Graviton partial waves and causality in higher dimensions

Do gravitational interactions respect the basic principles of relativity and quantum mechanics? We show that any graviton S-matrix that satisfies these assumptions cannot significantly differ from General Relativity at low energies. We provide sharp bounds on the size of potential corrections in terms of the mass M of new higher-spin states, in spacetime dimensions D>=5 where the S-matrix does not suffer from infrared ambiguities. The key novel ingredient is the full set of SO(D-1) partial waves for this process, which we show how to efficiently compute with Young tableau manipulations. We record new bounds on the central charges of holographic conformal theories.


I. INTRODUCTION
Relativity and quantum mechanics lie at the heart of particle physics. Notions such as relativistic causality ("signals cannot move faster than light") naturally lead to the concepts of waves, fields, and particles as force carriers [1]. Gravity challenges this unification; for example the precise meaning of causality in a fluctuating spacetime remains unclear. In this Letter we study a situation where causality can be unambiguously stated, and is in principle experimentally testable.
Our setup is 2 → 2 scattering between initially wellseparated objects in a flat Minkowski-like region of spacetime. A notion of causality is inherited from the flat background, and encoded in the mathematically precise axioms of scattering (S-matrix) theory. It can be used to constrain gravity itself. Consider higher-derivative corrections to Einstein's gravity at long distances: where C 2 , C 3 , C 3 are higher-curvature terms defined below. Weinberg famously argued that any theory of a massless spin-two boson must reduce to GR at long distances [2]. This was significantly extended in [3], who argued that the parameters α i must be parametrically suppressed by the mass M of new higher-spin states. In parallel, S-matrix dispersion relations have been used to constrain signs and sizes of certain corrections [4][5][6].
Recently, by combining these methods we showed how to bound dimensionless ratios of the form |α i M i | in any scenario where M M pl , such that corrections are larger than Planck-suppressed. However, these bounds featured the infrared logarithms that are well known to plague massless S-matrices in four dimensions.
In this Letter we present rigorous bounds in higherdimensional gravity, where infrared issues are absent. We overcome significant technical hurdles regarding the partial wave decompositions of higher-dimensional amplitudes. The resulting bounds have interesting applications to holographic conformal field theories.

A. Four-point S-matrices and local module
We treat the graviton as a massless particle of spin 2. The amplitude for 2 → 2 graviton scattering depends on the energy-momentum p µ j and polarization ε µ j of each. It can be written generally as a sum over Lorentz-invariant polynomials times scalar functions: We use conventions in which all momenta are outgoing and Mandelstam invariants, satisfying s + t + u = 0, are s = −(p 1 +p 2 ) 2 , t = −(p 2 +p 3 ) 2 , u = −(p 1 +p 3 ) 2 . (3) In kinematics where p 1 , p 2 are incoming, s and −t are respectively the squares of the center-of-mass energy and momentum transfer. The allowed polynomials in (2) are restricted by the fact that graviton polarizations are transverse traceless and subject to gauge redundancies [7]: Depending on the choice of spanning polynomials, the functions M (i) (s, t) may develop spurious singularities which would complicate their use. As explained in [8], there exist special generators of the "local module" such that any amplitude that is polynomial in polarizations and momenta leads to M (i) 's that are polynomial in s and t. These can be simply presented using gauge-and Lorentz-invariant building blocks: where F µ iν = p µ i ε iν − ε µ i p iν is proportional to the field strength . We define H's with other indices by permutation, and V i by cyclic permutations.
In this notation, any S-matrix involving four photons (thus homogeneous of degree 1 in each of the vectors ε µ j ) can be written as a sum of seven terms, involving three basic functions [8]: Here, we introduced the shorthands X and S: Thanks to Bose symmetry, all basic functions M (i) 4γ (a, b) are symmetrical in their two arguments, while the third one is further invariant under all permutations of s, t, u, since S is fully permutation symmetric. The combination X enjoys improved Regge behavior (discussed below).
The general four-graviton amplitude M can now be written using all products of the photon structures, supplemented by the element G equal to the determinant of all dot products between (p 1 , p 2 , p 3 , ε 1 , ε 2 , ε 3 , ε 4 ). The resulting 29 generators organize under permutations as two singlets, seven cyclic triplets, and one sextuplet [8]: These constitute a basis in generic spacetime dimension (D ≥ 8); lower dimensions are reviewed in appendix A.

B. Regge limit and dispersive sum rules
At low energies, the effect of quartic self-interactions in the effective theory (1) is to add polynomials in Mandelstam invariants to the amplitudes M (i) : this is a defining property of the local module [9]. We would like to use the assumption that graviton scattering remains sensible at all energies to constrain the size of these interactions.
Our axioms are best stated using smeared amplitudes: As argued in [10][11][12][13], for suitable wavefunctions Ψ, causality is interpreted as analyticity for s large in the upper-half plane, while unitarity further implies boundedness along any complex direction:  The essential conditions on Ψ(p) are: finite support in p (required for analyticity of M Ψ ), and normalizability at large impact parameters (ensuring boundedness). The bound (10) is assumed for polarizations that do not grow with energy. The behavior of the scalar functions M (i) can be deduced from the Regge scaling of the polarization structures they multiply; leading growth rates are recorded in table I. An important observation is that the leading terms are not all linearly independent, for example while both X 1234 , X 1324 ∼ s 2 , their difference grows more slowly. The coefficients of these structures inherit the opposite behavior. For example, the (smeared) photon amplitudes M (2) 4γ (s, t) ± M (2) 4γ (u, t) are bounded by constants times s −1 and s 0 , respectively.
We say that a dispersive sum rule has Regge spin k if it converges assuming that M/s k → 0; our axioms above state that sum rules with k > 1 converge. As can be seen from (8) and table I, M ∼ s k implies M (3) ∼ s k−4 , ensuring convergence of the following integral at fixed t = −p 2 (with u = p 2 − s): This identity yields a Kramers-Kronig type relation between scattering at low and high energies, by a standard contour deformation argument. Namely, one finds a low-energy contribution at the scale M M pl which is EFT-computable by assumption, plus a discontinuity at high energies s ≥ M 2 (see [12] for more detail). See appendix D for the low-energy amplitudes.
A salient feature of graviton scattering is that many sum rules, like B [1] 2 above, have no denominator: only the poles of M contribute at low energies. Acting on the low-energy amplitude (see (D4)), it yields: The dependence on p is exact up to EFT-computable contributions from other light poles (such as light Kaluza-Klein modes), which we account for in our analysis below, and Planck-suppressed loop corrections, which we neglect since M M pl . Thus (12) constitutes an infinite number of sum rules involving two EFT parameters α i . This "superconvergence" phenomenon is related to the graviton's spin and gauge invariance, which led to the energy growth of structures in (8). For other sum rules we construct improved combinations B imp k (p 2 ) which are designed to probe finite sets of EFT couplings. Our complete set of sum rules is detailed in appendix A 1.

III. CONSTRUCTION OF PARTIAL WAVES
Our assumptions about the right-hand-side of (12) and similar relations are minimal: Lorentz symmetry and unitarity with respect to the asymptotic states. The intermediate states that can appear in a scattering process in D = d + 1 dimensions form representations ρ under SO(d) rotations in the center-of-mass frame. Thus, the S-matrix can be written as a sum over projectors onto each representation. As far as the 2 → 2 S-matrix is concerned, unitarity is simply the statement that |S ρ | ≤ 1 for the coefficient of each projector.
The main technical complication in D > 4 is that many intermediate representations can appear. Furthermore, multiple index contractions can exist for a given representation. Listing them is equivalent to enumerating onshell three-point vertices between two massless and one massive particle. We introduce here an efficient method to construct structures and projectors in arbitrary D.

A. Partial wave expansion
Concretely, the partial wave expansion for a 2 → 2 graviton scattering amplitude takes the form where ρ runs over finite-dimensional irreps of SO(d), and the normalization n (D) ρ is in (C10). For completeness, a derivation of this formula is presented in appendix C.
The partial waves π ij ρ are functions of polarizations and momenta that transform in the representation ρ under the little group SO(d) preserving P µ = p µ 1 +p µ 2 . We build them by gluing vertices v i,a (n, e 1 , e 2 ), where a is an SO(d)-index for ρ, i labels linearly-independent vertices, and are natural vectors orthogonal to P . Note that n 2 = 1, and the e i are gauge-invariant, null, and orthogonal to n: In the center of mass frame, n and e i are simply the orientation and polarizations of incoming particles. Defining an outgoing orientation similarly, n µ ∝ (p 4 − p 3 ) µ , partial waves are defined by summing over intermediate indices: (16) where g ab is an SO(d)-invariant metric on ρ, and f denotes Schwarz reflection f (x) = (f (x * )) * .
Unitarity of S implies that the matrix S ρ (s) ≡ 1 + ia ρ (s) satisfies |S ρ (s)| ≤ 1, which implies 0 ≤ Im a ρ ≤ 2 (where an inequality of matrices is interpreted as positive-semidefiniteness of the difference). We illustrate these concepts in some examples in appendix C.

B. Review of orthogonal representations
A finite-dimensional irrep of SO(d) is specified by a highest weight ρ = (m 1 , . . . , m n ), where n = d/2 , see e.g. [14,15]. The m's are integers for bosonic representations and half-integers for fermionic representations, satisfying For tensor representations, |m i | are the row lengths of the Young diagram for ρ. Note that m n must be positive in odd-d, but can be negative in even-d -the sign of m n indicates the chirality of the representation. We omit vanishing m's from the end of the list, for instance denoting a spin-J traceless symmetric tensor by (J).
To manipulate tensors, we represent them as index-free polynomials in polarization vectors w 1 , . . . , w n ∈ C d , one for each row. The traceless and symmetry properties of a given irrep are captured by taking these to be orthogonal and defined modulo gauge redundancies [16]: The latter means that allowed functions of w must be annihilated by w 1 ·∂ w2 , etc.. Three-point vertices are then simply SO(d)-invariant polynomials v i (w 1 , . . . , w n ; n, e 1 , e 2 ) where the w's play the same role for a massive particle that the ε's play for gravitons.
Polynomials satisfying the gauge condition can be easily constructed by inscribing vectors in the boxes of a Young tableau, where each column represents an antisymmetrized product with w's. For example, given vectors a µ , . . . , e µ ∈ C d , we can define a tensor in the (3, 2) representation via Any tableau defines a valid tensor. Tableaux are not unique, since we can permute columns. Also, antisymmetrizing all the boxes in one column with another box (of not higher height) yields a vanishing polynomial, e.g.:

C. Vertices with two massless and one heavy state
With this technology, we can straightforwardly write all three-point vertices between two gravitons and an arbitrary massive state. Here we focus on generic dimensions D ≥ 8, relegating special cases in lower dimensions to appendix B. All we can write are the dot product e 1 ·e 2 and Young tableaux in which each box contains either n, e 1 or e 2 . Evidently, no tableau can have more than three rows, by antisymmetry.
As a warm-up, consider two non-identical massless scalars. Two-particle states form traceless symmetric Denoting by • an arbitrary (possibly zero) number of boxes containing n, the most general coupling between two scalars and a heavy particle is thus simply • .
Moving on to two spin-1 particles, one must add one power of each of e 1 , e 2 . These can appear either as e 1 ·e 2 or inside a tableau, giving the exhaustive list: A potential tableau e2 e1 • n was removed since it is redundant thanks to (20). Thus, there are six possible vertices. If the two particles are identical, e.g. photons, we get additional restrictions on the parity in n -for example the number of boxes in the first two structures must be even.
The analogous basis of couplings for gravitons in generic dimension D ≥ 8 are shown in table II. This basis agrees with [17]. Changes in lower dimensions are listed in appendix B.

D. Gluing vertices using weight-shifting operators
To glue vertices into partial waves we need to sum over intermediate spin states. This can be achieved efficiently using weight-shifting operators [18]. A general weightshifting operator D a is an SO(d)-covariant differential operator that carries an index a for some finite-dimensional representation of SO(d), such that acting on a tensor in the representation ρ it gives a tensor in the representation with shifted weights ρ + δ. We will be particularly interested in the operator D (h)µ that removes one box at height h from a Young diagram with height h: Conceptually, D (h)µ is a Clebsch-Gordon coefficient for ρ ⊂ ⊗ ρ: this ensures its existence and uniqueness up to normalization. Explicitly, D (h)µ is given by [19] where N Notice the shift by 1 in the last parenthesis: (24) is the familiar Todorov/Thomas operator that acts on traceless symmetric tensors [20].
For the definition (24) to be consistent, the following properties must hold: • D (h)µ preserves the gauge constraints: for all i < j, w i ·∂ wj D (h)µ X = 0 if X satisfies the same.
• D (h)µ sends traces to traces. By "traces" we mean index contractions in strictly gauge-invariant polynomials (not just products w 2 ·w 3 ) -for example, the following expression where µ denotes a unit-vector in the µ direction: These properties are nontrivial and determine D (h)µ up to an overall constant, which can be fixed by considering traces on height-h columns. For example, consider adjacent gauge transformations w i ·∂ wi+1 . Commuting across the i'th and (i + 1)'th parentheses one finds an unwanted term proportional to (N , whose vanishing recursively determines all N 's in terms of N (h) h as stated below (24). Effectively, D (h) recovers indices from index-free polynomials and enables one to evaluate the pairing (16) recursively in terms of simpler pairings, for example Such a formula holds for any choice of a column of maximal height h on the left factor, giving 1/m h times a sum with alternating sign over the boxes it contains, see (F1).
In practice, since D (h) sends tableaux to tableaux, it can be elegantly implemented as a combinatorial operation, as discussed in appendix F. By repeatedly applying (26) and its generalization (F1), any pairing can be reduced to a pairing between single-row tableaux of length m 1 = J: This can be computed efficiently by taking derivatives with respect to n and n of the basic scalar partial wave: where P J (x) is a Gegenbauer polynomial (see (C14)) and (a) n is the Pochhammer symbol. Thus, (26) and (28) allow us to glue the vertices from table II into partial wave expressions which hold for arbitrary J=m 1 , involving derivatives of P J (x) times dot products between graviton polarizations e j and directions n, n . This procedure can be straightforwardly and efficiently automated on a computer.
To limit the size of final expressions, we use the Gegenbauer equation (x 2 − 1)∂ 2 x P J (x) + . . . = 0 to remove any monomial of the form x a P (b) J (x) with a, b ≥ 2. We then insert a set of linearly independent polarizations to project onto the generators (8) of the local module and extract M (i) 's that are polynomials in x. Finally, we use the Gram-Schmidt method to find orthonormal combinations of vertices according to (C15). As a consistency check on our results, we verified that our partial waves are eigenvectors of the SO(d) quadratic Casimir.

IV. RESULTS AND INTERPRETATION
Dispersive sum rules like (12) express low-energy EFT parameters as sums of high-energy partial waves, times unknown positive couplings. The "bootstrap" game consists in finding linear combinations such that all unknowns contribute with the same sign. Such combinations yield rigorous inequalities that EFT parameters must satisfy if a causal and unitary UV completion exists.
To obtain optimal inequalities in a gravitational setting, we follow the numerical search strategy of [10,12]. Because of the graviton pole, it is not legitimate to expand around the forward limit; rather our trial basis consists of the improved sum rules B imp k (p 2 ) integrated against wavepackets ψ i (p) with |p| ≤ M . We ask for a positive action on every state of mass m ≥ M and arbitrary SO(d) irrep, as well as on light exchanges of spin J ≤ 2 and any mass. Full details of our implementation are given in appendix E. Figure 1 displays our main result: the allowed region for the dimensionless parameters (α 2 M 2 , α 4 M 4 ) which control the leading corrections to the action (1), in terms of the mass M of higher-spin states. For the purposes of illustration, we show the results for D = 5, 7, 10; other dimensions D lead to qualitatively similar plots. The parameters are defined more precisely in (D1), and enter the on-shell three-graviton vertex (D3). It would be interesting to compare these bounds with the explicit values of Wilson coefficients in "theory islands" arising from known UV completions [21].
The M -scaling of the bounds is significant: it implies that higher-derivative corrections can never parametrically compete with the Einstein-Hilbert term, within the regime of validity of a gravitational EFT. As soon as corrections become significant, new particles must be around the corner. Since we assume M M pl , graviton scattering is still weak at the cutoff. In gravity, unlike in other low-energy theories, the leading (Einstein-Hilbert) interactions cannot be tuned to zero without setting all other interactions to zero.
What happens at the scale M ? Since we allowed for exchanges of arbitrary light states of low spins, M is associated with the mass of J ≥ 3 states. The importance of higher-spin states was anticipated in [3]. In general, higher-spin states must come in towers that include all spins [22]. For instance, M could signal the begin-ning of a tower of higher-spin particles (as in weakly coupled string theory), that each couple to two gravitons with strength ∼ M 2 √ G. Alternatively, M could be the energy at which loops representing a large number N ∼ M 2−D /G of two-particle states that couple with weaker strength M D+2 2 G to two gravitons, become nonnegligible [23] [24]. Either way, graviton scattering must be profoundly modified at the scale M and above while remaining weak.
Our flat-space bounds have implications in curved spacetimes. As explained in [11], since the scattering processes under consideration take place in a region of small size ∼ 1/M , flat-space dispersive bounds uplift in AdS to rigorous bounds on holographic CFTs, up to corrections suppressed by 1/(M R AdS ) = 1/∆ gap .
Focusing on D = 5 (the AdS 5 /CFT 4 correspondence), stress-tensor two-and three-point functions are characterized by three parameters, including the central charges a and c that enter the conformal anomaly [25]. Their relation to higher-derivative couplings is particularly simple when the EFT action is expressed in terms of Weyl tensors, so that renormalization of the AdS radius is avoided. Using the field redefinition invariant formulas from [26] we find: which could potentially be improved at the ∼ 5% level.
In holographic theories, this result is stronger than the conformal collider bound 1 3 ≤ a c ≤ 31 18 [27] and establishes the parametric scaling anticipated in [3,28,29]. We stress that since ∆ gap is the dimension of the lightest higher-spin (non double-trace) operator, the bound holds even in the presence of light Kaluza-Klein modes (as in AdS 5 ×S 5 ) and is generally independent of the geometry of the internal manifold. Models with (a − c) of either sign are discussed in [30]; our results do not exclude either sign.
The leading contact interaction in D ≥ 7 is the 6derivative "third Lovelock term", which is related to α 4 in (1). Our bounds for this coefficient depend only weakly on its sign and on α 2 , α 4 , and yield the absolute limits in e.g. D = 7, 10: In analogy with scalar EFTs [22,[31][32][33][34][35] and fourdimensional gravitons and photons [12,[36][37][38], we expect this method to yield two-sided bounds on all higherderivative interactions that can be probed by fourgraviton scattering, and on many derivative couplings involving matter fields. In D ≥ 8, there are 19 independent sum rules with even spin k ≥ 2 that can be constructed from applying dispersion relations to coefficients of the local basis with independent Regge limits (8): where M ± ≡ M ± (s ↔ u) and t = −p 2 = −s − u is held fixed. We use multiple superscripts M (i1,. ..,i k ) to indicate a sequence of similar expressions involving the amplitudes M (i1) , . . . , M (i k ) . For odd k > 1, there are 10 independent sum rules: The Regge bound (10) implies that these sum rules converge for k > 1.

Sum rules in lower dimensions
In lower dimensions D ≤ 7, there are two novelties for local modules as noted in [8]. First, new identities can reduce the number of parity-even generators of the local module. This does not occur in D = 7. However, in D = 6 the generator G does not exist, thus we must remove the parity-even sum rules involving M (1) (s, u). Similarly in D = 5, we simply remove the parity-even sum rules involving M (1,6,7) (s, u).

(A3)
Correspondingly, we can construct more sum rules In D = 6, there are three parity-odd triplets: Here, we have introduced permutation operators σ to simplify the expressions: The corresponding parity-odd sum rules in D = 6 are given by Finally, in D = 5 there is one parity-odd triplet −iσ 34 12 which gives rise to three independent sum rules:

Improved sum rules
Eqs. (A1)-(A9) provide complete sets of dispersive sum rules in the considered dimensions. By "complete" we mean that any sum rule with spin-k convergence can be expressed as finite sum of the B ≤k up to corrections that vanish faster than spin-k at high energies. Generically, the action of B k (p 2 ) on the low-energy amplitude (D4) yields an infinite series of contact interactions. Following the method in [10], all but a finite number of contacts can be removed by adding an infinite series of higher-spin sum rules B (n) >k (0) expanded around the forward limit. As further discussed in [12], while it is not allowed to expand k = 2 sum rules in the forward limit (due to the graviton pole), there are no analogous problems for k > 2. Explicit formulas for the resulting B imp k (p 2 ) sum rules are recorded in ancillary files.

Appendix B: Vertices in lower dimensions
In the main text, we described three-point vertices for two gravitons and a massive state in dimensions D ≥ 8. In lower spacetime dimensions, the counting of three-point structures is modified, and we must take into account additional ingredients in the representation theory of the little group SO(d) (where d = D − 1). In this section, we describe these ingredients, and then discuss the individual cases D = 7, 6, 5 in turn. Detailed expressions can be found in the ancillary files included with this work. 1. Representation theory ingredients a. Self-duality and -symbols When d = 2n is even, representations with full-height Young diagrams split into self-dual or anti-self-dual cases, according to whether m n is positive or negative. Let us explain how to account for this in our index-free formalism. Recall that the polarization vectors w i satisfy the orthogonality conditions and gauge redundancies (18). When d is even, the variety defined by these conditions (called a "flag variety") splits into two irreducible components V ± , distinguished by whether w 1 ∧ · · · ∧ w n is self-dual or anti-self-dual. Specifically, we have To see why there are two components V ± , we can recursively solve the orthogonality conditions w i ·w j = 0. First, we use SO(d)-invariance and rescaling to set w 1 = (1, i, 0, . . . , 0). Using gauge-redundancies and w 1 ·w i = 0, the remaining w i must have the form w i = (0, 0, w ⊥ i ), where w ⊥ i ∈ C d−2 are null vectors. The w ⊥ i satisfy precisely the conditions and gauge redundancies for the flag variety of SO(d − 2). Repeating this process for the w ⊥ i 's, we eventually arrive at the flag variety for SO(2), parametrized by a single null vector w ⊥···⊥ n ∈ C 2 . Up to SO(2) transformations and rescaling, there are two possible null vectors w ⊥···⊥ n = (1, ±i), corresponding to the two components.
Furthermore, the product of (B2) with any polynomial in the w i 's is also self-dual (anti-self-dual), since it vanishes on V − (V + ). In general, we define a tableau with chirality ± by adding an term to any full-height column, for example: Note that it doesn't matter which full-height column we choose -the resulting polynomial is the same since it agrees on both components V + and V − ; this can be verified explicitly with Gram determinant identities.

b. Counting three-point structures
Using the methods of [17,40], one can show that possible three-point vertices for the representation ρ are classified by the following formula: Here, d−1 denotes the spin-2 representation of SO(d − 1). When we tensor an SO(d − 1) representation with ρ, we implicitly dimensionally reduce ρ to an SO(d − 1) representation. The notation (λ) • denotes the SO(d − 1)-singlet subspace of λ, and (λ) • ± denotes the SO(d − 1) singlet subspace with parity ±. Finally, |ρ| is the number of boxes in the Young diagram of ρ. The formula (B4) is useful for detecting linear dependencies between Young tableau in various spacetime dimensions.

c. Implications of CRT
CRT symmetry relates the SO(d) representation ρ to the dual reflected representation (ρ R ) * . When d ≡ 1, 2, or 3 mod 4, we have simply (ρ R ) * = ρ. In this case, we can choose conventions where three-point couplings for gravitongraviton-ρ vertices are real, simply by making the couplings invariant under p µ j → −p µ j , i → −i. In particular, when computing positivity bounds, we impose that the contribution of each type of partial wave to a sum rule is a positive-definite real symmetric matrix.
Meanwhile, when d ≡ 0 mod 4, dual reflection changes the sign of the weight m n , and hence exchanges self-dual and anti-self-dual representations ρ + ↔ ρ − . In this case, CRT implies that three-point coefficients of ρ + and ρ − are complex conjugates of each other. We discuss the implications of this for positivity bounds in D = 5 below.

Vertices in D = 7 (d = 6)
Because d = 6 is even, representations with height-3 Young diagrams split into self-dual and anti-self-dual cases. The only effect is to double the number of height-3 tableaux in table II by adding a ± chirality to each.
Let us denote a self-dual (anti-self-dual) representation by ρ + (ρ − ). In the absence of parity symmetry, the threepoint amplitudes g ggρ± between two gravitons and states in ρ + or ρ − need not be related. Consequently, we must sum over partial waves for each type of representation ρ + and ρ − independently. In bootstrap calculations, this requires including separate positivity conditions for ρ + -exchange and ρ − -exchange.
However, the contributions of ρ + -exchange and ρ − -exchange to parity-even sum rules are identical. Thus, when computing bounds using parity-even sum rules (such as our bounds on α 2 and α 4 ), positivity conditions associated to ρ + and ρ − are redundant, and it suffices to include only one of them (say ρ + ). 3. Vertices in D = 6 (d = 5) In spacetime dimension D = 6, SO(5) Young tableaux can have at most two rows. Since vertices are functions of five vectors (w 1 , w 2 , e 1 , e 2 , n), there is a unique way to use the Levi-Civita tensor. It is convenient to write it as a height-3 column: In spacetime dimension D = 5, SO(4) tableaux with two rows can have chirality ±. In addition, we can use the Levi-Civita tensor in the form (w 1 , a, b, c). Due to Gram determinant identities, this term can never be used if two-row columns are present, and it cannot be used twice. It is again convenient to draw it as a 3-row column:   (4). We use the same notation as in table II. The meaning of the height-3 column is given in (B6).
As discussed in section B 1 c, when d = 4, CRT implies that three-point coefficients of ρ + and ρ − are complex conjugates of each other. Given a pair of representations ρ + , ρ − with opposite chirality, let us denote the corresponding partial waves by π + , π − . The π ± are Hermitian matrices indexed by vertex labels i, j. Exploiting fact that generators of the local module are invariant under the Z 2 × Z 2 symmetry which includes the interchange between initial and final states, we can choose conventions where By choosing generators of the local module to be invariant under p j → −p j , i → −i, these relations automatically hold for all the coefficients of the projector on that basis. A contribution from ρ + -exchange to the discontinuity of the amplitude takes the form where M = g + g † + is a Hermitian matrix built from a vector of three-point couplings g + . The three-point couplings for ρ − are complex-conjugate to g + and can be grouped into the matrix g − g † − = g * + g T + = M * = M T . Together, ρ + and ρ − -exchange thus contribute Tr(M π + ) + Tr(M T π − ) = Tr(M (π + + π T − )) = 2Tr(M π + ).
So, summing the two opposite-chirality irreps simply gives a factor of 2. In parity-even sum rules, only the realsymmetric part of M and π contributes, while for parity-odd sum rules, only the imaginary part of both contributes. Thus, when computing bounds using parity-even sum rules (as we do in this work), we can essentially pretend that the three-point couplings are real and symmetrical. Furthermore, we need only include positivity conditions for one chirality (say ρ + ), since the contributions from ρ − are redundant.
Unitarity of S implies that each S ρ (s) is separately a unitary matrix S ρ (s)S ρ (s) † = 1. Taking a matrix element of M = −i(S − 1) in the basis states |n, e 1 , e 2 , we obtain the partial wave decomposition of the gravity amplitudes (13): where π ij ρ =v i,b g ba v j,a ≡ (v i , v j ) and S ρ (s) = 1 + ia ρ (s). From this derivation, the normalization can be fixed simply by taking the trace of (C5) and using the completeness relation (C3): where we have used rotational-invariance to perform the integral over n, and Tr indicates a sum over polarization states. (We detail the precise meaning of Tr for gravitons below in (C15).) We choose to normalize the vertices so that The normalization coefficient n is thus fixed to be dim ρ divided by essentially the phase space volume: The dimension dim ρ can be computed from standard formulas, see e.g. [15,39]. For spin-J traceless symmetric tensors, we have simply

Scalar scattering
Let us determine the precise expression for π ρ in the case of scalar scattering. Since each ρ = (J) appears with multiplicity 1, there is a unique vertex function up to a constant k J that we determine shortly. The partial waves are given by where x = n·n = 1 + 2t s , (a) n is the Pochhammer symbol, and P J (x) is a Gegenbauer polynomial given by Our normalization condition on vertices is equivalent to π J (n, n) = 1, which fixes since P J (1) = 1. We finally obtain π J (n , n) = P J (x), and (C7) recovers the familiar partial wave expansion for scalars, see e.g. [41].

Graviton scattering
In the case of graviton scattering, the orthonormality condition used in (C10) can be expanded as v i (n, e 1 , e 2 ) * , v j (n, e 1 , e 2 ) , functionals depend quadratically on the α j , we converge toward the boundary by optimizing a sequence of linearized quantities.
In practice, we fix the set of functionals and increase the number of states until the bounds do not change, keeping only those sets of functionals for which such convergence could be achieved. In going from 5 to 6 exponents, the bounds improved by no more than a few percent. We thus expect that the recorded bounds are conservatively correct, and likely within 5% of being optimal.
We anticipate that the partial waves computed in this work will serve in many other studies. We have thus prepared "process files" which contain the complete information used to bootstrap each of the graviton scattering process studied in this letter: GGGG5.m, GGGG6.m, GGGG7.m, GGGGd.m, for D = 5, 6, 7 and D ≥ 8 respectively, as well as a file GGGG4.m, which characterizes the D = 4 case studied in our earlier paper [12]. Each file contains: • The basis localbasis[GGGG [d]] of polarization structures used throughout the file, i.e. the L elements generated from (8) where L = 29 for D ≥ 8, written in terms of the H, V , X, S and G structures defined in section II (the latter two are denoted HS and HGram in the files).
• A list vertices[GG [d]] of three-point couplings v i between two gravitons and a massive state, written in the Young Tableau notation of sections III and B and divided by the scalar factor k J of (C13) (and e i denoted ep[i]).
• If an irrep allows n independent vertices, normalizations is an n × n matrix and matrix is n × n × L, such that their entry-wise product express the projector π ij in localbasis[amplitude]. The the a'th derivative P This constitutes the full information from which the bootstrap problem can be implemented in an automated way.