Multiparticle Factorization and the Rigidity of String Theory

Is string theory uniquely determined by self-consistency? Causality and unitarity seemingly permit a multitude of putative deformations, at least at the level of two-to-two scattering. Motivated by this question, we initiate a systematic exploration of the constraints on scattering from higher-point factorization, which imposes extraordinarily restrictive sum rules on the residues and spectra defined by a given amplitude. These bounds handily exclude several proposed deformations of the string: the simplest"bespoke"amplitudes with tunable masses and a family of modified string integrands from"binary geometry."While the string itself passes all tests, our formalism directly extracts the three-point amplitudes for the low-lying string modes without the aid of worldsheet vertex operators.

Introduction.-Thescattering amplitudes of string theory exhibit a litany of seemingly impossible features simultaneously.On the one hand, they offer a proof of principle for how to consistently extrapolate quantum gravity to ultrahigh energies, past the regime of validity of general relativity.Miraculously, they achieve all this while preserving causality, which mandates that high-energy elastic scattering is polynomially bounded in the center-of-mass energy at fixed impact parameter [1][2][3][4][5][6][7].Furthermore, string amplitudes make crucial use of higher-spin states, which are famously inconsistent in isolation but appear together in an infinite tower that elaborately conspires to maintain unitarity.
These conditions are so stringent that one might wonder whether string amplitudes are uniquely determined by this panoply of constraints.Recent years have witnessed a surge of interest in this question in the very simplest context of four-point scattering [6][7][8][9][10][11][12][13].Even in just the past few months, a dizzying array of such proposals has made an appearance in the literature [12][13][14][15][16][17][18][19][20][21][22], some revisiting the seminal work of Ref. [23].Rather surprisingly, the broad conclusion has been negative: it is possible to deform the four-point Veneziano amplitude [24] of the open string in many ways that still conform to these physical constraints.Even at four points, partial wave unitarity still allows for a range of consistent parameters [25], though the famous D = 26 of string theory still takes on special significance, as shown in Fig. 1.
It is natural to ask whether this immense freedom persists beyond four-point scattering.However, the n-point frontier of scattering has remained largely uncharted territory until the past year or so.
For example, a vast class of generalizations of n-point string amplitudes has been recently proposed [14] that exhibit a "bespoke," or completely tunable, arbitrary spectrum.Like in string theory, these bespoke amplitudes describe an infinite tower of higher spins exhibiting channel duality.At four points, there is evidence that specific models, at least in certain regions of parameter space, are causal and unitary.Furthermore, at n point these amplitudes are consistent with factorization on the  poles of the lowest-lying exchanged state.
Another salient development is a new approach to npoint scattering that applies to string theory but also quantum field theories like ϕ theory and Yang-Mills theory at all orders in the 't Hooft expansion.In this framework, n-point amplitudes are described by a simple integral over a set of auxiliary u variables that are the mathematical cousins of the usual moduli that define the Koba-Nielsen representation of string amplitudes.Conveniently, this formalism accommodates a natural deformation of the n-point string amplitude through the introduction of a multiplicative form factor P n (u) into the integrand.This function can be straightforwardly engineered so as to ensure factorization on massless poles.
The upshot of these results is a bit of whiplash: we have gone from the naive intuition that n-point string scattering is unique to an embarrassment of riches in putative consistent deformations!This is exciting and disturbing in equal parts.But is this freedom genuine?
In this paper we will argue that it is not, on account of the constraints imposed by factorization on massive poles onto a self-consistent set of three-point amplitudes.This condition is amazingly restrictive.Factorization in four-, five-, and six-point scattering is sufficient to vanquish the very simplest bespoke model as well as a large class of natural P n (u) constructions.These exclusions only require the consideration of scalars and vectors exchanged in the topologies depicted in Fig. 2. Along the way, we will introduce useful techniques for computing residues on massive poles and summarize a new set of simple sum rules that directly constrain the residues and mass spectrum of any putative n-point amplitude.
Of course, the n-point amplitudes of string theory satisfy the constraints from factorization on massive poles.But a corollary of this check is that we can directly extract the three-point amplitudes-coupling constants and all-directly from the higher-point amplitudes.
Warmup.-The constraining power of factorization is already evident at low point.As a simple illustration of this approach, let us briefly consider the tree-level tachyon amplitudes of the bosonic string.These amplitudes famously exhibit a linear spectrum m 2 k = k − 1, where k is a nonnegative integer labeling each level of the string, which supports modes of spin 0 ≤ ℓ ≤ k.Factorizing the four-point string amplitude onto levels k = 0 and 1, we obtain the residues where X i,j = (p i + p i+1 + • • • + p j−1 ) 2 are the planar Mandelstam invariants in mostly-plus signature.Here R 0 is generated by a scalar at level k = 0 and R 1 is generated by a scalar and a vector at level k = 1.Next, we compare these residues against an ansatz residue derived from a set of ansatz three-point on-shell amplitudes A ℓ1ℓ2ℓ3 k1k2k3 describing the three-particle interaction of the states at levels k 1 , k 2 , k 3 of spins ℓ 1 , ℓ 2 , ℓ 3 .Explicitly, A 000 000 = λ 000 000 , A 000 100 = λ 000 100 , and A 100 100 = λ 100 100 (ϵ 1 p 2 ).( 2) We then extract the corresponding Feynman vertices and glue them together to obtain the ansatz residues, Equating Eqs.
(1) and ( 3), we thus determine-up to unphysical signs-the coupling constants of the theory.
From squared coupling constants in Eq. ( 3), we see that the above construction is simply a restatement of the usual condition of positivity in four-point scattering.However, since we have determined the three-point amplitudes, we can feed these results back into higher-point residues to obtain new bounds.
For example, the five-point residue of the string amplitude, at level k = 1 for both internal propagators, is while the corresponding ansatz residue is ) ( Here A 000 100 and A 100 100 were fixed at four point, and we have defined the additional ansatz amplitudes, A 000 110 = λ 000 110 , A 100 110 = λ 100 110 (ϵ 1 p 2 ), and The residues in Eqs. ( 3) and ( 5) parameterize the consistent space allowed by factorization and our assumed spectrum.Hence, they can be used to fix the couplings of the theory or, conversely, rule out a putative amplitude that is incompatible with factorization.
General Procedure.-Theabove analysis generalizes to any spin and number of external particles.As we will see, however, factorization constraints from scalar and vector exchange in four-, five-, and six-point scattering are sufficient to dramatically reduce the space of consistent scattering amplitudes!Let us now describe the general procedure for constructing these ansatz residues.First, we take a chosen spectrum of masses and spins as an input.For concreteness, we assume a spectrum that is continuously connected to that of the string: a single scalar at level k = 0 and a single scalar and vector at level k = 1, with no degeneracy.However, we will assume nothing a priori about the precise values of the masses or the coupling constants of the theory.Throughout, we will consider amplitudes in which the external states are the colored scalars of mass m 2 0 at level k = 0. Second, we enumerate all three-point on-shell amplitudes for these states.The on-shell three-point amplitudes for zero, one, or two external vectors were given in Eqs. ( 2) and ( 6).The amplitude for three vectors is [26 Here the first and second terms correspond to the Yang-Mills cubic vertex and anomalous triple gauge boson vertex, respectively.From the above amplitudes it is easy to derive the corresponding Feynman vertices.
Finally, to build the ansatz residue we glue these Feynman vertices together with the Feynman propagators for scalars and vectors.The numerator for the latter is Π µν = η µν +p µ p ν /m 2 k , where p 2 = −m 2 k on the cut.Note that there is no dependence on the spacetime dimension D in the scalar and vector propagators.Furthermore, traces of the metric do not appear at tree level, so the bounds we encounter will be D-independent.
Examples.-In this section we compute the residues for various higher-point amplitudes and compare them directly to the residues constructed using the general procedure outlined above.As we will see, constraints from higher-point factorization are exceedingly powerful.
String Amplitudes.To begin, consider the standard treelevel n-point Koba-Nielsen factor [27][28][29], which we refer to as a "string amplitude" in a slight abuse of notation [30].Here p i are the external momenta, z i are moduli integrated over the disc, and z i,j = z i − z j are their differences.The spectrum of states is defined by m 2 k = k [31].Since the external legs of the string amplitude are planar-ordered, all factorization channels appear as simple poles in the planar Mandelstam invariants X i,j defined previously.Conveniently, these n(n − 3)/2 variables form a minimal basis for all n-point Mandelstam invariants.To make this dependence manifest, we recast Eq. ( 8) in terms of the SL(2,R) invariant crossratios, u i,j = (z i−1,j z i,j−1 )/(z i,j z i−1,j−1 ), in which case the string amplitude becomes Note that in going from Eq. ( 8) to Eq. ( 9) we have shifted the planar Mandelstam invariants by a constant Regge intercept α 0 , so the amplitude automatically has singularities at X i,j = −m 2 k and the spectrum is m 2 k = k + α 0 .The u variables satisfy the nontrivial relations, where (i ′ , j ′ ) corresponds to all chords of the n-gon that intersect the chord (i, j).These equations admit an (n−3)-dimensional space of solutions parameterized precisely by the moduli z i of the gauge-fixed worldsheet.
Using traditional z i variables, it is challenging to extract the residues at arbitrary levels.But the raison d'être of the u i,j variables is to transparently manifest all singularities, giving us a simple and explicit way to extract arbitrary residues.This is most easily done by a change of variables to positive coordinates, where the amplitude takes the form Here each y C is associated with a chord C appearing in some choice of triangulation T of the n-gon.A simple algorithm expresses u i,j in terms of y C [32].In this form the singularities X C + α 0 → 0 of chords in the triangulation T are associated with the logarithmic divergence of the integral near y C → 0. The integral diverges for X C + α 0 < 0 and must be defined by analytic continuation, indeed developing poles when X C + α 0 is a negative integer.For generic X C + α 0 the integrand has branch cuts in y C [33].However, when all X C + α 0 = −k C , the integrand instead has poles at y C → 0. The residue of the integral on the pole is computed by the residue of the integrand at y C → 0. In summary, the algorithm for computing the residue associated with any diagram or triangulation T is then simple: use the positive parameterization associated with T in Eq. ( 11), set the X C +α 0 → −k C , and compute the residue of the integrand at y C → 0.
With this procedure we straightforwardly compute all residues of the four-, five-, and six-point string amplitudes on the maximal factorization channels on which all internal legs are localized to levels k = 0, 1.We then compare these string residues to the ansatz residues constructed from gluing together arbitrary three-point amplitudes with real couplings.Remarkably, for any value of the Regge intercept α 0 ≥ −1, the string amplitude is perfectly consistent with these factorization conditions.As an immediate byproduct of this exercise, we extract the explicit three-point amplitudes of string theory for levels k = 0, 1 (i.e., masses m 2 = α 0 or 1 + α 0 ) and spins ℓ = 0, 1, given in the table below.We constructed these couplings uniquely from consistent factorization of higher-point string amplitudes, without making use of worldsheet vertex operators.It is straightforward to compute all three-point amplitudes of the string by generalizing this approach to higher levels, modulo considerations of degeneracy, which we leave to future work.
Bespoke Amplitudes.Next, let us demonstrate how consistent factorization imposes stringent constraints on the bespoke amplitudes defined in Ref. [14].For concreteness, we focus on the very simplest version of these amplitudes with a nonlinear spectrum, corresponding to h = 2 in the nomenclature of that paper.For this amplitude, the spectrum of resonances at level k is defined by the nonlinear rational polynomial m 2 k = (k 2 + p 1 k + p 2 )/(k + q 2 ), where p 1 , p 2 , q 2 are parameters satisfying p 1 q 2 − p 2 > 0. The corresponding npoint bespoke amplitude is simply a sum over the usual string amplitudes, (12) where I = 1, . .., n(n−3)/2 and in each term the planar Mandelstam X I has been composed with a nonlinear function ν The average over square root branches, for each index α 1 . . ., α n(n−3)/2 , ensures that the resulting amplitude is free of branch cuts and has polynomial residues.For appropriate choices of p 1 , p 2 , q 2 , Ref. [14] showed that the four-point amplitude possesses a dual resonant representation and is consistent with partial wave unitarity.The factor fixes the canonical normalization of level k = 0 external states, so the bespoke amplitude automatically factorizes on the exchange of those states.Proper factorization on states at higher k, however, is not yet guaranteed.The residues of the bespoke amplitudes are computed by extracting the residues of the string amplitude using Eq. ( 11) and inserting them into Eq.(12).Comparing the resulting bespoke residues against the ansatz residues at four, five, and six point for level k = 0, 1, we find that modulo some trivial theories with purely scalar interactions, there are no values of p 1 , p 2 , q 2 that are consistent with factorization.While a one-parameter family with p 1 /2 = 2q 2 = 1 + p 2 /q 2 does indeed satisfy the factorization constraints for the half-ladders, it fails for the six-point twisted half-ladder depicted in Fig. 2.
Thus, the very simplest bespoke amplitudes are inconsistent with factorization onto a single scalar at level k = 0 and a single scalar and vector at k = 1.It remains an open question whether these bounds can be evaded in the broader class of bespoke amplitudes [14], or perhaps by including degeneracy.
Deformed String Integrand.The critical feature of Eq. ( 11) is that it makes factorization on the level k = 0 poles obvious.When X i ⋆ ,j ⋆ + α 0 → 0 the integral develops a pole near u i ⋆ ,j ⋆ → 0. Because of the "binary" character of Eq. ( 10), the limit u i ⋆ ,j ⋆ → 0 sends u i ′ ,j ′ → 1 for the chords (i ′ , j ′ ) that cross (i ⋆ , j ⋆ ).Hence i,j u Xi,j +α0 i,j factorizes in a way that mirrors the amplitude.This suggests a more general class of functions that exhibit level k = 0 factorization: simply multiply the n-point integrand by a form factor P n (u) engineered to factorize into the product of lower-point factors when u X = 0.It is natural to restrict to a finite polynomial P (u) to preserve the Regge behavior characteristic of string amplitudes.
Let us define a simple family of deformations, parameterized by an integer m ≥ 3 that defines an m-gon Q (m)  inside the momentum n-gon.We then define where (i, j) sums over all pairs of vertices in Q (m) and we take u i,i+1 = 0.The parameter g controls the deformation away from the original string amplitude.Note that if a chord X intersects Q (m) , it must intersect at least one of the chords inside Q (m) .Therefore, on a factorization channel where u X → 0, the product of (1 − u i,j ) vanishes on account of Eq. ( 10), leaving us with the product over all the Q (m) that do not intersect X, so factorization at level k = 0 then follows.
For our analysis, we consider the P n (u) amplitudes, which we dub the "triangle" and "quadrilateral" deformations, respectively.In both cases, threepoint amplitudes at level k = 0, 1 are completely fixed by four-and five-point scattering.Feeding these couplings back into higher-point amplitudes, we find that if g ̸ = 0 then the six-point residues for both the triangle and quadrilateral are consistent for the four-, five-, and six-point half-ladders but inconsistent for the twisted half-ladder topology shown in Fig. 2. Hence, both the triangle and quadrilateral models are ruled out [34].
Amusingly, the triangle deformation can actually be ruled out at four point via partial wave unitarity, though at great cost.Computing the partial wave expansion on the Gegenbauer polynomials, we find that for nonzero g, the first negative partial wave is at spin 0 or 1, but at very high level k ∼ 1/g.This dramatic difference in effort illustrates the power of higher-point factorization.
Sum Rules.-Higher-pointfactorization imposes a set of nonlinear sum rules that depend on the masses and the numerical coefficients of the residues of a putative amplitude.These sum rules have the distinct advantage that they can be used to cleanly eliminate inconsistent theories without resorting to the laborious procedure of gluing together ansatz three-point amplitudes.
To this end, let us define a general parameterization of the residues output by the ansatz residue for arbitrary couplings.For example, the general four-point half-ladder residues at level k = 0, 1 are while at five point they are Here n-point residues are fully characterized by the numerical coefficients , where the subscripts k 1 , k 2 , . . .denote the levels on which each internal leg has been localized and the superscripts r 1 , r 2 , . . .denote the power of each Mandelstam invariant X I .The ordering of I is defined by the lexicographic order of the i, j subscripts in X i,j = X I that appear in a given residue [35].Note that the ω coefficients satisfy identities required by the flip isometries, so for example, ω r1r2r3 k1k2 = ω r3r2r1 k2k1 .Since the ansatz residue is constructed by gluing together arbitrary three-point amplitudes, it is a polynomial in X I with very precise relations linking the numerical coefficients of each kinematic structure.These conditions correspond to a set of nonlinear sum rules that relate the ω coefficients with the mass spectrum m 2 k , which we now summarize for four-, five-, and six-point residues computed at levels k = 0, 1.
Four-Point Sum Rules.For the general four-point residue in Eq. ( 14), we can compare to the ansatz residue to extract the corresponding three-point amplitudes, Immediately, the reality of the coupling constants imply inequalities on the residue coefficients, ω 0 0 > 0, ω 1 1 < 0, and where the first and second inequalities are not saturated because we assume that there are nonzero couplings to both the scalar and the vector.
See App.I for the analogous sum rules for consistent factorization at six point.
Discussion.-In this paper we have argued that consistent factorization is a remarkably stringent constraint on multiparticle scattering.Our analysis is by no means exhaustive, but rather intended as an invitation to explore this rich set of novel higher-point consistency conditions.There are many questions directly spurred by our preliminary observations.For example, it is well motivated to ask whether there are any known deformations of the n-point amplitude that actually satisfy consistent factorization.Some natural candidates include bespoke amplitudes with more highly nonlinear spectra, as well as other P (u) deformations of the string integrand.
A convenient byproduct of our analysis is that we can mechanically extract the three-point couplings of the string directly from the n-point scattering amplitudes.It would be very interesting to use this tool to learn about the structure of interactions and density of states of the string directly from the amplitudes themselves.These and related lines of investigation hold the promise of fundamental new insights into the question of whatif anything-makes string theory special.

I. SIX-POINT SUM RULES
We construct general sum rules for consistent factorization of all six-point diagrams at levels k = 0, 1.For ease of use, we also provide these results in a supplementary text file.
We first perform the factorization calculation for the six-point half-ladder diagram depicted in Fig. 2 of the Letter by computing the glued residues and comparing against an arbitrary six-point ansatz residue parameterized by the ω coefficients, For the final remaining topology, the six-point star diagram in Fig. 2 .Here, we are factorizing on the (X 1,3 , X < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 3 c 5 5 6 e 8 h k 2 M g XP M E E t J a Y + m C Y M = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l E t M e C F4 8 V 7 A e 0 o U y 2 m 3 b p Z p P u b o Q S + i e 8 e F D E q 3 / H m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g o b m 1 v b O 8 X d 0 t 7 + w e F R + f i k p e N U U d a k s Y h V J 0 D N B J e s a b g R r J M o h l E g W D s Y 3 8 3 9 9 h N T m s f y 0 U w T 5 k c 4 l D z k F I 2 V O j 0 U y Q j 7 b r 9 c c a v u A m S d e D m p Q

Figure 2 .
Figure 2. Consistent factorization in these channels is sufficient to rule out an enormous class of putative amplitudes.Counterclockwise from top left: four-and five-point halfladder; six-point twisted half-ladder, star, and half-ladder.