UV complete me: Positivity Bounds for Particles with Spin

For a low energy effective theory to admit a standard local, unitary, analytic and Lorentz-invariant UV completion, its scattering amplitudes must satisfy certain inequalities. While these bounds are known in the forward limit for real polarizations, any extension beyond this for particles with nonzero spin is subtle due to their non-trivial crossing relations. Using the transversity formalism (i.e. spin projections orthogonal to the scattering plane), in which the crossing relations become diagonal, these inequalities can be derived for 2-to-2 scattering between any pair of massive particles, for a complete set of polarizations at and away from the forward scattering limit. This provides a set of powerful criteria which can be used to restrict the parameter space of any effective field theory, often considerably more so than its forward limit subset alone.


Introduction
Low energy effective field theories (EFTs) are ubiquitous in modern physics, used to describe everything from fundamental particle interactions to phenomenological models of the Universe. The universal mechanism that underlies this is the decoupling of high energy physics from low energies whenever there is a clear hierarchy of scales. This decoupling is built into any local quantum field theory as a consequence of the uncertainty principle. In the modern perspective, non-renormalizable theories such as General Relativity, far from being inconsistent with the tenets of quantum mechanics, should be viewed as EFTs which can be consistently quantized at low energies to any desired order of accuracy up to, at a given order, a finite number of undetermined matching coefficients.
Although decoupling guarantees that we do not need to know the explicit UV completion of a given low energy EFT to make predictions, not all information about the high energy physics is lost. A given EFT will by definition break down in predictivity at some energy scale, the cutoff. The standard approach to UV completion introduces new degrees of freedom at energies at and above the cutoff in such a way that the S-matrix for the theory remains Lorentz invariant, analytic (causality), polynomially/exponentially bounded (locality) and unitary (predictive). If no such UV completion is possible, then the original EFT is describing something which is inherently incompatible with a local quantum field theory (QFT) description. In recent years it has been recognized that not all EFTs admit a local Lorentz invariant UV completion [1,2]. This can be demonstrated by showing that the requirement that the S-matrix is analytic imposes nontrivial constraints, positivity bounds, on the scattering amplitudes of the low energy effective theory, which in turn place constraints on the form of the EFT Lagrangian which could not have been determined by low energy symmetries and unitary alone. For spin-0 particle scattering the explicit constraints that apply to the 2-to-2 scattering amplitude are straightforward to derive in the forward scattering limit [2] and have recently been generalized by the authors to an infinite number of bounds that apply away from the forward scattering limit [3]. These bounds have for instance been applied to Galileon EFTs in [4].
More recently, a similar program is being brought to bear on gravity, i.e. the scattering of spin-2 particles. Remaining agnostic about the explicit form of a UV complete description of gravity, it is possible to derive constraints on gravitational EFTs by nevertheless demanding that such a UV completion should exist [5,6]. For example, this has been shown to severely constrain the parameter space of massive gravity [7], and pseudo-linear massive gravity [8]. However, the majority of bounds in the literature to date rely on one crucial assumption: a trivial crossing relation between the processes ps-channelq A`B Ñ C`D and pu-channelq A`D Ñ C`B.
(1.1) ward limit. A discussion of the bounds for spinning particles in the forward scattering limit, which applies to more general polarizations, addressing the nontrivial issues with analyticity and statistics for fermions has been given recently in [9]. Fermion-boson scattering includes additional branch cuts, which significantly simplify in the forward scattering limit, and [9] demonstrates that forward limit positivity bounds can be meaningfully extended to elastic scattering of particles of any spin. In addition to these extra branch cuts, another problem with extending these results away from forward scattering, is if the crossing relation is not sufficiently simple, it is not possible to guarantee positivity of the discontinuity of the scattering amplitude along the left hand branch cut which is a crucial ingredient 1 in the proof of the positivity bounds. For example, the real polarizations (used in the t Ñ 0 limit) have a non-trivial optical theorem when t ‰ 0 [10]. The more common helicity formalism, while having clear unitarity properties, transforms in a complicated way under crossing [11].
Historically, a number of approaches have been taken to deal with this problem. One approach is to expand a general spin scattering amplitude in terms of scalar invariant amplitudes which have simple crossing properties [12,13]. In practice however this approach is cumbersome for general spins. Closely related to the helicity amplitudes are the M -amplitudes which transform covariantly as tensor-spinors and so have more straightforward crossing properties [14][15][16][17] (for a general discussion on the relation of these approaches see [18]). Although simplifying the crossing relations helps, it is also necessary to deal with quantities which have positive discontinuities. A further approach that addresses this is to consider linear combinations of helicity amplitudes which respect positivity along both the left and right hand cuts [19].
In this work, we demonstrate how the transversity formalism [20], in which the crossing relation is (semi)-diagonal, resolves these issues most straightforwardly. The transversity formalism is simply a change of polarization basis in which the spin of the particle is projected onto the normal to the scattering plane (hence the name transversity). The simplicity of the crossing relation in the transversity basis will allow us to infer a dispersion relation which is positive on both the left and right hand branch cuts, for any choice of transversities, and in so doing derive for scattering of particles of general spin direct analogues of the scalar positivity bounds derived in [3].
In section 2 we briefly review the key properties of the helicity and transversity formalisms, and discuss the more complicated analyticity structure of spin scattering, and how to remove kinematic singularities. Then in section 3 we prove the positivity properties of the dispersion relation, which in turn allows us to prove general positivity bounds for particles of arbitrary spin. For the sake of clarity, in section 3 we will give the expressions for identical particles of mass m and spin S. In section 4, we discuss how this bound is altered in the case of different particle masses and spins. A summary and closing remarks are given in section 5. Much of the formalism is given in the appendices. After reviewing the connection between analyticity and causality in appendix A, we give a novel derivation of the crucial crossing formula for general spin scattering in appendix B, and illustrate this in explicit examples in appendix C which can be used to check the claimed analyticity properties. We connect this derivation with the historical approach in appendix D and finally we collect various technical properties of the amplitudes (appendix E) and Wigner matrices (appendix F) for convenience.

From Helicity to Transversity
In this section, we first review the commonly used helicity formalism for calculating the scattering amplitude. Our principal concern is the scattering of massive particles although statements about the analyticity of tree level scattering amplitudes will also apply in the massless limit. The helicity formalism is however not convenient to establish positivity bounds (except for the case of the forward scattering limit or the pure scalar interactions), as the crossing relations for nonzero spin particles are highly nontrivial. Fortunately, the crossing relations can be diagonalized by using the so-called transversity formalism, which we will introduce via a rotation from the helicity formalism. See Fig. 1 for a pictorial view of both approaches.
For concreteness, and to focus on the main points, we first look at the simple case where all the four particle masses are equal and the spin of particle C pS 3 q and D pS 4 q equal to that of particle A pS 1 q and B pS 2 q respectively: m 1 " m 2 " m 3 " m 4 " m, S 3 " S 1 , S 4 " S 2 . (2.1) We will then discuss extensions to more general cases where the masses can differ in Section 4.
As we are dealing with 2-to-2 scattering amplitudes, we will make great use of the Mandelstam variables: s "´pk 1`k2 q 2 , t "´pk 1´k3 q 2 , u "´pk 1´k4 q 2 " 4m 2´s´t . (2.2) For later convenience, it will also be useful to introduce the associated variables: which are both positive in the physical s and u-channel regions.

Helicity Formalism
When computing 2-to-2 scattering amplitudes, it is standard to consider plane wave 2-particle incoming and outgoing states. However, unitarity is better expressed using spherical waves as we can exploit angular momentum conservation. In what follows we shall review the standard spherical and plane wave states before relating them by means of the partial wave expansion. For definiteness, we consider the scattering plane to be the xz plane, and the y direction to be orthogonal to the scattering plane. We further fix coordinates so that the incoming particles move along the z-axis without loss of generality.
Spherical wave states: Irreducible representations of the SOp3q rotational symmetry provide the basis of the 'spherical wave' states J 2 |jmy " jpj`1q|jmy, J z |jmy " m|jmy , (2.4) where of course, here m is the spin projection along the z direction, rather than the particle mass m. Any three-dimensional rotation can be characterized by three Euler angles pα, β, γq, and implemented on a state via the operator Rpα, β, γq " e´i αJz e´i βJy e´i γJz , where J x , J y and J z are the angular momentum operators. The action of Rpα, β, γq on the spherical wave states can be expressed in terms of the Wigner D matrices [21] Rpα, β, γq|jmy " Explicit expressions for the small d matrix are given in Appendix F.
Plane wave states: On the other hand, one particle 'plane wave' states are eigenstates of momentum, with well-defined angular momentum in the rest frame J 2 |p " 0, S, λy " SpS`1q|p " 0, S, λy, J z |p " 0, S, λy " λ|p " 0, S, λy , where S is the spin of the particle. These transform into each other under boosts and rotations. For example, a nonzero momentum state is constructed from the rest frame as |p, S, λy " Rpφ, θ, 0qLppq|0, S, λy for p " pp sin θ cos φ, p sin θ sin φ, p cos θq , (2.8) where Lppq is the boost along the z direction to momentum pẑ. Note that a finite momentum state no longer has well-defined angular momentum, except along the momentum axis J¨p |p| |p, S, λy " λ|p, S, λy.
Physically, this is because the orbital angular momentum L " rˆp is zero along this axis. λ is called helicity, and is a good quantum number in all reference frames [10].
Two particle states are constructed simply as the tensor product of one particle states |p 1 p 2 λ 1 λ 2 y " |p 1 , S 1 , λ 1 y b |p 2 , S 2 , λ 2 y . The particle spins are some fixed, known quantities, so one usually omits them in the kets, and one can also factor out the center of mass motion and write [10,22] where P " p2 a m 2`p2 , 0, 0, 0q is the center of mass 4-momentum of the two particles, p is the 3-momentum value of particle A (or B), and the normalizations are xpθ 1 φ 1 λ 1 1 λ 1 2 |pθφλ 1 λ 2 y " δpcos θ 1´c os θqδpφ 1´φ qδ λ 1 1 λ 1 δ λ 1 2 λ 2 and xP |P 1 y " p2πq 4 δ 4 pP 1´P q. θ and φ are the angles of particle A in the center of mass system, and P contains the information of the total momentum and just goes along for the ride.
Partial wave expansion: Now, we want to relate the plane wave 2-particle states |pθφλ 1 λ 2 y, which we use when calculating scattering amplitudes, to the spherical wave 2-particle states |pJM λ 1 λ 2 y, which provide a convenient expression of unitarity. To do this, note that when the two particles collide along the z-axis, we have J z |p00λ 1 λ 2 y " pλ 1´λ2 q|p00λ 1 λ 2 y, which implies One may normalize the spherical wave states as , and this then fixes c J up to a phase |c J | 2 " p2J`1q{4π. We will choose this phase to be zero. Now we can use a rotation to go to any desired collision axis Consider scattering between initial state |iy " |p i 00λ 1 λ 2 y and final state |f y " |p f θφλ 3 λ 4 y that conserves the total energy momentum. Splitting the S matrix intoŜ " 1`iT , and remembering that we can factor out an overall momentum conserving delta function, we define the helicity amplitude H λ 1 λ 2 λ 3 λ 4 ps, θq via xf |T |iy " p2πq 4 δ 4 pP f´Pi qH λ 1 λ 2 λ 3 λ 4 ps, θq , (2.13) where and where s is the center of mass energy square defined in (2.2).
Then inserting the complete spherical wave basis, we have and T J λ 1 λ 2 λ 3 λ 4 psq is the partial wave helicity amplitude Since the angle φ is unimportant as the system is symmetric with respect to rotations about the collision axis, we may set φ " 0. Physically, T J λ 1 λ 2 λ 3 λ 4 psq is the scattering amplitude between two particles states of definite total angular momentum and definite individual helicities.
Kinematical singularities: In going from the s channel scattering angle θ to the Mandelstam variables 2 defined in (2.2), we have We see that additional singularities at s " 4m 2 may be introduced. In the physical region, t Ñ 0 whenever s Ñ 4m 2 , and so these residues vanish-i.e. these poles are unphysical. These are known as 'kinematical singularities' [11], and can be systematically removed [24,25]. We will return to this in more detail when we construct our transversity amplitudes.
Unitarity: Since angular momentum is conserved in the scattering process, the S-matrix can be block-diagonalized to different partial waves labelled by J. The partial wave unitarity condition then gives ipT J:´T J q "T J:T J , whereT J is the J component of the partial wave expansion of the transition matrixT . For the helicity amplitudes, this implies If the scattering process is time reversal invariant, the S matrix (and T matrix) is real analytic, so we further have T J λ 1 λ 2 λ 3 λ 4 " T J λ 3 λ 4 λ 1 λ 2 . Then the absorptive part is just the imaginary part From Eq. (2.19), we see that Abs s T J λ 1 λ 2 λ 3 λ 4 has the form ř α αλ 3 λ 4 α λ 1 λ 2 . If one regards tλ 1 λ 2 u and tλ 3 λ 4 u as two indices, then it is clear that Crossing: The amplitude associated with the s-channel (A`B Ñ C`D) is denoted H s λ 1 λ 2 λ 3 λ 4 , and the corresponding u-channel amplitude is H u λ 1 λ 4 λ 3 λ 2 , describing the process A`D Ñ C`B, whereB andD denote the antiparticles of B and D. Then under crossing particles B Ø D (when S 1 " S 3 and S 2 " S 4 and the particles have equal mass m), we have [11,[26][27][28] where the angle χ u is given by This result, as well as its inelastic generalization, are derived in Appendices B and D and are checked through examples in C. In the forward scattering limit t " 0, we have χ u " 0 and hence, which is consistent with the result of [9], where the helicities flip sign because the momenta effectively reverse. To avoid excessive notation, in what follows we will drop the ps, t, uqchannel sub/superscript notation when unimportant. Figure 1. The difference between the helicity and transversity formalism. The horizontal plane (xz plane) is the particle interaction plane. In the helicity formalism particle spins are projected onto the direction of motion, while in the transversity formalism particle spins are projected in the vertical direction, which is transverse to the interaction plane.

Transversity Formalism
Since H λ 1 λ 2 λ 3 λ 4 ps, t, uq contains a branch cut on the real axis of the complex s plane between s " 4m 2 and 8, the crossing symmetry implies that there is a second branch cut in the real axis between s "´t and´8. However, this second branch cut has no obvious positivity properties in the helicity formalism, due to the complicated crossing mixing of different helicity amplitudes as can be seen from Eq. (2.23) (unless χ u " 0, corresponding to the forward scattering limit t " 0, or unless all particles have zero spin). To go beyond the forward scattering limit for non-zero spins, we first need to simplify the crossing relation by going to the transversity basis, see Fig. 1.
Transversity Amplitudes: We define the transversity eigenstates [20,26] as a particular combination of the helicity eigenstates |p, S, τ y " ÿ λ u S λτ |p, S, λy , (2.26) where the unitary matrix u S λτ is simply the Wigner D S matrix associated with the rotation This unitary u S matrix has the virtue of diagonalizing any of the Wigner d S matrix, independently of their angles. See Appendix F for properties of the u S matrices.
The transversity amplitudes are thus related to the helicity amplitude via Physically, this corresponds to scattering particles with definite spin projection orthogonal to the scattering plane, i.e. eigenvalues of the operator [25] τ "´1 m w µ W µ pk i q with w µ "´2 µνρσ k ν 1 k ρ 2 k σ 3 ? stu , (2.29) where W µ pk i q is the Pauli-Lubanksi (pseudo)vector of particle i, µνρσ is the Levi-Civita tensor and k 1 , k 2 and k 3 are the respective momenta of particles A, B and C. In short, while the spin quantization axis of the helicity formalism is chosen to be the momentum direction of an incident particle, in the transversity formalism it is chosen to be transverse to the scattering plane (see Fig. 1). An important point to notice, which will become significant later, is that there is in general a ? stu " 0 branch point in the scattering amplitudes. Furthermore under ?
stu Ñ´?stu the transversity τ flips sign, something that will be clear in the properties of the transversity amplitudes. We will see below how to deal with this kinematic branch point.
Crossing: As already mentioned, the main motivation for considering the transversity amplitudes is that the unitary matrices u S diagonalize the Wigner matrices appearing in the helicity crossing relation (2.23), so the crossing relations in the transversity formalism are much simpler. Explicitly, for S 3 " S 1 and S 4 " S 2 we have where χ u is given in (2.24). Further considering elastic transversities τ 1 " τ 3 , τ 2 " τ 4 this is This is considerably simpler than the equivalent expression in the helicity basis. If we further take the forward scattering limit t " 0, we then have χ u " 0 and so T s τ 1 τ 2 τ 3 τ 4 ps, 0, uq " T ú τ 1´τ4´τ3´τ2 pu, 0, sq , (2.32) where the transversities flip sign because the scattering plane normal is reversed.
Kinematical Singularities: Note that at first sight the e iχu term appears to introduce additional poles/branch cuts in the complex s plane. For BB or FF scattering, ř j τ j is an even number, and the worst singularities are additional kinematic poles at s, u " 4m 2 and a kinematic branch point at stu " 0. For BF scattering, ř j τ j is odd 4 , and the crossing factor has an additional kinematic branch point at su " 0 [29]. We shall see below that these branch points and additional kinematical poles are removed by multiplying the scattering amplitudes by an additional regulating factor.
In general potential poles and branch cuts arise only at 5 4 In the special case of parity invariant amplitudes, Tτ 1 τ 2 τ 3 τ 4 vanishes unless τ1`τ2´τ3´τ4 is an even integer [26], but we will not assume parity invariance here. 5 These are known as 'thresholds' (at s " 4m 2 ), and 'pseudothresholds' (at s " 0), and the 'boundary of the physical space' (at ? stu " 0) in [25] and others.
• s " 0: The helicity amplitudes can be shown to be regular at s " 0 [25], and therefore so are the transversity amplitudes (by Eq. (2.28)).
• s " 4m 2 : These are factorizable singularities, which can be removed by multiplying an appropriate prefactor`?´u˘ξ´a where ξ " 1 if S 1`S2 is equal to a half integer (i.e. for BF scattering) and ξ " 0 otherwise. Essentially, this is subtracting all of the terms that go as negative powers of ps´4m 2 q 1{2 from the amplitude. On the physical space, s " 4m 2 requires t " 0, where it can be shown that the residue associated with these poles then vanish, and these terms are therefore not physical poles. However, their presence complicates the dispersion relation and so it is more convenient to subtract them. Recall that these come from replacing θ s with s, t, u by (2.18). To be consistent with the crossing relation 6 , we see that the maximal order of these poles is N " | ř i τ i | ď 2pS 1`S2 q for elastic amplitudes.
In practice it will prove convenient to utilize the prefactor where S " sps´4m 2 q as defined in (2.3). For elastic scattering, the prefactor in (2.34) has indeed the same analytic structure as that in (2.33). We emphasize however that (2.33) has the virtue of being applicable in the more general case of inelastic scattering, and preserves the positivity along both the left and right hand cuts.
• ? stu " 0: There is a potential branch point at stu " 0, which can be removed by taking an appropriate combination of the transversity amplitudes [29]. Since ? stu " S sin θ s { ? 4s, ? stu Ø´?stu corresponds to θ s Ø´θ s . Consequently any even function of θ s will not contain the branch cut. The two natural combinations are and In general as we go around ? stu " 0, we have [25] T τ 1 τ 2 τ 3 τ 4 | ?
(2.38) 6 The crossing relation (2.30) exchanges 1{ps´4m 2 q with 1{ps`tq. As there is originally no pole at u " 4m 2 (s "´t) in the scattering amplitude, the crossing relation must be turning every s " 4m 2 kinematical singularity into one at u " 4m 2 .

Positivity Bounds
In this section, we make use of the transversity amplitudes to derive an infinite number of positivity bounds for non-forward scattering amplitudes of arbitrary spins.

Unitarity and the Right Hand Cut
To begin with we consider the case of elastic scattering of particles of definite transversity, so that τ 3 " τ 1 and τ 4 " τ 2 . (3.1) The partial wave expansion for transversity eigenstates is rather complicated [26,30], in essence because one cannot define a rotationally invariant notion of transversity in a state with only two particles. Instead, we use the helicity partial wave expansion where we have set the interaction plane to lie along φ " 0 and in analogy with (2.15), we have definedT Using the properties of T τ 1 τ 2 τ 1 τ 2 ps, θq under θ Ñ´θ we find When considering Tτ 1 τ 2 τ 3 τ 4 all kinematic singularities are removed by construction, and so the remaining discontinuity along the right hand cut arises from the physical partial wave amplitudeT J λ 1 λ 2 λ 3 λ 4 psq. Consequently we can take the absorptive part 8 Using the Fourier series of the Wigner matrix (F.14), we can write d J µλ pθq`d J µλ p´θq as a sum over cospνθq with real coefficients Substituting in the discontinuity we find and By unitarity, we have established in (2.22) thatT J λ 1 λ 2 λ 3 λ 4 is a positive definite Hermitian matrix 9 , and it therefore directly follows that To proceed further we distinguish between the case of BB or FF scattering ξ " 0, and BF scattering ξ " 1.
e. the discontinuity comes from the physical part. 9 Strictly speaking unitarity only implies thatT is positive semi-definite. However following Ref. [3] we may use analyticity to guarantee that it is positive definite.
BB or FF scattering: For ξ " 0 the intermediate partial waves have integer angular momenta J and similarly ν is an integer. In the forward scattering limit t " θ " 0 we have the familiar optical theorem On the right hand cut, this property can be extended to any numbers of t derivatives by making use of the properties of the Chebyshev polynomials for integer ν. Indeed, the Chebyshev polynomials satisfy which is strictly positive for n ď ν and vanishes for all n ą ν for integer ν. We therefore have This means that, since the transversity amplitudes are analytic in t [19], we can analytically continue the optical theorem for forward scattering Abs s T τ ps, 0q ą 0 to the finite positive t case Abs s Tτ 1 τ 2 τ 1 τ 2 ps, t, uq ą 0, 0 ď t ă m 2 , s ě 4m 2 , (3.14) where we remind the reader that the absorptive part is related to the discontinuity across the real axis by Abs s " 1 2i Disc. Note that the analytic continuation cannot take us past the first pole in t. We have taken the first pole to be at t " m 2 , which occurs in generic cases. In special cases the one may be able to extend the range in t even further, for instance for purely scalar scatterings, one can go as far as t " 4m 2 [3,4].
BF scattering: For ξ " 1 we have to include an extra factor of ?´s u " ? S cospθ{2q in the definition of Tτ 1 τ 2 τ 1 τ 2 . The discontinuity is then It is then straightforward to see that Abs s Tτ 1 τ 2 τ 1 τ 2ˇθ"0 ą 0 and since ν˘1{2 is now integer, Once again this implies B n Bt n Abs s Tτ 1 τ 2 τ 1 τ 2 ps, t, uq ą 0 for 0 ď t ă m 2 , s ě 4m 2 , and @ n ě 0 . (3.16) Following an analogous procedure for Tτ 1 τ 2 τ 1 τ 2 ps, θq it is straightforward to show that Even though we are still dealing with a function of θ, we now have the difference of two cosines to deal with, 2 sin θ sin pνθq " cos ppν`1qθq´cos ppν´1qθq and so we cannot infer the same positivity properties of either the discontinuity or its derivatives. Nevertheless, the above expression will be important in determining the discontinuity along the left hand cut.

Crossing and the Left Hand Cut
Following the standard S-matrix paradigm, T sτ 1 τ 2 τ 1 τ 2 ps, tq is analytically continued to the whole complex Mandelstam plane in such a way as to ensure full crossing symmetry is respected. Since our positivity bounds will arise from fixed t dispersion relations, it is sufficient to consider the properties of the transversity amplitudes in the complex s plane, for fixed real t, accounting for s Ø u crossing symmetry. Once the kinematical singularities have been removed, the remaining physical singularities are the poles associated with physical particles, in this case a single pole at s " m 2 and the right hand (RH) branch cut s ě 4m 2 associated with multi-particle states. s Ø u crossing symmetry requires that there is a second pole at u " m 2 as well as a second left-hand (LH) branch cut for u " µ ě 4m 2 which corresponds to s " 4m 2´t´µ . In the vicinity of the LH cut the scattering amplitude can be determined by crossing symmetry and where S and U are defined in (2.3). Note that sin`χ u ř i τ i˘{ ? stu is even under θ s Ñ´θ s and hence contains no branch cut. This is as it should be since T sτ 1 τ 2 τ 1 τ 2 ps, t, uq is an even function of θ s by construction. Now, defining the u-channel scattering angle θ u via t "´1 2 pu´4m 2 qp1´cos θ u q, (3.21) then T uτ 1 τ 2 τ 1 τ 2 pu, t, sq has the same analyticity properties in terms of u and θ u as T sτ 1 τ 2 τ 1 τ 2 ps, t, uq has in terms of s and θ s . Similarly as in (3.2), T uτ 1 τ 2 τ 1 τ 2 ps, t, uq also has a partial wave expansion in terms of partial wave amplitudesT u,J λ 1 λ 2 λ 3 λ 4 describing the fixed total angular momentum J scattering process A`D Ñ C`B. Since all kinematical singularities have been removed, following the same argument as before, the remaining discontinuities can only arise from the partial wave scattering amplitudes themselves and so across the LH cut, Here we have defined the u-channel absorptive part Abs u as the discontinuity of the s-channel amplitude across the LH cut, which by crossing symmetry is related to the RH cut discontinuity of the u-channel amplitude Abs u T uτ where F u,Jν τ 1 τ 2 puq ą 0 is the u channel equivalent of F Jν τ 1 τ 2 psq defined in Eq. (3.7).
BB or FF scattering: To proceed, we first consider ξ " 0. Let us consider the combination It is clear that this is a sum of positive functions for u ą 4m 2 times e ipθu and the same can be said for any positive integer power of this quantity 10 . Furthermore S " sps´4m 2 q " pu´4m 2 qp1`cos θ u qpu`4m 2`p u´4m 2 q cos θ u q{4, is similarly a sum of positive functions for u ą 4m 2 times e ipθu . Since T sτ " T sτ we can without loss of generality focus on τ 1`τ2 ě 0. From this we conclude that
where C ν,p puq ą 0 for u ą 4m 2 and C ν,´p puq " C ν,p puq. Using the latter property, then finally we have It is then straightforward to see that which given the analyticity of the amplitude in t may in turn be extended to BF scattering: In the case ξ " 1, we include the additional factor ?´s u " ? U cospθ u {2q, and in this case ν and J are half integers, so Following the same arguments as previously we can express where dν ,p puq ě 0 for u ě 4m 2 and dν ,p puq " dν ,´p puq and since we have already shown this same property holds for S S 1`S2´p τ 1`τ2 q then where D ν,p puq ě 0 for u ě 4m 2 . Once again we infer that

Dispersion Relation
Following the famous result of Ref. [31], if the scattering amplitude satisfies a dispersion relation 11 for t 0 ă t ď 0, has an absorptive part which is analytic inside the Lehmann ellipse, is analytic in some small neighbourhood of s and t and if it satisfies positivity of all the derivatives of its absorptive parts along the LH and RH cuts, then the analyticity region in t may be extended to |t| ă R for finite R. In the case of a single mass we can push to R " 4m 2 . Assuming polynomial boundedness and analyticity in Martin's extended region, it will then follow from the partial wave expansion that the scattering amplitude will satisfy a t dependent extension of the Froissart bound [32] |T the additional powers arising from the p ?´s uq ξ S S 1`S2 prefactor. In the previous sections we have shown that the appropriately regularized transversity amplitudes Tτ 1 τ 2 τ 1 τ 2 ps, tq satisfy positivity of all derivatives along the LH and RH cuts, and so we may conclude that they satisfy identically the same analyticity properties as the scalar amplitudes [19]. Furthermore, following the arguments of Jin and Martin of Ref. [32] when these same conditions hold, the scattering amplitude at fixed t (0 ď t ă R) satisfies a dispersion relation with the same number of subtractions as for t " 0 which in turn implies ptq ă 1 and so We thus conclude that the regularized amplitudes Tτ 1 τ 2 τ 1 τ 2 ps, tq satisfy a dispersion relation in s for 0 ď t ă 4m 2 with N S subtractions. This condition is of course implicit in the usual assumption of 'maximal analyticity', however we have seen that the result is much stronger since we have proven the relevant positivity criteria which are the crucial ingredient in Martin's extensions of the Lehmann ellipse, the Froissart bound and the extension of the regime of the dispersion relation.
The transversity amplitude Tτ 1 τ 2 τ 1 τ 2 ps, tq contains a simple s-channel pole at s " m 2 and u-channel pole at s " 3m 2´t (or u " m 2 ), which appear already at tree level (although their residues are affected by loops). It proves useful to define an associated 'pole-subtracted' transversity amplitudeT τ ps, tq with these two poles removed. That is, we consider where Res denotes the residue. In the scalar case it proved convenient to also subtract the t-channel pole [3,4] although this subtraction did not actually play a significant role. For general spins, such a subtraction would not be convenient since the residue of the t-channel pole is itself a function of s, and subtracting it can modify the behaviour of the amplitude 12 at large s.
Consider a contour C forTτ ps, tq in the complex s plane, which encircles the poles at s 1 " m 2 and s 1 " 3m 2´t as well as a generic point s, as shown in Figure 2. By Cauchy's integral formula, we haveTτ We can deform this contour so that it runs around the branch cuts and closes with circular arcs at infinity (contour C 1 ). We emphasize that even when we are considering higher spins, a Froissart bound still applies [19] and |Tτ 1 τ 2 τ 1 τ 2 ps, tq| |s|Ñ8 ă |s| N S . This allows us to neglect the arcs at infinity by performing a sufficient number of subtractions. We can then obtain the following dispersion relation: where N S is given by Eq. (3.36).
The subtraction functions a n ptq in the dispersion relation are undetermined by analyticity and depend on the detailed information of the particular theory involved. To eliminate them, we simply take N s derivatives and consider the quantity Since we have already established that the absorptive parts are positive on both the RH and LH cuts in section 3.1 and 3.2, then our first positivity bounds is the simple statement that Figure 2. The scattering amplitude can be analytically continued to the entire complex s plane, with the poles at s " m 2 and 3m 2´t and branch cuts along the real axis from´t to´8 and from 4m 2 to 8.
where the range of real s is determined by the requirement that µ´s ą 0, µ´u ą 0 given µ ě 4m 2 . This is the direct generalization, for general spins, of the bound given in [2] applied in a larger range of s and t.

Positivity Bounds for Particles with Spin
In a recent work [3], we extended the simple positivity bounds of the form (3.44) for scalar theories, to an infinite number of bounds on the t and s derivatives of the scattering amplitude. The interpretation of these bounds depends somewhat on the context. In the case of an assumed weakly coupled UV completion, these infinite number bounds may be directly applied to the tree level scattering amplitudes of the EFT and put constraints on the coefficients in the effective Lagrangian. If we do not assume a weakly coupled UV completion, then the bounds may be applied to the full quantum scattering amplitudes and the contribution of the light loops may be further subtracted off to strengthen the bounds, as discussed in the example of the massive Galileon [4].
To begin with we will derive the bounds on the exact all-loop scattering amplitude. The procedure is identical to that discussed in [3] which we refer to for more details. The only distinction in the case of particles with spin is that we must use the regularized pole subtracted amplitudes,Tτ 1 τ 2 τ 1 τ 2 ps, tq, we have a larger number of subtractions N S ě 2, and the contributions from the LH and RH cut are not identical. Before getting to the general case, we can get a feel for how the bounds work by considering the first t derivative of (3.41). Defining new variables s " 2m 2´t {2`v, so that 45) then differentiating with respect to t gives Defining and using the integral inequality that for any positive definite function ρpµq ą 0 and evaluating at v " 0 we then infer that, Thus our second non-trivial bound is In practice, the above form of this bound is not so interesting since we have in mind M 2 " m 2 and so this will be dominated by the second term. Since f τ 1 τ 2 p0, tq is already positive from the lower bound, then there is little new content in this new bound. The situation is very different however if we imagine that the EFT has a weakly coupled UV completion. In this case, we expect the scattering amplitude already computed at tree level to satisfy all of the properties that we have utilized, specifically the Froissart bound. Given this, the above bound can be applied directly to the tree level scattering amplitudes. These amplitudes by definition do not include loops from the light fields and the branch cut will no longer be at µ " 4m 2 but rather at µ " Λ 2 th where the threshold energy Λ th is defined as the mass of the lightest state that lies outside of the regime of validity of the EFT. Assuming Λ th " m then the bound on the tree level scattering amplitude becomes In a typical EFT, both of these terms are of comparable order with the first potentially dominating, and so the bound becomes meaningfully independent of the existing requirement f tree τ 1 τ 2 p0, tq ą 0.
Even if we do not have a weakly coupled UV completion, we can use our knowledge of the light loops in the regime in which perturbation theory is valid (e.g. |k| ! εΛ th with ε ! 1) to subtract off their contribution from the amplitudes, thus removing part of the branch cut. This is achieved by defining where the RHS is computed using the knowledge of the light loops in the EFT. Assuming again εΛ th " m we may then impose Provided the hierarchy εΛ th " m is sufficiently great, then this will impose a bound independent of the leading order one f εΛ th τ 1 τ 2 p0, tq ą 0. We see that in all three cases, the form of the bound is the same, and the only distinction is the choice of M 2 .

General Higher Order Positivity Bounds
As discussed in the scalar case in [3] we can generalize the previous procedure to put bounds on all t derivatives and all even v derivatives of the scattering amplitude, provided the subtraction functions do not enter. In the scalar case, the triviality of the crossing relation implied that all odd v derivatives of the amplitude were zero. Following the notation of [3] and defining B τ 1 τ 2 pv, tq "Tτ 1 τ 2 τ 1 τ 2 ps " 2m 2´t {2`v, tq (3.54) and further defining ptq can be given in terms of the positive definite integrals Then following [3] verbatim, the general positivity bounds are and c r and β k are determined by c 0 " 1 and c k "´k´1 In the present case the scale M 2 is the minimum of µ`t{2´2m 2 which is 2m 2`t {2 or for the tree-level bounds applicable for a weakly coupled UV completion Λ 2 th`t {2´2m 2 « Λ 2 th .

Extensions
In previous sections, we considered the simple case where the mass of all the particles were the same m 1 " m 2 " m 3 " m 4 . Having laid out the strategy in that case, we can now move to more general scenarios.

Two Mass Eigenstates
We now consider a theory with two available mass states. Without loss of generality, we can order m 1 ą m 2 . In order to use the optical theorem at t " 0, we should consider transition amplitudes to final states with m 3 " m 1 , m 4 " m 2 , S 3 " S 1 and S 4 " S 2 . Note that in order for the heavy particle to remain stable against decay into two light particles, we further assume m 1 ă 2m 2 . This condition is not necessary when considering the tree level positivity bounds, since any associated branch cut arises at loop order. Defining the positive mass difference ∆ " m 2 1´m 2 2 , (4.1) we can generalize the analytic combinations S and U defined in (2.3) as follows which are positive on both the RH and the LH cuts. We also define the following quantity in terms of which we have a scattering angle given by Transversity amplitudes: We now define T τ 1 τ 2 τ 3 τ 4 as the scattering amplitude between eigenstates of the transversity operator, exactly analogous to (2.29), but with an overall normalization of ?´t Ψ in place of ? stu. Such an amplitude obeys the crossing relation (assuming S 3 " S 1 and S 4 " S 2 , i.e. assuming that the out-going particles are the same as the in-going ones but without necessarily the same transversity), with u channel crossing angles which clearly reduces to (2.24) when the mass difference ∆ Ñ 0. Note that to avoid cluttering notations we have omitted the u channel label for χ i .
Physical singularities: In the complex s plane, by unitarity the scattering amplitude must have poles at s " m 2 1 and s " m 2 2 , and a branch cut from s " pm 1`m2 q 2 to infinity. The crossing relation then tells us that there are also poles at s " 2m 2 1`m 2 2´t and s " m 2 1`2 m 2 2´t , and a branch cut from s " pm 1´m2 q 2´t to infinity. Note that since m 1 ă 2m 2 the poles and branch cuts are separated. We can hence proceed as before, providing that we have (4.10) The corresponding Re s axis (at fixed t) is shown in Figure 3.
Re s Figure 3. The scattering amplitude on the real s axis in a theory with two massive states. We can draw integration contours analogous to C and C 1 from Figure 2, providing that none of the poles overlap with the branch cuts.
Kinematical singularities: Just as in the case of a single mass, the transversity amplitudes may possess different types of kinematical singularities or branch points at: • S " 0: When writing cos θ and sin θ in terms of s, t, one introduces spurious singularities in 1{S. These are factorizable, and can be removed with a prefactor of S S 1`S2 . This corresponds to removing the s´4m 2 singularities in the equal mass limit.
• Ψ " 0: For boson-fermion scattering, cos θ{2 creates an unphysical branch point at Ψ " 0, which can be removed with a prefactor of`?Ψ˘ξ, where as before, ξ is 0 for BB or FF, and 1 for BF scattering. This corresponds to removing the branch point at u " 0 in the equal mass limit.
Positivity of the right hand cut: The relation between the transversity and helicity basis, as well as the partial wave expansion of the helicity amplitudes, are exactly as described earlier-even for unequal masses. Similarly, the Fourier series of the Wigner matrix (F.14) is unchanged. Therefore Tτ 1 τ 2 τ 1 τ 2 , as defined in (4.12), can be shown to obey the same positivity property (3.14) as its equal mass limit, and so as in the equal mass limit, using ? Ψ " ? S cospθ{2q, and the fact that all the Chebyshev polynomials N n,ν defined in (3.11) are positive, we can infer all the derivatives of (4.14) with respect to cos θ can be shown to be positive. Indeed, for ξ " 0, and for ξ " 1 (i.e. for BF scattering), It therefore immediately follows that B n Bt n Abs s Tτ 1 τ 2 τ 1 τ 2 ą 0 for 0 ď t ă m 2 2 and @ n ě 0 . It will also be useful to express the absorptive part of the difference as which is not necessarily positive, but will be useful in what follows.
Positivity of the left hand cut: Having defined the appropriate quantity (4.14) which is positive on the RH cut, we can now, exactly as before, establish the scattering amplitude on the LH cut by crossing symmetry. We proceed as in the equal mass limit and first relate the s channel transversity amplitude with the u channel one as in (3.18) T sτ 1 τ 2 τ 1 τ 2 ps, tq " c`T uτ 1 τ 2 τ 1 τ 2 pu, tq`c´T uτ 1 τ 2 τ 1 τ 2 pu, tq, with c`" S S 1`S2 U S 1`S2 cos p2χ 1 τ 1`2 χ 2 τ 2 q , (4.20) c´"´S and so Next we re-write this expression as a sum over cosppθ u q, after replacing ? Ψ " ? U cospθ u {2q. To express the previous relation as a sum of cos, we derive an exact analogue to (3.23), where the only subtlety to keep track off is the fact that we have now two different u channel angles χ 1 and χ 2 (defined in (4.8) and (4.9)) as opposed to only one χ u in the equal mass limit. This implies the existence of two separate sets of coefficients ? Se˘i χ i " ř p cȋ ,p puqe ipθ as opposed to the only one set in (3.23), but it nevertheless remains that cȋ ,p puq ą 0 for u ą pm 1`m2 q 2 .
Noting again that S is still a sum of cosppθ u q with positive coefficients when u ą pm 1m 2 q 2 , more precisely, then we are led to precisely the same relation as in (3.25) with now where C ν,p puq is analytic in u, and strictly positive for u ą pm 1`m2 q 2 .
Using this expression in (4.22) and taking derivatives of the Chebyshev polynomials at θ u " 0, we finally have where the coefficients C ν,p puq are all positive in the region u ą pm 1`m2 q 2 and so are the Chebyshev polynomials N n,p . This proves the following positivity relation: B n Bt n Abs u T sτ 1 τ 2 τ 1 τ 2 ps, tq ą 0 , for u ą pm 1`m2 q 2 , 0 ď t ă m 2 2 , @ n ě 0 . (4.27) We then have a dispersion relation with N S " 2`2S 1`2 S 2`ξ subtractions given bỹ Abs u Tτ 1 τ 2 τ 1 τ 2 p2m 2 1`2 m 2 2´t´µ , tq µ N S pµ´uq . (4.28) Then, we can follow the same steps as the equal mass case discussed in section 3.3 (see the steps following Eq. (3.40)) to derive positivity bounds.

Multiple Mass Eigenstates
We can even generalize the previous procedure further and suppose we had n possible intermediate states, m i , which we order m 1 ě m 2 ě ... ě m n ą 0. Then all we need in order to construct positivity bounds as we have done is to require i.e. all of the particles are kinematically stable against two-body decay (once again this is not necessary when considering the tree level bounds). Then all of the poles lie within the s and u channel branch points, and so are contained within a contour like C in Fig. 2. In this case, we are led to the same positivity bound, providing t is restricted to the region The proof is identical to the preceding subsection, with m 1 and m 2 replaced with any desired pair of particle masses m i and m j .

Discussion
The utility and universality of effective field theories is both a blessing and a curse. They successfully describe the low energy world, and decoupling ensures that predictions from a low energy EFT can be made without any precise knowledge of its explicit high energy completion. On the other hand this suggests that all EFTs are from a theoretical point of view equal, and only experiments/observations can distinguish them. In certain fields, for example in constructing EFTs for inflation, dark energy or physics beyond the standard model, observations and experiments are sufficiently limited that there remain a large wealth of candidate EFTs that can describe the same data. In this article we have discussed theoretical tools by which this impasse may be partly broken. We have seen that the very existence of a standard UV completion leads to powerful constraints on the scattering amplitudes of the low-energy EFTs. Violating any of these constraints directly implies an obstruction of the EFT from ever admitting a standard UV completion. Establishing the IR implications of UV completion is thus essential in segregating between different types of EFTs. For 2-to-2 elastic scattering amplitude, these constraints were previously known in the forward scattering limit for arbitrary particles. In this work we have derived an infinite number of constraints that span beyond the forward scattering limit for particles of arbitrary spins.
To this aim, given the 2-to-2 elastic scattering amplitude for particles of arbitrary spins, first we have shown how to construct a regularized transversity amplitude which is free of kinematic singularities, has the same analyticity structure as a scalar scattering amplitude, and has a positive discontinuity along both its left hand and right hand cuts. This generalizes an approach given for a special case in [19] (for helicity amplitudes) to the case of general spins. This result is far from straightforward, due in particular to the complex analyticity structure for fermion scattering. Crucially, in order to achieve positivity along both cuts we must work with the transversity amplitudes and not helicity amplitudes. In appendix B we have given an independent derivation of the s´u crossing formula for scattering amplitudes of arbitrary spin away from the forward scattering limit which is central to these arguments. This uses the multispinor framework [33][34][35][36] where particles of general spin are viewed as being made up of tensor products of spin 1/2 states.
Once this has been done, the positivity bounds that have been derived for scalars [2] and in particular their non-forward limit extensions [3,4] can immediately be passed over to particles of general spin, with the only caveat being that the number of subtractions increases with the number of spins for the regularized amplitudes, i.e. those combinations which are free of kinematic singularities.
These bounds will apply to scattering of massive states in any low energy effective theory that arises from an analytic, Lorentz invariant UV completion. We also expect them to apply in the massless limit for tree level scattering amplitudes, since the usual obstruction to the massless limit comes from the branch cut that begins at s " 4m 2 , but this branch cut only arises when light loops are included. The massless pole is itself harmless since it can be subtracted out. This is true provided that the Froissart bound holds. For massless particles, Froissart could be violated; nevertheless we would expect that the fixed t dispersion relations remain bounded by a polynomial, and this is sufficient to derive positivity bounds with the only difference being that the number of subtractions N S should be increased sufficiently to account for the growth of fixed t amplitude at large s.
In general, the bounds apply away from the forward scattering limit; however, whether they will be stronger away from this limit will depend on the model. At least the example of Galileon EFTs given in [4] is a proof of principle that in certain cases the non-forward limit bounds are stronger. We will give explicit examples of the application of these bounds to particular classes of EFTs elsewhere [37].

A Analyticity and Causality
The connection between the analyticity of the scattering amplitude for fixed momenta transfer, and causality is well established, but sufficiently forgotten that we will review here the essential details of the proof without dwelling on the lengthy mathematical subtleties. We shall do so for scalar particles, but this can easily be extended to general spin by utilizing the appropriate helicity wavefunctions and accounting for statistics. Denoting a : A the creation operators for a particle of type A and a :Ā that for the associated anti-particle with relativistic normalization ra A pkq, a : B pk 1 qs " p2πq 3 δ 3 p k´ k 1 q2ω k δ AB , then the s-channel A`B Ñ C`D scattering amplitude is (Ŝ " 1`iT ) x0|a C pk 3 qa D pk 4 qT a : B pk 2 qa : A pk 1 q|0y " xk 3 |a D pk 4 qT a : B pk 2 q|k 1 y "´ixk 3 |a D pk 4 qrŜ, a : B pk 2 qs|k 1 y , where in the last step we have used the fact that xk 3 |a D pk 4 qa : B pk 2 q " 0 unless B is identical to C or D in which case it will correspond a non-scattering process which drops out ofT . Denoting the (in general complex) free fields by then it is straightforward to show as a consequence of Wick's theorem rŜ, a : A pkqs " which just follows from the elementary fact that .4) and the fact that the commutator respects the Leibniz rule. Then using the property of the stability of the one-particle statesŜ|ky " |ky "Ŝ : |ky x0|a C pk 3 qa D pk 4 qT a : B pk 2 qa : A pk 1 q|0y "´i where we have defined the current operatorĴ B pxq " i δŜ δφ B pxqŜ : "´iŜ δŜ : δφ B pxq . Similarly using we have x0|a C pk 3 qa D pk 4 qT a : B pk 2 qa : Causality is encoded in the statement that (this is expounded on in [38], section 17.5) Putting this together we have δĴ B pxq δφDpyq " 0 " θpy 0´x0 qrĴ B pxq,ĴDpyqs`contact terms (A. 13) where the contact terms vanish for x 0 ‰ y 0 , i.e. they are (derivatives of) delta functions. The precise form of these contact terms cannot be determined by causality (since they are instantaneous) or unitarity. In momentum space, the contact terms correspond to polynomial functions of energy/momenta and so their addition is equivalent to modifying the subtraction terms in the dispersion relation. Consequently it is sufficient to derive a dispersion relation assuming no subtractions are needed (i.e. no contact terms), and include the subtractions at the end of the calculation. With this in mind the scattering amplitude can be taken to be x0|a C pk 3 qa D pk 4 qT a : B pk 2 qa : A pk 1 q|0y "´ż d 4 x ż d 4 ye´i k 4 .y`ik 2 .x θpy 0´x0 qxk 3 |rĴ B pxq,ĴDpyqs|k 1 y .
(A.14) Using the translation properties of momentum eigenstates to remove an overall momentum conserving delta function, the stripped s-channel scattering amplitude is A A`BÑC`D pk 1 , k 2 ; k 3 , k 4 q " ż d 4 x e´i pk 2`k4 q.x{2 θpx 0 qxk 3 |rĴDpx{2q,Ĵ B p´x{2qs|k 1 y . (A.15) An identical calculation for the u-channel amplitude would give A A`DÑC`B pk 1 ,´k 4 ; k 3 ,´k 2 q " ż d 4 xe ipk 2`k4 q.x{2 θpx 0 qxk 3 |rĴ B px{2q,ĴDp´x{2qs|k 1 y "´ż d 4 xe´i pk 2`k4 q.x{2 θp´x 0 qxk 3 |rĴDpx{2q,Ĵ B p´x{2qs|k 1 y. (A. 16) In the latter form, it is clear that the only difference between the u-channel scattering amplitude and the s-channel is the choice of retarded θpx 0 q versus advanced´θp´x 0 q boundary conditions (up to a sign). As a consequence their difference is Inserting a complete set of positive energy multi-particle momentum eigenstates |p n y this can be written as where q " pk 2`k4 q{2. As a function of q, this is only non-zero if either pq˘pk 1`k3 q{2q 2 ě m 2 L where m L is the mass of the lightest particle (corresponding to the lightest p n ) or if the pole terms corresponding to single particle intermediate states are subtracted out, then 4m 2 L . In terms of Mandelstam variables this is s ě 4m 2 L or u ě 4m 2 L . The region where this is not satisfied is precisely the Mandelstam triangle and so we conclude that the s-channel and u-channel functions are identical in the analyticity window of the Mandelstam triangle and are elsewhere analytic continuations of each other. The full rigorous proof of this is lengthy [38][39][40] but relies only on the above physical considerations.
Focussing on elastic scattering, m 1 " m 3 and m 2 " m 4 we may choose the Breit coordinate system where e. p " 0 and | e| " 1. In terms of Mandelstam variables we have t "´4 p 2 and E " a p 2`λ2`m2 2 " ps`t{2´m 2 1´m 2 2 q{p a 4m 2 1´t q. Then the s-channel amplitude is Analyticity in s at fixed t ă 4m 2 1 corresponds to analyticity in E at fixed t. From the above expression, to define an analytic continuation we must carefully deal with the convergence of the integral associated with the analytic continuation of the square root a E 2´m2 2`t {4. However, if we focus purely on the high energy regime we may approximate this as Since the integrand vanishes for | x| 2 ą px 0 q 2 and x 0 ă 0, we conclude that the domain of integration is x 0´ e. x ě 0 and so the s-channel scattering amplitude may be extended to an analytic function of E in the upper half complex energy plane since e iEpx 0´ e. xq θpx 0´ e. xq Ñ 0, as x 0´ e. x Ñ 8 , for fixed ImpEq ą 0 .
Similarly at high energies the u-channel amplitude at high energies takes the form where now the integrand has support for x 0´ e. x ď 0 and so may be taken as an analytic function of E in the lower half complex energy plane. Once these results are extended to low energies as well (e.g. [38][39][40]) modulo poles, then using the property that the s-channel and uchannel amplitudes coincide in the Mandelstam region, we infer that the scattering amplitude is analytic in the whole complex E plane, at fixed t modulo poles and the branch cuts along the real axis gapped by the Mandelstam triangle. Since E " ps`t{2´m 2 1´m 2 2 q{p a 4m 2 2´t q this is equivalent to analyticity in s at fixed real t ă 4m 2 2 . The original rigorous proofs [38,39] only applied for t ď 0 or some intermediate positive value [40], however the results of Martin which make additional use of unitarity of the scattering amplitude (that do not immediately follow from the above integral representation) extend these to t ă pm 1`m2 q 2 [31] where m L is the lightest particle mass.

B Crossing Relations from Multispinors
In this section we will derive the crossing formula by calculating tree level scattering amplitudes for arbitrary spin particles, To do this we will make significant use of transversity spinors, and so we first set notation. Throughout we will consider scattering in the xz-plane. Using Lorentz invariance to fix the total 3-momentum to be zero, on-shell conditions to fix the energies in terms of the 3momenta, and overall momentum conservation, one can write the scattering amplitude as a function of just two variables namely the 3-momentum |p| " p and the scattering angle θ.
The four-momentum of each particles is then given by with a respective angle for each particles in the scattering process A`B Ñ C`D: θ 1 " 0 , θ 2 " π , θ 3 " θ and θ 4 " π`θ .

(B.3)
Before giving any explicit amplitudes, it will be useful to derive precisely the transversity spinor states and polarizations. For that we start in the standard helicity basis and convert to transversity.

B.1 Transversity states
The helicity spinorsũ λ that satisfy the Dirac equation r´iC`msũ λ e ip.x " 0 arẽ The helicity anti-spinor statesṽ λ can then be derived by charge conjugation, with the charge conjugation matrix C given by C "´iγ 0 γ 2 and where we work in the standard Dirac convention for the γ matrices.

B.2 Multispinors
Although our interest is in the scattering of particles which are irreducible representations of the Poincaré group, the crossing formula apply equally for scattering of unphysical states which sit in reducible representations 13 , specifically those that are given as tensor products of spin 1/2 representations. This is because all that is relevant is the transformation of states under complex Lorentz transformations. Since it is significantly easier to derive the crossing formula for arbitrary reducible spin states we shall do so here. The wavefunction for such a state of maximal spin 2S will be given by a multispinor with 2S components [33][34][35][36]41] and the associated quantum field for particle A takes the form Hereâ A,τ 1 ...τ 2S pkq annihilates a particle A with transversity string τ 1 . . . τ 2S corresponding to a reducible representation of total transversity τ " τ 1`. . . τ 2S , andâ :Ā ,τ 1 ...τ 2S pkq creates the associated anti-particleĀ and the canonical normalization is such that râ A,τ 1 ...τ 2S pkq,â : Consider an interaction process A`B Ñ C`D in which each particle is represented by a multispinor field. By angular momentum conservation, S 1`S2 " S 3`S4`i nteger. Let us assume for now that the overall integer is positive, so that there are more incoming spinor indices than outgoing. In this case, to construct a scalar interaction we will need to contract incoming indices, which is naturally achieved with the charge conjugation matrix since u τ 1 α pk 1 qC αβ u τ 2 β pk 2 q "v τ 1 pk 1 qu τ 2 pk 2 q (B.15) To begin with, let us consider the case S 3 ě S 1 , S 2 ě S 4 and define L " S 1`S2´S3´S4 . With this choice, we can consider the following interaction Lagrangian 13 Although reducible representations will necessarily contain ghost states, e.g. 1{2 b 1{2 " 3 ' 1(ghost) these states will still have to respect crossing symmetry in the same way and so this need not concern us in deriving the crossing relations.
The s-channel scattering amplitude is schematically given by (we take the initial state in the formâ :

Bâ
: A |0y and the final stateâ : where η DC η DA are the usual statistics factors arising from reordering creation and annihilation operators (η ij "´1 if both particles i and j are fermions and`1 otherwise). More precisely η DC η DA " x0|â CâDâ In writing this short hand, we understand pūpk 3 qupk 1 qq 2S 1´κ´L to mean a product of 2S 1κ´L factors in which each u andū has a distinct transversity, similarly for the other factors. The total transversity for particle A is the sum of all the individual τ 1 's for each element u τ 1 pk 1 q in the string, and similarly for the other particles τ i .
To compute F 5 we needv τ 1 pk 1 qu τ 2 pk 2 q "´2τ 1 1 2m a s´4m 2 δ τ 1 τ 2 (B.37) andv´τ 1 pk 1 qv´τ 2 pk 4 q " 2τ 1˜p Again analytically continuing this into the region s ă 4m 2 along an anticlockwise contour that avoids the right hand branch cut then Finally after some algebra (checking all 2 4 combinations) In the above we have calculated each ratio for a given spinor factor. The full amplitude is a product of such factors, however we have written each factor so that the product is straightforward to take 14 with the result that where τ i now denotes the total transversity which is a sum of all the τ i transversities for each element in the string.
Thus the crossing relation takes the form Although this was derived under the assumption S 3 ě S 1 , S 2 ě S 4 and S 1`S2 ě S 3`S4 , the final result cannot depend on this fact, which can be demonstrated by recomputing for interactions in the opposite case following the same procedure. Focusing on the elastic (for spins but not necessarily transversities) case S 1 " S 3 and S 2 " S 4 this becomes simply Finally for elastic transversities as well τ 1 " τ 3 , τ 2 " τ 4 this is which is the result needed in the main text.

C Explicit Examples
In this Appendix we give some of the simplest scattering amplitudes between spin-0, spin-1/2 and (now irreducible!) spin-1 particles, and show that they behave exactly as expected with regards to their kinematic singularities and crossing properties. We consider different types of four-point interactions and confirm their analyticity and crossing relations.

C.1 Scalar-Scalar
The simplest four-scalar interaction in this context is a λφ 4 interaction, which gives a trivial tree-level scattering amplitude A s ps, t, uq " λ. The analyticity and crossing property are then manifestly trivial, A s ps, t, uq " A t pt, s, uq " A u pu, t, sq . (C.1)

C.2 Scalar-Spinor
Next we turn to scalar-spinor interactions and start with the four-point of the form λψψφφ interaction, where for simplicity we consider scattering between four distinct particles (i.e. the two scalars are distinct -even if they carry the same mass -and so are the two fermions).
‚ ψφ Ñ ψφ scattering: Let us consider the following interaction in the Lagrangian, Then the s channel transversity amplitudes for the scattering process where the angles θ i are given in (B.3).
Analyticity: First we can clearly see that the combinations T˘defined in (2.41) and (2.42) (with ξ " 1 since we are dealing with BF scattering) are explicitly analytic in s, t and u and free of all kinematical singularities as argued in section 2.2. Indeed for the u channel scattering process ψφ Ñ ψφ, Crossing: Next, when it comes to the crossing relation, we can see that the u channel transversity amplitude is identical to the s channel, T u τ 1 0τ 3 0 ps, t, uq " T s τ 1 0τ 3 0 ps, t, uq , (C. 6) and so for any τ 1 "˘1{2 and any τ 3 "˘1{2 the following relation is identically satisfied, which is precisely the crossing relation (B.42) derived in appendix B with in this case, η 1 u " p´1q 2S 3 "´1, e iπ ř τ i " e 2iπτ 1 "´1 and with the angle χ u precisely as in (D. 38) or (2.24) .

C.3 Spinor-Spinor
Next, we may look at spinor-spinor four-point interactions. The simplest one is of the form λpψψq 2 where we consider again distinct spinors more specifically, The scattering process ψ A ψ B Ñ ψ C ψ D has amplitude, In this case we manifestly have η 1 u " 1 and when τ 1 ‰ τ 2 we have ř τ i " 0 (since the amplitude is proportional to δ τ 1 ,τ 3 δ τ 2 ,τ 4 ) so the crossing and analyticity relations are trivially satisfied in that case.
To make things slightly more interesting, we can also consider the following interaction pψ C γ µ ψ A qpψ D γ µ ψ B q, where again we consider all the fields to be distinct even though they carry the same mass and spin. Such an interaction then leads to an s channel transversity amplitude which is non-trivial when tτ 3 , τ 4 u ‰ tτ 1 , τ 2 u. Using the expression of the anti-spinor in terms of the spinors (B.5)ψ D γ µ ψ B "´ψBγ µ ψD, we can easily see that the transversity amplitude in the s and u channels associated to the scattering process ψ A ψ B Ñ ψ C ψ D are simply related as T s τ 1 τ 2 τ 3 τ 4 ps, t, uq " pū τ 3 pθ 3 q.γ µ .u τ 1 pθ 1 qqpū τ 4 pθ 4 q.γ µ .u τ 2 pθ 2 qq "´T u τ 1 τ 2 τ 3 τ 4 ps, t, uq . (C.24) 15 To be more specific, we define ?´u as ?´u " ? s´x, with x " 4m 2´t ą 0. The physical region being defined for s ą 4m 2 , as the limit from the upper half of the complex plane for s. Then defining s " x`ρe iθ , we have ? s´x " ? ρe iθ{2 . Then analytically continuing from θ " 0`to θ " π´, we have ?´u • Starting with the cases where ř τ i " 0, the simplest one is when τ 1 "´τ 2 " τ 3 "´τ 4 . This case we have T s¯˘¯p s, t, uq " u´s 4m 2 , (C. 25) so the associated combination T`given in (2.41) is clearly analytic, while the Tv anishes. Crossing symmetry is also trivial in that case, and satisfies (D.8) with η 1 u " 1, T s¯˘¯p s, t, uq "´T ū˘¯˘p s, t, uq " T ū˘¯˘p u, t, sq . (C.26) • Still considering the case where ř τ i " 0, one can look at τ 1 "´τ 2 "´τ 3 " τ 4 or the crossing equivalent τ 2 Ø τ 4 which simply leads to T s¯¯˘p s, t, uq "´T s˘¯¯p s, t, uq " t 4m 2 . (C.27) The analyticity of the associated combinations This concludes all the different cases one can consider for the scattering amplitude associated with the 4-spinor interaction pψγ u ψq 2 .

C.4 Scalar-Vector
Finally, we shall consider a scalar-vector example. The simplest four-point scalar-vector interaction in this context is A µ where again we consider all particles to be distinct and the two vectors A µ carry the same mass m.
• Starting again with the case where ř τ i " 0, the non-trivial amplitudes are either for τ 1 " τ 3 " 0 (for which T s " T u " 1) or τ 1 "´τ 3 "˘1 (for which T s " T u " t{4m 2 ). In either cases the analyticity and crossing relations are trivially satisfied.
(C. 38) In this case S 3 " 0 and since we are only dealing with bosons, η 1 u " 1. Moreover since T s 1000 " T s 0˘100 " 0, for the relevant cases we always have e iπ ř τ i " 1. These transversity amplitudes therefore satisfying the appropriate crossing relations derived in (B.42).

D Crossing Relations from Lorentz Rotations
To make contact with previous analyses, we now compare the transversity crossing relations derived in Appendix B with the 'historical' approach found in the literature [11,25,27,28].
In these works the crossing relations are given in terms of a complex Lorentz transformation which for the helicity amplitudes takes the form where η t and η u are statistics factors which are difficult to determine [27,28]. Their value depends on the choice of branches for the phase of the arguments of each of the Wigner matrices (which are only periodic in 4π for fermions).
It is apparent from the above equations for the helicity amplitudes that the crossing relations do not relate sign definite amplitudes to other sign definite amplitudes, simply because the Wigner matrices will contain negative signs terms. They are therefore of limited interest when it comes to establishing positivity bounds. In the remainder of this appendix, we shall first describe how to recast the helicity crossing relations (D.1) and (D.2) in the more useful transversity basis (see also section 2.2), and then provide a review of the historical derivation of (D.1) and (D.2).

D.1 Transversity Crossing Relations
As can be seen from equations (D.1) and (D.2) for helicity amplitudes, the crossing relations are highly non-trivial and positivity of the amplitude of the s channel on the right hand cut implies no particular information on the sign of the u channel amplitude to be evaluated on the left hand cut. To be able to derive the relevant positivity bounds, we should therefore work in terms of scattering amplitudes which are (semi)-diagonal under crossing. This is possible by working in the transversity basis. Indeed the Wigner (small) d matrices that enter the crossing relations for definite helicities are related to the Wigner D matrices arising from the rotation operator, D S ab pα, β, γq " e´i aα d S ab pβqe´i bγ and the u S ab defined as are precisely what transforms a helicity state into a transversity state.

D.2 Helicity Crossing Relations
To make connection with older literature [11,25,27,28] we will now summarize the steps in deriving the s´t and from it the s´u crossing relations.

B Ø C Crossing
The scattering amplitude A`B Ñ C`D depends on the 4-momenta of the external states, and can be defined in any reference frame, Similarly as in appendix B, we fix the four-momenta in terms of the 3-momentum and the scattering angle θ, leading to the expressions for the 4-momenta (B.2), with respective angles θ i given in (B.3).
Return to center of mass frame: The amplitude of the new channel A`C ÑB`D is no longer evaluated in the center of mass frame. To remedy this, we perform a Lorentz transformationL given bŷ L k s 0 " k t 0 ,´L k s θs " k t π ,´L k s π " k t θt andL k s θs`π " k t π´θt .
(D. 17) and similarly for the t-channel.
In the scattering plane, there are only three independent Lorentz generators, so any three of these relations uniquely specifiesL (and the fourth is guaranteed by momentum conservation). This transformation brings us to the t-channel center of mass frame, which differs from the s-channel frame by a reversal of the normal to the scattering plane. The explicit form for this Lorentz transformation is well-known, and it can be written [11,25,27] L " B t 0 Rχ 1 pB s 0 q´1 " B t π Rχ 2`B s θs˘´1 " B t θt Rχ 3 pB s π q´1 " B t π´θt Rχ 4`B s π`θs˘´1 (D. 18) where B s θ is a boost along p s θ of magnitude | p s θ |, where the spatial part of p s θ is given in (B.2), i.e. B s θ pm, 0, 0, 0q T " k s θ . Rχ is a rotation of angleχ about the axis perpendicular to the scattering plane withχ 1 "χ 4 " π´χ 2 " π´χ 3 " χ t , (D. 19) where, cos χ t "´s t ?
ST , (D. 20) and care has been taken to ensure that the square roots are well-defined in the s and t physical regions, in which ST " sps´4m 2 q tpt´4m 2 q ą 0, stu ą 0 .

(D.21)
It is then unambiguous that cos χ t ą 0 and sin χ t ă 0, which implies that the range for the angle χ t is 0 ď χ t ď π{2 in the physical s and t regions.
This tensorial Lorentz transformation only determines the angles χ i up to a shift of 2π. Note that when fermions are involved, one should also consider the spinorial Lorentz transform, generated by´1 4 rγ µ , γ ν s, in place of the usual Lorentz generators. However, note that shifting any of the χ i by 2π will only introduce an overall sign, p´1q 2S i , which can be absorbed into the overall statistics prefactor η.
This can be written as (D.2) by relabelling t and u (at this stage they are just complex arguments of a function; note that this does not include changing the subscripts of η t and χ t from t to u), and absorbing the p´1q λ i´λ 1 i into the Wigner matrices. Explicitly, this gives us the relations η u " η BC η CD´ηtˇC ØD¯, That is, the χ u crossing angle is defined in both the s and u physical regions as cos χ u "´s u ?

E Discrete Symmetries of Helicity and Transversity Amplitudes
This appendix documents the properties of the amplitudes under C, P and T. These are operators which act on the Hilbert space, and throughout we will adopt the notation, for such an operator X X : A " xf |T |iy Ñ xXf |T |Xiy " A 1 (E.1) i.e. the amplitude A is mapped under X to another function A 1 , related to A by the relevant operator acting on each of the incoming and outgoing states.

Parity
We define our rest frame helicity states as eigenstates of P P|0λy " η P |0λy (E.2) where η P is a phase which is independent of λ (because P commutes with J) and must be˘1 (because P 2 " 1). Parity commutes with rotations, but not with boosts. Applying a boost in the z direction gives P|p 0 , λy " η P p´1q S`λ |p π ,´λy . (E. 3) The two-particle state then transforms as 18 , P|p θ , λy " η P 12 p´1q S 1´S2`λ |p π`θ ,´λ 1´λ2 y . (E.5) 18 Note that this can also be written as, P|p θ,φ λy " η P e´i πS |p π´θ,π`φ ,´λy (E.4) where the momentum undergoes a standard inversion. We will not use this expression however, because it sends our φ " 0 states to φ " π states, and would require keeping track of the azimuthal phase.

Charge Conjugation
Under charge conjugation, the quantum numbers are conjugated, but the kinematics (momenta and helicities) are unaffected. This means it is particularly trivial to write down the action of C on our states. Introducing a label a for the particle species (i.e. all of the quantum numbers excluding spin and momentum) C|p θφ , λ, ay " η C |p θφ , λ,āy (E. 18) whereā is the antiparticle of a, with inverse charges. η C is an overall phase, which must bȇ 1 (because C 2 " 1) and cannot depend on the helicity (because C commute with J). C acts on each particle in a multi-particle state C|p θφ , λ 1 λ 2 , a 1 a 2 y " η C 12 |p θφ , λ 1 λ 2 ,ā 1ā2 y (E. 19) and does not affect the kinematics or relative phases. On the helicity amplitude, we therefore have a trivial replacement of particles with antiparticles . (E.20) The action on transversity amplitudes is analogous.

CPT
Now it is simply a matter of combining the previous results. For the underlying QFT to be consistent, the amplitudes must respect CPT invariance. The action of CPT on a 1-particle state is CPT|p θ λay " η CP T p´1q S´λ |p θ ,´λ,āy (E. 21) i.e. both P and T effectively invert the momentum, P inverts the helicity, and C replaces particle with antiparticle. η CP T is the product η C η P η T , which we can set to unity. Then, for two particles, CPT|p θ , λ 1 λ 2 , a 1 a 2 y " p´1q S 1˘S2´λ |p θ ,´λ 1´λ2 ,ā 1ā2 y (E. 22) So under CPT, we find As an antiunitary, antilinear operator, CPT invariance requires xCPTβ|S|CPTαy "`xβ|pCPTq : SCPT|αy˘˚"`xβ|S : |αy˘˚" xα|S|βy and so we arrive at, where the sign is`for B`B Ñ B`B or F`F Ñ F`F and´otherwise, and arises from permuting the creation operators. The corresponding transversity condition is T τ 1 τ 2 τ 3 τ 4 "˘p´1q 2S 1`2 S 4 T τ 1 τ 2 τ 3 τ 4 (E.25) The combined sign˘p´1q 2S 1`2 S 4 is`for BB Ñ BB, BF Ñ BF , F B Ñ F B and F F Ñ F F , and´otherwise-i.e. CPT prevents a particle from changing its total spin.
In particular, the Wigner d S matrices can be diagonalized with the u S ab matrices ÿ b,c u S ab d S bc pθqpu S dc q˚" e iaθ δ ad . (F.36) These relations are particularly relevant when deriving the relations between the helicity and transversity amplitudes.