Triple crossing positivity bounds for multi-field theories

We develop a formalism to extract triple crossing symmetric positivity bounds for effective field theories with multiple degrees of freedom, by making use of su symmetric dispersion relations supplemented with positivity of the partial waves, st null constraints and the generalized optical theorem. This generalizes the convex cone approach to constrain the s2 coefficient space to higher orders. Optimal positive bounds can be extracted by semi-definite programs with a continuous decision variable, compared with linear programs for the case of a single field. As an example, we explicitly compute the positivity constraints on bi-scalar theories, and find all the Wilson coefficients can be constrained in a finite region, including the coefficients with odd powers of s, which are absent in the single scalar case.


Introduction and summary
Recently, there has been significant progress in understanding the consistent parameter spaces of effective field theories (EFTs) in the form of positivity bounds.These are constraints on the Wilson coefficients of an EFT, or more generally on some physical observables derived from the EFT scattering amplitudes, and arise from simply assuming the EFT's UV completion is consistent with fundamental principles of S-matrix such as causality/analyticity and unitarity.The positivity bounds can be quite stringent, often eliminating large chunks of the naive parameter space, which highlights the fact that not everything consistent with the symmetries can be a valid EFT.
Applying the optical theorem to the dispersion relation for an identical scalar amplitude, one can derive the forward positivity bound which states that the Wilson coefficient in front of s 2 should be positive [1] (see also earlier works [2,3]), s, t, u being the standard Mandelstam variables.There are a few directions to generalize the scope and strength of positivity bounds for 2-to-2 scattering.First of all, the same argument for the s 2 bound can also be used to infer that the coefficients in front of all even powers of s are all positive, if the theory is weakly coupled at least in the IR.But there is more to the forward dispersion relation than meets the eye of the optical theorem's positivity.In fact, the Hankel matrix of all the coefficients of even s powers are also positive definite [4], as can be seen from the connection to the (mathematical) moment problems [4,5].Once the t expansion of the dispersion relation is also included, the structure of positivity bounds becomes much richer.By using the positivity of derivatives of the amplitude's absorptive part and relaxing the UV scale of the dispersive integrand, Ref [6] derived an infinite tower of positivity bounds away from the forward limit that can be cast as a recurrence relation involving s and t derivatives (see [7][8][9] for earlier works on going beyond the forward limit).These recurrence bounds have also been generalized to massive particles with spin using the transversity basis for the external polarizations in which su crossing relations are (semi-)diagonalized [10].
However, these recurrence positivity bounds only utilized the su crossing symmetry that is inherent in the twice subtracted dispersion relation.It has been realized that huge gains can be profited if one imposes st crossing symmetry on the su symmetric dispersion relation, which allows us to bound all the coefficients in the s, t expansion both from above and below [11,12].Specifically, the st crossing symmetry implies that certain linear combinations of the Wilson coefficients in the su symmetric expansion of the EFT amplitude have to vanish, which, by substituting in the sum rules from the su symmetric dispersion relation, gives rise to an infinite series of null constraints on the dispersive integrals.These null constraints can be added into the sum rules of the Wilson coefficients to form linear programs to extract the optimal bounds from the positivity of the UV spectral function.These linear programs involve a continuous decision variable, which nevertheless can be efficiently solved by the publicly available SDPB package [13].For the s 2 coefficient, Ref [12] was also able to derive an upper bound in terms of the cutoff, thanks to the upper bound of partial wave unitarity and the first null constraint.For the case of a single scalar field, these new triple crossing symmetric bounds can restrict the dimensionless Wilson coefficients to be parametrically order O (1).This is of course something one naturally expects, as can be seen in many examples by explicitly integrating out the heavy fields from the UV models, but these new bounds put the naturalness argument on a rigorous and concrete footing, barring the possibility of a potential accidental large coupling.
An alternative way to use the triple permutation symmetry of a scalar amplitude is to start with a dispersion relation that is triple symmetric [14] (based on an earlier work [15]), and then locality enforces an alternative set of null constraints on the triple symmetric dispersion relation.Yet, another way to extract the same positivity bounds [16] is to first perform a general linear rotation to simplify the partial wave expansion in the dispersion relation and then convert the problem to a bi-variate moment problem that is well studied in mathematics.Whether a point in the parameter space satisfies the positivity bounds can then be determined by checking positive definiteness of a series of coefficient matrices, and the null constraints in this way can be imposed at the level of Wilson coefficients, that is, by slicing out the triple crossing symmetric subspace of the allowed bounds.
An interesting application of these triple crossing symmetric bounds is that, while the forward positivity bound can marginally rule out [1] massless Galileon theory [17], which arises as decoupling limits of several gravitational models (see e.g., [18]), the new bounds can now effectively rule out Galileon theories where the Galileon symmetry is weakly broken, as these bounds dictate that the weakly broken scale must be parametrically close to the cutoff scale in these theories [11].On the other hand, full-blown applications of positivity bounds in gravitational theories requires a judicial treatment of the t channel pole, which survives the twice subtraction and whose singularity in the forward limit is balanced out by the divergence in the dispersive integral.By transforming to the impact parameter space, it has been shown that the forward singularity can be overcome, carving out some sharp boundaries for the swampland [19].See  for some other interesting discussions of positivity bounds in gravity and cosmology.
While the above positivity bounds for a single scalar field presents a significant step towards a better understanding of the structure of the parameter space of EFTs, our universe is more complex than just a single scalar field.We typically encounter EFTs with many degrees of freedom.In the absence of any new particle signals at the LHC, the EFT approach to parameterize possible Beyond the Standard Model physics has become increasingly popular.In the Standard Model EFT (SMEFT), for example, there are a large number of low energy modes.How to optimally handle many degrees of freedom adds a whole new dimension to the problem of extracting the positivity bounds in a generic EFT.Recently, progress has been made in this direction for the lowest order s 2 coefficients, the positivity bounds on which are of course important phenomenologically.
If there are many modes in a low energy EFT, a simple generalization of the elastic positivity bounds is to linearly superpose the different modes to get elastic positivity bounds for the superposed states.However, this does not produce the strongest positivity bounds, as it misses positivity bounds coming from considering elastic amplitudes between "entangled states" [43,44].By using generalized optical theorem, one can see that the s 2 coefficients of the multi-field amplitudes form a convex cone, whose extremal rays correspond to irreps of the EFT symmetries or one particle UV states projected down to the EFT symmetries [43].This highlights the importance of positivity bounds in inverse-engineering the UV completion of SMEFT and the importance of higher order operators in Beyond the Standard Model phenomenology.Furthermore, the dual of the amplitude cone is a spectrahedron, and thus finding the strongest positivity bounds can be turned into a (normal) semi-definite program (SDP) [44], which can be efficiently solved by many widely available algorithms.Other applications of positivity bounds in SMEFT can be found in [43][44][45][46][47][48][49][50][51][52][53][54][55][56].Also, see [57][58][59][60][61][62][63][64][65] for a few other generalizations of positivity bounds, and [66][67][68][69] and reference therein for recent progress in S-matrix bootstrap, which overlaps the development of positivity bounds.
In this paper, we initiate the study of positivity bounds on higher order Wilson coefficients for theories with multiple degrees of freedom, incorporating both the triple crossing and convex cone approach, to better the understanding of the parameter space structure of multi-field EFTs.We will focus on theories with scalar fields only, which have simpler partial wave expansions.For the multi-field case, the dispersion relation contains a spectral tensor that is indexed by the four external particles and is not necessarily positive for generic combinations of the particles.To extract the optimal bounds, we take a convex geometry approach to view this spectral tensor as living in a convex cone.A key observation is that the dual cone of the spectral tensor cone is a spectrahedron.This allows us to formulate an SDP with only one continuous decision variable to extract the optimal positivity bounds, which can again be efficiently solved by the SDPB package.This is compared to the single scalar case which can be formulated as linear programs with one continuous variable.For both cases, the continuous variable comes from the dispersive integral and is associated with possible scales of the UV states.On the other hand, this is also similar to the s 2 convex cone, but now the spectrahedron depends on the partial waves, the UV scale and the orders of the v = s + t/2 and t expansion.
Due to the multi-field structure, the generic st null constraints are now more complex, as they are now indexed by the external particles and the external particle indices are swapped for some terms in the constraints.Nevertheless, a convenient way to add them to the SDP is essentially to invoke something similar to the dual cone for the Wilson coefficients.As the null constraints are equalities, instead of inequalities that are used to define the dual cone in convex geometry, the structure that is introduced is really just the boundary of the dual cone.Another way to use our formalism, which is not explicitly demonstrated in the paper, is to perform the SDPs without adding the dispersive null constraints.Instead, the null constraints are to be imposed as equalities on the Wilson coefficients directly.That is, one restricts the outcomes of these SDPs to the subspace defined by the null constraints in the coefficient space.However, this typically involves computing SDPs with quite a few coefficients, as even the lowest null constraints in a multi-field theory already contain quite a few coefficients.If we are faced with a problem that involves only a couple of particular coefficients, the approach explicitly implemented in this paper is much more efficient.
As an illustration, we apply the formalism to constrain the Wilson coefficients of bi-scalar theories endowed with some discrete symmetries.We explore the geometric shapes for the v 2 and v 2 t coefficients and also compute two-sided bounds for higher order coefficients, which are obtained agnostic about the other coefficients.To obtain these two-sided bounds, it is essential to take into account the finiteness of the v 2 coefficients.Different from the single scalar case, we now also have coefficients with odd powers of s, and we find that these coefficients are also bounded from both sides, despite that its associated (raw) spectral tensor does not form a salient cone.All in all, the triple crossing bounds can be used to constrain the Wilson coefficients of a multi-field EFT in a finite region near the origin.
The paper is organized as follows.In Section 2, we derive the su symmetric dispersion relation and expand the relation to get sum rules for all the Wilson coefficients, and then we impose st crossing symmetry to get the null constraints on the dispersion relation.In Section 3, we formulate the extraction of triple crossing symmetric positivity bounds as SDPs, show how these programs can be implemented with SDPB.
In Section 4, for a simple example, we explicitly calculate the triple crossing positivity bounds for bi-scalar theory with the double Z2 symmetry and the Z2 symmetry, with the numerical results presented in five figures and one table.In Appendix A, we list the explicit expressions for a few quantities used in the first few null constraints for a quick reference.A briefly discussion on generalization to the case with massive fields is presented in Appendix B.

Sum rules for multi-fields
In many circumstances, EFTs contain multiple low energy modes in their spectrum so as to reproduce our phenomenal world.Let us suppose that there are N light modes in a D dimensional low energy EFT.Consider an EFT with a large hierarchy between the cutoff and the masses of the modes, Λ ≫ mi, so that it is a good approximation to take the massless limit mi → 0, and therefore we have the following simple kinematic relations where s = −(p1 + p2) 2 , t = −(p1 − p3) 2 , u = −(p1 − p4) 2 with p1, p2, p3, p4 being the external momenta, and θ is the scattering angle between particle 1 and 3.For simplicity, we will consider multi-scalar theories as our examples, as their partial wave expansion is easier to perform explicitly.For massive scalars, whose crossing relations are still trivial, a generalization to the case with the same mass is also straightforwardsee Appendix B. We will assume that the EFT is weakly coupled and loop contributions are suppressed below the cutoff, so we can focus on the leading tree level amplitudes.For tree level amplitudes, the positivity bounds can be directly expressed in terms of the Wilson coefficients, while for the loop amplitudes it might be more convenient to express the bounds in terms of observables derived from the amplitude.

su symmetric dispersion relation
To derive positivity bounds for an EFT, we make use of the su symmetric dispersion relation, whose existence reflects unitarity, analyticity, locality and crossing symmetry of the underlying UV amplitude.Consider UV scattering amplitude A ijkl (s, t) for process ij → kl, where i, j, k, l = 1, 2, ..., N label different low energy modes.Following the same steps as the case of a single scalar (see, eg, [6]), particularly utilizing Cauchy's integral formula for A ijkl (s, t) in the complex s plane for fixed t along with the Froissart-Martin bound [70,71] and su crossing symmetry, we can express the amplitude as a dispersive integral of the absorptive part of the amplitude over µ > Λ 2 : where Λ is the scale at which the lowest heavy modes come in, identified with the cutoff here for simplicity and the absorptive part is defined as Abs A(µ, t) = 1 2i Disc A(µ, t) = 1 2i A(µ+iϵ, t)−A(µ−iϵ, t) with ϵ → 0 + .When A ijkl is time reversal invariant, the absorptive part reduces to the imaginary part Abs A ijkl (µ, t) = ImA ijkl (µ, t).This is the so-called twice subtracted dispersion relation with the subtraction terms a (0) ijkl (t) and a (1) ijkl (t), which are needed because the Froissart-Martin bound, derived from unitarity, analyticity and locality/polynomial boundedness, can only constrain the UV behavior of Abs A ilkj (s, t) to diverge slower than s 2 .In the last line of the equation above, we have chosen the subtraction point at µp = −t/2 so that (s + t 2 ) 2 = (u + t 2 ) 2 can factor out of the whole dispersive integral, which allows us to define a convenient SDP later.Since B ijkl (s, t) is su symmetric, we have B ijkl (s, t) = B ilkj (u, t), so a We have chosen the scalar fields to be in a self-conjugate basis for simplicity, but the formalism below can be easily generalized for a general basis, in which one should additionally conjugate the particle after crossing.For example, the crossing symmetry would be B ijkl (s, t) = B i lk j (u, t) and the u channel part of the dispersion relation would have Abs A i lk j (µ, t), where j and l stands for the conjugate particle of j and l respectively.For our later convenience, it is useful to introduce a new momentum invariant v to replace the s and then the s ↔ u crossing symmetry simply becomes v ↔ −v: where we have defined Bijkl (v, t) ≡ B ijkl (s, t).Then we can write the dispersion relation as For elastic scattering, i.e., when i = k, j = l, we can also factor out Abs Aijij(µ, t) in the integrand and have a (1) ijij (t) = 0, which means that there will be only even powers of v on the both sides of the dispersion relation, as is the case for scattering between identical particles.For inelastic scattering amplitudes, odd powers of v are generally present, and we will see that the Wilson coefficients of these terms also have two-sided bounds, despite their individual spectral tensors not forming salient cones.The remarkable feature of the dispersion relation is that on the left hand side the quantity B ijkl (s, t) can be well approximated by the EFT computations for s, t ≪ Λ 2 , while the right hand side relies on the absorptive part of the amplitude from the high energy UV theory, noting that the integration goes all the way up to infinity.In other words, the dispersion relation is a tool for us to use some salient properties of UV physics to constrain the EFT in the IR.
The absorptive part of the amplitude can be expanded by partial waves where the Gamma function Γ(α) is positive for D ≥ 3 and is the Gegenbauer polynomial, the D dimensional generalization of the Legendre polynomial in 4D.In a scattering process, angular momenta being conserved implies that unitarity can be applied to individual partial waves.A direct consequence of partial wave unitarity is the generalized optical theorem, which for partial wave ℓ means Abs where X denotes all possible intermediate states.For later convenience, we can absorb the positive factor in the expansion into the ij → X partial wave amplitude and define (We have included the constant C ℓ (1 + 2t/µ) in terms of t.)As we want to extract positivity bounds that are independent of the UV models, we shall take the amplitude from ij to X, m ij ℓ (µ), to be arbitrary for every µ and ℓ, except for certain symmetries between the indices i and j that are already known in the low energy EFT -the exact values of m ij ℓ (µ) are ultimately determined by specifics of the UV completion.Defining a short hand notation the dispersion relation can be written as (2.12) We have subtracted the v 0 and v 1 terms (i.e., the ã(0) ijkl (t) and a (1) ijkl (t) terms) from Bijkl (s, t), so the expansion on the right hand side starts from v 2 .As we shall see shortly, ã(0) ijkl (t) and a ijkl (t) can also be expressed in terms of dispersive integrals by imposing the st crossing symmetry B ijkl (s, t) = B ikjl (t, s) on the su symmetric dispersion relation.Expanding both sides of Eq. (2.11) on v and t and re-ordering the summations appropriately, we can get where we have defined the Taylor expansion notation for the Gegenbauer polynomials and the Taylor expansion notation for (µ . (2.16) Matching the coefficients of powers of v and t on the two sides of Eq. (2.13) gives a set of sum rules for the Wilson coefficients where we have further defined The first few C m,n ℓ in 4D are explicitly given by From these sum rules, we can see that even m = 2h and odd If the theory is time reversal invariant, then we can restrict to the case where m ij ℓ are real numbers.To see this, let us split m ij ℓ into m ij ℓ,R + i m ij ℓ,I , and we have If the amplitude is invariant under time reversal, then we have m ij ℓ (m kl ℓ ) * + (j ↔ l) = m kl ℓ (m ij ℓ ) * + (j ↔ l), which means that the imaginary part of Eq. (2.21) vanishes.For this case, if we re-define to also include the summation over the real and imaginary part of the original complex m ij ℓ (hence the notation X ′ ), which does not really incur any computation burden for our purposes, we get the sum rules where now m ij ℓ (µ) are real numbers for any µ and ℓ.For the bi-scalar example to be explicitly considered later, we will focus on the time reversal invariant case.

st null constraints
The sum rules (2.17) come from directly expanding the su symmetric dispersion relation.However, the amplitude actually contains stu triple crossing symmetries.(For scattering between two identical particles, they are really symmetries, the amplitude being invariant under permutations of stu; for the case of multifield scattering, they probably could be more appropriately called crossing relations, as crossing generally links different amplitudes.)Additional set of sum rules, which will be referred to the st null constraints, can be extracted by imposing the st crossing symmetry on the su symmetric dispersion relation [11,12].
Having obtained the sum rules (2.17), the easiest way to obtain the null constraints is to use the st symmetry to impose null equalities on c m,n ijkl , and by replacing c m,n ijkl with dispersive integrals via Eq.(2.17), we also get null constraints in terms of summation over the UV states.To this end, we expand the amplitude in s and t B ijkl (s, t) = p,q a p,q ijkl s p t q . (2.24) Note that the expansion coefficients a m,n ijkl are different from c m,n ijkl , the latter being the expansion coefficients of v = s + t 2 and t.For scalars, we have the st crossing symmetry B ijkl (s, t) = B ikjl (t, s). (2.25) (For massless fields with spin, we will still have this crossing symmetry, but the partial wave expansion will be more complicated; for example, in the helicity formalism, the Wigner d-matrices should be used instead.)Plugging Eq. (2.24) into Eq.(2.25), we see that the null constraints are simply a p,q ijkl = a q,p ikjl . (2.26) To convert to the relations between c m,n ijkl , the s, t expansion can be matched to the v, t expansion Bijkl (v, t) = m,n≥0 c m,n ijkl s + t 2 m t n .Note that here the v expansion starts from m = 0, and we have defined some new coefficients c 0,n ijkl and c 1,n ijkl , which are just the Taylor expansion coefficients of ã(0) ijkl (t) and a (1) By binomially expanding (s + t 2 ) m and relabeling the summations, we get null constraints in terms of the su symmetric coefficients c m,n ijkl : As mentioned above, the st crossing symmetry further allows us to expand c 0,n ijkl and c 1,n ijkl in terms of the c m≥2,n ijkl coefficients, which is not surprising as we are using twice subtracted dispersion relation.To see this, we impose the conditions B ijkl (0, t) = B ikjl (t, 0) and B ijkl (−2t, t) = B ikjl (t, −2t), which are respectively given by ã(0) ijkl (t) + a (1) Combining these two equations allows us to eliminate a ijkl (t) on the left hand side and get an equation going like 4ã Adding the two equations, we get 4ã ), which can be used to solve for c 0,n ijkl .After this, we can substitute the result of ã(0) ijkl (t) back into Eq.(2.29), which then allows us to solve for c 1,n  ijkl .The coefficients c 0,0 ijkl , c 0,1 ijkl , c 1,0 ijkl , c 1,1 ijkl and c 0,2 ijkl will not affect non-trivial null constraints.The first few c 0,n ijkl and c 1,n ijkl that do go into the non-trivial null constraints are explicitly given by Substituting the above relations into Eq.(2.28), we get null constraints in terms of c m≥2,n ijkl np,q ijkl = (a p,q ijkl − a q,p ikjl )| c 0,n , c 1,n →c m≥2,n = 0, (2.38) the first few of which are listed in Appendix A. We find the first nontrivial null constraint comes in at p + q = 4, which is similar to the single scalar case and related to the fact that the dispersion relation needs twice subtraction.Not surprisingly, there are more null constraints in the multi-field case than in the single field case at each order.
In practice, we can further impose jl symmetry for np,q ijkl = 0 and have As su crossing symmetry is built-in in the su dispersion relation, imposing jl symmetry for np,q ijkl = 0 does not incur any loss in any meaningful null constraints.This is also consistent with what we find in our numerical SDP studies later.The first few nontrivial and independent null constraints from n ijkl = 0 are given by: These null constraints restrict the viable Wilson coefficients to be in a linear subspace of the original linear space spanned by c m,n ijkl .Put it another way, the viable Wilson coefficients live in the null space of the homogeneous linear system of the null constraints.
There are at least two ways to impose these null constraints.The direct way is to first extract the (potential) positivity region in the parameter space of c m,n ijkl from the su symmetric dispersion relation, which can be done with the SDP method that will be described momentarily, and then use the null constraints (2.39) to slice out the linear subspace that satisfies the st crossing symmetry, as emphasized by [16] for the case of a single scalar.That is, the fully crossing symmetric positivity bounds are given by the intersection between the su symmetric positivity bounds and the null space of the st null constraints.In practice, the first step of this method can often be cumbersome to implement with the SDP method, as one may need to extract the su positivity bounds for quite a few Wilson coefficients before one can use a number of the st null constraints to reduce to the fully crossing symmetric subspace -there are typically quite a few coefficients in these null constraints, especially in a multi-field EFT.The alternative way is to first use the sum rules (2.17) to convert the null constraints (2.39) to a set of null dispersive integrals, and then linearly add these null dispersive integrals to the sum rules (2.17), which again can be turned into an SDP, as we shall see shortly.The upshot of this method is that one can efficiently constrain some selected Wilson coefficients that are of concern for a particular problem.This is the approach we will take for the examples in this paper.
To cast the null constraints in terms of the dispersive integrals, we substitute the sum rules (2.17) into the above constraints and end up with null constraints of the form where E ± p,q and F ± p,q are polynomials of ℓ and the first few of them that appeared in (2.40-2.46)are given in Appendix A for a quick reference.In the next subsection, we will formulate the positivity bounds as the outcome of a tractable SDP solvable with SDPB.In order to do that, as will become clearly shortly, viable null constraints should be put in a form with only m ij ℓ (m kl ℓ ) * in dispersive integral.This can be achieved by contracting the above sum rule with a general tensor N ijkl p,q and we get That is, to get all available null constraints, we should survey all possible forms for the constant tensor N ijkl p,q .This means that in a generic multi-field EFT there can be many more null constraints than in the single scalar case.If the external particles i, j, k, l are endowed with some symmetries, however, the form of N ijkl may be more restricted.Note that as expected the constraints |m 11 ℓ | 2 E + p,q + F + p,q /µ p+q+1 = 0, which can be obtained by choosing N ijkl p,q = δ i 1 δ j 1 δ k 1 δ l 1 , are exactly the null constraints from identical scalar scattering.

Multi-field positivity bounds with full crossing
With the sum rules (2.17) and the null constraints (2.48) established, we are now ready to formulate our problem as a convex optimization program to obtain the optimal positivity bounds.For the single scalar case, the optimal bounds can be obtained via linear programing.For the multi-field case, full-blown semidefinite programing with one continuous decision variable µ is generally needed, which however is still directly solvable via the powerful package SDPB, by now a standard tool in CFT bootstrap [72].

Semi-definite program
In the sum rules for c m,n ijkl , i.e., Eq. (2.17), we see that the expression inside the angle bracket splits into two parts: the part with m ij ℓ (m kl ℓ ) * + (−1) m m il ℓ (m kj ℓ ) * , which depends on the UV theory and is mostly unknown, and the part with C m,n ℓ /µ m+n+1 , where C m,n ℓ is a known polynomial of ℓ and µ is the unknown scale of the UV modes.A naive approach would be to formulate an optimization problem with all of µ and m ij ℓ (µ) as decision variables, which, however, is intractable as, even if we cut ℓ at some finite integer ℓmax and approximate the continuous µ with Nµ discrete points, m ij ℓ (µ) still represents about N 2 ℓmaxNµ continuous decision variables.
To proceed, we first note that there is a sign difference in m ij ℓ (m kl ℓ ) * + (−1) m m il ℓ (m kj ℓ ) * between sum rules c m,n ijkl with even m and odd m So, to construct an SDP to constrain multiple c m,n ijkl , we can introduce a decision variable tensor Q ijkl m,n that is symmetric for even m and anti-symmetric for odd m when exchanging j and l.That is, if there exists a constant tensor Q ijkl m,n with the symmetries such that the following conditions i,j,k,l;m,n are satisfied, then we have a positivity bound i,j,k,l;m,n As it stands, the constraint conditions (3.5) are rather complicated, as it still contains m ij ℓ (µ).However, it is easy to see that, if we view m ij ℓ as a vector (viewing ij as one index), the inequality (3.5) is simply a quadratic form, and the condition simply means that m,n Q ijkl m,n C m,n ℓ /µ m+n+1 as a Hermitian matrix with indices ij and kl is positive semi-definite positive.Thus, whether a set of Wilson coefficients satisfy the positivity bounds can be determined by checking whether the objective of the following SDP, i,j,k,l;m,n Q ijkl m,n c m,n ijkl , has a non-negative minimum: given a set of Wilson coefficients c m,n ijkl , minimize i,j,k,l;m,n where X ij,kl ⪰ 0 denotes that the matrix X with indices ij and kl is positive semi-definite.Now, this program only has one continuous variable µ, which as we shall see in the next subsection is manageable.
The collection of all feasible sets of Wilson coefficients c m,n ijkl to the SDP forms the positivity region in the c m,n ijkl parameter space.The boundaries of the positivity region, i.e., the positivity bounds, are where the objective of the SDP vanishes where, to avoid clutter in the indices, we have suppressed the summation over ijkl and adopted the shorthand notation X ijkl c ijkl , and m.X.m * ≡ i,j,k,l m ij X ijkl (m kl ) * . (3.10) Let us recapitulate from the point view of convex geometry.We can view c m,n ijkl as elements of a convex cone, which is generated by conical combinations of [m ij ℓ (m kl ℓ ) * + (−1) m m il ℓ (m kj ℓ ) * ]C m,n ℓ /µ m+n+1 .The positivity bounds m,n Qm,n • c m,n ≥ 0 define a dual cone whose elements are Q ijkl m,n .The optimal positivity bounds are given by the boundaries of the c m,n ijkl cone or the extremal rays of Q ijkl m,n .However, from Eq. (3.8), we see that it is m,n Q ijkl m,n C m,n ℓ /µ m+n+1 that lives in a spectrahedron, the viable space of the decision variables of an SDP.So, at technical level, we can view the (partial) "spectral tensor" m ij ℓ (m kl ℓ ) * as forming a convex cone whose elements are indexed by i, j, k, l and whose dual cone is the spectrahedron ℓ /µ m+n+1 .In the SDPs above to obtain the positivity bounds, we have only made use of the su symmetric dispersion relation and have yet to take into account the st crossing symmetry.As we mentioned previously, one way to impose the st crossing symmetry is to intersect the positivity bound region with the st symmetric null constraints (2.39), if we have computed bounds for sufficiently many Wilson coefficients.In the following, however, we will take an alternative approach, to impose null constraints with dispersive integrals (2.48).As we often want to know positive bounds for a few specific Wilson coefficients, the latter approach is usually more convenient for this purpose.
The essential idea is that we can add the null constraints with N (N ) p,q from the previous section to the SDP above.Since the null constraints are null within the average ⟨• • •⟩, the objective function of the SDP is unchanged (still m,n Qm,n • c m,n ), but the null constraints do alter the linear matrix inequalities of the SDP via adding terms with N (N ) p,q , alongside with new decision variables, as this is done without imposing the average ⟨• • •⟩.Thanks to the fact that we can add the null constraints with decision variables of opposite signs, this new SDP will bound the Wilson coefficients from different directions, leading to the EFT couplings to be constrained in an enclosed convex region.More concretely, if there exist constant tensor N ijkl p,q and constant tensor Q ijkl m,n with the symmetries then we have a positivity bound i,j,k,l;m,n In the language of convex optimization, the linear matrix inequalities now construct a cone in a larger space with both the c m,n ijkl and n p,q ijkl coefficients, but in the objective function it is projected down to the subspace of the c m,n ijkl coefficients.Clearly, with the null constraints included, whether a set of Wilson coefficients satisfy the positivity bounds can be determined by checking whether the objective of the following stronger SDP has a non-negative minimum: given a set of Wilson coefficients c m,n ijkl , minimize to have a non-negative minimum.We emphasize that we have suppressed the ijkl indices for Qm,n and N (N ) p,q , which are considered as matrices when evaluating the positive semi-definiteness "⪰ 0".The collection of all feasible sets of Wilson coefficients c m,n ijkl to the SDP form the positivity region in the c m,n ijkl space and the boundaries of the positivity region are again where the objective Qm,n • c m,n vanishes.
We want to emphasize that, by adding the null constraints to the SDP, we have utilized more information of the amplitude, i.e., the triple crossing symmetry, in extracting the constraints on the Wilson coefficients.The importance of these extra terms in the linear matrix inequalities (3.15) is that it generally allows us to constrain the coefficients from opposite directions.This is possible because the contributions from the null constraints can dominate the left hand side of the linear matrix inequalities (3.15), which makes it possible for Q ijkl m,n to have both signs [11,12].More concretely, suppose the SDP (3.14 and 3.15) gives rise to a bound of the form m,n Qm,n • c m,n ≥ 0 for a given set of decision variables Q ijkl m,n and N ijkl p,q ; thanks to the null constraints, the SDP may also be feasible for Qijkl m,n = −Q ijkl m,n , which will be facilitated by a different set of N ijkl p,q , and thus the SDP gives rise to another bound of the form m,n Qm,n • c m,n ≤ 0.

Implementation
As we have seen, generally, our SDP problem is defined in the complex domain with complex linear matrix inequality constraints.Efficient SDP algorithms are usually implemented in the real domain.The reason is that a complex SDP problem can be easily transformed to a real SDP problem by using the fact that a Hermitian matrix H is positive semi-definite if and only if a doubly enlarged matrix constructed from ReH and ImH is positive semi-definite where ReH/ImH is the real/imaginary part of the complex Hermitian matrix H.However, as we mentioned previously, in the following, we will focus on exemplary theories with time reversal invariance, for which, according to the argument around Eq. (2.21), m ij ℓ (µ) can be taken as real matrices With this simplification, it is already an SDP in the real domain.The special feature of this SDP is that it has a continuous decision variable µ that is in the denominators and goes from Λ 2 to ∞.This is hardly a problem, as it can be easily transformed to a polynomial matrix program, solvable by SDPB [13].
For readers unfamiliar with SDPB, this package is able to efficiently solve the following polynomial matrix program (by transforming it into a standard SDP): given A + 1 real numbers ba and J(A + 1) real symmetric matrices of dimension nj × nj . . .P a j,1n j (x) . . . . . . . . .P a j,n j 1 (x) . . .P a j,n j n j (x) with each element P a j,uv (x) being a polynomial of x, the package can survey all possible real numbers ya to maximize b0 such that M 0 j (x) + A a=1 yaM a j (x) ⪰ 0 for all x ≥ 0 and 1 ≤ j ≤ J.
Note that here the linear matrix inequality conditions are required to satisfy for 1 ≤ j ≤ J and for all continuous x ≥ 0. To convert our SDP problem (3.14) into the standard form above, we can choose a basis Q ijkl (a) for the Q ijkl m,n .The form of Q (a) can be mapped from the symmetries of c m,n ijkl , and specifically is restricted by crossing symmetries ,n and as well as the internal symmetries of the theory, which will be further explained with an example shortly.Other choices of Q ijkl m,n are simply redundant.In an appropriate basis, we then have where {y a m,n } are a set of constants.Similarly, the form of N ijkl p,q can be derived from the symmetries of n p,q ijkl , specifically the crossing symmetries N ijkl p,q = N ilkj p,q = N kjil p,q = N jilk m,n and again the internal symmetries of the theory.Expanded in an appropriate set of basis, we have So, after these, the decision variables of the SDP are y a m,n and z b m,n , together with the continuous variable µ.Since the UV state mass scale µ ranges from Λ 2 to ∞, we can define a dimensionless variable which takes values from 0 to ∞. Multiplying by a common factor of µ M M with MM being the greatest power of 1/µ in (3.14) for a given problem, the linear matrix inequality (3.15) becomes m,n,a where we have defined . In the following, the coefficient components projected to the basis matrices will be denoted as Using these notations, our SDP can be formulated as follows: a positivity bound is found if the following polynomial matrix program can minimize a,m,n subject to m,n,a and the minimum of a,m,n y a m,n c m,n is semi-positive, with zero being a boundary point of the positivity bounds.Practically, of course, we can only include a finite number of null constraints and partial waves, and one should include sufficient numbers of them so that the results converge to the required accuracy.We emphasize that the strength of using the null constraints in the dispersive integral form is that we can selectively constrain a small number of Wilson coefficients c m,n ijkl that are of concern in a particular problem.

Upper bounds of the s 2 coefficients
Using the upper bound of partial wave unitarity and the first null constraint, Ref [12] was able to derive an upper bound for the Wilson coefficient of identical scalar scattering c 2,0 iiii .In a multi-field theory, we also want to derive upper bounds on the s 2 coefficients other than identical scalar scattering.As with the single scalar case, we will see later that the existence of the upper bounds for all these coefficients is essential to conclude that all the 2-2 scattering coefficients are bounded in a finite region in a multi-field theory.
Let us first briefly review how it works for the c 2,0 iiii coefficient.In our notation, after incorporating the first null constraint, we get the sum rule where α is an arbitrary constant, If we choose α > 0, at sufficiently large ℓ, µ can have a range from Λ 2 to Ξ 2 ℓ,α = max Λ 2 , (α Ñ (1) 1,3 ) 1/2 ) where the integrand of the above dispersive integral is negative.If we take the upper bound of partial wave unitarity and subtract the negative part of the dispersive integral, we can get an inequality Let us assume that the negative part of the dispersive integrand emerges when ℓ > ℓ * α for a given α * , and then the integration on the right hand side can be computed explicitly.Note that when ℓ increases, Ξ 2 ℓ,α also increases so that the sum over ℓ actually converges.The results depend on ℓ * α and α * , which can be evaluated numerically, and it is found that the optimal upper bound is given by which occurs at α * = 0.025 and ℓ * α = 2.In a multi-field EFT, while Im a ijkl ℓ is still bounded for generic ijkl, thanks to partial wave unitarity, it may not be positive definite.So the simple inequality method above does not generically apply.However, extracting the upper bounds on c 2,0 ijkl can be readily formulated as an optimization problem if one discretizes the integrals in the c 2,0 ijkl sum rules and the null constraints by sampling only discrete values of µn [73? ]: After the discretization, we impose partial wave unitarity conditions on Im a ijkl ℓ (µn) for a finite number of ℓ and all the discretized µn.Along with the null constraints, this becomes a semi-definite program [74], or a simpler linear program if the linearized unitarity conditions are used [73], which sometimes approximates the full unitarity conditions very well.Readers are referred to [73,74] for the details of the strategy and the numerical implementation.While the above inequality method is computationally cumbersome to include more null constraints, it is easy to use this discretization method to improve the bound (3.30) with more null constraints.Combining these upper bounds with the lower bounds, which form a salient convex cone for the v 2 (or s 2 ) coefficients [44], the v 2 coefficients can be constrained to be a "capped" salient cone.

Full crossing bounds on bi-scalar theory
With the formalism established, in this section we shall illustrate how to obtain the triple crossing symmetric bounds in practice.The multi-field case is different from the single field case in a number of aspects.With multiple light fields, the number of Wilson coefficients proliferates very quickly with the increase of the number of fields.This does not present any particular difficulty in determining whether a given set of coefficients are within the positivity bounds per se.However, for illustration purposes, we shall limit ourselves to the simple case of bi-scalar theory endowed with some discrete symmetries, for which it is easier to visualize the geometric shapes of the bounds.We will start with the simpler version where the two scalars are invariant under a double Z2 symmetry in 4 dimensional spacetime.We will see how the v 2 t coefficients are bounded in a finite region, and then compute the two-sided bounds for the higher order coefficients.We then relax the symmetry slightly, to let the theory only have one Z2 symmetry.We find that the geometric shapes now become more complex, but still one can bound the coefficients in a small finite region.

Bi-scalar theory with double Z 2 symmetry
We shall start with a simple case of two scalars with a double Z2 symmetry in 4D, that is, a scalar field theory invariant under two discrete transformations • ϕ1 → −ϕ1 (the ϕ1 ↔ ϕ2 symmetry implies that we also have ϕ2 → −ϕ2) To perform the SDP optimization, we can first find a basis for Q ijkl (a) and N ijkl (a) , which are determined by crossing symmetries and internal symmetries of the theory, as mentioned in Section 3.2.To be precise, the forms of Q ijkl (a) and N ijkl (b) can be used, but that is just redundancy and can significantly increase the computational costs when there are many light modes.)For our particular example, the reflection symmetry ϕi → −ϕi requires that Q iiij (a) (i ̸ = j) and its cyclic permutations vanish.The (time reversal invariant) su crossing symmetry requires Q 1122 , where we have used the sign of integer a to differentiate the basis vector for Q ijkl 2h,n (with a > 0) and Q ijkl 2h+1,n (with a < 0).Also, for negative a, we have Q ijij (a) = −Q ijij (a) = 0 (i and j can be the same).The i ↔ j and k ↔ l crossing symmetries require Q 1212 (a) = Q 2121 (a) .After these considerations, for the case of Z2 symmetry ϕ1 → −ϕ1, we find that a generic Qm,n takes the form where the rows and columns of matrix Q ijkl m,n ≡ Q ij,kl m,n are ordered such that ij and kl take the sequence of 11, 22, 12, 21, as noted in the equation above.This result will be used in the next subsection for the single Z2 theory.For the case of double Z2 theory, we have an additional symmetry of ϕ1 ↔ ϕ2, which implies . Therefore, a simple basis for Q ijkl m,n with the double Z2 symmetries ϕ1 ↔ ϕ2 and ϕ1 → −ϕ1 is given by For N ijkl (a) , the conditions N ijkl p,q = N ilkj p,q = N kjil p,q = N jilk m,n and the double Z2 symmetries require that N ijkl p,q have the same symmetries as Q ijkl 2h,n , so we shall choose N ijkl p,q to have the same basis as Q ijkl 2h,n .This means that we will use the following null constraints Our empirical numerical explorations also confirm that including additional null constraints does not affect the positivity bounds.For the single scalar, we would only have the null constraint (2E + p,q + 2F + p,q )Q (1) at any given p and q, but for the double Z2 bi-scalar case we have two other null constraints at the same p and q. (Note that this is not at any given p + q, which would also have multiple null constraints even for the single scalar case.)Since all the basis matrices are block diagonal, the linear matrix inequality of this SDP takes a simple block diagonal form where A and B are 2 × 2 matrices.This is obviously equivalent to the lower dimensional conditions A ⪰ 0 and B ⪰ 0.
With the bases established, we can run the SDP (3.27) with SDPB to decide whether a given set of Wilson coefficients in a particular problem can satisfy the triple crossing positivity bounds.To carve out the geometric shapes of the bounds for a given set of Wilson coefficients, we search for places where the objective function vanishes to find the boundaries of the positivity bounds.
v 2 order coefficients Let us first determine the finiteness of the v 2 coefficients.The optimal lower positivity bounds for the v 2 coefficients in a multi-field EFT can be identified as the extremal rays of the dual cone of the v 2 coefficient cone, and generally these optimal bounds can be obtained by normal SDPs with no continuous decision variable [44].For double Z2 bi-scalar theory, these optimal bounds can also be obtained analytically and are described by the three inequalities [44] .
On the other hand, the 3D upper bounds on the c 2,0 (i) coefficients of a double Z2 bi-scalar theory are computed in Ref [] by formulating it as an SDP problem, as mentioned above around (3.31).The results are obtained using null constraints of the lowest orders and can be further improved.Nevertheless, combining with the lower bounds and projected to 1D bounds, the two-sided bounds on the c 2,0 (i) coefficients are given by

order coefficients
Now, we move away from the forward limit and consider coefficients with a t derivative.In this subsection, for simplicity, we shall focus on the subspace where all the coefficients associated with Q (3) vanish: c m,n (3) = 0, which allows us to visualize the bounds.This means that we still have Q (3) and associated y in the SDP, but we simply set c m,n (3) = 0 in the objective function.Generally, there are two independent Wilson coefficients for any given n and even m = 2h in the c m,n (3) = 0 subspace of the double Z2 theory: c 2h,n = 2c m,n 1111 , c 2h,n = 4c 2h,n 1122 .So we shall investigate the positivity bounds on the coefficients: c 2,0 (1) , c 2,0 (2) , c 2,0 (3) , c 2,1 (1) and c 2,1 (2) .For this SDP problem, the components of the A and B matrices are explicitly given by p,q (2E + p,q + F + p,q − F − p,q ) + z 3 p,q (F + p,q + F − p,q )] (4.8) p,q (2F + p,q ) + z 3 p,q (2E + p,q )] (4.9) = 4c 2,1 1122 ) for bi-scalar theory with double Z 2 symmetry in the cm,n (3) = 2c m,n 1221 = 0 subspace, agnostic about all higher order coefficients.c m,n ijkl is the Wilson coefficient in front of the (s + t 2 ) m t n term in the ij → kl amplitude, and the tilded coefficients are defined as cm,n (a) ≡ c m,n (a) /c 2,0 (1) .We choose units such that the cutoff Λ = 1.We see that the triple crossing positivity bounds form a closed, finite region in the parameter space, numerically of order O(1).We will see that the Wilson coefficients can still be constrained to a small finite region.
For the Z2 bi-scalar theory, Q ijkl m,n must take the form of Eq. (4.1), and so its basis can be chosen as With this basis, we have c m,n = 4c m,n 1122 , c m,n (4) = 2c m,n 1212 , and other Wilson coefficients are related to the above ones by crossing and internal symmetries.Again, we choose N ijkl p,q to have the same basis as Q ijkl 2h,n .From these bases, we see that the linear matrix inequality can still be cast in a block diagonal form First, we shall explore the shape of the full crossing symmetric bounds on the coefficients of the v 2 and v 2 t terms in the Z2 theory.Truncated to this next leading order, we already have 7 parameters: c2,0 (2) , c2,0 (3) , c2,0 (4) , c2,1 (1) , c2,1 (2) , c2,1 (3) and c2,1 (4) , if we measure them in terms of c 2,0 (1) .For simplicity, we shall restrict to the subspace c2,0 (4) = c2,1 (4) = 0, for which case, the B ⪰ 0 condition again can be neglected.Then, the SDP problem additionally contains a scaling invariance in the remaining 5D parameter space: c (2) → α 2 c (2) , c (3) → αc (3) .To see this, note that the A matrix is explicitly given by A Expressions for np,q ijkl , E ± p,q and F ± p,q While the jl symmetrized null constraints n p,q ijkl = 0 are sufficient for our approach with the su symmetric dispersive relation, it is conceivable that the unprojected null constraints np,q ijkl = 0 can be useful in other approaches.Here we also list the first few np,q ijkl null constraints for a comparison: They are mostly the same as n p,q ijkl = 0 except for (p, q) = (1, 3), (1,4), (1,5), ... .The corresponding explicit expressions of E ± p,q and F ± p,q for the first few np,q ijkl null constraints in 4D (remove the parts in the square brackets to get E ± p,q and F ± p,q for the n p,q ijkl null constraints) are given by where in the last step we have shifted the integration variable by 2m 2 .Note that if desirable we could also subtract out the known low energy contribution of the dispersive integral from 4m 2 to Λ 2 as we did in the main text.The absorptive part of the amplitude in the massive case can be expanded by partial waves as follows Abs A ijkl (µ, t) = 2 4α+2 π α Γ(α)µ With these established, we can follow the same steps as the massless case to obtain the positivity bounds with SDPB.
+ 6ℓ 5 − 37ℓ 4 − 84ℓ 3 + 179ℓ 2 + 222ℓ − 180) + 1 2 (−2ℓ 2 − 2ℓ + 5) (A.15)expansion depends on the particle masses mi, mj, m k , m l highly nonlinearly.Nevertheless, if all the modes are of the same mass m such as those in a symmetry multiplet, it is straightforward to generalize our formalism to include the mass corrections.We can essentially follow the same steps as the massless case.The differences are that now we define the v variable and the subtraction point µp as