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

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.


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 GR = (Gauge 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 doublecopy 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 KKinspired 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] nat-urally 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 S 1 /Z 2 . 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 KKmodes. 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 doublecopied 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 1 , 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 ampli-tude 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 zeromode 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 fourpoint 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 stringbased 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 O(M −1 Pl ). 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 α → 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 R 1,24 × S 1 . We can first compactify the extra spatial dimensions of coordinates {X 4 , · · ·, X 24 } with their radii r j = O(M −1 Pl ) R (j = 4, · · ·, 24), so they are fully decoupled at energy scales much below the reduced Planck scale M Pl = (8πG) −1/2 . Thus, we only need to study the toroidal compactification of the single extra spatial dimension of coordinate X 25 on a circle S 1 with radius R , which is much larger than the Planck length M −1 Pl 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 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 (≡ X ) as a scalar field on the string worldsheet and it obeys the periodic boundary condition on the circle S 1 : For the closed strings, without loss of generality, the periodic boundary condition further takes the following form: where w ∈ Z is the winding number describing how many times a closed string winds around the circle S 1 . Thus, the eigenvalues of generators of Virasoro algebra (L 0 , L 0 ) for the oscillation modes of closed string are derived as follows: where α is the Regge slope,n ∈ Z denotes the Kaluza-Klein (KK) level, and (N, 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 L 0 =L 0 = 0 : withn ∈ Z and sgn ≡ sign(n) = ±1. In the above, M n = |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 = N − N . We can further decompose the mass spectrum (2.4) as follows: where the right-hand-side (RHS) contains the contributions from the squared KK-mass M 2 n = n 2 /R 2 and a squared topological mass (wR/α ) 2 (related to the string winding number). For studying the low energy limit of string theory as a field theory, we will set w = 0 and thus N = N = 1 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: where q ∈ Z labels the D-branes transverse to the compactified circle S 1 with equal distance L. We make the following mode expansion for open string: where we set the two ends of the open string at the q-th brane and (q + q )-th brane: Thus, the mass spectrum of open strings is derived as follows: For studying the low energy limit of string theory as a field theory, we only need to consider the w = 0 case and thus N = 1 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): and make the rescaling (α , R, L) → 1 4 (α , R, L), 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:

12a)
V cl (ζ, k,n) = ig cl d 2 z ζ µν : ∂X µ∂ X ν e ik·X e ipnX : , where k denotes the momentum in the noncompactified spacetime and pn =n/R (n ∈ Z) is the quantized momentum in 26d. The compactified 26d string coordinates (X , Y) are defined as: where X L (z) and X R (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: where the Lorentz indices µ, ν = 0, 1, . . . , 24 . While for a compactified spatial dimension, we have the Green functions of open strings under the Dirichlet boundary condition: (2.15b) And the Green functions for closed string is given by where the 26d Lorentz indices M, N = (0, 1, . . . , 25) .
In string theory, imposing the orbifold compactification S 1 /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 ∆M 2 op = 1 16α , and the mass of each KK closed-string state receives an increase ∆M 2 cl = 1 4α . Thus they will be decoupled in the field-theory limit of α → 0 . Hence, we will first make our analysis by using the periodic boundary condition on S 1 without imposing the Z 2 . Then, we can construct the vertex operators for KK open strings and KK closed strings with specified Z 2 parity: where n ∈ Z + and the superscript "+(−)" stands for the Z 2 -even (Z 2 -odd) state. We note that for the zero-modes (n = 0), vertex operators for open and closed strings, V op (ζ, k, 0) and V cl (ζ, k, 0), are always Z 2 -even. In particular, we will be interested in the above vertex operators having Z 2 -even parity because their scattering amplitudes will give, in the low energy fieldtheory 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 S 1 /Z 2 . 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 S 1 /Z 2 .

Open and Closed String Amplitudes for Massive KK States
For a N -point KK string amplitude, the conservation of the compactified momentum (KKnumber) is achieved by the neutral condition [28] of the vertex operators on the string worldsheet, i.e., the sum of KK numbers (n j ) of the external states should vanish: wheren j ∈ Z is the KK number of the j-th external state, n j =|n j | ∈ Z + , and the sign ofn j is denoted as sgn j = sign(n j ). Thus we can write pn j = sgn j ×M n j . Hence, the tree-level N -point open-string amplitude can be written as follows: where λ 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 Z 2 even (odd), corresponding to its vertex operator V ± op (ζ j , k j , n j ) being Z 2 even (odd) as indicated by its superscript + (−).
[An external state of the amplitude A (N ) op (ζ, k, n) can also be chosen as non-eigenstate of Z 2 parity and thus its corresponding vertex operator is V op (ζ j , k j , ±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.
wherek j = (k µ j , sgn j ×M n j ) is the 26d momentum and the notation T [1, α(2 · · · N )] = Tr(T 1 T α(2) · · · T α(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: where the coefficient C D 2 is given by . The constants C g D 2 and C X D 2 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 A (N ) 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 closedstring amplitude at tree level: . In the above Eq.(2.22b), the polarization tensor ζ µν of closed strings is expressed as a sum of the products of polarization vectors of two open strings: where the coefficient ab ∈ R . In Eq.(2.22b), the coefficient a j b j is defined as the product of ab for all external graviton states: The string momentum kernel S α connects the two open-string amplitudes and takes the following explicit form [27]:

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 α → 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: With these, we can derive the following low energy N -point graviton scattering amplitude: where the gravitational coupling κ and the closed-string coupling g cl are connected by the relation κ = 2πα g cl . In the above, is the momentum kernel in the field theory limit and takes the following form [26] [27]: (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 , y 2 , y 3 ) = (0, 1, ∞): where the superscript "(4)" in each amplitude A op is not displayed for simplicity and the relation In the above, the function F (y 4 ) is defined as:

Elastic Amplitudes of KK Gauge Bosons from KK Open Strings
In this subsection, we study the four-point elastic KK scattering process (n, n) → (n, n) with all external states being Z 2 -even. We observe that Eq. op can be decomposed into a sum of the six sub-amplitudes, as presented in Fig. 1. We note that the compactification under S 1 respects the 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.(3.3) are connected to the other three corresponding combinations in the second row by Z 2 parity transformation. From the above, our key insight is that even though the external states on the LHS of Eq.(3.3) are all Z 2 -even, the external states on the RHS contain two Z 2 -even states and two Z 2 -odd states such that the condition (2.18) is obeyed. This is because our string compactification of 26d is under S 1 with the periodic boundary condition (2.1) (without having Z 2 orbifold). Hence, even for a scattering amplitude with 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.
and the sub-amplitudes with color ordering [1243]: where we have defineds = s/M 2 n . 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 Z 2 even: where the polynomials {P j } are given by The above elastic KK gauge boson amplitudes (3.6a)-(3.6b) 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: where the kinematic factors {K el j } 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
Next, we analyze the four-point color-ordered partial amplitude for the inelastic channel (n, n) → (m, m) with all external states being Z 2 -even. We inspect Eq. Then, we derive the four-point inelastic scattering amplitudes of longitudinal gauge bosons with color-ordering [1234]: where we have used the notations, For the color-ordering [1243], we derive the four-point inelastic scattering amplitudes of longitudinal gauge bosons as follows: 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 Z 2 even: where the polynomials {P j } take the forms: (3.14) Inspecting the string-based KK gauge boson amplitudes (3.13), we can re-express them in the following forms: where the kinematic functions {K in j } are summarized in Eq.(B.15) of Appendix B. We find that these {K in j } 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 (0, 0) → (n, n). We find that the condition (2.18) allows only two combinations of the external KK states (as originally defined under the S 1 compactification of 26d bosonic strings), (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: wheres andq 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 Z 2 even: We can re-express the above string-based inelastic amplitudes in the following forms: where the kinematic functions {K in j } 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 colorordered massive KK sub-amplitudes in d-dimensions, such as T [1 +n 2 +n 3 −n 4 −n ], 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 S 1 . Namely, we can express the (d+1)-dimensional momentumkμ in terms of the d-dimensional momentum k µ plus an extra discretized (d+1)-th component:kμ = (k µ ,n/R). For a given polarization vector ζ µ of the on-shell gauge boson in d-dimensions, we can symbolically express it as a (d +1)-dimensional polarization vectorζμ = (ζ µ , 0). Thus, we haveζ i ·ζ j = ζ i · ζ j andζ i ·k j = ζ i · k j , where the subscripts (i, j) denote the particle numbers of the external states. 2 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 relationŝ 20) and the relation between the two sets of Mandelstam variableŝ where the (d+1)-dimensional momenta obey the on-shell conditionsk 2 i =k 2 j = 0 . Optionally, we can first write a d-dimensional massless (zero-mode) scattering amplitude 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 T sub KK (s ij ) as follows: with each d-dimensional external state having its momentum obey the on-shell condition of the massive KK gauge boson (k 2 j = −M 2 j ) and its polarization vector replaced by the polarization vector ζ 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 α → 0 of the open-string scattering amplitude (3.1): Our later explicit calculations of the KK scattering amplitudes will be always performed in the effective (3+1)-dimensional spacetime with d = 4, and with a single compactified extra spatial dimension of coordinate X 25 . The other extra spatial dimensions of coordinates {X 4 , · · ·, X 24 } have much smaller radii r j = O(M −1 Pl ) (j = 4, · · ·, 24), so they are fully decoupled at energy scales much below the reduced Planck scale M Pl , 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 S 1 .
For the other massless color-ordered amplitudes, such as the one with color ordering [1243], they can be obtained through the relation: For instance, we consider the elastic KK scattering (n, n) → (n, n) as discussed in section 3.1. We can obtain the color-ordered sub-amplitude T [1 +n 2 +n 3 −n 4 −n ] by the replacement s → (s−4M 2 n ) in T [1 0 2 0 3 0 4 0 ]. In general, the color-ordered sub-amplitude T [1n 1 2n 2 3n 3 4n 4 ] can be obtained by the following replacements: 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.
where the basis amplitudes {T el 1 , T el 2 , T el 3 } are given by the following sub-amplitudes: .

(3.27c)
We note that the KK numbers of the sub-amplitude T [1 ±n 2 ∓n 3 ±n 4 ∓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) → (m, m): where the basis amplitudes {T in 1 , T in 2 } are obtained by the relations: .

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: where we have replaced the closed-string coupling by the relation g cl = κ/(2πα ). For the two massive KK open-string amplitudes inside {· · · }, the Regge slope should be rescaled as α → α /4 . This is equivalent to considering the N -point low energy field theory formula (2.27) and derive the four-longitudinal KK graviton amplitude for the case of N = 4 . 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.

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 α → 0 for the closed-string amplitude (4.1), we derive the four-longitudinal KK graviton scattering amplitude for the elastic channel (n, n) → (n, n). 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: where in the above second equality we have defined the short-hand notation for the partial amplitudes as T a j [1 +n 2 −n 3 +n 4 −n ] = T [1 +n a 1 2 −n a 2 3 +n a 3 4 −n a 4 ], and so on. Note that a massive KK graviton in 4d has 5 physical helicity states and their polarization tensors are given by Thus, in Eq.(4.2) the helicity indices of each partial amplitude are {a j , b j } = {±1, L}. For the graviton polarization tensor ζ µν ±2 , its coefficient a j b j = ±1,±1 = 1; for the polarization tensor ζ µν ±1 , its coefficient a j b j = ±1,L = L,±1 = 1 √ 2 ; and for the polarization tensor ζ µν L , its coefficient a j b j takes the following values:  where the coefficients {X j } in the numerator are given by X 0 = −2(255s 5 + 2824s 4 −19936s 3 + 39936s 2 − 256s +14336), X 2 = 429s 5 − 10152s 4 + 30816s 3 − 27136s 2 − 49920s + 34816, X 4 = 2(39s 5 − 312s 4 − 2784s 3 − 11264s 2 + 26368s −2048), (4.6) 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: We stress that the string-based double-copy formula (4.2) for 4-point amplitudes or (2.27) 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 stringbased double-copy formula (2.27) or (4.2), 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 O(E 0 ) and 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 0 n ), 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 O(M 2 n /E 2 ) and O(M 2 n 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 ζ µν L of the (helicity-zero) longitudinal KK graviton in Eq.(4.3b) contains the sum of three products of two gauge boson polarization vectors , while the simple double-copy by using the longitudinal KK gauge boson amplitude alone could only provide the polarization-vector product of ζ µ L ζ ν L , which does not include the other two products (ζ µ +1 ζ ν −1 , ζ µ −1 ζ ν +1 ) of ζ µν L 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 (4.2) is automatically built in from the beginning. Another key feature of our stringbased construction (4.2) is that it intrinsically includes a set of KK gauge boson sub-amplitudes with different KK-number combinations as allowed by the condition (3.3) due to the original string compactification under S 1 (without the orbifold 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 ζ µν L also contains additional products of transverse polarization vectors (ζ µ . We note that the KK Goldstone boson A an 5 is just a scalar field without any polarization vector, and A an 5 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 a5 n -amplitude is uniquely defined via the correspondence A a5 n ⊗ A a5 n → h 55 n . Hence, the LO gravitational KK Goldstone amplitude M 0 [4h 55 n ] as given by the double-copy of the LO KK Goldstone amplitude T 0 [4A a5 n ] 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 M 0 [4h n L ] to its corresponding LO gravitational KK Goldstone amplitude M 0 [4h n 55 ]: (4.10) We have verified this insight by explicitly computing the four-point elastic scattering amplitude M 0 [4h n L ] 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],

Constructing Inelastic Scattering Amplitudes of Four KK Gravitons
In this subsection, we study the four-point inelastic scattering channels (n, n) → (m, m) and (0, 0) → (n, n) for the massive KK closed-string amplitudes (4.1). We will take the low energy field theory limit α → 0 and derive the inelastic scattering amplitudes of four-longitudinal KK gravitons.
With the above and using our extended massive double-copy formula (4.16), we construct the mixed four-point inelastic graviton amplitudes of (0, 0)→ (n, n) and derive following form: We can further derive the following LO and NLO amplitudes from Eq.(4.17) under high energy expansion:

Constructing Multi-Point Scattering Amplitudes of KK Gravitons
Our above analyses of the four-point scattering amplitudes can be further extended to the Npoint amplitudes with N 5 . We first consider a five-point inelastic scattering process (2n, n) → (n, n, n), where all the external KK states are set to be Z 2 even. We find that the condition where we assign the KK number of particle-1 as +2n for convenience.
Thus, using the general formula (2.27) of the N -point KK graviton amplitudes, we derive the five-point longitudinal KK graviton scattering amplitude as follows: (4.20) + all permutations of (+n, −n) .
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 O(M −1 Pl ) 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 S 1 with relatively larger radius R . 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 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 S 1 /Z 2 . 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 (2.27) 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 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- 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 (4.2), (4.12) 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)  center-of-mass frame: Next, we consider the inelastic KK scattering process of (n, n) → (m, m). Thus, the 4momenta of the external states in the center-of-mass frame can be defined as follows: With these, we can define the Mandelstam variables: from which we deduce s + t + u = 2(M 2 n +M 2 m ). The corresponding longitudinal polarization vectors of the KK gauge bosons are given by

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 S 1 /Z 2 .

B.1 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: where {K el j } denote the kinematic factors for KK gauge bosons, and { K el j } denote the kinematic factors fro KK Goldstone bosons, with the functions {Q j , Q j } expressed as Q 0 = 8s 3 −63s 2 +72s+80 , Q 0 = 15s 2 +24s−80 ,

(B.4)
Making the high energy expansion of 1/s, we derive the following LO scattering amplitudes: Note that the above expansion of 1/s differs from the expansion of 1/s 0 [cf. Eq.(A. 3)] as adopted in Ref. [14]. We also note that in each channel of (s, t, u) the LO longitudinal KK gauge boson amplitude differs from the LO KK Goldstone boson amplitude by the same amount: K el 0 j − K el 0 j = 4c θ . Hence, due to the Jacobi identity the elastic KK longitudinal gauge boson amplitude and KK Goldstone boson amplitude are equal at the LO, T 0L [4A a n L ] = T 05 [4A a n 5 ], which verifies the KK gauge boson equivalence theorem (KK GAET) [18] [21]. 3 Then, we further define the BCJ-type numerators: where j ∈ (s, t, u) , and we have decomposed the numerators {N el j , N el j } into the LO and NLO parts under high energy expansion. With these, we can reformulate the scattering amplitudes (B.1) as follows: Then, we find that the LO numerators {N el 0 j , N el 0 j } and the NLO numerators {δN el j , δ N el j } are both mass-dependent and their sums violate the kinematic Jacobi identity by terms of O(E 0 M 2 n ) and smaller: We note that all the O(E 2 ) terms in the LO amplitudes are mass-independent and obey the kinematic Jacobi identity as shown in Eq.(B.9a), while all the Jacobi-violating terms in the LO/NLO amplitudes are mass-dependent and have O(E 0 M 2 n ) or smaller. Because of these Jacobi-violating terms, the conventional BCJ double-copy method of the massless gauge theories cannot be naively applied to the case of the elastic scattering amplitudes of KK gauge (Goldstone) bosons. However, we note that the amplitudes (B.8) are invariant under the generalized gauge transformations of the kinematic numerators: In the above, the gauge parameters (∆ el , ∆ el ) can be solved by requiring the gauge-transformed numerators to satisfy the Jacobi identities: j N el j = 0 and j N el j = 0 . Thus, we derive the following general solutions: Expanding both sides of Eq.(B.11), we derive the gauge parameters (∆ el , ∆ el ) = (∆ el 0 +∆ el 1 , ∆ el 0 + ∆ el 1 ) at the LO and NLO: (B.12) Then, we can extend the conventional BCJ method and apply the color-kinematics duality to the following gauge-transformed scattering amplitudes: We find that this extended BCJ-type double-copy construction gives the correct LO KK graviton (Goldstone) amplitudes at O(E 2 M 0 n ), and also gives the correct structure of the NLO KK graviton (Goldstone) amplitudes at O(E 0 M 2 n ) although the coefficients do not exactly match that of the original KK graviton (Goldstone) amplitudes at the NLO. So, we need proper modifications on the extended double-copy construction of the massive NLO KK gauge/gravity amplitudes, as shown in Ref. [14]. In the current study, we have demonstrated in sections 3-4 that the doublecopy construction for the massive KK gauge/gravity amplitudes can be successfully realized by using the KK string-based formulation of the extended massive KLT-like relations, which hold for the exact N -point tree-level amplitudes without making the high energy expansion.
Then, we make the high energy expansions for the above amplitudes at the LO and NLO: T [A a n L A b n L → A c m L A d m L ] = T 0L + δT L , T [A a n 5 A b n 5 → A c m 5 A d m 5 ] = T 05 + δ T 5 , (B.18a) δT L = g 2 (C s δK in s + C t δK in t + C u δK in u ), δ T 5 = g 2 (C s δ K in s + C t δ K in t + C u δ K in u ).
We further define the following LO and NLO inelastic numerators: Then, we compute their sums at the LO and NLO. We find that the sums of these numerators violate the kinematic Jacobi identities, j N in j =0 and j N in j = 0 . Thus, to recover the kinematic Jacobi identity, we make the following generalized gauge transformations for the inelastic numerators: under which the scattering amplitudes (B.14a)-(B.14b) are invariant. Then, imposing the kinematic Jacobi identities on the gauge-transformed amplitudes j N in j = 0 and j N in j = 0 , we derive the general solutions of the gauge parameters (∆ in , ∆ in ) as follows: where we have set r = 2 for illustration.

B.2.2 Inelastic Scattering
Amplitudes of (0, 0) → (n, n) Next, we study another inelastic channel (0, 0) → (n, n). We compute the following full tree-level scattering amplitudes of KK gauge bosons and of KK Goldstone bosons: where the sub-amplitudes {K in j } and { K in j } are given by Then, we make the high energy expansions of the above amplitudes at the LO and NLO: δT L = g 2 (C t δK in t + C u δK in u ) , δ T 5 = g 2 (C t δ K in t + C u δ K in u ) .

(B.30c)
Since the s-channel sub-amplitudes vanish, K in s = K in s = 0, we derive the following LO inelastic sub-amplitudes of (t, u) channels, K in 0 u = K in 0 u = 1+ c θ , (B.31) and the following NLO inelastic sub-amplitudes, To recover the kinematic Jacobi identities, we make the generalized gauge-transformations (B.24) on the numerators such that j N in j = 0 and j N in j = 0 . Thus, we derive the following general solutions of the gauge parameters (∆ in , ∆ in ): Finally, under high energy expansion, we derive the LO and NLO gauge parameters as follows: (B.35)