of the external-state gauge bosons for the sub-amplitudes:

$\displaystyle\\{+2n,+n,-n,-n,-n\\}\hskip 0.85358pt,\hskip
5.69054pt\\{+2n,-n,+n,-n,-n\\}\hskip 0.85358pt,\hskip
5.69054pt\\{+2n,-n,-n,+n,-n\\}\hskip 0.85358pt,$
$\displaystyle\\{+2n,-n,-n,-n,+n\\}\hskip 0.85358pt,\hskip
5.69054pt\\{\mathrm{all\ permutations\ of\ \left(+,-\right)}\\}\,,$ (4.19)

where we assign the KK number of particle-1 as $+2n$ for convenience.

Thus, using the general formula (2.4) of the $N$-point KK graviton amplitudes,
we derive the five-point longitudinal KK graviton scattering amplitude as
follows:

$\displaystyle\mathcal{M}[1^{2n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt3^{n}_{L}\hskip 0.85358pt4^{n}_{L}\hskip
0.85358pt5^{n}_{L}]=\frac{\kappa^{3}}{\,128\sqrt{2\,}\,}\\!\sum_{\\{a_{j},b_{j}\\}}$
$\displaystyle\widehat{\varrho}_{ab}\,\Big{\\{}\hskip 5.69054pt$
$\displaystyle\big{\\{}(s_{12}\\!-\\!9M_{n}^{2})(s_{13}\\!-\\!M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+2n}2^{+n}3^{-n}4^{-n}5^{-n}]\hskip
1.42262pt\mathcal{T}_{b_{j}}[1^{+2n}3^{-n}2^{+n}5^{-n}4^{-n}]$
$\displaystyle+(s_{12}\\!+\\!s_{23}\\!-\\!9M_{n}^{2})(s_{13}\\!-\\!M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+2n}2^{+n}3^{-n}4^{-n}5^{-n}]\hskip
1.42262pt\mathcal{T}_{b_{j}}[1^{+2n}2^{+n}3^{-n}5^{-n}4^{-n}]\big{\\}}$
$\displaystyle+\big{\\{}(s_{12}\\!-\\!M_{n}^{2})(s_{13}\\!-\\!9M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+2n}2^{-n}3^{+n}4^{-n}5^{-n}]\hskip
1.42262pt\mathcal{T}_{b_{j}}[1^{+2n}3^{+n}2^{-n}5^{-n}4^{-n}]$
$\displaystyle+(s_{12}\\!+\\!s_{23}\\!-\\!M_{n}^{2})(s_{13}\\!-\\!9M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+2n}2^{-n}3^{+n}4^{-n}5^{-n}]\hskip
1.42262pt\mathcal{T}_{b_{j}}[1^{+2n}2^{-n}3^{+n}5^{-n}4^{-n}]\big{\\}}$ (4.20)
$\displaystyle+\big{\\{}(s_{12}\\!-\\!M_{n}^{2})(s_{13}\\!-\\!M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+2n}2^{-n}3^{-n}4^{+n}5^{-n}]\hskip
1.42262pt\mathcal{T}_{b_{j}}[1^{+2n}3^{-n}2^{-n}5^{-n}4^{+n}]$
$\displaystyle+(s_{12}\\!+\\!s_{23}\\!-\\!5M_{n}^{2})(s_{13}\\!-\\!M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+2n}2^{-n}3^{-n}4^{+n}5^{-n}]\hskip
1.42262pt\mathcal{T}_{b_{j}}[1^{+2n}2^{-n}3^{-n}5^{-n}4^{+n}]\big{\\}}$
$\displaystyle+\big{\\{}(s_{12}\\!-\\!M_{n}^{2})(s_{13}\\!-\\!M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+2n}2^{-n}3^{-n}4^{-n}5^{+n}]\hskip
1.42262pt\mathcal{T}_{b_{j}}[1^{+2n}3^{-n}2^{-n}5^{+n}4^{-n}]$
$\displaystyle+(s_{12}\\!+\\!s_{23}\\!-\\!5M_{n}^{2})(s_{13}\\!-\\!M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+2n}2^{-n}3^{-n}4^{-n}5^{+n}]\hskip
1.42262pt\mathcal{T}_{b_{j}}[1^{+2n}2^{-n}3^{-n}5^{+n}4^{-n}]\big{\\}}$
$\displaystyle+\,\mathrm{all\leavevmode\nobreak\
permutations\leavevmode\nobreak\ of}\leavevmode\nobreak\ (+n,-n)\Big{\\}}\,.$

Then, we consider the six-point scattering process with all the external
states being $\mathbb{Z}_{2}$ even. The KK numbers of the six external states
are chosen as
$(\hat{n}_{1},\hat{n}_{2},\hat{n}_{3},\hat{n}_{4},\hat{n}_{5},\hat{n}_{6})\\!=\\!(n,n,n,n,n,n)\hskip
0.85358pt$. From the condition (2.18) of the KK number conservation, we deduce
the following allowed combinations of the KK numbers of the external-state
gauge bosons of the sub-amplitudes:

$\displaystyle\\{+n,\hskip 0.85358pt+n,\hskip 0.85358pt+n,\hskip
0.85358pt-n,\hskip 0.85358pt-n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,\hskip
5.69054pt\\{+n,\hskip 0.85358pt+n,\hskip 0.85358pt-n,\hskip 0.85358pt+n,\hskip
0.85358pt-n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,\hskip 5.69054pt$ (4.21)
$\displaystyle\\{+n,\hskip 0.85358pt+n,\hskip 0.85358pt-n,\hskip
0.85358pt-n,\hskip 0.85358pt+n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,\hskip
5.69054pt\\{+n,\hskip 0.85358pt+n,\hskip 0.85358pt-n,\hskip 0.85358pt-n,\hskip
0.85358pt-n,\hskip 0.85358pt+n\\}\hskip 0.85358pt,\hskip 5.69054pt$
$\displaystyle\\{+n,\hskip 0.85358pt-n,\hskip 0.85358pt+n,\hskip
0.85358pt+n,\hskip 0.85358pt-n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,\hskip
5.69054pt\\{+n,\hskip 0.85358pt-n,\hskip 0.85358pt+n,\hskip 0.85358pt-n,\hskip
0.85358pt+n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,\hskip 5.69054pt$
$\displaystyle\\{+n,\hskip 0.85358pt-n,\hskip 0.85358pt+n,\hskip
0.85358pt-n,\hskip 0.85358pt-n,\hskip 0.85358pt+n\\}\hskip 0.85358pt,\hskip
5.69054pt\\{+n,\hskip 0.85358pt-n,\hskip 0.85358pt-n,\hskip 0.85358pt+n,\hskip
0.85358pt+n,\hskip 0.85358pt-n\\}\hskip 0.85358pt,\hskip 5.69054pt$
$\displaystyle\\{+n,\hskip 0.85358pt-n,\hskip 0.85358pt-n,\hskip
0.85358pt+n,\hskip 0.85358pt-n,\hskip 0.85358pt+n\\}\hskip 0.85358pt,\hskip
5.69054pt\\{+n,\hskip 0.85358pt-n,\hskip 0.85358pt-n,\hskip 0.85358pt-n,\hskip
0.85358pt+n,\hskip 0.85358pt+n\\}\hskip 0.85358pt,\hskip 5.69054pt$
$\displaystyle\\{\mathrm{all\ permutations\ of\ \left(+,-\right)}\\}\hskip
0.85358pt.$

Using the above decomposition and Eq.(2.4) of the general $N$-point KK
graviton amplitudes, we obtain the six-point longitudinal KK graviton
scattering amplitude as follows:

$\displaystyle\mathcal{M}[1^{n}_{L}\hskip 0.85358pt2^{n}_{L}\hskip
0.85358pt3^{n}_{L}\hskip 0.85358pt4^{n}_{L}\hskip 0.85358pt5^{n}_{L}\hskip
0.85358pt6^{n}_{L}]=\frac{\kappa^{4}}{\,1024\,}\\!\sum_{\\{a_{j},b_{j}\\}}$
$\displaystyle\widehat{\varrho}_{ab}\,\Big{\\{}\hskip 7.11317pt$
$\displaystyle
s_{14}\big{\\{}(s_{12}\\!-\\!4M_{n}^{2})(s_{13}\\!-\\!4M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+n}2^{+n}3^{+n}4^{-n}5^{-n}6^{-n}]\hskip
0.85358pt\mathcal{T}_{b_{j}}[1^{+n}4^{-n}3^{+n}2^{+n}6^{-n}5^{-n}]$
$\displaystyle+(s_{12}\\!-\\!4M_{n}^{2})(s_{13}\\!+\\!s_{34}\\!-\\!4M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+n}2^{+n}3^{+n}4^{-n}5^{-n}6^{-n}]\hskip
0.85358pt\mathcal{T}_{b_{j}}[1^{+n}3^{+n}4^{-n}2^{+n}6^{-n}5^{-n}]$
$\displaystyle+(s_{13}\\!-\\!4M_{n}^{2})(s_{12}\\!+\\!s_{23}\\!-\\!8M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+n}2^{+n}3^{+n}4^{-n}5^{-n}6^{-n}]\hskip
0.85358pt\mathcal{T}_{b_{j}}[1^{+n}4^{-n}2^{+n}3^{+n}6^{-n}5^{-n}]$
$\displaystyle{-(s_{25}\\!+\\!s_{26})(s_{13}\\!-\\!4M_{n}^{2})}$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+n}2^{+n}3^{+n}4^{-n}5^{-n}6^{-n}]\hskip
0.85358pt\mathcal{T}_{b_{j}}[1^{+n}2^{+n}4^{-n}3^{+n}6^{-n}5^{-n}]$ (4.22)
$\displaystyle+(s_{12}\\!+\\!s_{24}\\!-\\!4M_{n}^{2})(s_{13}\\!+\\!s_{34}\\!-\\!4M_{n}^{2})$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+n}2^{+n}3^{+n}4^{-n}5^{-n}6^{-n}]\hskip
0.85358pt\mathcal{T}_{b_{j}}[1^{+n}3^{+n}2^{+n}4^{-n}6^{-n}5^{-n}]$
$\displaystyle{-(s_{25}\\!+\\!s_{26})(s_{13}\\!+\\!s_{34}\\!-\\!4M_{n}^{2})}$
$\displaystyle\hskip
0.85358pt\mathcal{T}_{a_{j}}[1^{+n}2^{+n}3^{+n}4^{-n}5^{-n}6^{-n}]\hskip
0.85358pt\mathcal{T}_{b_{j}}[1^{+n}2^{+n}3^{+n}4^{-n}6^{-n}5^{-n}]\big{\\}}$
$\displaystyle+\,\mathrm{all\ permutations\ of}\ (+n,-n)\Big{\\}}\,.$

The above five-point and six-point KK gauge boson amplitudes and KK graviton
amplitudes are worth of further systematic studies and we will pursue these in
the future works.

##  Conclusions

The Kaluza-Klein (KK) compactification [1] of higher dimensional spacetime is
a fundamental ingredient of the major directions for new physics beyond the
standard model (SM), including the string/M theories [2] and extra dimensional
field theories with large or small extra dimensions [3]. Studying the double-
copy construction of graviton scattering amplitudes from gauge boson
scattering amplitudes has pointed to profound deep connections between the
gauge forces and gravitational force in nature.

So far substantial efforts have been made to formulate and test the double-
copy constructions between the massless gauge theories and massless general
relativity (GR) [8]. But the extensions of conventional double-copy method to
massive gauge/gravity theories are generally difficult, because most of such
theories (including the massive Yang-Mills theory and massive Fierz-Pauli
gravity) violate explicitly the gauge symmetry and diffeomorphism invariance
(which are the key for successful double-copy construction). The two important
candidates with promise include the compactified KK gauge/gravity theories and
the topologically massive Chern-Simons gauge/gravity theories. The extended
BCJ-type double-copy construction for realistic massive KK gauge/gravity
theories was found [14] to be highly nontrivial even for the four-point KK
scattering amplitudes at tree level, and proper modifications of the
conventional BCJ method are generally needed for the KK scattering amplitudes
at the next-to-leading order (NLO) of the high energy expansion [14].

In this work, we studied the scattering amplitudes of massive KK states of
open and closed bosonic strings under toroidal compactification. The essential
advantage of the compactified KK string theory is that the connection between
the KK closed-string amplitudes and the proper products of KK open-string
amplitudes can be intrinsically built in from the start. For the present
study, we take the bosonic string theory as a computational tool for
establishing the massive KLT-like relations of KK string states and for
deriving the low energy KK graviton scattering amplitudes.

In section 2.1, we set up the toroidal compactification for the 26d bosonic
string theory where the 21 of the extra spatial dimensions have very small
compactification radii of $\mathcal{O}(M_{\text{Pl}}^{-1})$ and get decoupled
in our effective string theory below the Planck scale. Thus, we only deal with
the KK strings in a single compactified 25th spatial dimension under $\hskip
0.85358pt\SS^{1}$ with relatively larger radius $R\hskip 1.42262pt$. With
these, in section 2.2 we studied vertex operators of the KK open and closed
strings. In particular, these include a class of vertex operators having
$\mathbb{Z}_{2}$-even parity whose scattering amplitudes will give, in the
field-theory limit, the corresponding scattering amplitudes of the KK gauge
bosons and of the KK gravitons in the KK gauge/gravty theories under the 5d
compactification of $\hskip 0.85358pt\SS^{1}\\!/\mathbb{Z}_{2}\hskip
0.85358pt$. Then, we formulated the $N$-point scattering amplitudes (2.22) of
the KK open and closed strings in section 2.3, and further derived the
corresponding scattering amplitudes (2.4) of the KK gauge bosons and gravitons
in the low energy field-theory limit in section 2.4. We observed that any KK
amplitude with external states being $\mathbb{Z}_{2}$ even (odd) should be
decomposed into a sum of relevant sub-amplitudes whose external states have KK
numbers obey the conservation condition (2.18).

In section 3, using the formulas of section 2 we computed explicitly the four-
point color-ordered scattering amplitudes of KK open strings and derived the
corresponding gauge boson scattering amplitudes in the field theory limit. The
four-point elastic KK gauge boson scattering amplitudes are given in
Eqs.(3.4)-(3.7) of section 3.1, while the four-point inelastic KK gauge boson
scattering amplitudes are given in Eqs.(3.10)-(3.12) and Eq.(3.17) of section
3.2. Then, in section 3.3, we analyzed the structure of the color-ordered
scattering amplitudes of massive KK gauge bosons. We demonstrated that the
tree-level massive KK gauge boson amplitudes can be obtained from the
corresponding color-ordered amplitudes of the massless zero-mode gauge bosons
by making proper shifts of the Mandelstam variables. This serves as an elegant
and efficient method to compute any color-ordered massive KK gauge boson
amplitudes.

In section 4, we applied the massive KLT-like relation (4.1) of four-point KK
string amplitudes by taking the field theory limit, and derived the double-
copy formulas (4.1), (4.2) and (4.16) for constructing the four-point KK
graviton scattering amplitudes. These give an explicit prescription on how to
construct the exact four-point KK graviton amplitudes from the sum of relevant
products of the corresponding color-ordered KK gauge boson amplitudes. With
these, we computed the exact tree-level four-point elastic KK graviton
scattering amplitudes (4.5)-(4.6) in section 4.1, and the exact tree-level
four-point inelastic KK graviton scattering amplitudes (4.13)-(4.14) and
(4.17) in section 4.2. Finally, in section 4.3, we used our general string-
based double-copy construction formula (2.4) to obtain the five-point and six-
point scattering amplitudes of massive KK gravitons as given by Eq.(4.20) and
Eq.(4.22).

Acknowledgments:
We thank Henry Tye for valuable discussions. We also thank Ziqi Yan for
discussing Ref. [23]. The works of HJH, YFH and YL were supported in part by
National Natural Science Foundation of China (under grants Nos. 11835005,
12175136), by National Key R & D Program of China (under grant No.
2017YFA0402204), and by the CAS Center for Excellence in Particle Physics
(CCEPP). The research of SH was supported in part by National Natural Science
Foundation of China (under Grant Nos. 11935013, 11947301, 12047502, 12047503).

Appendix:

##  Appendix A Kinematics of Four-Point Scattering Amplitudes of KK States

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

For the four-particle elastic scattering of KK states $(n,\hskip
0.85358ptn)\\!\rightarrow\\!(n,\hskip 0.85358ptn)$, we define the 4-momenta of
the external KK states in the center-of-mass frame as follows:

$\displaystyle p_{1}^{\mu}$ $\displaystyle=-E\hskip
0.85358pt(1,0,0,\beta)\hskip 0.85358pt,\hskip 8.53581pt$ $\displaystyle
p_{2}^{\mu}=-E\hskip 0.85358pt(1,0,0,-\beta)\hskip 0.85358pt,$ (A.1)
$\displaystyle p_{3}^{\mu}$ $\displaystyle=E\hskip 0.85358pt(1,\beta
s_{\theta},0,\beta c_{\theta})\hskip 0.85358pt,$ $\displaystyle
p_{4}^{\mu}=E\hskip 0.85358pt(1,-\beta s_{\theta},0,-\beta c_{\theta})\hskip
0.85358pt,$

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

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

where $\hskip 0.85358pt\beta\\!=\\!(1\\!-\\!M_{n}^{2}/E^{2})^{1/2}\hskip
0.85358pt$. With the on-shell condition
$\,E^{2}\\!=\\!E^{2}\beta^{2}\\!+\\!M_{n}^{2}\,$, we may define the following
set of Mandelstam variables:

$\displaystyle s_{0}\,=\,4E^{2}\beta^{2}\,,\quad\leavevmode\nobreak\
\leavevmode\nobreak\
t_{0}\,=\,-\frac{1}{2}s_{0}(1\\!+\\!c_{\theta})\,,\quad\leavevmode\nobreak\
\leavevmode\nobreak\ u_{0}\,=\,-\frac{1}{2}s_{0}(1\\!-\\!c_{\theta})\,,$ (A.3)

where we have introduced the notation $\hskip 0.85358pts_{0}=s-4M_{n}^{2}$ ,
and thus $(s_{0},\,t_{0},\,u_{0})=(s\beta^{2},\,t,\,u)$. Summing up the
Mandelstam variables (A.2) and (A.3) gives the following relations:

$s+t+u=4M_{n}^{2}\,,\hskip 22.76219pts_{0}+t_{0}+u_{0}=0\,.$ (A.4)

Then, we define the following polarization vectors for the external KK gauge
bosons in the center-of-mass frame:

$\displaystyle\zeta^{\mu}_{1,\hskip 0.85358pt\pm 1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(0,1,\pm\text{i},0\right)\\!,\hskip
22.76219pt$
$\displaystyle\zeta^{\mu}_{1,L}=\frac{E}{M_{n}}\left(\beta,0,0,1\right)\\!,$
(A.5) $\displaystyle\zeta^{\mu}_{2,\hskip 0.85358pt\pm 1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(0,1,\mp\text{i},0\right)\\!,\hskip
22.76219pt$
$\displaystyle\zeta^{\mu}_{2,L}=\frac{E}{M_{n}}\left(\beta,0,0,-1\right)\\!,$
$\displaystyle\zeta^{\mu}_{3,\hskip 0.85358pt\pm 1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(0,\mp\text{i}\hskip
0.85358ptc_{\theta},1,\pm\text{i}\hskip 0.85358pts_{\theta}\right)\\!,\hskip
22.76219pt$
$\displaystyle\zeta^{\mu}_{3,L}=\frac{E}{M_{n}}\left(\beta,s_{\theta},0,c_{\theta}\right),$
$\displaystyle\zeta^{\mu}_{4,\hskip 0.85358pt\pm 1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(0,\pm\text{i}\hskip
0.85358ptc_{\theta},1,\mp\text{i}\hskip 0.85358pts_{\theta}\right)\\!,\hskip
22.76219pt$
$\displaystyle\zeta^{\mu}_{4,L}=\frac{E}{M_{n}}\left(\beta,-s_{\theta},0,-c_{\theta}\right)\\!.$

Next, we consider the inelastic KK scattering process of $\hskip
0.85358pt(n,\hskip 0.85358ptn)\\!\\!\rightarrow\\!\\!(m,\hskip
0.85358ptm)\hskip 0.85358pt$. Thus, the 4-momenta of the external states in
the center-of-mass frame can be defined as follows:

$\displaystyle p_{1}^{\mu}$ $\displaystyle=-E(1,0,0,\beta)\hskip
0.85358pt,\hskip 8.53581pt$ $\displaystyle p_{2}^{\mu}=-E(1,0,0,-\beta)\hskip
0.85358pt,$ (A.6) $\displaystyle p_{3}^{\mu}$
$\displaystyle=E(1,\beta^{\prime}s_{\theta},0,\beta^{\prime}c_{\theta})\hskip
0.85358pt,$ $\displaystyle
p_{4}^{\mu}=E(1,-\beta^{\prime}s_{\theta},0,-\beta^{\prime}c_{\theta})\hskip
0.85358pt,$

where $\hskip 0.85358pt\beta\\!=\\!(1\\!-\\!M_{n}^{2}/E^{2})^{1/2}\hskip
0.85358pt$ and $\hskip
0.85358pt\beta^{\prime}\\!=\\!(1\\!-\\!M_{m}^{2}/E^{2})^{1/2}$. With these, we
can define the Mandelstam variables:

$\displaystyle s$
$\displaystyle=-\left(p_{1}\\!+p_{2}\right)^{2}\\!=4E^{2},\qquad$ (A.7)
$\displaystyle t$
$\displaystyle=-\left(p_{1}\\!+p_{4}\right)^{2}\\!=-\frac{1}{4}s\hskip
0.85358pt(\beta^{2}\\!+\\!\beta^{\prime
2}\\!+\\!2\beta\beta^{\prime}c_{\theta})\hskip 0.85358pt,$ $\displaystyle u$
$\displaystyle=-\left(p_{1}\\!+p_{3}\right)^{2}\\!=-\frac{1}{4}s\hskip
0.85358pt(\beta^{2}\\!+\\!\beta^{\prime
2}\\!-\\!2\beta\beta^{\prime}c_{\theta})\hskip 0.85358pt,$

from which we deduce $\,s+t+u=2(M_{n}^{2}+\\!M_{m}^{2})\hskip 0.85358pt$. The
corresponding longitudinal polarization vectors of the KK gauge bosons are
given by

$\displaystyle\zeta^{\mu}_{1,L}$
$\displaystyle=\frac{E}{M_{n}}({\beta},0,0,1)\hskip 0.85358pt,$
$\displaystyle\zeta^{\mu}_{2,L}=\frac{E}{M_{n}}({\beta},0,0,-1)\hskip
0.85358pt,\quad$ (A.8) $\displaystyle\zeta^{\mu}_{3,L}$
$\displaystyle=\frac{E}{M_{m}}{({\beta^{\prime}},s_{\theta},0,c_{\theta})}\hskip
0.85358pt,\quad\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\zeta^{\mu}_{4,L}=\frac{E}{M_{m}}({\beta^{\prime}},-s_{\theta},0,-c_{\theta})\hskip
0.85358pt,$

while their transverse polarization vectors are mass-independent and remain
the same as in Eq.(A.5). For another inelastic scattering channel $\hskip
0.85358pt(0,\hskip 0.85358pt0)\\!\rightarrow\\!(m,\hskip 0.85358ptm)\hskip
0.85358pt$, we just set the initial state masses to be zero
($M_{n}\\!\\!\rightarrow\\!0$) in the above setup and only allow transverse
polarizations for the massless initial state gauge bosons.

##  Appendix B Full Scattering Amplitudes of KK Gauge and Goldstone Bosons

In this Appendix, for the sake of comparison we present systematically the
four-point elastic and inelastic scattering amplitudes of KK gauge bosons in
the 5d KK Yang-Mills gauge theories under the orbifold compactification of
$\,\SS^{1}\\!/\mathbb{Z}_{2}\hskip 0.85358pt$.

###  Elastic KK Gauge and Goldstone Boson Scattering Amplitudes

According to Ref. [14], we summarize the four-point elastic scattering
amplitudes of longitudinal KK gauge bosons and of KK Goldstone bosons as
follows:

$\displaystyle\mathcal{T}[A_{L}^{a\hskip 0.85358ptn}A_{L}^{b\hskip
0.85358ptn}\\!\rightarrow\\!A_{L}^{c\hskip 0.85358ptn}A_{L}^{d\hskip
0.85358ptn}]\,=\leavevmode\nobreak\ \,$ $\displaystyle
g^{2}(\mathcal{C}_{s}\mathcal{K}_{s}^{\mathrm{el}}+\mathcal{C}_{t}\mathcal{K}_{t}^{\mathrm{el}}+\mathcal{C}_{u}\mathcal{K}_{u}^{\mathrm{el}})\,,$
(B.1a) $\displaystyle\widetilde{\mathcal{T}}[A_{5}^{a\hskip
0.85358ptn}A_{5}^{b\hskip 0.85358ptn}\\!\rightarrow\\!A_{5}^{c\hskip
0.85358ptn}A_{5}^{d\hskip 0.85358ptn}]\,=\leavevmode\nobreak\ \,$
$\displaystyle
g^{2}(\mathcal{C}_{s}\widetilde{\mathcal{K}}_{s}^{\mathrm{el}}+\mathcal{C}_{t}\widetilde{\mathcal{K}}_{t}^{\mathrm{el}}+\mathcal{C}_{u}\widetilde{\mathcal{K}}_{u}^{\mathrm{el}})\,,$
(B.1b)

where $\\{\mathcal{K}_{j}^{\mathrm{el}}\\}$ denote the kinematic factors for
KK gauge bosons,

$\displaystyle\mathcal{K}_{s}^{\mathrm{el}}$
$\displaystyle\displaystyle=-\frac{\,(4\bar{s}^{2}\\!-\\!5\bar{s}\\!-\\!8)c_{\theta}\,}{2\bar{s}}\,,$
(B.2a) $\displaystyle\mathcal{K}_{t}^{\mathrm{el}}$
$\displaystyle\displaystyle=-\frac{\,Q_{0}\\!+Q_{1}c_{\theta}\\!+Q_{2}c_{2\theta}\\!+Q_{3}c_{3\theta}\,}{\,2(\bar{s}\\!-\\!4)[(3\bar{s}\\!+\\!4)\\!+\\!4\bar{s}\hskip
0.85358ptc_{\theta}\\!+\\!(\bar{s}\\!-\\!4)c_{2\theta}]\,}\,,$ (B.2b)
$\displaystyle\mathcal{K}_{u}^{\mathrm{el}}$
$\displaystyle\displaystyle=\frac{\,Q_{0}\\!-Q_{1}c_{\theta}\\!+Q_{2}c_{2\theta}\\!-Q_{3}c_{3\theta}\,}{\,2(\bar{s}\\!-\\!4)[(3\bar{s}\\!+\\!4)\\!-\\!4\bar{s}\hskip
0.85358ptc_{\theta}\\!+\\!(\bar{s}\\!-\\!4)c_{2\theta}]\,}\,,$ (B.2c)

and $\\{\widetilde{\mathcal{K}}_{j}^{\mathrm{el}}\\}$ denote the kinematic
factors fro KK Goldstone bosons,

$\displaystyle\widetilde{\mathcal{K}}_{s}^{\mathrm{el}}$
$\displaystyle\displaystyle=-\frac{\,(3\bar{s}\\!-\\!8)c_{\theta}\,}{2\bar{s}}\,,$
(B.3a) $\displaystyle\widetilde{\mathcal{K}}_{t}^{\mathrm{el}}$
$\displaystyle\displaystyle=\frac{\,\widetilde{Q}_{0}\\!+\widetilde{Q}_{1}c_{\theta}\\!+\widetilde{Q}_{2}c_{2\theta}\,}{\,2(\bar{s}\\!-\\!4)[(3\bar{s}\\!+\\!4)\\!+\\!4\bar{s}\hskip
0.85358ptc_{\theta}\\!+\\!(\bar{s}\\!-\\!4)c_{2\theta}]\,}\,,$ (B.3b)
$\displaystyle\widetilde{\mathcal{K}}_{u}^{\mathrm{el}}$
$\displaystyle\displaystyle=-\frac{\,\widetilde{Q}_{0}\\!-\widetilde{Q}_{1}c_{\theta}\\!+\widetilde{Q}_{2}c_{2\theta}\,}{\,2(\bar{s}\\!-\\!4)[(3\bar{s}\\!+\\!4)\\!-\\!4\bar{s}\hskip
0.85358ptc_{\theta}\\!+\\!(\bar{s}\\!-\\!4)c_{2\theta}]\,}\,,$ (B.3c)

with the functions $\\{Q_{j},\widetilde{Q}_{j}\\}$ expressed as

$\displaystyle
Q_{0}=8\bar{s}^{3}\\!-\\!63\bar{s}^{2}\\!+\\!72\bar{s}\\!+\\!80\,,\hskip
22.76219pt$
$\displaystyle\widetilde{Q}_{0}=15\bar{s}^{2}\\!+\\!24\bar{s}\\!-\\!80\,,$
(B.4) $\displaystyle
Q_{1}=2(7\bar{s}^{3}\\!-\\!44\bar{s}^{2}\\!+\\!80\bar{s}\\!-\\!64)\,,\hskip
22.76219pt$
$\displaystyle\widetilde{Q}_{1}=4(3\bar{s}^{2}\\!-\\!20\bar{s}\\!+\\!32)\,,$
$\displaystyle
Q_{2}=8\bar{s}^{3}\\!-\\!45\bar{s}^{2}\\!+\\!8\bar{s}\\!+\\!48\,,\hskip
22.76219pt$ $\displaystyle\widetilde{Q}_{2}=-3(\bar{s}\\!-\\!4)^{2}\,,$
$\displaystyle Q_{3}=2\bar{s}(\bar{s}^{2}\\!-\\!10\bar{s}\\!+\\!24)\,.$

Making the high energy expansion of $\,1/s\hskip 0.85358pt$, we derive the
following LO scattering amplitudes:

$\displaystyle\mathcal{K}_{s}^{{\mathrm{el}}\hskip 0.85358pt0}$
$\displaystyle=\frac{\,5c_{\theta}\,}{2}\,,\hskip 34.1433pt$
$\displaystyle\widetilde{\mathcal{K}}_{s}^{{\mathrm{el}}\hskip 0.85358pt0}$
$\displaystyle=-\frac{\,3c_{\theta}\,}{2}\,,$ (B.5a)
$\displaystyle\mathcal{K}_{t}^{{\mathrm{el}}\hskip 0.85358pt0}$
$\displaystyle=\frac{\,13\\!+\\!5c_{\theta}\\!+\\!4c_{2\theta}\,}{2(1\\!+\\!c_{\theta})\,}\,,\hskip
34.1433pt$ $\displaystyle\widetilde{\mathcal{K}}_{t}^{{\mathrm{el}}\hskip
0.85358pt0}$
$\displaystyle=\frac{\,3(3\\!-\\!c_{\theta})\,}{2(1\\!+\\!c_{\theta})\,}\,,$
(B.5b) $\displaystyle\mathcal{K}_{u}^{{\mathrm{el}}\hskip 0.85358pt0}$
$\displaystyle=-\frac{\,13\\!-\\!5c_{\theta}\\!+\\!4c_{2\theta}\,}{2(1\\!-\\!c_{\theta})\,}\,,\hskip
34.1433pt$ $\displaystyle\widetilde{\mathcal{K}}_{u}^{{\mathrm{el}}\hskip
0.85358pt0}$
$\displaystyle=-\frac{\,3(3\\!+\\!c_{\theta})\,}{2(1\\!-\\!c_{\theta})\,}\,,$
(B.5c)

and the NLO scattering amplitudes:

$\displaystyle\delta\mathcal{K}_{s}^{\mathrm{el}}$