Scattering Massive String Resonances through Field-Theory Methods

We present a new method, exact in $\alpha'$, to explicitly compute string tree-level amplitudes involving one massive state and any number of massless ones. This construction relies on the so-called twisted heterotic string, which admits only gauge multiplets, a gravitational multiplet, and a single massive supermultiplet in its spectrum. In this simplified model, we determine the moduli-space integrand of all amplitudes with one massive state using Berends-Giele currents of the gauge multiplet. These integrands are then straightforwardly mapped to gravitational amplitudes in the twisted heterotic string and to the corresponding massive amplitudes of the conventional type-I and type-II superstrings.


INTRODUCTION
The historical origin and the discovery of key features of string theory can be attributed to the study of its scattering amplitudes. Computations and structural properties of string amplitudes rely on exactly solvable correlation functions of vertex operators in a two-dimensional conformal field theory (CFT). For closed strings, the CFT approach leads to a factorization of the correlators into holomorphic and antiholomorphic building blocks, socalled chiral correlators. This property underlies the treelevel double-copy relation between perturbative gravity and gauge theories obtainable from string theory [1][2][3], and inspired loop-level generalizations [4,5].
While the tree-level CFT prescription has long been textbook material [6,7], recent discoveries of powerful double-copy structures within the chiral correlators have dramatically changed our perspective. Tree-level amplitudes of n massless states of the open superstring [8,9] and the open bosonic string [10,11] can be factorized into scalar integrals over moduli spaces of punctured disk worldsheets and quantum field theory (QFT) building blocks carrying all the dependence on the external polarizations. In hindsight, this striking structure can be traced back to a decomposition of chiral correlators into a basis of integrals in the twisted cohomology defined by the moduli-space integration [12,13]. This cohomology decomposition is a general feature of string theory, its applicability to massless closed-string amplitudes was demonstrated in [11,[14][15][16].
In this letter, we present the first all-multiplicity instance of double-copy structures and cohomology decompositions of string amplitudes with massive external states. More specifically, we describe a simple QFT setup that computes the necessary building blocks for open and closed superstring amplitudes with n−1 massless and a single massive level-1 state. These are derived from Feynman diagrams of 10D super-Yang-Mills (SYM) theory deformed by a cubic operator involving two gauge multiplets and one spin-2 multiplet analogous to the first massive level of the open superstring.
Our QFT construction stems from the heterotic version of the chiral or twisted string theories [17,18]. They differ from conventional strings by a relative sign flip of the inverse string tension α ′ between the holomorphic and anti-holomorphic sectors. The level-matching condition is then flipped, leading to a finite physical spectrum. Accordingly, the moduli-space integrals in their amplitudes encode the exchange of a finite set of internal states. The chiral correlators and their cohomology decompositions, however, can be freely translated between twisted and conventional strings [18,19].
Because of the finite spectrum, interactions among massless and massive states of the heterotic twisted strings can be exactly described by a Lagrangian, making calculations simpler. According to [11,20], the α ′ → ∞ limit of the theory is related to a four-derivative massless supergravity that becomes conformal in four dimensions. A massless 4D Lagrangian was derived in [21,22], and this theory is equivalent to Witten's twistor string, containing both N = 4 SYM and conformal supergravity [23,24].
Here we will use a subsector of the 10D Lagrangian of the twisted string to reverse-engineer the chiral correlator for n-point scattering of gauge multiplets and a single massive state. This correlator can then be exported to conventional string theories to obtain the fully simplified open-and closed-string tree amplitudes involving a mass-level-one state, with manifest double-copy structure and their exact α ′ -dependence. As a byproduct, the chiral n-point correlators also determine the gravitational couplings of a single massive state in the twisted string. Further details will appear in a longer paper [25].

BASICS OF HETEROTIC STRINGS
We begin by reviewing the twisted heterotic string, comparing it to the conventional heterotic string.
A. Vertex operators: Physical states of both twisted and conventional closed strings are represented via vertex operators of the form where the polarization data factorizes into holomorphic and antiholomorphic pieces, respectively V R andV L . The plane waves involve spacetime momenta k m (with vector indices m, n, p, . . . = 0, 1, . . . , 9) subject to the mass-shell condition k 2 + M 2 = 0. We leave implicit the normal ordering with respect to the Wick contractions with signature η mn = diag(−1, 1, 1, . . . , 1). The relative ± sign is positive for conventional and negative for twisted strings, and it propagates to the Koba-Nielsen factors The physical spectrum of twisted heterotic strings is described by (1) with the following chiral halves in canonical superghost pictures depending on ϕ: The bosonic side involves Kac-Moody currentsJ a with adjoint indices a, b, . . . = 1, 2, . . . , dim(G) of an unspecified gauge group G with generators T a satisfying [T a , T b ] = c abc T c . The supersymmetric side contains the matter variables λ m , S α , S β m of the Ramond-Neveu-Schwarz (RNS) superstring [26][27][28][29], with Weyl-spinor indices α, β, . . . = 1, 2, . . . , 16. The SO (1,9) Pauli matrices satisfy {γ m , γ n } = 2η mn , and we are using k αβ ≡ k m γ m αβ . The massless states depend on the transverse polarization vectors ǫ m ,ǭ m and a chiral spinor satisfying k αβ χ β = 0. The massive states are given by a symmetric traceless tensor φ mn , a 3-form e mnp and a γ-traceless vector-spinor ψ α m subject to k m e mnp = k m φ mn = k m ψ α m = 0. The physical vertex operators are organized into three multiplets of 10D N = 1 supersymmetry: • a gauge multiplet involving gluon (A) and gluino (X ), • a supergravity multiplet involving graviton, B-field and dilaton (Vǭ ⊗ V ǫ ) as well as gravitino and dilatino (Vǭ ⊗ V χ ), • a massive multiplet with k 2 = − 4 α ′ comprising a spin-2 field Φ mn , a 3-form E mnp and a spin- 3 2 field Ψ α m , The massive states can be viewed as a double copy of a tachyon,V T = 1, with the first mass level of the open superstring [30]. This construction hinges on the twisted level-matching condition.
B. Tree-level amplitudes: n-point tree-level string amplitudes are given by an integral over the moduli space M 0;n of n-punctured Riemann spheres. The integrand is the CFT correlator of n string vertices, with the freedom to fix any triplet of punctures via SL 2 (C). The conventional and twisted string amplitudes, respectively M + and M − , only differ in the Koba-Nielsen factor (3) and can be cast as Both explicitly factorize the main quantities of interest here: the chiral correlators I R (Ī L ). They are rational functions of z j (z j ) and multilinear in the polarizations of the chiral halves V R (V L ), therefore a key origin of double-copy structures. The integrals (6) can be expressed in terms of the Kawai-Lewellen-Tye (KLT) formula [1,31,32] as bilinears in disk integrals, with a sign flip of α ′ in one of the factors to describe M − [18]. The sphere integrals in M + feature an infinite number of poles for integer values of the generalized Mandelstam variables, In contrast, the sphere integrals with KN − evaluate to rational functions of s ij...p and match the pole structure of a QFT with finite mass spectrum: M 2 = 0 and M 2 = 4 α ′ .

FIELD-THEORY PERSPECTIVE
We here translate the three-point amplitudes M − of one massive vertex (5) and two gauge multiplets into the corresponding QFT Feynman vertices. Their gaugecovariant completion deforms the Lagrangian of 10D N = 1 SYM, and the combined Feynman rules suffice to determine the chiral correlators I R for one massive state and any number of massless ones.
A. Three-point amplitudes: The prescription above yields the well-known three-point SYM amplitudes while the amplitudes with one massive state are simply Their kinematic factors are identical to those found in the open-superstring amplitudes at the corresponding mass levels [33][34][35].

B. Lagrangian:
The finite spectrum of the twisted heterotic string motivates to investigate a Lagrangian description of the massive amplitudes (8). We will discuss three kinds of contributions starting with the standard Lagrangian of N = 1 SYM, where we set g YM and the gravitational coupling κ to 1 throughout this letter. The second term L linear in (9) contains all the gauge interactions linear in the massive fields, and we will argue that they are exhausted by the gauge covariantized three-point interactions (8), The third term L quad in (9) contains the kinetic terms of the massive fields, which are not explicitly needed here. They are uniquely specified and can be mapped to a Kaluza-Klein multiplet of 11D supergravity. The ellipsis in (9) features also a standard N = 1 supergravity sector with couplings to any combination of gauge multiplets and massive states. Moreover, we are omitting interaction terms of more than one massive state.
A central claim of our proposal is that L − het has no further operators involving one massive state and an arbitrary number of gauge multiplets. As a first consistency check, we have reproduced all four-point and bosonic fivepoint string amplitudes with a single massive state from the Lagrangian terms given here.
A more general argument can be made to rule out higher-point interactions of the form √ α ′ ΦTr{F 2 (α ′ F ) N }. Based on previous work [11,20], the tensionless limit α ′ → ∞ should be well behaved and result in a four-derivative supergravity theory that is classically conformal after dimensional reduction to 4D. In this limit, we may redefine √ α ′ Φ mn → Φ mn to get a dimensionless and massless field that recombines with the standard graviton into the gravitational field, see e.g. [11,21,36,37]. But the field-strength factor (α ′ F ) N cannot absorb α ′ since F mn must have dimension two in a conformal theory. Therefore, interactions of the form ΦTr(F ≥3 ) and their supersymmetric completions would obstruct a well-defined tensionless limit.
On these grounds, the ellipsis in (9) does not refer to a higher-derivative expansion of an effective Lagrangian. In conventional string theories, in turn, the description of massive spin-two scattering through an effective action is under investigation [38].
C. All-multiplicity single-trace computation: Next we describe an efficient recursive procedure to compute tree amplitudes of (n−1) gauge multiplets and one massive state from the Lagrangian of the twisted heterotic string. The terms given in (9)  Tr(T a1 T a ρ (2) . . . T a ρ(n−1) ) × A(1, ρ(2, 3, . . . , n−1)|n) + multi-trace . (12) Here 1, 2, . . . , n−1 refers to gauge-multiplet states, and the last leg n is taken to be massive. The color-ordered single-trace amplitudes A are cyclic in 1, 2, . . . , n−1 and only receive contributions from Feynman diagrams involving propagating gauge multiplets which are completely determined by L SYM +L linear . The omitted terms in (9) only affect multi-trace contributions to (12).
Since the Lagrangian L − het only features traces over nested commutators of adjoint fields, the traces in the first line of (12) must recombine into color factors that are products of n−3 structure constants × A(1, ρ(2, . . . , n−2), n−1|n) + multi-trace , in direct analogy with the Dixon-Del Duca-Maltoni decomposition of gauge-theory amplitudes [40].

D. Perturbiners:
We use the perturbiner method [41][42][43][44][45][46][47][48][49] to organize the diagrammatic computation of the color-ordered amplitudes in (12). To each ordered word P = 12 . . . in external-particle labels (letters), we associate multi-particle momenta k P = k 1 + k 2 + . . . and multi-particle polarizations such as ǫ m P , f mn P , χ α P which are identified with Berends-Giele currents [50]. The gauge-multiplet recursions in the Lorenz gauge involve sums over all order-preserving deconcatenations of P = QR into non-empty words Q and R. The recursion ends with single-particle labels, defined by the on-shell polarizations. For the massive fields, (9) leads to similar recursions with the following single-trace contributions of gauge multiplets: The notation +cyc P instructs to add cyclic permutations of the letters in P . In this way, the n-point amplitudes

STRING AMPLITUDES FROM QFT
The introduced QFT description implies new results for a variety of string amplitudes that we now describe.
C. Implications for other twisted heterotic string amplitudes: Using the above ingredients, it is straightforward to compute multi-trace or gravitational amplitudes. For example, the four-point amplitude of two gauge multiplets 1, 3, a gravitational multiplet 2 h and a massive multiplet 4 follows from (19) along with I ∞ R in (25) and The sphere integrations then yield which exhibits the expected massless poles from gaugemultiplet exchange in the s 12 , s 23 channels and poles in s 13 and 1+s 13 from graviton-and massive-state exchange. The same techniques lead to all-multiplicity results for multi-trace and gravitational amplitudes with a single massive state: The underlying I ∞ L are straightforward to obtain from Wick contractions ofJ ai (z i ) & ǫ j ·∂zX − (z j ), and their Parke-Taylor decompositions are well-known from conventional strings [54][55][56].
D. Implications for type-I superstrings: We can also export our method to conventional strings. Treelevel amplitudes of the open type-I superstring with only one massive mutiplet n boil down to I R KN 1/2 + integrated over a disk boundary, where we have fixed z 1 = 0 and z n−1 = 1 in The rescaling α ′ → 4α ′ characteristic to open strings applies to the entire right-hand side of (28). In this case, n is Lie-algebra valued, i.e. (28) is the coefficient of the n-trace Tr(T a1 T a2 . . . T an ). The open-string incarnation of the massive spin-2 field has been related to conformal supergravity in the massless limit [57]. As functions of the n 2 (n−3) Mandelstam invariants s n in (24), the disk integrals F ρ in (29) coincide with the basis in the massless open-string amplitudes of [8]. However, the relations between s n and s 1,n−1 or s jn with j = 1, 2, . . . , n−1 depend on the external masses, i.e. the denominator of the four-point example [33,35] A type I (1, 2, 3, 4) = Γ(1+s 12 )Γ(1+s 23 ) Γ(1+s 12 +s 23 ) A(1, 2, 3|4) (30) equals Γ(−s 13 ) in the massive case rather than Γ(1−s 13 ) as in the massless one. Also the five-point instance of (28) for external states Φ 5 [58] or E 5 and four gluons has been verified via explicit integral reduction in the chiral correlator which also crosschecks fermionic component amplitudes via supersymmetry [30,59].
the prefactor of s 12 s 13 s 23 cancels the massless double poles from A(1, 2, 3|4)A(1, 2, 3|4). It is striking that the QFT computation of the kinematic factors A completely fixes the polarization dependence of the string amplitudes (28) and (31) where the propagation of the complete massive spectrum is reflected by well-studied scalar integrals F ρ and G ρ|σ .

CONCLUSIONS AND FURTHER DIRECTIONS
We have developed here a new method combining QFT and string-theory techniques to obtain all-multiplicity tree amplitudes (exact in α ′ ) with a massive external state. Our results readily apply to the gauge and gravity sectors of the twisted heterotic string as well as type-I and type-II superstrings. The backbone of our construction is a cohomology decomposition of the moduli-space integrands, which is known [52] to directly generalize to string amplitudes with several massive states, as well as higher mass levels of conventional string theories. It is remarkable that the currently known all-multiplicity coefficients have a QFT interpretation for every type of string theory. For several massive states, or higher mass-level states (see for instance [60,61] for detailed studies of mass level 2), one may expect that the coefficients of the cohomology decompositions continue to exhibit structural simplicity, which hopefully stems from a QFT perspective. Since the twisted heterotic string does not admit higher masslevel states, such a construction goes beyond the scope of the current treatment.
A more direct generalization is to formulate the obtained Lagrangian and amplitudes in pure-spinor superspace, based on massive vertex operators [62][63][64]. Besides manifest spacetime supersymmetry, this gives access to BRST-cohomology methods. Similarly, we expect our techniques to be useful at loop level: The Lagrangian description of massive states may shed light on the open questions on loop amplitudes of twisted strings, and feed into conventional-string amplitudes, for instance, via unitarity cuts.
Potential physics applications of massive string amplitudes include exploring chaos in the scattering of excited string states [65], motivated by their correspondence with black-hole microstates [66]. The relevance of excited string states for black-hole physics, causality and unitarity led to a regained interest in their scattering amplitudes [67][68][69]. Moreover, massive strings resonances may become relevant at colliders in the case of a low string scale [35,70,71].
An interesting generalization of our work is to consider amplitudes with two massive modes: Such higherspin massive amplitudes were recently used for describing classical Kerr black-hole scattering [72], needed for binary inspiral and gravitational wave physics. Similarly, there has been a revival of string-amplitude methods for black-hole eikonal scattering [73,74], which can benefit from better knowledge of massive string amplitudes. Finally, massive string amplitudes in flat spacetime also carry relevant information for the AdS/CFT correspondence, as for instance showcased in [75][76][77].