Positivity in Multi-Field EFTs

We discuss the general method for obtaining full positivity bounds on multi-field effective field theories (EFTs). While the leading order forward positivity bounds are commonly derived from the elastic scattering of two (superposed) external states, we show that for a generic EFT containing 3 or more low-energy modes, this approach only gives incomplete bounds. We then identify the allowed parameter space as the dual to a spectrahedron, constructed from crossing symmetries of the amplitude, and show that finding the optimal bounds for a given number of modes is equivalent to a geometric problem: finding the extremal rays of a spectrahedron. We show how this is done analytically for simple cases, and numerically formulated as semidefinite programming (SDP) problems for more complicated cases. We demonstrate this approach with a number of well-motivated examples in particle physics and cosmology, including EFTs of scalars, vectors, fermions and gravitons. In all these cases, we find that the SDP approach leads to results that either improve the previous ones or are completely new. We also find that the SDP approach is numerically much more efficient.

Introduction Positivity bounds are constraints on the Wilson coefficients of an effective field theory (EFT) that can be bootstrapped from fundamental properties of the S-matrix of the UV theory [1][2][3]. Recently, there has been a lot of interest in extending the strength and scope of the positivity bounds [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], as well as applying the bounds to constrain EFTs in various contexts (see for example ). In many situations, and particularly for constraining the parameter space of the Standard Model Effective Field Theory (SMEFT) [4, 20-22, 29, 35, 43], the leading positivity bounds for the s 2 terms (s, t being the standard Mandelstam variables) in the amplitude are phenomenologically the most relevant ones. The most widely used positivity bounds so far are based on the forward (t = 0) elastic scattering of two factorized states, each of which can be an arbitrary mixture of various particle modes. However, it has been shown that this approach does not always give the best bounds [4]. In addition, obtaining the complete set of superposed elastic bounds is known to be NP-hard [30].
In this letter, we will establish a geometric method for obtaining the full set of leading forward positivity bounds for EFTs with multiple low-energy modes. It applies not only to the SMEFT, but also to all other EFTs that involve multiple particles or multiplet particles. We will compare with the previous results and show how the new/non-elastic bounds arise from scattering entangled states.
Notations We will use capital calligraphy letters to denote rank-4 tensors (e.g. T ∈ R n 4 ). The inner product of tensors is defined by T 1 ·T 2 ≡ ijkl T ijkl 1 T ijkl 2 . We say that T is positive semidefinite (PSD) if T ijkl is a PSD matrix when ij is viewed as one index and kl another, which is denoted by T 0. The null space of this matrix is denoted as Null(T ). S n×n + is the set of n × n PSD matrices. We denote by − → S n 4 the set of rank-4 n-dimensional tensors, T , that satisfy the following crossing symmetries T i(j|k|l) ≡ T ijkl + T ilkj . The set of extremal rays (ERs) of some convex cone X is denoted as ext(X). An ER is an element of X that cannot be split into two linearlyindependent elements inside X. We shall consider the t → 0 limit of a two-to-two amplitude, M ij→kl (s) = M ij→kl (s, t = 0), which is only a function of s, and we define the M tensor Here i, j, k, l are indices for the low energy degrees of freedom, enumerating particle species, polarization and other quantum numbers. We will simply call this M tensor "amplitude". Dispersion relation Axiomatic principles of the UV amplitude, such as analyticity, unitarity and crossing symmetry, lead to a dispersion relation which expresses M ijkl in terms of an integral of the discontinuity of the amplitude along the positive real s axis (see e.g. [4]) where (j ↔ l) denotes the previous term with j and l swapped. This assumes that a self-conjugate particle basis is chosen, which is always possible by replacing |i and |ī by (|i + |ī )/2 and (|i − |ī )/(2i). Λ is the subtraction scale for improved positivity, below which the EFT is valid: we have slightly changed the definition of M ijkl by subtracting the dispersive integral below Λ, see more explanations in Ref. [4]. Upon using the generalized optical theorem, this relation implies that M ijkl is a convex cone generated from positive linear combinations of elements of the form m ij m kl + m il m kj [4], i.e., The elements of C n 4 are invariant under (j ↔ l) and (i ↔ k) exchanges. We will also assume that m ij is either symmetric or antisymmetric, which is simply Bose symmetry for scalars, but implies parity-conservation for vectors. This is equivalent to further requiring C n 4 ⊂ − → S n 4 . Positivity bounds arise as the boundary of C n 4 . All components of M can be computed in terms of Wilson coefficients, so bounds on M can be converted to bounds on these coefficients. Conventionally, these bounds are derived by the elastic scattering of a pair of factorized but arbitrarily superposed states, |u = i u i |i They constrain the signs of the elastic components in M ijkl , and also set upper and lower bounds on inelastic scattering amplitudes [21,22,29,43,44]. We will however show that these bounds are non-optimal.
The goal of this work is to understand the exact boundary of C n 4 , which is in general beyond superposed elastic bounds. In the presence of sufficient symmetries in the theory, an efficient way to do this is through the extremal positivity approach presented in Refs. [4], which determines the ERs of C n 4 using the symmetries of the EFT, and constructs C n 4 from the ERs (see [51] for similar ideas). However, if operators that involve states not connected by any symmetries are considered, or if the theory possesses no symmetry at all, the number of ERs can become infinite, and this approach may not apply [22]. In this work, we propose a more general approach that does not rely on the symmetries of the theory, and is thus immediately applicable to all multi-field EFTs.
General bounds from spectrahedron Let us briefly outline this general approach. First, notice that because cone C n 4 is convex, the dual cone of C n 4 , defined as is also convex and all bounds Q · M ≥ 0 for all Q ∈ C n 4 * exactly describe the original cone C n 4 . That is, the dual of dual cone C n 4 * equals to the original cone C n 4 . Therefore, instead of finding the C n 4 cone of amplitudes M, we can equivalently work with the dual cone C n 4 * . To determine salient cone C n 4 * , we can simply find all its ERs, as positive linear combinations of these ERs generate the whole C n 4 * [52].
More precisely, since C n 4 is contained in the − → S n 4 subspace, it is convenient to define the duality within − → S n 4 : We now need to find Q n 4 . For any Q ∈ Q n 4 , Q · M ≥ 0 ⇔ Q ijkl m i(j m |k|l) ≥ 0 ⇔ 2Q ijkl m ij m kl ≥ 0 for any m (thanks to Q ∈ − → S n 4 ), which is equivalent to Q 0. Therefore we have Q n 4 = S n 2 ×n 2 + ∩ − → S n 4 which is known as a spectrahedron. Geometrically, a spectrahedron is the intersection of the cone of PSD matrices (S n 2 ×n 2 + ) with an affine-linear space (in our case, − → S n 4 ), and is a well studied geometric object, intimately linked to SDPthe latter is simply an optimization on a spectrahedron [53]. The complete and independent positivity bounds are simply Q · M ≥ 0 for all Q ∈ ext(Q n 4 ).
We have essentially converted the problem of finding positivity bounds to a geometric problem: finding the ERs of a spectrahedron. Note that these ERs are in the dual space Q n 4 , and are to be distinguished from the ERs of the physical amplitude space C n 4 . The latter have been used in Refs. [4] to directly construct the boundary of C n 4 . As we have mentioned, this procedure becomes cumbersome to use for theories with large n but insufficient symmetries to determine the ERs. On the contrary, we will see that the new approach presented here does not have this limitation.
How do we search for the ERs in Q n 4 ? Just like a polyhedron, a spectrahedron has many (flat) faces of different dimensions. It has been shown in Ref. [54] that for any point Q in a spectrahedron, there exists a unique face F (Q) that contains Q with the lowest possible dimension and where Null(Q) is constant (independent of where Q is on face F (Q)). This provides a characterization of the faces, and in particular the ERs (which are 1-dimensional faces) of a spectrahedron. Let u 1 , u 2 , · · · u k be a basis of Null(Q) and Q 1 , Q 2 , . . . , Q m be a basis of − → S n 4 , then the null space of the following (n 2 k) × m matrix gives the linear subspace that contains F (Q). If Null(B) is 1-dimensional, then Q is an ER. The positivity bounds are simply Q · M ≥ 0 for all such Q's. Toy model: multi-scalar Consider an EFT of n scalar modes φ i=1,...,n . At the tree-level, the relevant operators are dim-8, and a basis can be chosen as Let us consider simply a Z 2 symmetric model (φ i → −φ i ). The amplitude can be computed straightforwardly. We find M iiii = 4C iiii , M iijj = M ijji = M jiij = M jjii = C iijj ≡ C iijj + 1 2 C ijij , and M ijij = M jiji = C ijij . All other elements vanish.
The same Z 2 symmetry can be applied to its dual space, the spectrahedron Q n 4 . For n = 2, a general element in Q n 4 can be parameterized as: where the rows (columns) correspond to the i, j (k, l) pairs taking (1, 1), (2, 2), (1, 2), (2, 1). The 2 × 2 blockdiagonal structure is due to the Z 2 symmetry. Crossing symmetry is reflected the common matrix elements, while Q 0 leads to the inequalities. Writing Q ≡ x i Q i , each Q can be represented by a x = (x 1 , . . . , x 4 ). From these inequalities, we can find the ERs: x e1 (r) = (1, r, r 2 , |r|) and x e2 = (0, 0, 0, 1), where r is an arbitrary real number, and x e1 (r) is extremal for any r. They are complete because any other x can be written as x = Each ER corresponds to an independent positivity bound. The second ER, x e2,i Q i · M ≥ 0, simply gives C 1212 ≥ 0. The r-dependent ER, x e1 (r), gives 4C 2222 r 2 + 4C 1122 r + 2C 1212 |r| + 4C 1111 ≥ 0. Together they are equivalent to: As a quick application of this result, it improves the previous positivity bounds on the parameters of the Higgs-Dilaton inflationary model [50]. 3-dimensional slice of C 2 4 (left) and Q 2 4 (right) for the bi-scalar toy example with Z2 symmetry. The three axes in the left plot are taken to be (x, y, z) = 3C 1122 , normalized to 4C1111 + C1212 + 4C2222 = 1. Those in the right plot are the same but with C ( ) ijkl → Q ijkl .
To illustrate the relation between C 2 4 and its dual, in Figure 1 we display the 3D cross sections of the physical amplitudes C 2 4 and the spectrahedron Q 2 4 , which are both 4D cones. The two types of ERs of Q 2 4 are highlighted by the red and green extreme points, respectively. The boundary of the C 2 4 are dual to these ERs: a vertex in Q 2 4 corresponds to a facet in C 2 4 and vice versa, as implied by duality. Finding the full bounds is therefore equivalent to finding ext(Q 2 4 ). On the other hand, the ERs of the physical amplitudes C 2 4 are also highlighted. They can be of special physical interest, and we refer to Ref. [23] for potentially interesting phenomenological consequences. (More general cases with more modes and without Z 2 symmetry are presented in Supplementary Material.) Our approach always gives the complete bounds available from the dispersion relation. In contrast, the conventional positivity approach based on elastic scattering can be incomplete for a model with multiple modes. The elastic bounds are complete iff all elements of ext(Q n 4 ) can be written in form of This can always be done for bi-scalar models, even without the Z 2 symmetry (see Supplementary Material). However, this ceases to be true when there are 3 and more scalars. To see this, it suffices to give an example of Q being extremal in Q 3 4 but not of the form of Q uv . One explicit example is Q ex = 4 α=1 U ij α U kl α , with the following four U α matrices: Q ex is a rank-4 matrix, so it cannot be written as some Q uv , which is at most rank-2 by definition. We will explain the physics interpretation of Q ex later, using the SM flavor operators as an example. We see that in the most general case, elastic positivity is incomplete for EFTs with more than 2 low-energy modes. In practice, however, the existence of symmetry relations can delay the appearance of non-elastic bounds. For example, the 4-W operators presented in Ref. [4] contain non-elastic bounds. The W -boson carries 2 helicities and is charged under the adjoint of SU(2), which is equivalent to the fundamental of SO(3), thus the number of independent components in this case is 6. However, if reducing the SO(3) to SO (2), which leads to 4 independent components left, there is no non-elastic bound any more.
General numerical method For a model with many low energy modes, the optimal positivity bounds can be efficiently obtained numerically. To see this, note that M being in C n 4 is equivalent to Q · M ≥ 0 for all Q ∈ Q n 4 . This means we can get the optimal bounds by requiring the following semi-definite program (SDP) has a non-negative minimum. This solves the problem in polynomial time complexity, and always gives the best bounds within given numerical accuracy, in contrast to the elastic positivity approach, which is NP-hard and leads to incomplete bounds.
It is sometimes useful to explicitly describe the boundary of C n 4 . To this end, an MC approach can be adopted in order to obtain a random sampling of linear bounds. To find an ER, one simply: 1. Pick a random point Q in Q n 4 , and compute F (Q) using Eq. (5).
2. If F (Q) is 1-dimensional, then Q is on an ER; otherwise, take a random straight line in F (Q), and find its intersection(s) with the boundary of Q n 4 (which is an SDP problem).
3. Let Q be one of the intersection points and iterate, until an ER is found.
The iteration will take Q to a random ER. If the problem only has a finite number of bounds, this iteration will capture all bounds. This is often the case if one considers the self-interactions of some multiplet particle (see examples in Ref. [4,44]). For non-polyhedral cones, we will get a sampling of bounds with a finite number of iterations.
Our new approach in principle captures all the information from the forward and twice-subtracted dispersion relation, and improves many previous results based on elastic scattering. We now demonstrate this in subspaces of SMEFT.
can also contribute through diagrams with two insertions. The amplitude M can then be mapped to Using the MC approach, we find 45 linear inequalities, which we have also verified with the symmetric extremal approach [4]. They can be written in the form of x· c ≥ 0, and the first 6 x vectors are Previous results on parity-conserving operators based on selected elastic scattering in Ref. [29] can be reproduced already by the 3rd to the 6th x vectors. We emphasize that this is a new result and an important step towards the full set of SMEFT positivity bounds.
The new approach is most powerful when multiple gauge-boson fields are incorporated, where the positivity cone is no longer polyhedral. A phenomenologically relevant case is the operators that characterize the anomalous quartic-gauge-boson couplings (QGCs), which is an essential part of the electroweak program at the LHC (see Refs. [56][57][58] for recent results). Knowing positivity bounds for these operators will provide guidance for future experimental searches. For operators sourcing only the transversal modes, using the SDP approach, we find that the coefficient space is cut down to 0.681% of the total. This agrees with Ref. [22], where the same number is obtained by approximating the amplitude space by a polyhedral cone with a large number (N ≈ O(10 3 )) of edges and extrapolating N → ∞, which is much less efficient. The full set of aQGC bounds can also be determined by the SDP approach. We will present it in a future work.
SM flavor sector A perhaps more relevant example is the SMEFT operators in the flavor sector. The SM fermions come with 3 generations, so full positivity bounds cannot be derived from elastic scattering of mixed flavors; flavor symmetry needs not be a symmetry of the SMEFT, so the symmetric extremal approach [4] does not apply. The SDP approach solves this problem. Consider one fermion species f , say the right-handed electron f = e R , but for all 3 generations. Using the Fierz identity, the dim-8 four-fermion operators can always be written  To illustrate the improvement of the new approach, we pick a set of coefficients C 0 that saturates the non-elastic bound Q ex given in Eq. (8), and display both elastic and the exact bounds in Figure 2. These bounds are obtained by varying one operator at a time, while keeping the others fixed at C 0 , whose values are indicated with red dots. Elastic amplitudes are only bounded from below, while others are bounded from both sides. Since C 0 is chosen to saturate the Q ex bound, the exact bounds could often uniquely fix the coefficients, so some exact bounds are not visible in the plot.
The new bound from Q ex can be interpreted as coming from combining four channels between initial and final states |I α = |F α = U ij α |i ⊗ |j , for α = 1, 2, 3, 4. The U matrices are given in Eq. (8) and are at least rank-2, implying the two incoming particles are entangled. The U 1 matrix, for example, describes the scattering of the entangled state |I 1 = |F 1 = |e ⊗|e +|τ ⊗|τ . Individually, these states cannot be used to construct positivity bounds, because the u-channel contribution in the dispersion relation, U ij α U kl α m il m kj , is not positive semidefinite. However, the Q ex tensor combines these channels together such that 4 α=1 U ij α U kl α ∈ − → S n 4 is crossingsymmetric, which guarantees that both s-and u-channels are positive.
Positivity bounds for the flavor operators of the SMEFT are phenomenologically relevant, as the existence of flavor-violating effects (e.g. µ → 3e) would set lower bounds on the flavor-conserving ones (e.g. e + e − → e + e − ), providing important guidance for future experiments [43]. While dim-6 contributions potentially give the dominant contribution, future precision measurements are likely to have sufficient precision to simultaneously determine both dim-6 and dim-8 effects through global fits [23]. Novel observables have also been designed to extract dim-8 information without being affected by the dim-6 ones [59]. Phenomenological studies for dim-8 SMEFT have started in the recent years [23,33,35,47,[59][60][61][62][63][64][65], and their interplay with positivity bounds may reveal crucial information about UV physics. Our new approach guarantees the best positivity bounds at dim-8, and is thus crucial for fully capturing this information.
Summary We have shown that the full s 2 positivity bounds for EFTs with n low-energy modes are given by the ERs of the spectrahedron Q n 4 . We have formulated the problem of finding the optimal bounds as a semi-definite program, which can be efficiently solved in polynomial times. We have presented realistic examples and improved previous results in the areas of cosmology, LHC and flavor physics (see Supplementary Material for more details, with Refs. [66][67][68][69] included there), which are all useful physical results by themselves. Our approach is straightforwardly applicable to all multi-field EFTs, and represents a crucial step towards fully extracting the positivity constraints for realistic EFTs with many degrees of freedom.
Here we provide more details about the multi-scalar model we have used to illustrate the basic ideas of our new approach. Apart from illustration purposes, the multi-scalar model is a theory with little symmetry, and from the viewpoint of the positivity cones, other theories can be obtained by appropriate symmetric projections of this cone. Scalar fields are also widely used in model building in cosmology, and as briefly mentioned in the main text, these results can readily be used to improve the physical bounds in the literature.
Bounds for multi-scalar EFT with Z 2 . In the main text, we have constructed the Q n 4 cone for n = 2 with Z 2 symmetry, which can be easily done by directly implementing crossing symmetry and Z 2 symmetry. Here we will construct the Q n 4 cone for higher n. We shall explicitly illustrate the procedure for the case of n = 3, and an explicit construction for higher n is also possible but more involved.
For scalar EFTs that are invariant under φ i → −φ i for all i, an element Q ijkl of the Q n 4 spectrahedron, which satisfies the same symmetries as M ijkl , is a n 2 × n 2 block diagonal matrix, with the first block being followed by So all off-diagonal elements in the first block, (b 0 ) ij , are equal to the off-diagonal element of b ij . Q being positive semi-definite requires that all the blocks be individually positive semi-definite. Using the fact that the ERs of S n×n + are the rank-1 symmetric matrices, one can show that Q n 4 has two sets of ERs, defined by the following conditions, respectively, where Q ex1 is a function of an arbitrary n-dimensional vector x, and Q ij ex2 depends on two integers, 1 ≤ i < j ≤ n. One can easily check that these are indeed extremal using Eq. (5). (For n = 2, we have shown that they are the only ERs.) In the following, we will prove that for n = 3 there are no more ERs in addition to Q ex1 (x) and Q ij ex2 .
First, let us set up the notation. An element of Q 3 4 can be parameterized as The null space of Q( x) is a direct sum of the null spaces of the diagonal blocks: and so on, where v (0) and v (kl) are the basis vectors of Null(b 0 ) and Null(b kl ) respectively. 0 m is an m-dimensional zero vector. The necessary and sufficient condition for Q( x 0 ) to be extremal in Q 3 4 is that the B matrix defined in Eq. (5) has a one-dimensional null space. This means the following system, uniquely determines x up to normalization. This requirement is simply that Q( x) has the same null space as Q( In the following we consider rank(b 0 ) ≥ 1.
2. rank(b kl ) = 2 for some 1 ≤ k < l ≤ 3. The null space for b kl does not exist, and v (kl) vanishes. Consider Eq. (19). Since there is no u i vector constructed from v (kl) , Q z,kl u i = 0 for all u i , and so there is no equation from (19) that can be used to constrain z kl . The only possible ER in this case is that all components in x 0 vanish except for z kl . This again gives the second type in (13).
In the following we consider rank(b kl ) ≤ 1 for all 1 ≤ k < l ≤ 3.
3. rank(b kl ) = 0 for all 1 ≤ k < l ≤ 3 (i.e., all b kl 's vanish). Each b kl has a two-dimensional null space, and so Eq. (19) would simply force y kl = z kl = 0, so that Null(b kl ) will not change. We are left with three parameters, x 1 , x 2 , x 3 ≥ 0. For Q( x 0 ) to be extremal, only one of them can be nonzero. So we have rank(b 0 ) = 1, a special case for the first type in Eq. (13).
4. rank(b kl ) = 1 for all 1 ≤ k < l ≤ 3, i.e., z kl = |y kl |. Each b kl has a one-dimensional null space, and so in Eq. (19), the corresponding v (kl) vector, being (1, 1) T or (1, −1) T , will enforce z kl = −y kl or z kl = y kl through equation (y kl Q y,kl + z kl Q z,kl ) v (kl) = (y kl ± z kl )(±1, 1) T = 0. We are left with 6 free parameters in x 0 : three x k 's and three y kl 's. To constrain them up to normalization, we need 5 additional independent equations coming from Eq. (19). Since each null vector v (0) gives at most 3 independent ones, we need at least two null vectors, which means rank(b 0 ) = 1. This leads to the first type in Eq. (13). The above exhausts all possibilities. We thus conclude that Eq. (13) covers all ERs for n = 3. One can similarly construct the ERs for higher n, but it can have more possibilities. For example, for n = 4, one may have ERs with rank(b 0 ) = 2. This is possible if, for example, b 12 and b 34 vanish altogether. After taking into account (y kl Q y,kl + z kl Q z,kl ) v (kl) = 0, we are left with 8 free parameters: x 1 , . . . x 4 , y 13 , y 14 , y 23 , y 24 , and thus two null vectors from v (0) are sufficient to constrain them.

One
We now derive the corresponding bounds using the ERs Q ex1 (x) and Q ij ex2 , and remove the x-dependence that appears in the ERs of the first type. These bounds would be the exact ones for n = 3, but are only conservative for n > 3. For the first class of ERs, b 0 can be written as xx T , so the bounds are where |x| ≡ |x 1 | |x 2 | . . . |x n | T . This means that the following 2 n−1 matrices 4C 1111 2s 1 s 2 C 1122 + C 1212 · · · 2s 1 s n C 11nn + C 1n1n 2s 1 s 2 C 1122 + C 1212 4C 2222 · · · 2s 2 s n C 22nn + C 2n2n . . . . . . . . .
ERs of a generic bi-scalar EFT. Now consider a generic multi-scalar model, that is without Z 2 symmetry. We will focus on the case of n = 2. For n = 2, the amplitude M is given by We find that ext(Q 2 4 ) can be parameterized by three parameters: where a, b, c are free real numbers satisfying c 2 ≥ ab, which are the equivalence of the r parameter in the Z 2 symmetric case. In the following, we will prove that these ERs are complete. It is sufficient to show that all elements of Q 2 4 can be written as a positive linear combination of elements in Eq. (26). With crossing symmetry, an arbitrary element Q can be parameterized as a matrix According to the Sylvester's criterion, Q 0 implies that |Q| ≥ 0, D ≥ 0, as well as the determinant of the bottom-right 2 × 2 block also being positive, i.e., D ≥ |B|. Defining the matrix we have |M | = 2|Q|/(D − B) ≥ 0. Since Q 0 also implies A B B C 0 and A ≥ 0, we conclude that M 0. This allows us to write down the decomposition of M as follows and then Q can be written as the sum of the following ERs provided c 2 i ≥ a i b i for all i. To show this is possible, defining ∆ i = c 2 i − a i b i , we have i ∆ i = (D − B)/2 ≥ 0. Note that the Cholesky decomposition allows us to set a 2 = a 3 = c 3 = 0, so that ∆ 3 = 0 and ∆ 2 = c 2 2 ≥ 0. If ∆ 1 is also non-negative, then we have ∆ i ≥ 0 for all i. Otherwise, one can always replace a i , b i , c i by a i , b i , c i for i = 1, 2, the latter being defined as and similar for b i , c i . This changes E 1 and E 2 but will leave the sum of E 1 , E 2 and E 3 invariant. Now ∆ 1 = c 1 2 − a 1 b 1 is a function of θ, and we have The latter condition is valid because we have set a 2 = 0. Therefore we can find a θ such that ∆ 1 = 0 and ∆ 2 = (D − B)/2 ≥ 0. With this θ, Q can be written as a sum of E 1 , E 2 and E 3 , which are extremal in Q 2 4 . The full positivity bounds are then given by Q(a, b, c) · M ≥ 0. Further removing the a, b, c dependence is possible thanks to a theorem proved by Hilbert about PSD quartic forms [68], which will be presented shortly. Before that, we want to point out that for n = 2, all the ERs of M are just ERs of elastic amplitudes between u i |i and v i |i : To see this, simply take u = (a, c − √ c 2 − ab) and v = (a, c + √ c 2 − ab), and we find that Q uv is then proportional to Q(a, b, c) of Eq. (26). Now, we derive the explicit expressions for positivity bounds on the generic bi-scalar model. First, note that the Q(a, b, c) · M ≥ 0 can be re-written as for all c 2 ≥ ab. This can be equivalently cast as that is, the quartic form f must be PSD. It was proved by Hilbert that a PSD quartic form is a sum of squares, iff it has no more than 3 variables [56]. In our current case, f only has 3 variables (r, w, s), so whether it is PSD can be determined by whether one can write it as a sum of complete squares. Note that r 2 and s 2 cannot appear within the squares, as f (r, s, w) is at most quadratic in r, s respectively. Let f = α x α · w 2 rs rw sw and X = α x α x T α ∈ S 4×4 + . Comparing with Eq. (35), X can be written as where 2d = α x 1 α x 2 α − x 3 α x 4 α is undetermined. The bounds can be derived by the existence of a real parameter d such that X is a 4 × 4 PSD matrix. For the non-degenerate case without any bound saturation, using the Sylvester's criterion, this is equivalent to the following conditions: These are polynomial inequalities for d, and one has to find the constraints on coefficients such that at least one solution for d exists. This can be carried out analytically, which leads to the explicit positivity bounds: and

More details about SM gauge bosons
Here we provide more details about the positivity bounds on the SMEFT anomalous gluon couplings. We first list the relevant gluon operators in the gg → gg scatterings in the SMEFT. The P -conserving dim-8 operators are as follows [57] Q (1) while the dim-6 operator O G = f ABC G Aν µ G Aρ ν G Aµ ρ also enters through double insertion diagrams. Thus, we define the coefficient vector As mentioned in the main text, the optimal positivity bounds can be efficiently obtained by the SDP approach numerically, which can then be uplifted to bounds consisting of integers. The results agree with those from the symmetric extremal approach. The full set of 45 bounds can be presented in the form of x · c ≥ 0. The x vectors are Finally, for illustration purposes, we take a further simplification by imposing the condition M B = 0 9×9 , so that M A can be rewritten as: To further illustrate the improvement of our new approach, in Figure 3, we show the lower bounds on the flavorconserving coefficient |C 1111 | as a function of C 1112 , with other coefficients fixed at C 0 . This plot indicates that FIG. 4. Improvement in constraining the dRGT parameters c3 and d5 (left) and the Z2 bi-field spin-2 EFT parameters d and λ (right; see Ref. [66]). "Elastic" denotes the superposed elastic positivity bounds, while "Exact" bounds are obtained by SDP.
flavor-conserving signal of new physics (such as e + e − scattering) is bounded by the flavor-violating ones from below (such as µ → 3e), and our new approach is crucial for fully capturing this information.

Applications in constraining spin-2 EFTs
Here we present another interesting application of our formalism for spin-2 EFTs. General relativity, when viewed as an EFT on Minkowski space, is a theory with a massless spin-2 particle, the graviton. The discovery of the late time cosmic acceleration has prompted the proposal that the graviton may have a finite, cosmological scale Compton wavelength, i.e., a small, Hubble scale mass. The theory of massive gravity has traditionally been regarded as illdefined due to theoretical inconsistencies such as the ghost instabilities until the dRGT model [67] was discovered which eliminates the ghost degree of freedom and is unique up to 2 free parameters κ 3 and κ 4 (or c 3 and d 5 with the relations κ 3 = 2 − 4c 3 , κ 4 = 1 − 4c 3 − 8d 5 ). The dRGT model consists of the standard Einstein-Hilbert term plus a nonlinear potential for the graviton, and can be compactly written down with the help of the square root of the metric, but for our purposes the leading Lagrangian is simply given by where R is the Ricci scalar, M P is the Planck mass, m is the graviton mass, h µν = g µν − η µν and Ihhh = − µνρσ αβγδ δ µ α h ν β h ρ γ h σ δ and so on. There has been suggestions that the dRGT model might not have an analytical UV completion (See, for example, [69]), thus violating positivity bounds. However, Ref. [30] has applied elastic positivity bounds to the dRGT model and found that c 3 and d 5 are compatible with positivity bounds within a finite island in the parameter space. A massive graviton has 5 independent polarization modes, and the exact bounds can be obtained by a SDP on Q 5 4 . We find a small improvement in the left bottom corner of the 2D finite region, improving the minimum d 5 value; see the left plot of Figure 4. This suggests that dRGT theory is robust even beyond elastic positivity.
Elastic positivity has also been used to constrain multi-field spin-2 EFTs [38,27], which are relevant in various contexts (see [66] and reference therein). With more modes, the improvements from our approach can become more prominent. In the right plot of Figure 4, we have shown an exemplary comparison between the full bounds and the elastic bounds for the Z 2 bi-field cycle theory [66]: where now we have two spin-2 fields h µν = g (1) µν − η µν and f µν = g (2) µν − η µν . The results are similar for other cross sections of the parameter space. For example, see Figure 5 for the bounds on the subspace of c and d. Apart from improving the bounds, evaluations of the elastic bounds with the ODE method [30] can be extremely inefficient for bi-field spin-2 EFTs: the results converge slowly with the number of the random initial conditions needed to seed the ODE evolution. In comparison, the SDP approach is far more efficient.