Heterotic and bosonic string amplitudes via field theory

Previous work has shown that massless tree amplitudes of the type I and IIA/B superstrings can be dramatically simplified by expressing them as double copies between field-theory amplitudes and scalar disk/sphere integrals, the latter containing all the $\alpha'$-corrections. In this work, we pinpoint similar double-copy constructions for the heterotic and bosonic string theories using an $\alpha'$-dependent field theory and the same disk/sphere integrals. Surprisingly, this field theory, built out of dimension-six operators such as $(D_\mu F^{\mu \nu})^2$, has previously appeared in the double-copy construction of conformal supergravity. We elaborate on the $\alpha' \rightarrow \infty$ limit in this picture and derive new amplitude relations for various gauge-gravity theories from those of the heterotic string.


Contents
3 Bosonic string amplitudes from the double copy 12 4 Heterotic string amplitudes from the double copy 1 Introduction this decomposition of open-string amplitudes follows the pattern of the field-theory KLT formula [51]. This unexpected double-copy structure of the open superstring suggests an interpretation of the moduli-space integrals over disk worldsheets as scattering amplitudes in a putative effective theory of scalars. The latter has been dubbed Z-theory and investigated in view of its low-energy limit [52,53] where couplings between non-linear sigma models and bi-adjoint scalars are recovered as well as the α ′ -expansion of its non-linear field equations [54]. The Z-theory approach [52,53] underpins the color-kinematics duality of non-linear sigma models [15,55] from a string-theory perspective and gives rise to the same amplitude representations as the cubic action of Cheung and Shen [56]. According to the evidence presented in ref. [57], the double-copy structure of the open superstring is expected to extend to loop level. Tree-level amplitudes of the open bosonic string are conjectured to admit the same doublecopy organization as observed for the superstring [58]. In a universal basis of disk integrals or Z-theory amplitudes, the only difference between bosonic and supersymmetric strings resides in the rational kinematic factors; these are α ′ -independent gauge-theory trees in the case of the superstring [50], and more complicated rational functions of α ′ in the case of the bosonic string [58]. As a main result of this work, these α ′ -dependent kinematic factors of the bosonic string will be reproduced from a specific field-theory Lagrangian, thus pinpointing the missing double-copy component of the open bosonic string. Surprisingly, this field theory, dubbed (DF ) 2 + YM, turns out to be a massive deformation of the (DF ) 2 theory entering the double-copy construction for conformal supergravity [20]. More precisely, the (DF ) 2 + YM theory, depending on α ′ through its mass parameter, interpolates between pure Yang-Mills (YM) theory (α ′ → 0) and the undeformed (DF ) 2 theory (α ′ → ∞).  In addition to the double-copy representations for open-string amplitudes (first row of table 1), we will identify similar field-theory constituents for gravitational amplitudes of the closed bosonic and the heterotic string (second and third row, table 1). This will generalize a simplified representation of massless tree amplitudes of the closed superstring where the cancellations of various multiple zeta values in the α ′ -expansion are taken into account: The selection rules applied to the multiple zeta values in passing from the open to the closed superstring [59] are mathematically formalized by the so-called single-valued "sv" projection [60,61]. The closed superstring emerges from the double copy between SYM and the svprojected open superstring [62]. As pointed out in ref. [58], gravitational amplitudes of superstrings, heterotic strings and bosonic strings only differ in their field-theory kinematic factors. Hence, different double copies involving one of SYM or (DF ) 2 + YM as well as a single-valued open-string theory gives rise to the closed bosonic string or the gravity sector of the heterotic string, see table 1.
As a second major novel result, we extend the collection of field-theory double copies to tree-level amplitudes involving any number of gauge-and gravity multiplets in the heterotic string. For this purpose, bi-adjoint scalars are incorporated into the α ′ -dependent (DF ) 2 + YM theory, resulting in the so-called (DF ) 2 +YM+φ 3 theory whose Lagrangian will be given below. Upon double copying with the single-valued open superstring, this completes the massless sector of the heterotic string, see the last column of table 1. The remaining double-copy options involving (DF ) 2 +YM+φ 3 theory and one of Z-theory or the single-valued open bosonic string yield compactified versions of bosonic string theories 1 .
In the low-energy limit α ′ → 0, the YM+φ 3 theory, known from the double copy of EYM [12,13], can be obtained from the (DF ) 2 +YM+φ 3 theory. Hence, our double-copy construction of the heterotic string encodes all EYM amplitude relations that reduce multitrace couplings of gluons and gravitons to pure gauge-theory amplitudes and should reproduce the results of ref. [31]. The opposite limit α ′ → ∞ of the (DF ) 2 +YM+φ 3 theory, on the other hand, gives access to conformal supergravity coupled to gauge bosons upon double copy with SYM. Indeed, in this regime, the double-copy amplitudes built from the (DF ) 2 +YM+φ 3 theory and SYM agree with those coming from the heterotic ambitwistor string, which were studied in detail in ref. [65].
Finally, we will deduce from the CHY formulae, given in that reference, that amplitude relations of conformal supergravity coupled to SYM are recovered in the α ′ → ∞ limit of the corresponding amplitude relations of the (conventional) heterotic string [43]. An alternative way of arriving at conformal-gravity amplitudes via an α ′ → ∞ limit involves the twistedstring construction of Huang, Siegel and Yuan [66]. We show that taking this limit for the twisted heterotic-string amplitudes one obtains precisely a KLT formula with the undeformed (DF ) 2 theory and SYM amplitudes. This paper is organized as follows: A review in section 2 sets the stage for introducing the (DF ) 2 + YM theory and for explaining its significance for the open and closed bosonic string in section 3. The generalization of these results to the heterotic string is discussed in section 4 which allows to explore various corollaries for different types of gauge-gravity theories in the α ′ → ∞ limit in section 5.
In this section, we review the known double-copy structure of tree-level amplitudes in various string theories and conformal supergravity. This will set the stage for sections 3 and 4 where we identify the field theory that governs the polarization dependent kinematic factors of the bosonic and heterotic string amplitudes. For simplicity, throughout this paper the fieldand string-theory amplitudes will be given without overall coupling-dependent prefactors. Appropriate powers of the gauge coupling g and the gravitational coupling κ can be restored following appendix A.

Open superstrings
The manifestly supersymmetric pure-spinor formalism [67] has been used to determine all tree amplitudes of massless open-superstring states [50] in terms of simpler building blocks, The sum runs over (n−3)! permutations ρ, and the factor of A SYM refers to color-ordered tree amplitudes of ten-dimensional SYM [68] which carry all of the polarization dependence and supersymmetry. The α ′ -dependence, by contrast, is entirely carried by the integrals [50] over moduli spaces of punctured disks which are taken to be in the SL(2, R) gauge with (z 1 , z n−1 , z n ) = (0, 1, ∞). As is well known, the A SYM obeys BCJ relations [2] and A s obeys monodromy relations [37,38]. The permutations ρ, π ∈ S n−3 are chosen such that the disk integrals in (2.2) form independent (n−3)!-element bases under the BCJ and monodromy relations, respectively. Thus, the F π ρ form an (n−3)! × (n−3)! matrix with entries indexed by integration domains π and permutations ρ of integrand labels. The permutations ρ are understood to act on the insertion labels as 2 where ρ acts trivially on the labels 1, n−1, n of the fixed punctures. Alternatively, one can organize the disk integrals in terms of the so-called Z-theory am- n i<j |z ij | 2α ′ s ij ρ {z 12 z 23 · · · z n−1,n z n,1 } .
tion. Both (2.8) and later double-copy statements on string amplitudes are understood to only apply to massless external states, and we will suppress the specifier "massless" henceforth. Whether the Z-theory amplitudes (2.8) define a sensible theory or not is an interesting open question. Evidence supporting an interpretation as a theory stems from the observation that, at leading order in α ′ , the disk integrals (2.4) reproduce the doubly-partial amplitudes of non-linear sigma models coupled to bi-adjoint scalars upon (partial) symmetrization in π [52,53]. Hence, the Z(π|ρ) integrals are interpreted as the doubly-partial amplitudes in a collection of effective scalar theories dubbed Z-theory which captures all the α ′ -dependence of open-string amplitudes. The low-energy expansion of the Z-theory equations of motion can be found in ref. [54].
First evidence for a similar correspondence between one-loop amplitudes of open and closed strings has been given in [79]. Given that the factors A SYM in the open-string amplitude (2.1) are unaffected by the sv operation, the closed-string amplitude (2.9) can be cast into a double-copy form [62] M II = τ,ρÃ SYM (1, τ, n, n−1)S[τ |ρ] 1 svA s (1, ρ, n−1, n) , (2.12) where sv only acts on the open-string amplitudes. The double copy for the closed-string amplitude is then summarized by the word equation which is understood to apply to massless external states according to our earlier disclaimer. Alternatively, the double copy can be written more symmetrically by substituting eq. (2.8) into eq. (2.13), giving a "triple copy" (closed superstring) = (SYM) ⊗ sv(Z-theory) ⊗ (SYM) . (2.14) The corresponding KLT-like formula is (2.15) However, note that this "triple copy" is not a new concept, it simply amounts to applying the double copy twice. Note that svZ(π|σ) can be interpreted as closed-string versions of Z-theory amplitudes [54], and they are given by integrals [80] sv Z(τ |σ) = 2α ′ π n−3 d 2 z 1 d 2 z 2 . . . d 2 z n vol(SL(2, C)) n i<j |z ij | 4α ′ s ij τ {z 12z23 · · ·z n−1,nzn,1 }σ {z 12 z 23 · · · z n−1,n z n,1 } , (2.16) over the moduli space of punctured Riemann spheres 5 .

Bosonic and heterotic strings
For the open bosonic string, n-point tree amplitudes were conjectured 6 to have the same expansion in terms of a basis of disk integrals (2.2) as for the open superstring [58], The kinematic factors B, which will soon be identified with partial amplitudes of a specific gauge theory, capture the entire dependence on the external polarizations, and they additionally depend on α ′ . We give the lowest-multiplicity expressions in terms of gluon polarizations e µ i and linearized field strengths f µν i = k µ i e ν i − e µ i k ν i with D-dimensional Lorentz indices (D = 26 for the critical string) and i = 1, 2, . . . , n labelling the external legs. We use While B(1, 2, 3, 4, 5) at five points can be obtained from ref. [58], extracting the highermultiplicity generalizations 7 from string theory is an open problem. Instead, we will later on employ field-theory techniques to obtain an all-multiplicity description of the kinematic factors B in the bosonic-string amplitude (2.17). Given that the geometric-series expansion of the factor (1 − 2α ′ s ij ) −1 in (2.19) leads to purely rational coefficients for (α ′ s ij ) w , the amplitudes (2.17) of the open bosonic string violate uniform transcendentality. Instead, the combination of the purely rational B and the uniformly transcendental integrals F π ρ in (2.2) sets the upper bound that the order of α ′ w involves MZV of weight ≤ w. In particular, given the leading low-energy behavior imposed by the field-theory limit of (2.17), the pieces of maximal transcendentality in bosonic open-string amplitudes, i.e. the contributions of weight-w MZVs at the order of α ′ w , coincide with those of the superstring (2.1) [58]. Of course, the worldsheet integrand of the open bosonic string underlying (2.17) can be exported to the gravitational sector of the heterotic string. Hence, the n-point tree amplitude of supergravity states h in the heterotic theory [58]  From the absence of MZVs in the α ′ -expansion of B, the piece of maximal transcendentality at a given order in α ′ is universal to gravitational amplitudes in the closed superstring, the heterotic string and the closed bosonic string [58]. The structure of the bosonic-and heterotic-string amplitudes (2.17), (2.22) and (2.23) matches the open-and closed-superstring results for A s and M II . Hence, the results reviewed in this section imply that the gravity sector of the bosonic and heterotic string exhibits a double-copy structure. Indeed, we will pinpoint a gauge theory in section 3 which furnishes the bosonic double-copy component and takes the role of SYM in (2.8) and (2.13).

Field-theory BCJ relations for string amplitudes
As a common property of the constituents in the above string double copies, they are all required to obey the BCJ relations of gauge-theory tree amplitudes [2], e.g. n−1 j=2 k 1 · (k 2 +k 3 + . . . +k j )A SYM (2, 3, . . . , j, 1, j+1, . . . , n) = 0 . (2.24) Given that total derivatives in the worldsheet punctures integrate to zero, the same relations hold for different choices of the right argument ρ in the Z-theory amplitudes Z(π|ρ) in (2.4) with a fixed integration domain π [51]. Accordingly, the single-valued Z-theory amplitudes (2.16) obey BCJ relations for either the left or right argument. For open-string amplitudes A s (π) or A bos (π), different disk orderings π are constrained by the generalization of the BCJ relations (2.24) to monodromy relations [37,38]. They augment the BCJ relations by a series expansion in ζ 2 (α ′ s ij ) 2 which preserve the uniform transcendentality of the superstring. As firstly pointed out in ref. [82], monodromy relations can be projected separately to any order in α ′ and to any MZV which is (conjecturally) independent over Q. Truncating the monodromy relations among (2.17) to zero transcendentality leads to the BCJ relations [58] n−1 j=2 k 1 · (k 2 +k 3 + . . . +k j ) B(2, 3, . . . , j, 1, j+1, . . . , n) = 0 , (2.25) valid to all orders in α ′ , as can be easily verified for the four-point example in (2.19). The same reasoning implies that also single-valued open-string amplitudes satisfy the BCJ relations [80] n−1 j=2 to all orders in α ′ , ensuring consistency of the double-copy formulae (2.12), (2.22) and (2.23).

Conformal supergravity as a double copy
Conformal-supergravity (CSG) amplitudes of the non-minimal Berkovits-Witten-type theory [83] are constructed from the double copy [20] Here, in the second factor, SYM stands for any pure super-Yang-Mills theory in dimensions D ≤ 10, and the first factor is a bosonic gauge theory with the following Lagrangian in any dimension: The field strength and covariant derivatives are defined as The vector A a µ transforms in the adjoint representation of a gauge group G with indices a, b, c, . . ., the scalar ϕ α transforms in a real representation R of the gauge group (with generators T a R ), and g is the coupling constant 8 . The Clebsch-Gordan coefficients C αab and d αβγ are implicitly defined through the two relations Requiring that the tensors C αab , d αβγ and T a R transform covariantly under infinitesimal gaugegroup rotations implies three additional Lie-algebra relations. Together, these five tensor relations are sufficient to reduce any color structure appearing in a pure-gluon tree amplitude to contractions of f abc tensors. See appendix B in ref. [20] for all of these identities.
Tree-level gluon amplitudes of the so-called (DF ) 2 theory defined by the Lagrangian (2.28) have been shown to obey BCJ relations (2.24) up to at least eight points. The colorkinematics duality guarantees [14] that the double copy will give amplitudes that are invariant under space-time diffeomorphisms, i.e. amplitudes in a gravitational theory. In terms of treelevel amplitudes, the double copy (2.27) for conformal supergravity can be expressed using the standard KLT formula, where the external states of the (DF ) 2 amplitude are restricted to gluons in order to obtain graviton amplitudes in the double copy. A curious feature of the tree-level amplitudes of (DF ) 2 theory is that the product between any two polarization vectors e i · e j always cancels out through non-trivial identities [20], e.g.
It implies that when considering dimensional reduction, the scalars that come from the extradimensional gluon components will automatically decouple from the theory. In the CHY formulation of (DF ) 2 theory given in [65] the e i · e j are manifestly absent. The double poles ∼ s −2 12 in (2.32) reflect a gluon propagator ∼ k −4 due to the four-derivative kinetic term in the Lagrangian (2.28), and the unusual s 13 -channel pole in A (DF ) 2 (1, 2, 3, 4) that defies the naive expectation from the color ordering, is due to an exchange of the ϕ α scalar. This exchange follows from the coupling ∼ C αab ϕ α F a µν F bµν in the Lagrangian (2.28) along with the structure constants generated from the contraction C αab C αcd , see (2.30).

Bosonic string amplitudes from the double copy
In this section, we will consider the extended gauge theory (DF ) 2 + YM and argue that massless tree amplitudes (2.17)  Note that the closed bosonic string can be alternatively viewed as a triple copy involving single-valued Z-theory see the discussion around (2.14) for the analogous triple-copy structure of closed superstrings. The bosonic field theory, (DF ) 2 +YM entering (3.1) is defined by the following Lagrangian which augments (2.28) by mass-dependent terms also considered in ref. [20]. This (DF ) 2 +YM gauge theory smoothly interpolates between the previously introduced (DF ) 2 theory (m = 0) and pure Yang-Mills theory (m = ∞). Its field content comprises a massless gluon, a massive gluon and a massive scalar (see appendix C.1 for the propagators). A priori, the mass is a free parameter, but we will find that it has to be related to the inverse string tension α ′ by Since m 2 is negative, it gives rise to tachyonic modes, i.e. the massive gluon as well as the massive scalar ϕ α are tachyonic. In the context of the open and closed bosonic strings, this reflects the appearance of scalar tachyon modes that propagate internally. Note that the Lagrangian (3.3) is normalized such that the limit m → 0 (α ′ → ∞) is well behaved; for a well-behaved limit m → ∞ (α ′ → 0) we need to multiply the Lagrangian by m −2 = −α ′ , which explains such overall normalization factors appearing in some of the subsequent amplitudes.
Based on a few reasonable assumptions, we can argue that the Lagrangian (3.3) is the unique field theory that describes the kinematic factors B in the bosonic-string amplitudes, which results in the double-copy statements given in eq. (3.1). These assumptions are: (i) In the low-energy limit α ′ → 0, the factors B become YM tree amplitudes.
(ii) In the high-energy limit α ′ → ∞, the B/α ′ are finite and have at most poles s −2 i...j .
(iii) B are partial amplitudes coming from a bosonic gauge theory that obeys the colorkinematics duality in (at least) D ≤ 26 dimensions.
The first assumption (i) follows from the field-theory limit of the construction of ref. [58], see the discussion around (2.21). Assumption (ii) is an observed property of the explicit examples given in the same paper. Moreover, the general validity of (ii) appears plausible since the limit Since tachyon propagators usually line up with massless propagators in moduli-space integrals 9 the limit α ′ → ∞ introduces double poles ∼ s −2 ij as for instance seen in (2.32). Additional poles s −n ij of higher order n ≥ 3 do not occur since the OPE among gluon vertex operators does not introduce any singularities |z ij | 2α ′ k i ·k j −n with n ≥ 3 into the worldsheet integrand. Finally, assumption (iii) is consistent with the observation [58] that the B obeys the well-known relations of gauge-theory partial amplitudes: cyclicity, Kleiss-Kuijf and BCJ relations.
Under the above three assumptions, both limits α ′ → 0, ∞ of B result in gauge-theory amplitudes that have no dimensionful parameter. It is thus a simple task to list all possible operators of a given dimension in the underlying gauge-theory Lagrangians and to constrain unknown couplings using color-kinematics duality. From dimensional analysis, it follows that the limit α ′ → 0 is dominated by dimension-four operators (in D = 4 counting), and α ′ → ∞ is dominated by dimension-six operators (in D = 6 counting). In ref. [20] it was shown that there is a unique color-kinematics satisfying dimension-six Lagrangian that is built out of parity-invariant 10 operators involving F µν 's and the scalar ϕ α , and where the kinetic terms have at most four derivatives. Once the dimension-four YM term, (F a µν ) 2 , is introduced, there exists a one-parameter solution to the constraints of color-kinematics duality [20], this free parameter is the mass m and the resulting Lagrangian the one in eq. (3.3). By mutual consistency of the three assumptions, m 2 has to be proportional to (α ′ ) −1 .
Color-stripped amplitudes A (DF ) 2 +YM up to multiplicity eight are straightforward to compute from the Feynman rules of the Lagrangian (3.3). We have confirmed that they reproduce all known kinematic factors B coming from the open-bosonic-string amplitudes (2.17), given that the mass is identified with α ′ as in eq. (3.4). More precisely we have derived (2.18) and (2.19) at three and four points as well as the expression for B (1,2,3,4,5) in the arXiv submission of ref. [58] from the gauge-theory Lagrangian (3.3). These checks and the plausibility of the assumptions (i) to (iii) provide strong evidence for our general conjecture (3.5) on the field-theory description of B. The latter in turn implies the doublecopy structures (3.1) of the massless sector of bosonic open-and closed-string theories. 9 A typical disk integral in the opening line for the four-point amplitude of the open bosonic string reads We have used integration by parts to relate it to the basis function F (2) (2) in (2.2) which in turn defines as α ′ → ∞ and thereby introduces a double pole ∼ s −2 12 into A (DF ) 2 (1, 2, 3, 4). 10 Requiring that the theory is defined in D ≤ 26 dimensions rules out the appearance of a Levi-Civita tensor in the Lagrangian. Given that the amplitude structure is analogous to (2.15) and therefore symmetric under the exchange of the left-and right-moving kinematic factors, one can also rewrite (2.22) as along the lines of (2.14) and (3.2).
In this section, we extend this construction to heterotic-string amplitudes involving arbitrary combinations of gauge-multiplet and supergravity-multiplet states. We will spell out a field-theory Lagrangian for the first double-copy component in (heterotic string) = (DF ) 2 + YM + φ 3 ⊗ sv open superstring (4.5) which is equivalent to the triple copy

The structure of heterotic string amplitudes
Given that the gauge and gravity multiplets of the heterotic string couple via genus-zero worldsheets, already the gauge sector exhibits structureful tree-level amplitudes with multiple traces of the color factors. Single-trace amplitudes A het can be obtained from the single-valued projection [80] A het (1, 2, . . . , n) = svA s (1, 2, . . . , n) (4.7) of the type-I results (2.1). Double-trace amplitudes of gauge multiplets in turn have been reduced to linear combinations of their single-trace counterparts (4.7) [43]. The expansion coefficients in these double-trace results exemplify that multi-trace contributions generically violate uniform transcendentality, and it is an open problem to generalize the explicit relations of ref. [43] to higher numbers of traces. In fact, the Lagrangian given in the next subsection will implicitly reduce any multi-trace sector to linear combinations of (4.7). Mixed amplitudes M g+h het involving external gauge-multiplets g and gravity-multiplets h can also be organized according to the trace structure of their color dependence. The singletrace sector with ≤ 3 gravitons as well as the double-trace sector with ≤ 1 graviton has been expressed in terms of the gauge amplitudes [43]. The amplitude relations of this reference include 11 A het (1, 2, . . . , n; p) = n−1 j=1 (e p · x j ) A het (1, 2, . . . , j, p, j+1, . . . , n) (4.8) A het (1, 2, . . . , n; p, q) = A het (1, 2, . . . , i−1, q, i, . . . , j, p, j+1, . . . , n) + (p ↔ q) , (4.9) with gluon legs 1, 2, . . . , n, graviton momenta p, q, graviton polarizations e p , e q as well as region momenta x j ≡ k 1 +k 2 + . . . +k j . By analogy with the field-theory result [31], amplitude relations of this type are expected to exist for all multi-trace sectors of the heterotic string and any number of gluons and gravitons [43,80]. This would allow us to reduce the massless tree-level S-matrix of the heterotic string to the same basis of sphere integrals seen in the gravity sector (2.22) or the type-II string (2.9) and leads us to conjecture that for a suitable choice of non-supersymmetric kinematic factors A (DF ) 2 +YM+φ 3 (1, τ, n, n−1). The factors of A (DF ) 2 +YM+φ 3 (1, τ, n, n−1) capture the color degrees of freedom of the gluons and the extra polarization vectors of the gravitons such as e p and e q in (4.8) and (4.9). Given that the factors of A SYM within A s contribute gauge-multiplet polarizations, the external states in (4.10) are gauge and gravity multiplets if A (DF ) 2 +YM+φ 3 (1, τ, n, n−1) involve external scalars s and gluons g, respectively. Denoting their appearance in the i th and j th leg by i s and j g , respectively, the independent three-and four-point instances read as well as We recall the notation f 24 = s 24 (e 2 · e 4 ) − (e 2 · k 4 )(e 4 · k 2 ), and the five-point analogues of (4.12) can be found in appendix C.2. The subscript of A (DF ) 2 +YM+φ 3 as well as the notation A 1 , A 2 , . . . andf A 1 A 2 A 3 for the adjoint indices and the structure constants will become clear below. Similarly, the coupling λ controls the relative weight of single-trace and multi-trace contributions 12 , and the conventions for the overall normalization of (4.10) are detailed in appendix A. Note that the heterotic-string amplitudes (4.10) are understood to comprise at least two external gluons, and the purely gravitational cases requiring a different bookkeeping of normalization factors are captured by (2.22). Using a combination of the arguments in ref. [58,82], it will be argued in appendix C.3 that the kinematic factors in (4.10) obey BCJ relations n−1 j=2 k 1 · (k 2 +k 3 + . . . +k j ) A (DF ) 2 +YM+φ 3 (2, 3, . . . , j, 1, j+1, . . . , n) = 0 , (4.13) for arbitrary number of external gluons and adjoint scalars.

Scalar extension of the (DF ) 2 theory
As a main result on the heterotic-string amplitudes, the kinematic factors A (DF ) 2 +YM+φ 3 in (4.10) are conjectured to follow as the color-ordered amplitudes in a field theory involving scalars and vectors with Lagrangian [20] 12 The coupling λ can be inserted into the Kac-Moody currents of the gluon vertex operators, which then with F 3 defined in (2.29). This augments the Lagrangian (3.3) by massless 13 scalars φ aA that are charged under a global group G het as well as the local gauge group G. We treat it as a single field that carries adjoint gauge-group indices a, b, c, . . . and global-group adjoint indices A, B, C, . . ., and this allows for a cubic scalar self-interaction with an unconstrained and dimensionless coupling λ. Note that the global group G het becomes the gauge group of the heterotic string after the double copy (4.5) is performed, and in the tree-level context of this work, G het can be generalized beyond the rank-sixteen cases realized by compactification of sixteen spacetime dimensions [63,64]. As for the (DF ) 2 +YM theory, the Lagrangian (4.14) is normalized such that the limit m → 0 (α ′ → ∞) is well behaved. For a well-behaved limit m → ∞ (α ′ → 0) in turn, we need to multiply the Lagrangian by m −2 = −α ′ , and rescale the bi-adjoint scalar φ aA → mφ aA and redefine its self-coupling mλ = λ ′ , which explains why such steps are needed in some of the subsequent formulas. The kinematic factors A (DF ) 2 +YM+φ 3 appearing in the massless heterotic-string amplitudes (4.10) we claim to arise from the partial tree-level amplitudes of the Lagrangian (4.14) after color-decomposing with respect to the local gauge group G. As exemplified by (4.11) and (4.12), these color-ordered amplitudes still depend on the degrees of freedom of the global group G het through the appearance of δ AB andf ABC tensors.
That the identification of the kinematic factor in eq. (4.10) is correct can be argued through similar considerations as in section 3. By considering the A (DF ) 2 +YM+φ 3 as unknown factors appearing in the massless heterotic-string amplitudes, we assume the following minimal properties (in addition to the properties already considered for the B amplitudes in section 3), (i) In the low-energy limit −1/α ′ = m 2 → ∞ combined with the abelian limit of G het (λ → 0), the factors A (DF ) 2 +YM+φ 3 become tree amplitudes in 26-dimensional YM theory dimensionally reduced to D = 10, where the scalars arise as internal components of the gluons.
(iii) A (DF ) 2 +YM+φ 3 are partial amplitudes coming from a bosonic gauge theory that obeys the color-kinematics duality in (at least) D ≤ 10 dimensions.
The first property comes from the knowledge that heterotic-string amplitudes in the fieldtheory limit and abelian limit become double copies between 26-dimensional YM theory and ten-dimensional SYM [63,64]. The second property comes from considering gluon scattering at tree level in the heterotic string and taking the gauge-group coupling to be large in comparison to the gravitational coupling in units of α ′ . This limit can be implemented via λ → ∞ which gives single-trace amplitudes of uniform transcendentality, consistent with the singlevalued projection (4.7) of the open-string amplitudes [80]. A double copy with φ 3 amplitudes on one side is equivalent to the identity operation, and thus property (ii) is consistent with eq. (4.5). Assumption (iii) follows from (4.13) based on the arguments in appendix C.3. The amplitudes obtained from the Lagrangian (4.14) obey all of the above properties, and have the right behaviour in the limits α ′ → 0 and α ′ → ∞ (see section 3 for the analogous α ′ → 0, ∞ properties in case of the kinematic factors of the bosonic string). From the analysis in ref. [20], L (DF ) 2 +YM+φ 3 given in (4.14) is the unique Lagrangian with these properties.
More on property (i): Property (i) deserves some further explanation as it is not obvious from simply staring at the Lagrangian (4.14). After the rescaling φ aA → mφ aA , the limit m → ∞ and λ → 0 can be taken by keeping all terms proportional to m 2 and dropping the rest. The ϕ α is now an auxiliary field that can be integrated out, this gives the φ 4 interaction that is typical of dimensionally reduced YM theory: where the overall m 2 has been removed by a trivial rescaling. Note that since we obtain dimensionally reduced YM theory, we can simply think of the φ aA scalars as being the extradimensional gluons coming from the 26-dimensional theory. This interpretation works even for finite m, but in the limit m → 0 the interpretation breaks down (as previously mentioned, dimensional reductions of the pure (DF ) 2 theory yield scalars that automatically decouple [20]). More on property (ii): To elaborate on a stronger version of property (ii), we may take the m → ∞ limit of the Lagrangian (4.14) while keeping the product λ ′ ≡ mλ finite (which is now a dimensionful coupling constant). This yields the YM + φ 3 Lagrangian introduced in ref. [12], where n s is the number of external scalars, and thus this generalizes the low-energy limit of the pure-gluon amplitudes (2.21). Note that the purpose of inserting an overall factor of m ns−2 and redefinition of λ is to make the dimensions agree on the two sides of the equality. (Accordingly, the α ′ → 0 limit of the heterotic amplitude (4.10) needs the analogous rescaling.) Consistency as α ′ → ∞: The compatibility of the double copy (4.10) with the Lagrangian (4.14) in the opposite limit m → 0 can also be seen. Correlation functions of the heterotic string again yield worldsheet integrals whose expansion in a uniform-transcendentality basis of (svF ) ρ σ involves coefficients ∼ 2α ′ 2α ′ s i...j −1 . As explained in section 3, the latter usually line up with massless propagators s −1 i...j such that the limit α ′ → ∞ introduces double poles s −2 i...j into A (DF ) 2 +YM+φ 3 as expected from the massless limit of (4.14).
Relating α ′ to the mass parameter of (DF ) 2 + YM + φ 3 theory: At this point, one can also see that the mass parameter m 2 = −(α ′ ) −1 of (4.14) needs to be interlocked with α ′ in the Koba-Nielsen factor of the disk integrals (2.2) and their single-valued projection. Our argument is based on the level-matching condition for left-and right-moving degrees of freedom which removes the tachyon from the heterotic-string spectrum. Tachyon propagation in massless four-point amplitudes of the heterotic string is suppressed by the zeros of the Virasoro-Shapiro factor . This cancellation crucially depends on having the same value of α ′ in the two sides of the double copy (4.10). These arguments can be repeated in the context of the closed bosonic string: The double copy of the kinematic factors B in the tree amplitudes (2.23) of the closed bosonic string introduces double poles at 2s ij = −m 2 . Once the mass parameter of the underlying (DF ) 2 + YM theory is adjusted to α ′ , these double poles conspire with the zeros of the accompanying sphere integrals at 2s ij = α ′ −1 . Hence, the field-theory mass parameter must be related to α ′ in order to arrive at the kinematic pole structure for tachyon propagation.
The above discussion is tailored to open-string conventions for the normalization of α ′ . The conventional closed-string normalization of α ′ in (4.19) and related equations can be attained by replacing α ′ → α ′ 4 which identifies the mass of the closed-string tachyon to be twice the mass of the open-string tachyon.
Explicit checks: Amplitudes up to multiplicity eight can be straightforwardly computed from the Feynman rules of the (DF ) 2 + YM + φ 3 theory (see appendix C.1); we have explicitly checked that these agree with all the corresponding kinematic factors extracted from heterotic string amplitudes up to multiplicity five, as well as the triple-trace color structure 14 in the six-point heterotic-string amplitude (4.10).

Gauge-gravity amplitude relations and CSG from α ′ → ∞
This section is dedicated to field-theory implications of our double-copy representations of string amplitudes. We discuss some connections between the α ′ → ∞ limit of heteroticstring amplitudes and those of conformal gravity, including its coupling to gauge theories. In this limit, the double-copy structure of the heterotic string is explained to imply amplitude relations in more exotic examples of gauge-gravity theories, see section 5.2 below.

Amplitude relations for conformal supergravity coupled to gauge bosons
As reviewed in section 2.5, the α ′ → ∞ limit of the ((DF ) 2 + YM)-theory amplitudes can be double copied with SYM to yield conformal-gravity amplitudes [20] of Berkovits-Witten type [83] (see also [84,85]). In ref. [65], it was shown that these conformal-gravity amplitudes can also be obtained from the heterotic ambitwistor string. Given the relation between the tensionless limit of usual string theory and ambitwistor strings [86][87][88], it is reasonable to expect the α ′ → ∞ limit of the heterotic-string relations (4.8) and (4.9) to apply to colorstripped amplitudes A CSG+SYM in conformal supergravity coupled to gauge bosons.
Indeed, it will be shown in appendix D.1 through the CHY formalism that this is the case for a small number of external gravitons. While the amplitude coefficients in (4.8) for the one-graviton case do not depend on α ′ and allow for the immediate translation − (e q · p) A SYM (1, 2, . . . , i−1, q, i, . . . , j, p, j+1, . . . , n) + (p ↔ q) , see appendix D.1 for a detailed derivation from the CHY formalism. Note that CSG + SYM is a ten-dimensional generalization of Witten's twistor string theory [89] (including the conformal gravity sector that is often discarded). The generalization of (5.1) and (5.2) to three gravitons, which can also be derived from the α ′ → ∞ limit of the corresponding heteroticstring relations [43], can be found in appendix D.2. Further generalizations to arbitrary numbers of gravitons and traces are encoded in the double-copy (4.10) along with the Lagrangian in section 4.2.
Since the CSG + SYM amplitudes can be obtained from the α ′ → ∞ limit of the double copy ((DF ) 2 + YM + φ 3 ) ⊗ SYM they correspond to a double copy between (DF ) 2 + φ 3 and SYM, where the Lagrangian for the (DF ) 2 + φ 3 theory is obtained by setting m = 0 in (4.14). From this fact it is clear that (5.2) can be used to straightforwardly identify BCJ numerators of the (DF ) 2 + φ 3 theory after writing the A SYM factors in a Kleiss-Kuijf basis, in similar fashion to the method introduced in ref. [14] where corresponding EYM relations employ BCJ numerators of the YM + φ 3 theory; see ref. [14] for further details.
The multiparticle generalization of these examples and the CHY integrand associated with the double-trace sector can be found in appendix D.3.

More exotic examples of gauge-gravity theories
In this subsection, we will investigate more exotic gauge-gravity theories that can be obtained by double-copying (DF ) 2 with massless gauge theories coupled to a bi-adjoint scalar. These additional gauge-gravity theories will be argued to share the amplitude relations of either EYM or CSG + SYM. As shown in [65], the double copy of (DF ) 2 with itself produces the amplitudes of R 3 (six-derivative) gravity coming from bosonic ambitwistor strings. If one instead considers the double copy of (DF ) 2 with (DF ) 2 + φ 3 , one then obtains the amplitudes of R 3 gravity coupled to (DF ) 2 gauge bosons. Interestingly, given the similarity of the CHY integrands of both theories [65], it is easy to show that the amplitude relations in this section and appendices D.2 and D.3 can be carried over to this R 3 + (DF ) 2 theory. For this purpose, the amplitude subscripts in (5.1), (5.2) and their generalizations need to be adjusted according to CSG + SYM → R 3 + (DF ) 2 and SYM → (DF ) 2 , respectively. Since we show in this paper that the amplitude relations for CSG + SYM are related to an α ′ → ∞ limit, it is reasonable to suspect the same is true for R 3 + (DF ) 2 . It would be interesting to identify the finite-α ′ version of these relations.
In table 2 below we give an overview of possible double-copy constructions involving gauge theories coupled to a bi-adjoint scalar. While the entries of R 3 gravity and its couplings to the (DF ) 2 theory have just been discussed, it remains to introduce one last gauge-gravity theory: The entry "CG + (DF ) 2 " in table 2 refers to conformal gravity coupled to (DF ) 2 gauge bosons. Table 2. Overview of the double-copy constructions discussed here involving pure gauge theories, or gauge theories coupled to a bi-adjoint scalar.
Similarly, for a higher number of gauge bosons and gravitons as well as for the multi-trace sectors, one can export the amplitude relations of EYM [28,30,31,42] by adjusting A EYM → A CG+(DF ) 2 and A SYM → A (DF ) 2 . In appendix D.4, we introduce the CHY integrand for the CG + (DF ) 2 theory and justify the agreement of its amplitude relations with those of EYM.
Since this cancellation of all MZVs can be traced back to properties of the sphere integrals, the analogous combination with one of A s replaced by A bos reduces to the kinematic factors i.e. to the zero-transcendentality piece of the gravitational heterotic-string amplitude (2.22). Following the discussion of section 3, the kinematic factors B of the bosonic string degenerate to the double-copy constituent A (DF ) 2 of conformal supergravity when taking the limit α ′ → ∞. Then, the right hand side of (5.9) reduces to the KLT representation (2.31) of the tree amplitudes of conformal supergravity [20], i.e. we are led to the following new representation: A bos (1, π(2, . . . , n−2), n, n−1)S α ′ [π|σ] 1 A s (1, σ(2, . . . , n−2), n−1, n) . (5.10) Since the tensionless limit of these "twisted" heterotic string amplitudes gives the same result as the ambitwistor strings, this is another way of seeing that the amplitudes coming from the gravity sector of the heterotic ambitwistor string correspond to conformal-supergravity amplitudes.
(5.11) Then, taking the high-energy limit α ′ → ∞ of the "twisted" heterotic string yields conformal supergravity coupled to SYM, 15 (5.12) and reproduces the amplitude relations in section 5.1 upon color decomposition. When translating these ten-dimensional amplitudes (5.12) to four dimensions, we obtain Witten's twistor string theory [89], and the corresponding scattering equations present in the Roiban-Spradlin-Volovich (RSV) formula [92] are now a natural consequence of the α ′ → ∞ limit, as originally considered by Gross and Mende [93].

Summary and Outlook
In this work, we have proposed field-theory double-copy structures for all massless tree-level amplitudes of bosonic and heterotic strings. One side of the double copy comprises the same Z-theory amplitudes or moduli-space integrals over punctured disk and sphere worldsheets as seen in their superstring counterparts. The additional, non-supersymmetric double-copy components specific to bosonic and heterotic strings are conjectured to descend from massive gauge theories whose Lagrangians are spelt out in the main text. These gauge-theory Lagrangians are combinations of dimension-four and dimension-six operators whose relative scaling is controlled by the mass parameter α ′ = −m −2 . The dimension-six part which dominates in the α ′ → ∞ limit is known as the (DF ) 2 gauge theory in the double-copy construction of conformal supergravity [20]. Our results give particularly striking examples for the ubiquity of field-theory structures in string amplitudes and universal properties across different string theories. Apart from the universal α ′ -dependent moduli-space integrals, tree-level amplitudes in different string theories are characterized by gauge theories which specify the kinematic factors. Properties of the moduli-space integrals require these field theories to obey the duality between color and kinematics in the same way as the Z-theory amplitudes do. Given that the respective string-theory context fixes the mass dimensions and the particle content of the field-theory operators, the α ′ → 0, ∞ behaviour and the color-kinematics duality uniquely specify the extensions of the (DF ) 2 theory relevant to bosonic and heterotic strings.
We also showed that relations analogous to those of EYM hold for the amplitudes involving gauge and gravity multiplets in conformal (super-)gravity, and that they can be obtained by taking the α ′ → ∞ limit of the corresponding heterotic-string amplitudes. Since the theory of gluons coupled to conformal gravity is precisely the one coming from the heterotic ambitwistor string, our results add to the claims that the latter is more naturally associated with the tensionless limit of ordinary strings.
While the results of this work only concern massless external states, the organization of the moduli-space integrals into (n−3)! × (n−3)! bases should extend to scattering of massive string resonances [58]. It is conceivable that the accompanying kinematic factors depending on the higher-spin polarizations can also be traced back to field theories that generalize the (DF ) 2 + YM + φ 3 Lagrangian.
Another important line of follow-up research concerns loop amplitudes. For massless one-loop amplitudes of maximally supersymmetric open strings, a double-copy structure resembling matrix elements of gravitational R 4 operators has been recently proposed in ref. [57]. The (generalized) elliptic functions of the worldsheet punctures in the reference are tailored to maximal supersymmetry, but the results of ref. [48] on one-loop open-string amplitudes in orbifold compactifications might pave the way to generalizations with reduced supersymmetry. It would be interesting to investigate similar double-copy structures in one-loop amplitudes of bosonic and heterotic strings as well as higher-loop generalizations.
While the double-copy structure of string amplitudes is perturbative in the current context, the fact that kinematic factors of type I, IIA/B and heterotic theories uniformly arise from a field-theory perspective, with a simple Lagrangian interpretation, suggests a connection to non-perturbative string dualities. Understanding the role of the (DF ) 2 + YM + φ 3 gauge theory and Z-theory at loop level may shed new light on these dualities, such as the heterotic/type-I duality [94], or even M-theory [95].
where n g and n h denote the numbers of external gluons and gravitons, respectively, with total n = n g + n h . The coupling constants κ and g appear in the ten-dimensional low-energy effective action as G −1/2 L GR ∼ 4 κ 2 R and G −1/2 L YM ∼ 1 4 (F a µν ) 2 , where the metric and the field strength are expanded as G µν = η µν + κh µν and F a Note that the color-stripped heterotic single-trace amplitude A het has a slightly different overall coupling-dependent factor relative to the color-dressed amplitude M g+h het ; the relative factor is (2g/κ) ng −2 which is proportional to λ ng−2 .
The corresponding field-theory amplitudes are normalized in analogous manner, e.g. where the "gauge" subscript stands for YM, SYM or (DF ) 2 theory, including the deformations of the (DF ) 2 theory, and "gravity" stands for Einstein, EYM, conformal supergravity, or deformations thereof.

B Single-valued MZVs in the α ′ -expansion
For tree amplitudes (2.2) and (2.9) of the open and closed superstring, the α ′ -expansion gives rise to MZVs (2.10) in a uniformly transcendental pattern [59,[74][75][76][77]: The w th order in α ′ is exclusively accompanied by MZVs (2.10) of weight w = n 1 + n 2 + . . . + n r . Moreover, the (n−3)! × (n−3)! matrix form of (2.2) is suitable to relate coefficients of different MZVs that are conjecturally independent over Q: Let P w and M w denote (n−3)! × (n−3)! matrices whose coefficients are degree-w polynomials in α ′ s ij with rational coefficients, then (upon suppressing the indices π, ρ ∈ S n−3 in (2.2)) [59] For instance, the coefficient of ζ 2 ζ 3 is a matrix product P 2 M 3 involving the coefficients P 2 and M 3 of ζ 2 and ζ 3 , and similarly, ζ 3,5 is accompanied by a matrix commutator of M 3 and M 5 (see [59] for the continuation to higher weights and explanation of the rational prefactor 1 5 ). The state-of-the-art-methods 16 to obtain the explicit form of the P w and M w at n points involve the Drinfeld associator [69] (also see [101]) or a Berends-Giele recursion for Z-theory amplitudes [54]. Results up to and including seven points are available for download via [102].
Once the closed-string tree amplitude is cast into the form (2.9), the matrix svF can be obtained by imposing certain selection rules on the MZVs in the analogous open-string expansion (B.1) which are obscured by the string-theory KLT formula [1]. These selection rules have been pinpointed to arbitrary weights [59] (see [77] for earlier work on weights ≤ 8) and were later on identified [62] with the single-valued projection [60,61]. As exemplified by the leading orders the closed-string α ′ -expansion follows from (B.1) by acting with the single-valued projection (2.11) [60,61] on the MZVs, see the references for sv(ζ n 1 ,n 2 ,...,nr ) at higher weight and depth.

C.1 Feynman rules
As a gentle warning to the reader, we note that the gauge-theory Lagrangians throughout the paper are given using a Lorentzian metric of mostly-minus signature. Amplitudes obtained from standard Feynman-rule calculations in this metric are then compared with the stringtheory results after flipping the metric signature. Consequently, all amplitudes found in the paper use a mostly-plus metric. We now collect the Feynman rules employed for calculations in the mass-deformed scalarcoupled (DF ) 2 + YM + φ 3 theory with Lagrangian (4.14). In mostly-minus signature, the propagators for the gluon A a µ , scalars ϕ α and φ aA are as follows: The three-and four-vertices in mostly-minus signature are: Expressions for five-and six-vertices can be found in ref. [20]. Partial amplitudes are obtained introducing gauge-group generators (canonically) normalized as where f a 1 a 2 a 3 are real structure constants.
For the theory of conformal supergravity coupled to gauge bosons, there is an expression analogous to (D.2) which, as is implicit in [65], can be obtained from the heterotic ambitwistor string and is given by n+r (1, 2, . . . , n; p 1 , . . . , p r ) = C(1, 2, . . . , n) W 11 · · · 1 r ({p, e p , z}) Pf ′ Ψ n+r ({k, p, e k , e p , z}) , where W {I} denotes a product of the Lam-Yao cycles. The latter have been introduced in ref. [105] and were later used in ref. [81] in the study of the CHY formulation of amplitudes coming from theories involving higher-dimensional operators, such as F 3 for gauge theory and R 2 for gravity. In the particular case appearing in (D.3), it can be written in terms of the diagonal elements of the C-block of Ψ n+r as We are now ready to prove that (4.8) literally translates into (5.1) for conformal gravity.
In fact, since Pf Ψ r=1 = W 1 = C pp , the proof is identical to the one given in ref. [28], and amounts to showing that, in this case, both the first and the third lines of (4.9) apply in this case. Once more, this could be expected from the absence of explicit α ′ -dependence in those terms. The last term in (D.6) has a new tensor structure w.r.t. the Pfaffian. However, it is very similar in form to the last term in (D.8). In fact, the former can be obtained from the latter by replacing (e p · e q ) with s −1 pq (e p · q)(e q · p). This in turn ultimately implies that one can perform such a substitution directly in the expression relating the (single-trace) scattering amplitudes of two gravitons and n gluons in EYM to that of n+2 gluons, and thus obtain the analogous expression in conformal supergravity. Doing so in the expression given in ref. [28], we get an additive contribution of − (e p · q)(e q · p) 2s pq which has precisely the same form as the α ′ → ∞ limit of the second line of (4.9). This reproduces the second line of (5.2) and completes the proof that the α ′ → ∞ limit of the heterotic-string relations (4.8) and (4.9) applies to conformal supergravity coupled to gauge bosons.
The methods of appendix D.1 allow for a direct derivation of (D.11) from the CHY formalism.
Note that the ordered set {r+1, τ, ℓ} in summation range for σ in (D.15) and (D.16) is understood as inserting the permutation τ from the previous one in between legs r+1 and ℓ.