PPositivity Bounds without Boosts
Tanguy Grall and Scott Melville DAMTP, Centre for Mathematical Sciences, University of Cambridge, CB3 0WA, United Kingdom (Dated: February 12, 2021)We derive the first positivity bounds for low-energy Effective Field Theories (EFTs) that are notinvariant under Lorentz boosts. “Positivity bounds” are the low-energy manifestation of certainfundamental properties in the UV—to date they have been used to constrain a wide variety ofEFTs, however since all of the existing bounds require Lorentz invariance they are not directlyapplicable when this symmetry is broken, such as for most cosmological and condensed mattersystems. From the UV axioms of unitarity, causality and locality, we derive an infinite family ofbounds which (derivatives of) the 2 → From the early ages of our universe to present day con-densed matter physics, the natural world favours solu-tions to its fundamental laws which break Lorentz in-variance. For instance, both the inflationary spacetimebackground [1–4] and many-body systems at high den-sity [5–11] effectively provide a preferred reference frame,and low-energy fluctuations in these systems are accu-rately captured by Effective Field Theories (EFTs) whichare not invariant under (linearly realised) Lorentz boosts.The EFT framework has proven remarkably successful,even in systems with broken boosts: by providing a sim-ple parametrisation of the possible low-energy interac-tions (in which coupling constants are fixed by compar-ing with data), it has allowed us to make great progressin analysing our low-energy observations even in caseswhere we do not fully understand the underlying UVphysics (such as inflation).However, care must be taken when adopting an EFTparametrisation, since not all of the EFT parameterspace is equally viable. Over the past two decades, a pow-erful connection between low-energy Lorentz-invariantEFTs and their underlying high-energy UV completions,known as positivity bounds , have been developed [12–22]. Using the causal properties of the 2 → → I. PROPERTIES OF THE AMPLITUDE
Let us begin by listing the properties of A π π → π π , the2 → π , that we will use to derive positivity bounds. (a) Spacetime Symmetry. We will assume that thePoincar´e invariance of the system is broken by a singlepreferred (time-like) direction n µ . This breaks enoughsymmetry (i.e. Lorentz boosts) to capture condensedmatter systems such as fluids and cosmological systems a r X i v : . [ h e p - t h ] F e b such as inflation, but preserves enough symmetry (i.e.spatial rotations and translations in both space and time)to retain a well-defined S -matrix description of the scat-tering between asymptotic states. We will additionallyassume that, in the absence of interactions, the equationof motion for the scalar π at low energies takes the form, ω = c π | k | + m , (1)where c π is a fixed constant speed (which may differfrom the invariant speed c of the broken Lorentz boosts), m is an invariant mass, and we are working in momen-tum space so that i∂ µ acting on π becomes k µ = ( ω, k ),where ω = n µ k µ is the time-like component (i.e. fre-quency) and k are the remaining spatial components (i.e.wavenumbers). The asymptotic states for our scatteringprocess are therefore labelled by their spatial momen-tum, k j , with their frequencies ω j fixed by (1), wherewe adopt a convention in which ω > <
0) for in-coming (outgoing) fluctuations. The low-energy effectiveinteractions are built from both invariant contractions ∂ µ ∂ µ as well as time-like derivatives n µ ∂ µ acting on π ,and a natural basis of kinematic variables is therefore s ij = ( ω i + ω j ) − c π | k i + k j | and ω ij = ω i + ω j . Theseare not all independent, since e.g. time translations sets ω = ω and similarly spatial translations set s = s .A complete basis is given by the five variables [81], s = s , t = s , ω s = ω , ω t = ω , ω u = ω (2)where u = s is fixed by s + t + u = 4 m . Forthe interaction, π π → π π , which we refer to asthe s -channel process with amplitude A π π → π π =: A s ( s, t, ω s , ω t , ω u ), real physical momenta correspond tothe domain s > − t > ω s > (cid:112) s + ( ω u − ω t ) , whichwe refer to as the s -channel region . (b) Crossing. In addition to the s -channel region, thereare five other regions in which all of the momenta canbecome real. These correspond to the 2 → s : π π → π π , t : π ¯ π → ¯ π π , u : π ¯ π → π ¯ π , ¯ s : ¯ π ¯ π → ¯ π ¯ π , ¯ t : ¯ π π → π ¯ π , ¯ u : ¯ π π → ¯ π π , where ¯ π is the charge-conjugate field (= π for a realscalar). For example, the u -channel process corre-sponds to the u -channel region u > − t > ω u > (cid:112) u + ( ω s − ω t ) , and the associated amplitude is A u ( u, t, ω u , ω t , ω s ). Since we are considering the scat-tering of identical scalar fluctuations, all six channels arephysically equivalent. This leads to relations between theamplitude at different kinematics, e.g. [82], A s ( s, t, ω s , ω t , ω u ) = A u ( u, t, ω u , ω t , ω s ) . (3) (c) Unitarity. Unitarity of the S -matrix leads to the well-known optical theorem for scattering amplitudes,2Disc A s = (cid:88) n A π π → n A ∗ π π → n (4)which relates the discontinuity, Disc A s = i ( A s − A ∗ ¯ s ),to a sum over amplitudes for the inelastic processes2 → n . We will require that the underlying high-energyphysics which UV completes the EFT is unitarity—in particular, the optical theorem then guarantees thatDisc A s ≥ π π coincides with π π ) in the UV. (d) Analyticity. Causal interactions correspond to an-alytic response functions [40, 83]. For Lorentz-invariantscattering amplitudes, causality translates into the con-dition that A s ( s, t ) is analytic in the complex s -plane atfixed t (for all Im s (cid:54) = 0). This is a key ingredient in de-riving Lorentz-invariant positivity bounds, used to relatethe EFT amplitude (at small s ) to the underlying UVamplitude (at large s ) via Cauchy’s residue theorem.However, when boosts are broken the amplitude is alsoa function of { ω s , ω t , ω u } and one must specify how theseadditional variables are to be held fixed when analyticallycontinuing into the complex s -plane. For instance, onechoice is to fix the scattering kinematics so that k + k =0, enforcing the so-called centre-of-mass condition ,CM condition: ω s = √ s , ω t = ω u = 0 , (5)since this corresponds to an incoming 2-particle statewhich is rotationally invariant, allowing the optical the-orem (4) to be simplified using the usual partial waveexpansion. (5) is the choice made in previous work [67,77, 78]. However, the resulting amplitude, A (CM) s ( s, t ) = A s ( s, t, √ s, ,
0) is not suitable for positivity argumentsfor two reasons:(i) A (CM) s ( s, t ) is not analytic in s at fixed t due to thisnon-analytic choice of ω s , which can be seen already inperturbation theory [84],(ii) the condition (5) is not preserved under s ↔ u cross-ing (3), A (CM) s ( s, t ) = A u ( u, t, , , √ s ) (cid:54) = A (CM) u ( u, t ),and in fact (5) is mapped to an unphysical kinematics(complex values of momenta).These two obstacles to constructing positivity boundswere pointed out as early as [67], and have thus far pre-vented the construction of robust bounds in EFTs with-out boost invariance.To overcome these issues, in this work we have identi-fied a way to fix the energies which is analytic in s andinvariant under s ↔ u crossing, so that the resulting am-plitude is amenable to positivity arguments. We beginby noting that the conventional proofs of analyticity forLorentz-invariant amplitudes (see e.g. the Appendix of[16]) take place in the frame, k − k = 0, which corre-sponds to enforcing the Breit condition ,Breit condition: (6) ω s + ω u = (cid:112) m − t , ω s − ω u = s − u √ m − t , ω t = 0 , at small masses. This condition fixes the particle en-ergies as analytic functions of s , and is preserved un-der s ↔ u crossing (which swaps s with u and ω s with ω u ) so the resulting amplitude A (Breit) s ( s, t ) enjoys a triv-ial crossing relation, A (Breit) s ( s, t ) = A (Breit) u ( u, t ). ForLorentz-invariant theories, (5) and (6) simply correspondto two different choices of Lorentz frame—one can there-fore move freely from the centre-of-mass frame (in whichunitarity is simplest) to the Breit frame (in which causal-ity and crossing are simplest)—but when boosts are bro-ken, A (CM) s ( s, t ) and A (Breit) s ( s, t ) are very different phys-ical processes (different particle energies { ω s , ω t , ω u } ).For general kinematics, A s ( s, t, ω s , ω t , ω u ), the energiesdo not typically satisfy the Breit condition (6), howeverwe can always parametrise them in the following way:Breit parametrisation: (7) ω s + ω u = 2 M γ, ω s − ω u = s − u M , where ω s and ω u have been written in terms of two newvariables γ and M [85]. Mathematically, (7) fixes the en-ergies in a way which is analytic in s (at fixed t, M, ω t , γ )and invariant under s ↔ u crossing [86], and so the result-ing amplitude ˜ A s ( s, t, M, ω t , γ ) (= A s with energies fixedas in (7)) enjoys all of the good properties of A (Breit) s ( s, t ),in particular a trivial crossing relation,˜ A s ( s, t, M, ω t , γ ) = ˜ A u ( u, t, M, ω t , γ ) . (8)Physically, varying s with { γ, M, ω t } held fixed corre-sponds to changing the interaction energy while main-taining a constant velocity relative to the Breit frame (bycontrast, the velocity of the centre-of-mass will vary with s ). While it is clear that ˜ A s is an analytic function of s in the EFT, we further require that it is also analyticin the UV. In Appendix A, we show that this require-ment is related to the causality condition that operatorscommute outside of the π -cone. (e) Polynomial Boundedness. Finally, we require abound on the high-energy growth of the amplitude:lim s →∞ | ˜ A s ( s, t, M, ω t , γ ) | < s , (9)where s is taken large at fixed { t, M, ω t , γ } . In theLorentz-invariant case, the classic results of Froissart andMartin [87, 88] show that polynomial boundedness of theamplitude, lim s →∞ |A s ( s, t ) | < s , follows from locality (see e.g. [89] for a modern textbook derivation). In Ap-pendix B, we argue that the analogous steps which leadto the Froissart-Martin bound in the Lorentz-invariant case can also be performed without Lorentz boosts (us-ing the spherical wave expansion of [73]), and so we willrefer to property (9) as locality of the UV physics. II. POSITIVITY BOUNDS
Armed with the above properties (a–e), we cannow construct a dispersion relation for the amplitude˜ A s ( s, t, M, ω t , γ ) and derive a number of new positivityrelations which can be used to constrain low-energy EFTsin which boosts are broken.In the complex s -plane (with { t, M, ω t , γ } held fixed),˜ A s ( s ) has the same analytic structure as in a Lorentz-invariant theory—it is analytic everywhere except for thepoles and branch cuts on the real s -axis that are requiredby unitarity. We can therefore follow the standard pro-cedure [12]: evaluate the integral (cid:72) C dµ ˜ A s ( µ ) / ( µ − s ) n +1 first along a contour C which closely encircles the poleat µ = s (giving ∂ ns ˜ A s by Cauchy’s residue theorem),and then along a pair of semicircular contours in the up-per/lower half-plane (closed above/below the real axis)whose radii go to infinity. These are equal by property(d) analyticity, which implies that,1 n ! ∂ ns ˜ A s ( s ) = C ∞ + (cid:90) ∞−∞ dµπ Im ˜ A s ( µ )( µ − s − i(cid:15) ) n +1 (10)where we have suppressed the dependence on { t, M, ω t , γ } and replaced ˜ A s ( µ + i(cid:15) ) − ˜ A s ( µ − i(cid:15) )with 2 i Im ˜ A s ( µ ) using the real analyticity of ˜ A s . C ∞ is the contribution from the arcs at infinity, C ∞ = (cid:72) | µ |→∞ dµπ ˜ A s ( µ ) /µ n +1 , and vanishes by property(e) locality for all n ≥
2. Property (b) crossing can beused to relate the branch cut on the negative real axisto that along the positive real axis, so that there is asingle integral (cid:82) ∞ m − t/ dµ over the kernel, P n ( µ, s ) = Im ˜ A s ( µ )( µ − s ) n +1 − Im ˜ A u ( µ )( u − µ ) n +1 . (11)Following the philosophy of [15, 59] (see also [47]), thestrongest positivity bounds are obtained by subtractingas much of the IR (EFT) information as possible, so thatthe bounds represent only our ignorance of the UV. Thiscorresponds to subtracting the portion of the branch cutup to the scale s b at which the EFT is no longer reliable,˜ A ( n ) s ( s ) := ∂ ns ˜ A s ( s ) n ! − (cid:90) s b m − t/ dµπ P n ( µ, s ) . (12)˜ A ( n ) s is the n th derivative of the EFT amplitude withlow-energy branch cuts removed, and is related to theunderlying UV completion (i.e. Im ˜ A s ( µ ) at large µ > s b )by the dispersion relation,˜ A ( n ) s ( s ) = (cid:90) ∞ s b dµπ P n ( µ, s ) . (13)In the forward limit, t → ω t → A s coincides with Disc ˜ A s , andtherefore using the final property (c) unitarity we estab-lish that P n ( µ, s ) ≥ µ ≥ s b and any even n ≥ | s | < s b is within the EFT’s regime of va-lidity and that M > γ ≥ A (2 N ) s (cid:12)(cid:12) t =0 ω t =0 ≥ N ≥ s , M > γ ≥ ∂ ns A ( s, t =0) > not required any notion of weakcoupling to construct this dispersion relation—the EFTmay be arbitrarily strongly coupled, as long it is able tocapture the amplitude up to some scale s b which is largerthan 2 m − t . At low energies, loop corrections to theEFT amplitude are suppressed by powers of s/ Λ and ω/ Λ, where Λ is the EFT cutoff. In practice, the exactpositivity bound (14) therefore implies that,12 ∂ s ˜ A tree s | s =0 t =0 ≥ π (cid:90) s b m − t dµµ Im ˜ A − loop s | s =0 t =0 (15)providing s b is chosen sufficiently low that higher orderloop corrections can be ignored. We stress that subtract-ing up to s b with M held fixed corresponds to energies ω b / Λ ∼ s b /M Λ, and so the effective cutoff for the EFTamplitude ˜ A s in the complex s plane is set by the prod-uct Λ M . If the theory is weakly coupled, then we cantrust the loop expansion even at energies near the cutoff(since there is an additional small parameter which sup-presses these corrections), and so ω b can be taken closeto Λ, subtracting the entirety of the EFT branch cut (to any desired order in this weak coupling).For illustration we have focussed here on the forwardlimit ω t = 0 and t = 0, which produces the infinite familyof bounds (14) that constrain every ∂ Ns ˜ A s with N ≥ t derivative of A s ( s, t ) was devisedin [15]. We show in Appendix C that, when boosts arebroken, one can similarly go beyond the forward limitand place bounds on not just every even s derivative of˜ A s beyond ∂ s , but also on every t derivative, ( ∂ t ) i , andon every energy derivative of the form (cid:0) ∂ /∂ω ∂ω (cid:1) j .These bounds place highly non-trivial constraints on thelow-energy EFT. In terms of the convenient variables v = s − u and ω − = ω s − ω u , a general low-energy EFTamplitude can be expanded as, A s ( s, t, ω s , ω t , ω u ) = (cid:88) a, b C ab (cid:18) ω , ω , vω − (cid:19) ω a − t b (16)where we have made crossing symmetry (8) manifest.What we have achieved in equation (14) is a bound on ev-ery C N ( ω , ω , M ) | ω = ω except N = 0. Going beyondthe forward limit in Appendix C produces further boundson every symmetric derivative C ( j,j, Ni ( ω , ω , M ) | ω = ω . III. APPLICATIONS
The positivity bound (14) and its perturbative form (15)(as well as the bounds derived in Appendix C) can beapplied to any low-energy EFT with rotation and trans-lation invariance, and must be satisfied if the UV com-pletion is to obey properties (c) unitarity, (d) causalityand (e) locality (as defined above). To provide a con-crete example of the power of these bounds, we will nowfocus on the simple EFT of a single scalar field π , toleading order in derivatives and up to quartic order inthe field (we further assume an approximate shift sym-metry so that the mass and potential interactions aresmall corrections). The action and corresponding 2 → S [ π ] = (cid:90) d x c − π (cid:16) − ( ∂π ) + α Λ ˙ π − α Λ ˙ π ( ∂π ) + β Λ ˙ π − β Λ ˙ π ( ∂π ) + β Λ ( ∂π ) (cid:17) , (17)Λ A s = 2 β ( s + t + u ) + (2 β − α )( s ω s + t ω t + u ω u ) + 24 ω ω ω ω (cid:20) ( β − α α ) − α (cid:18) ω s s + ω t t + ω u u (cid:19)(cid:21) , where ˙ π = n µ ∂ µ π and ( ∂π ) = − ˙ π + c s δ ij ∂ i π∂ j π iscontracted using the effective metric which determinesthe free propagation (1), { α , α , β , β , β } are constantWilson coefficients and the overall factor of the soundspeed c π ensures canonical normalisation [73, 80]. De- spite its simplicity, this action captures the low-energydegree of freedom of a superfluid, and also the scalarfluctuations produced during inflation in the slow-roll de-coupling limit. Positivity Bounds.
The only non-zero ˜ A (2 N ) s in theforward limit at this order is N = 1, which we write as f = Λ ˜ A (2) s | ω t =0 ,t =0 . This is given explicitly by, f ( γ ) = 4 β + 2( β − α ) γ + ( β − α α ) γ − α γ . (18)The simplest positivity bound comes from (14) with noneof the EFT branch cut subtracted, which requires that f ≥ γ ≥ γ = 1 happens to coin-cide with the centre-of-mass frame bound of [67], how-ever we have derived the bound (18) in a different waywhich guarantees the required analyticity and positivityproperties of the amplitude. In [73], it was shown usingperturbative unitarity that γ cannot be made arbitrarilylarge when α (cid:54) = 0, and here (18) provides an alternativedemonstration of this from perturbative analyticity [92].The one-loop contribution to the positivity bound (15)can be evaluated from the tree-level amplitude in (17)using the optical theorem, and is a simple polynomial in s/M when γ = 1 and ω t = t = 0. At large s ,Im ˜ A − loop s = β π s Λ M (cid:18) O (cid:18) M s (cid:19)(cid:19) , (19)where β = (cid:0) β + β − (cid:0) α + α (cid:1) (cid:0) α + α (cid:1)(cid:1) . Sub-tracting this branch cut up to an energy scale ω b = s b / M (cid:29) M gives a loop-improved positivity bound, f | γ =1 ≥ β π ω b Λ (cid:18) O (cid:18) Mω b (cid:19)(cid:19) , (20)which is strictly stronger than the previous bound. Notethat if the EFT is strongly coupled, one cannot take ω b close to Λ without perturbation theory breaking down,and so (20) is only ever a small improvement. Howeverfor weakly coupled theories there is an additional smallcoupling which suppresses higher order corrections evenwhen ω b is taken close to Λ, and in that case (20) can be asignificantly stronger bound, as we will now see explicitlyin the case of a superfluid. Superfluid.
When describing the low-energy fluctua-tions of a superfluid, in which boosts are broken spon-taneously by the preferred spacetime direction n µ , thescalar π corresponds to a Goldstone boson which trans-forms non-linearly under the broken boost symmetry [93], n µ → n µ + b µ , π → π + Λ b µ x ν η µν (21)where b µ n ν η µν = 0 is an infinitesimal space-like shiftand Λ is the energy scale associated with the symmetry-breaking [94]. Note that while the low-energy action(17) depends only on the effective metric Z µν whichdetermines the π propagation, the UV boost symme-try introduces a new metric η µν . They are related by Z µν = c s η µν − (1 − c s ) n µ n ν , where c s = c π /c is the di- mensionless ratio of the low-energy sound speed and theinvariant UV speed preserved by (21). This non-linearsymmetry fixes all but one Wilson coefficient at each or-der, and the coefficients appearing in (17) can all be fixedin terms of { c s , α , β } only, α = 1 − c s c s , β = 32 c s α + (1 − c s ) c s , β = 1 − c s c s . (22)The simplest positivity bound f ≥ β ≥ β min1 ( c s , α ), where, β min1 = 3 α γ α c s (cid:18) − c s − γ (cid:19) − − c s )12 c s γ . (23)Physically, this means that if the low-energy system has asound speed c s and cubic interaction α , then the under-lying UV physics must also produce a sufficiently largequartic interaction β if it is to be consistent (i.e. unitary,causal, local). It is not possible, no matter how ingeniousour UV model-building, to arrange for large effects atquadratic and cubic order in π without also generatinglarge effects at quartic order (unless we sacrifice one ofthe above fundamental properties).Including the one-loop correction to the positivitybound gave (20), which crucially depends non-linearly on β . In particular, there are some values of α and c s forwhich there are no allowed values for β — the improvedbound (20) can only be satisfied if, (cid:18) − c s + 32 α c s (cid:19) ω b Λ + 30 π c s ≥ , (24)which allows us to bound α in terms of c s only—forinstance for sufficiently small c s and large α ∼ /c s ,this bound requires α c s (cid:38) / α = β = 0, the positivitybound (24) can only be satisfied if c s ≤
1, in which caseit requires that new physics must become important ator below the scale ω b = 320 π Λ c s / (1 − c s ) . Inflation.
During inflation, the expanding spacetimebackground spontaneously breaks temporal diffeomor-phisms. Following the seminal work of [1], these timediffeomorphisms can be restored (non-linearly realised)by introducing a single scalar degree of freedom, π . Theresulting low-energy action for scalar and metric fluctu-ations is invariant under a gauged version of (21) and inthe decoupling and subhorizon limits,Decoupling: ω (cid:29) Λ M P , Subhorizon: ω (cid:29) H , (25)where H is the background Hubble rate, the scalar sec-tor of the theory is described by (17) with the tuning(22). In this context, c s is the sound speed of scalar per-turbations produced during inflation, and α ( β ) con-trols the primordial bi- (tri-)spectrum. The Planck [95]measurements of the primordial power spectrum fix thesymmetry-breaking scale to be Λ = (58 . ± . H (at68% confidence), and the observed limits on the primor-dial bispectrum (i.e. on { c s , α } ) are shown in Figure 1.The positivity bounds derived here provide a qual-itatively new way to analyse these primordial signals.For instance, equation (23) for β min1 is a robust lowerbound on the size of the trispectrum g NL , given measure-ments of the bispectrum f NL and c s . Taking for illustra-tion the central values of the Planck 2018 contours [95], f eqNL ≈ −
26 and f orNL ≈ −
38 ( c s ≈ .
031 and α ≈ − f ≥ g NL ≥
12 400 ( β ≥ +6 . × )[96]. Comparing this with the currently allowed observa-tional range, − . × < g NL < +1 . × , we see thatpositivity arguments can be used to rule out over 70% ofthe observationally allowed parameter space! Of course,since these numbers are sensitive to the relatively largeuncertainties in c s and α , they should be taken only asan illustration: the point is that, in future, as our mea-surements of the bispectrum improve, positivity boundswill provide a new way to “bootstrap” information aboutthe trispectrum (without the need to directly detect g NL ).Finally, our bound (24) can be used to place a restric-tion on the values of α and c s which are compatible withthe UV properties listed in Section I (namely unitarity,causality and locality). In order to compare this boundwith Planck observations, we must specify a value for ω b ,the energy scale up to which we can reliably subtract theIm ˜ A s from the dispersion relation. Rather than sim-ply fix a value for ω b , instead we use (24) to define ascale ω NP ( c s , α ), which is the largest possible energy atwhich (24) can be satisfied. This is the scale at whicheither New Physics (new degrees of freedom beyond π )or Non-Perturbative effects (e.g. a resummation of π loops) must become important, if we are to preserve uni-tarity, causality and locality. We show in Figure 1 howthis scale compares with current Planck constraints on { c s , α } . The majority of the 68% confidence region re-quires new physics within an order of magnitude of H . Closing Remarks.
In summary, we have constructedthe first set of positivity bounds—constraints placed ona low-energy Effective Field Theory by the consistency ofits underlying UV completion—when boosts are broken.The key to overcoming previous obstructions, namely alack of analyticity and crossing symmetry, lay in iden-tifying a new parametrisation for the amplitude—in ourBreit variables, the 2 → A s ( s ) en-joys the same structure as a Lorentz-invariant amplitude,and hence its dispersion relation can be used to bridge be-tween the EFT and the UV completion. The new positiv-ity bounds, which constrain all s , t and ω ω derivativesof the EFT amplitude, can be applied to a much widerclass of systems than their Lorentz-invariant predeces-sors, and in particular are applicable to condensed matterand cosmology. As an example, we have shown how thesebounds can be used to infer the scale of new physics whichmust be present in the early Universe, given the current Planck constraints on primordial non-Gaussianity. In ad-dition to placing the positivity arguments on a more rig-orous footing and deriving a number of new bounds, byovercoming the lack of crossing symmetry in [67] we havebeen able to subtract the 1-loop EFT branch cut for thefirst time using the optical theorem (this is not possibleusing centre-of-mass kinematics since then the u -channelcut is not physical). The resulting bound (20) is not onlyquantitatively stronger, but it gives a qualitatively newconstraint on α in terms of c s alone (24), providing arobust upper bound on the scale of new physics which isinsensitive to the trispectrum.Although we focussed on examples in which Lorentzboosts were broken spontaneously, a UV Lorentz symme-try was not required to translate the properties (a-e) intobounds on the EFT. Our positivity bounds should stillapply in cases where the UV completion does not recoverLorentz invariance providing that property (d) still holds(this could be the case when there is still some maximumspeed limiting the transfer of information, but which maynot be the same for all observers). At the other extreme,in the event that the new physics which UV completesthe EFT at its Λ completely restores Lorentz invariance,then at weak coupling one could simply subtract all of theLorentz-violating effects in the EFT and assume that theremaining ˜ A (2 n ) s exhibits Lorentz-invariance: this couldsalvage kinematics such as the centre-of-mass frame, andpotentially produce further positivity bounds. We havenot pursued this strategy here because assuming sym-metry restoration exactly at the EFT cutoff is a ratherstrong assumption about the UV completion (certainlymuch more restrictive than the assumptions of unitarity,causality and locality) [97].The results presented here open up several avenues forfuture work. First and foremost, there are a number ofsystems in which Lorentz boosts are broken and yet thefree propagation at low energies has the form ω = c π k ,and our bounds can be immediately applied to the cor-responding low-energy EFTs (see e.g. [98–105])—in par-ticular for the EFT of inflation, where we have exploredbut the simplest positivity bound, in future a system-atic application of all of the available bounds should becarried out. Second, in covariant theories which possessboth a Lorentz-invariant and a Lorentz-breaking back-ground (e.g. (cid:104) φ (cid:105) = 0 and (cid:104) φ (cid:105) ∝ t ), it would be inter-esting to compare the bounds one would infer from thestandard Lorentz-invariant arguments about the Lorentz-invariant background from the bounds derived here aboutthe Lorentz-breaking background [106]. Third, on theconceptual side, while we have focussed on the scatteringof scalar fields, the approach put forward here (first iden-tifying the correct frame in which analyticity and crossingare manifest) can also be extended to the scattering offields of arbitrary spin. Fourth, in the context of infla-tion, in order to go beyond the subhorizon limit and fullyaccount for the expanding FLRW background, one needsto move away from scattering amplitudes (which are nolonger well-defined)—a natural candidate with which to - - - - - c s α ω NP < < < < Λ - -
100 0 100 200 - - f NLeq f NLor c s = ω N P < H < H < H FIG. 1. Assuming that the UV completion is unitary, causal and local, our positivity bound (24) sets a scale ω NP beyond whichinflation can no longer be single-field and weakly coupled. The red shaded regions correspond to ω NP < H, H, H andΛ = 58 . H , and the grey contours show the 68% ,
95% and 99 .
7% confidence intervals from the Planck 2018 observations of theequilateral and orthogonal bispectrum ( f eqNL and f orNL ), which are determined by the Wilson coefficients ( c s and α ) appearingin the EFT of Inflation (17). For the majority of the 68% confidence interval, the leading-order EFT requires the inclusion ofnew UV physics beyond the scalar fluctuations π (or a non-perturbative treatment of π loops) at an energies below 10 H . connect UV and IR are the wavefunction coefficients , andrecent work has begun to shed light on how these objectsare constrained by unitarity and locality [107–113]. ACKNOWLEDGMENTS
We would like to thank Simon Caron-Huot, Claudia deRham, Daniel Green, Johannes Noller, Enrico Pajer,David Stefanyszyn and Andrew Tolley for useful discus-sions and comments, and also the organisers of the work-shop
Cosmology 2021: the Rise of Field Theory . TG issupported by the Cambridge Trust, and SM by an UKRIStephen Hawking Fellowship. This work was partiallysupported by STFC consolidated grant ST/P000681/1.
Appendix A: Microcausality and Analyticity
Analyticity of the scattering amplitude is fundamentallytied to the principle of causality. For Lorentz-invarianttheories this connection is well-understood and analytic-ity of the Lorentz-invariant scattering amplitude A s ( s, t )at fixed t is well-established [114–121].When boosts are spontaneously broken, far less isknown about the analytic structure of A s ( s, t, ω s , ω t , ω u ).In this appendix, we will show how the requirement of mi-crocausality is closely related to the analyticity property(d) described in Section I. In particular, we will show that whenever the UV interactions preserve/contract thecausal cone (i.e. c UV ≤ c π ), then A s ( s, t, ω s , ω t , ω u ) isguaranteed to be analytic for certain domains in { s, ω s − ω u } at fixed { t, ω s + ω u , ω t } . When UV interactions ex-pand the caucal cone (i.e. c UV > c π ), then the conven-tional microcausality arguments are not strong enough toguarantee analyticity, but we can nonetheless show that A s ( s, t, ω s , ω t , ω u ) is analytic in the { ω s , ω t , ω u } at fixed { s, t } at any order in perturbation theory. Microcausality.
In a quantum theory, causality (sig-nals cannot propagate faster than some speed c max ) isencoded in the operator relation,[ O ( x ) , O ( y )] = 0 when c max | x − y | < | x − y | , (A1)which is the familiar statement that operators (observ-ables) at space-like separated points commute. In orderto connect this requirement to the amplitude, we definethe position-space current operator, J A ( x ) ≡ i δ ˆ Sδπ A ( x ) ˆ S † , (A2)so that the amplitude can be written as, A s = (cid:90) d x e − i k µ x µ Θ( x ) (cid:104) k | (cid:104) J ¯4 (cid:16) x (cid:17) , J (cid:16) − x (cid:17)(cid:105) | k (cid:105) , (A3)where k µ = k µ + k µ (where now we use conventionsin which ω >
0, in contrast to the main text). When themomenta k and k labelling the one-particle states areboth real, then causality (A1) requires that this matrixelement vanishes unless x is time-like, i.e. the integra-tion region can be restricted to c max x > | x | . If k µ isanalytically continued into the complex plane (with k and k fixed), this integral representation for the ampli-tude converges whenever Im [ k µ x µ ] < x µ in this range. The strongest requirement correspondsto ˆ x aligning with Im k , and the amplitude is thereforeanalytic for any complexification of k µ which obeys,Im ω ≥ c max | Im k | > , (A4)as a consequence of (A1). This is the connection betweencausality (i.e. space-like separated operators must com-mute) and the analyticity of the amplitude for complexvalues of momenta which we will exploit—when writ-ten in terms of the variables { s, t, u, ω s , ω t , ω u } , (A4) de-scribes a domain in which A s must be analytic for causalinteractions. With Boosts.
Before discussing theories without boostinvariance, let us briefly review how (A4) is used to es-tablish analyticity in s at fixed t in the Lorentz-invariantcase. For a Lorentz-invariant scattering amplitude, onecan use boosts to go to the Breit frame, in which k = 0(i.e. k = − k ). Conservation of momentum then re-quires that k · k = 0 and so k is orthogonal toboth k and k . For instance, using rotations to align k t = k − k along the z -axis, the particle momenta canbe parametrised in this frame by, k = k t , k = k − k t , k = − k t , (A5)where k = k + k − k is fixed by momentum conserva-tion. The two independent variables, k t = | k − k | and k = | k + k | , are related to the Mandelstam variablesby c k t = − t and ω (cid:112) m − t = s + t − m − m ,where ω = (cid:112) c k − m − t is the energy appearingin (A4). These are often known as Breit coordinates , andhave the advantage that analytically continuing k intothe complex plane with k t held fixed (i.e. the complex s plane at fixed t ) deforms particles 2 and 4 into thecomplex plane while keeping k and k real. Analysingthe precise domain of analyticity is an involved prob-lem, and in particular requires careful treatment of thesquare roots in k ( s, t ). However, when considering t , m and m to be much smaller than | s | , we have that ω ≈ s/ (cid:112) m − t ≈ ck , and so the condition (A4)becomes simply Im s > c max ≤ c (i.e. when themaximum speed at which signals can propagate is con-tained within the lightcone which determines the propa- gation of free fields). Causality, for a Lorentz-invariantamplitude, therefore requires that A s be an analytic func-tion of s in the upper half-plane when t (and the masses)are fixed to real physical values. Without Boosts.
Without invariance under Lorentzboosts, it is no longer possible to set k = 0 in general.Instead, using only invariance under rotations, we canexpress the momenta in terms of three magnitudes andthree angles, k t = k t ˆ e , , k = k ˆ e θ , , k = k ˆ e θ ,φ , (A6)where ˆ e θ,φ = (sin θ cos φ, sin θ sin φ, cos θ ) is a spatial unitvector. To express { k t , k , k , θ , θ , φ } compactlyin terms of { s, t, u, ω s , ω t , ω u } , we will now assume thatthe masses are negligible. Then the variables ω t and t correspond to specifying k t and fixing the angles θ and θ in terms of the magnitudes k and k , c π k t = (cid:113) ω t − t , cos θ = (cid:118)(cid:117)(cid:117)(cid:116) − tc π k − tω t , cos θ = (cid:118)(cid:117)(cid:117)(cid:116) − tc π k − tω t . (A7)The variables ω s and ω u then correspond to k and k , c π k = (cid:112) ( ω s + ω u ) + t , c π k = (cid:112) ( ω s − ω u ) + t , (A8)and finally s determines the remaining parameter φ ,cos φ = − s + t − tc π k t (cid:0) ω s − ω u (cid:1) c π k k sin θ sin θ . (A9)The advantage of this parametrisation is that k and k can be kept real by holding { t, ω t , ω s + ω u } fixed,while condition (A4) then places a restriction on the com-plex values which { s, ω s − ω u } may take. Just as in theLorentz-invariant case, precisely characterising this re-gion requires careful treatment of square roots of the form (cid:112) ( ω s − ω u ) + t − ω t . To illustrate the key features, wewill consider both t and ω t to be fixed at values muchsmaller than s , ( ω s − ω u ) and ( ω s + ω u ) . This simplifies(A4) to, c π Im [ ω s − ω u ] ≥ c max (cid:113) Im [ a ] + Im [ b ] > a ≈ ω t √− t sω s + ω u , b ≈ (cid:112) ( ω s − ω u ) − a . (A11)Once t, ω t and ω s + ω u are fixed to real physical values,then (A10) specifies the domain analyticity for ω s − ω u and s . In particular, note that when ( ω s − ω u ) (cid:29) s is large (or when ω t →
0) this becomes simplyIm [ ω s − ω u ] > c max ≤ c π [122], and so it is ω s − ω u which plays the role of s from the Lorentz-invariant case. For our purposes, fixing M and γ to realvalues gives, a ≈ ω t √− t sγM , b ≈ s M (cid:115) ω t γ t (A12)and therefore (A10) becomes Im s > c max ≤ c π . When making s complex at fixed (real) values of { t, M, ω t , γ } , causality (A1) requires that A s is analyticin the upper half-plane, just as in the Lorentz-invariantcase. Crucially however, the upper bound required of c max is set by the the π -cone (i.e. the free propagation(1)). If one imagines a UV completion in which thereis some maximum speed c UV , then causality in that UVcompletion is only strong enough to guarantee analyticityof the amplitude if c UV ≤ c π . In cases where c UV > c π ,a more sophisticated argument, which accounts for howthe operators behave in the region between the two cones(i.e. 1 /c UV < x / | x | ≤ /c π , where (A1) no longer guar-antees commutation) is needed. Crossing.
To establish analyticity in the full complex s -plane, one needs to repeat the above argument in the u -channel. To see this, consider the u -channel amplitudecorresponding to the process π ¯ π → π ¯ π : A u = (cid:90) d x e i k µ x µ Θ( x ) (cid:104) k | (cid:104) J (cid:16) x (cid:17) , J ¯4 (cid:16) − x (cid:17)(cid:105) | k (cid:105) , = − (cid:90) d x e − i k µ x µ Θ( − x ) (cid:104) k | (cid:104) J ¯4 (cid:16) x (cid:17) , J (cid:16) − x (cid:17)(cid:105) | k (cid:105) . (A13)Analytically continuing k µ into the complex plane, theabove integral representation converges if Im[ k µ x µ ] < − c max x > | x | , i.e. providing that − Im ω ≥ c max | Im k | > . (A14)In the Lorentz invariant setting, for t , m and m muchsmaller than | s | , this implies Im s < A s is re-quired to be analytic also in the lower half-plane s -plane.When boosts are broken the condition (A14) reads − c π Im [ ω s − ω u ] ≥ c max (cid:113) Im [ a ] + Im [ b ] > , (A15)which, in the limit of large ( ω s − ω u ) (or ω t →
0) be-comes Im [ ω s − ω u ] < c max ≤ c π ), so that A s isanalytic in the whole ( ω s − ω u ) plane (except for possi-ble branch cuts on the real axis Im [ ω s − ω u ] = 0, whichare fixed by unitarity in the physical regions). In ourboosted Breit coordinates, (A15) describes analyticity inthe lower-half of the complex s plane (Im s <
0) with { s, t, M, ω t , γ } fixed. ˜ A s is therefore analytic in the wholecomplex s plane, modulo possible poles/branch cuts onthe real s axis (which are fixed by unitarity), whenever c max ≤ c π . Since this standard causality argument is notstrong enough to prove analyticity non-perturbativelywhen c max > c π , for that case we turn to perturbativearguments. Perturbation Theory.
Within perturbation theory,there can be no poles or branch cuts in the energies { ω s , ω t , ω u } at fixed s and t . To see this, note that wecan always write ω i = n µ p i,µ and then strip-off from theamplitudes all factors of n µ , A L -loop s = n µ . . . n µ j A L -loop µ ...µ j . (A16)The remaining A L-loop µ ...µ j is then a series of L loop inte-grals over covariant integrands (with respect to the Z µν which determines the free propagator), which can onlyever produce non-analyticities in the invariants s and t . An Example.
We illustrate this point by explicitly com-puting the 1-loop diagram generated by two ˙ π inter-actions. The s -channel diagram can be stripped as in(A16), leaving behind, A µ ...µ = 4! β ω ω ω ω (cid:90) d k (2 π ) q µ q µ k µ k µ ( k − m ) ( q − m )where q µ = p µ + p µ + k µ . The t and u -channel diagramsare the same up to a permutation of the external legs. Indimensional regularisation, this integral evaluates to, A s = 31280 β π Λ F s ( ω j ) log ( − s ) (A17)+ t, u permutations + local counter-termswhere F s ( ω j ) = ω ω ω ω (cid:0) ω s + s ω s + s (cid:1) is ananalytic function of the energies. As claimed above, theonly branch cuts in this expression are in the invariantvariables s, t and u . This feature, that the energies ap-pear only as an analytic factor in each term, is generic.The explicit computation (A17) also provides a usefulcross-check of our earlier equation (19). Using the opticaltheorem and the tree level amplitude (17), the one-loopimaginary part from the β interaction is found to be,Im ˜ A − loop s | γ =1 ω t =0 t =0 = 3640 β π s Λ M + 384 M s + 256 M s + 24 M s + s ˜Λ , (A18)where ˜Λ = M Λ. This agrees with the discontinuityacross the logarithms in (A17).
Appendix B: Spherical Wave Expansion
In this appendix we briefly review properties of thespherical wave expansion for amplitudes without Lorentz0boosts introduced in [73], the positivity properties ofDisc A s and the Froissart-Martin bound (9).The key idea is to express the amplitude in a partic-ular channel (e.g. the s channel) not in terms of linearmomentum eigenstates, | k (cid:105) = | ω ϑ ϕ (cid:105) , but in termsof angular momentum eigenstates | ω (cid:96) m (cid:105) (where m is the angular momentum conjugate to ϕ and (cid:96) repre-sents the total angular momentum, (cid:96) ( (cid:96) + 1)). Mathe-matically, this amounts to the decomposition, A s ( k , k , k , k ) = (cid:88) (cid:96) ,(cid:96) m ,m f m (cid:96) (ˆ k ) f m ∗ (cid:96) (ˆ k ) a m m (cid:96) (cid:96) ( ω s , s )(B1)where f m(cid:96) are a complete set of functions on the sphere.This makes rotational invariance manifest and in particu-lar greatly simplifies the optical theorem (4). Comparingwith the corresponding spherical wave expansion for the¯ s -channel gives,Disc A s = (cid:88) (cid:96) ,(cid:96) m ,m f m (cid:96) (ˆ k ) f m ∗ (cid:96) (ˆ k )Disc a m m (cid:96) (cid:96) (B2)where Disc a m m (cid:96) (cid:96) = i (cid:0) a m m (cid:96) (cid:96) − a m m ∗ (cid:96) (cid:96) (cid:1) . The opticaltheorem then takes the form,Disc a m m (cid:96) (cid:96) = (cid:88) n z n(cid:96) m z n ∗ (cid:96) m (B3)where z n(cid:96) m = (cid:82) d ˆ k ¯ f m (cid:96) (ˆ k ) A π π → n (where ¯ f m(cid:96) sat-isfies (cid:82) d ˆ p ¯ f m(cid:96) (ˆ p ) f m (cid:48) (cid:96) (cid:48) (ˆ p ) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) ) and the sum runsover all intermediate n particle state (and includes inte-grals over their kinematics). Unitarity therefore requiresthat Disc a m m (cid:96) (cid:96) is a positive definite matrix in (cid:96) m and (cid:96) m . Forward Limit.
Taking the forward limit (ˆ k = ˆ k )in (B2) therefore immediately establishes Disc A s ≥ f m(cid:96) can bereplaced with their moduli, A s | ˆ k =ˆ k = (cid:88) (cid:96) ,(cid:96) m ,m | f m (cid:96) (ˆ k ) | | f m (cid:96) (ˆ k ) | a m m (cid:96) (cid:96) (B4)which proves that Disc A s and Im A s coincide in theforward limit. To prove the positivity of Disc A s be-yond the forward limit, one needs to specify how theangles in ˆ k and ˆ k are related to the analytic variables { s, t, ω s , ω t , ω u } . Angular Variables.
Once the total incoming energy ω s and momentum k s = | k + k | are fixed, the al-lowed frequencies for ω lie on an ellipse with semimajoraxis ω s / ρ s = c π k s /ω s . These can beparametrised with either the angle θ between k and k s (the true anomaly of the ellipse), or alternatively by theangle ϑ (the eccentric anomaly of the ellipse), which for small masses are related by [123], ω = ω s − ρ s − ρ s cos θ = ω s − ρ s cos ϑ ) . (B5)The spherical wave expansion (B1) can then be written, A s ( s, t, ω s , ω t , ω u ) (B6)= 64 π (cid:88) (cid:96) (cid:96) m m Y m (cid:96) ( ϑ , ϕ ) Y m ∗ (cid:96) ( ϑ , ϕ ) a m m (cid:96) (cid:96) ( ω s , ρ s )as described in [73] [124], where ϕ j is the azimuthal angleof ˆ k j when k s is aligned with the z -axis. We refer thereader there for the explicit a m m (cid:96) (cid:96) from the EFT (17).These angles are related to the analytic variables by, ω t = − k s ϑ − cos ϑ ) , t = − s − cos ϑ ) ω u = − k s ϑ + cos ϑ ) . (B7)where cos ϑ = cos ϑ cos ϑ + sin ϑ sin ϑ cos( ϕ − ϕ )is the relative angle between k and k . Each choiceof { ω s , ρ s , ϑ , ϑ , ϕ − ϕ } corresponds to a unique { s, t, ω s , ω t , ω u } in the s -channel region. Symmetries.
The spherical wave coefficients in (B6) areconstrained by various symmetries. For example, despitethe centre-of-mass motion k s breaking two of the rota-tional symmetries, there is one remaining rotation (rota-tions about k s ) which guarantees that A s depends onlyon the relative azimuthal angle ϕ − ϕ . Furthermore, aspatial parity transformation corresponds to ϕ j → π − ϕ j ,and a time inversion corresponds to sending A s → A ¯ s ,which has the analogous spherical wave expansion withthe roles of k k and k k exchanged. Finally, since weare scattering identical scalar fields, whether we use k and k or k and k to define the partial waves shouldmake no difference, and consequently (B6) should be in-variant under replacing cos ϑ with cos ϑ = − cos ϑ andcos ϑ with cos ϑ = − cos ϑ . Altogether, these symme-tries require,Rotational symmetry: a m m (cid:96) (cid:96) = 0 unless m = m Bose symmetry: a m m (cid:96) (cid:96) = 0 unless (cid:96) + (cid:96) evenParity: a m m (cid:96) (cid:96) = a − m − m (cid:96) (cid:96) Time Reversal: a m m (cid:96) (cid:96) = a m m (cid:96) (cid:96) (B8) Elastic Unitarity Bound.
Since every term in (B3)is positive, we can write an inequality involving only the n = 2 term, Disc a m m (cid:96) (cid:96) ≥ (cid:88) Jm a m m(cid:96) J a m m ∗ (cid:96) J (B9)1This is now a non-linear bound which the 2 → v m(cid:96) , we can resolve theidentity as δ JJ (cid:48) = ˆ v m ∗ J ˆ v m (cid:48) J (cid:48) + ... and write (B9) as,Im (cid:88) (cid:96) ,(cid:96) m ,m ˆ v m (cid:96) a m m (cid:96) (cid:96) ˆ v m ∗ (cid:96) ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) (cid:96) ,(cid:96) m ,m ˆ v m (cid:96) a m m (cid:96) (cid:96) ˆ v m ∗ (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (B10)Since | z | ≥ Im z for any complex z , and the unit vectorˆ v m(cid:96) can be written as a general vector v m(cid:96) divided by itsnorm, then (B10) implies an upper bound on the spher-ical wave coefficients, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) (cid:96) ,(cid:96) m ,m v m (cid:96) a m m (cid:96) (cid:96) v m ∗ (cid:96) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:88) (cid:96)m (cid:12)(cid:12) v m(cid:96) (cid:12)(cid:12) (B11)for any complex vector, v m(cid:96) . This is the analogue of theunitarity bound | a (cid:96) | < Convergence and Boundedness.
The partial waveexpansion can also be used to derive the Froissart-Martinbound on the high energy growth of the amplitude inthe absence of Lorentz boosts. This proceeds analo-gously to the Lorentz invariant proof, see for instance[89]. Since the large (cid:96) behaviour of the spherical har-monics is Y m(cid:96) ( θ, φ ) ∼ e (cid:96) | Im θ | , convergence of the spheri-cal wave expansion (B6) requires that the spherical wavesfall off at large (cid:96) like,lim (cid:96) ,(cid:96) →∞ a m m (cid:96) (cid:96) ∼ e − (cid:96) η − (cid:96) η , (B12)for some η j ( ω s , ρ s ) ≥ | Im ϑ j | . This allowed domain ofthe η j defines a region analogous to the Lehmann ellipse[125] in the Lorentz invariant case. Using the definitionof ϑ j above, at large s we have, cos ϑ = 1+ z + /s + ... andcos ϑ = 1+ z − /s + ... , where z ± = 4 M (2 M ( γ − ± ω t ) − t are held fixed [126]. Consequently, convergence of thespherical wave expansion up to some fixed thresholds z th ± requires,lim (cid:96) ,(cid:96) →∞ s →∞ a m m (cid:96) (cid:96) ∼ e − ( (cid:96) √ z th+ + (cid:96) √ z th − ) / √ s . (B13)Further assuming polynomial boundedness (i.e. thatDisc A s does not grow faster than s n for some n ), thismeans that the sums in (cid:96) and (cid:96) can be truncated ata maximum (cid:96) ∗± ∼ (cid:113) s/z th ± log( s n ) (since the neglectedterms provide an exponentially small correction). Thismeans that at large s (with t, M, ω t , γ held fixed),lim s →∞ A s ∼ π (cid:96) ∗ + (cid:88) (cid:96) =0 (cid:96) ∗− (cid:88) (cid:96) =0 (cid:112) (cid:96) + 1 (cid:112) (cid:96) + 1 a (cid:96) (cid:96) (B14) where we have used that Y m(cid:96) = δ m √ (cid:96) + 1 / √ π as ϑ →
0. From the unitarity bound (B11) (with v (cid:96) chosenas √ (cid:96) + 1 if (cid:96) ≤ (cid:96) ∗ = max( (cid:96) ∗ + , (cid:96) ∗− ) and 0 otherwise),we have that lim s →∞ A s (cid:46) (cid:80) (cid:96) ∗ (cid:96) =0 (2 (cid:96) ∗ + 1) ∼ (cid:96) ∗ , andtherefore, lim s →∞ A s (cid:46) s log s (B15)when t, M, ω t , γ are fixed. This coincides with theusual Froissart-Martin bound which constrains Lorentz-invariant amplitudes, i.e. the presence of a symmetry-breaking direction n µ does not affect the very high energy(small-scale) behaviour of the amplitude, which remainsbounded as if fully Poincar´e-invariant. Other Channels.
Above we have focussed on the s -channel, in which s > − t > | ω u ± ω t | < (cid:112) ω s − s .When we consider the analytically continued amplitude A s ( s, t, ω s , ω t , ω u ) for values outside of this region, wecan cross over to other channels. While (B6) can only beused in the s -channel region, there is an analogous partialwave expansion for each of the other five channels. Forinstance, for the u -channel, in which u > − t > | ω s ± ω t | < (cid:112) ω u − u , we can write, A s ( s, t, ω s , ω t , ω u ) = A u ( u, t, ω u , ω t , ω s ) (B16)= 64 π (cid:88) (cid:96) (cid:96) m m Y m (cid:96) ( ϑ u , ϕ u ) Y m ∗ (cid:96) ( ϑ u , ϕ u ) b m m (cid:96) (cid:96) ( ω s , ρ s )using the crossing (3), where ϑ uj are the eccentric anglesof k j relative to k u = k − k , ϕ uj are the correspondingazimuthal angles, and b m m (cid:96) (cid:96) are the spherical waves forthe u -channel process (and are equivalent to a m m (cid:96) (cid:96) whenscattering identical scalars). There is a spherical waveexpansion for each of the six channels, and to which thepreceding arguments apply. Examples.
In order to illustrate the properties of thespherical wave coefficients described above, we providehere the a m m (cid:96) (cid:96) for two simple interaction terms. Weconcentrate on quartic interactions, for which there arefinitely many non-zero spherical wave coefficients [127],and in particular a β ˙ π interaction (which appears in(17) at leading order) as well as a β t ˙ π ¨ π interaction(which is the first non-trivial, non time reversal invariantinteraction at lowest order in derivatives). The ampli-tudes corresponding to each interaction are: A [ β ] = 4! β (cid:89) i =1 ω i , A [ β t ] = 3 ω s ω t ω u (cid:89) i =1 ω i , (B17)2and the corresponding spherical wave coefficients are: a [ β ] = ω s π (3 − ρ s ) , (B18) a [ β ] = a [ β ] = ω s π ρ s ( − ρ s )48 √ , (B19) a [ β ] = ω s π ρ s , (B20)and, a [ β t ] = − a [ β t ] = ω s π ρ s ( − ρ s + 30 ρ s − √ , (B21) a [ β t ] = − a [ β t ] = ω s π ρ s (3 − ρ s )13440 , (B22) a [ β t ] = − a [ β t ] = − ω s π ρ s √ , (B23)where ρ s = 1 − s/ω s . Notice that these respect the rele-vant symmetries (B8) we mentioned earlier. In particularthe spherical waves for the β t interaction, which is oddunder time reversal, are anti -symmetric under (cid:96) ↔ (cid:96) (unlike the β spherical waves which are symmetric, sincethe ˙ π interaction is even under time reversal). Appendix C: More Positivity Bounds
In the main text we have derived the forward limit posi-tivity bounds (14). In this Appendix, we describe a strat-egy which can be used to generate a further infinite fam-ily of bounds on the t and ω ω derivatives of the EFTamplitude. Beyond the Forward Limit.
In the Lorentz-invariantcase, positivity of Disc A s ( s, t ) can be extended beyondthe forward limit to include any number of t derivativesby exploiting properties of the partial wave expansion[15, 56]. To go beyond the forward limit when boosts arebroken, the partial wave expansion must be replaced withthe spherical wave expansion recently derived in [73], andwe have proven in Appendix B that the optical theorem(4) implies, (cid:18) ∂∂t (cid:19) i (cid:18) ∂ ∂ω ∂ω (cid:19) j Disc A s (cid:12)(cid:12)(cid:12)(cid:12) t =0 ω t =0 ≥ , (C1)for any integer i and j and for any forward-limit s -channelkinematics. Note that the ω ω derivative can be writtenas ∂ ω u − ∂ ω t and so these bounds correspond to goingbeyond the function evaluated at t = 0 and ω t = 0 toinclude arbitrary derivatives ( ∂ t ) i and ( ∂ ω t ) j at thesepoints. Dispersion Relation.
The goal is then to use a dis-persion relation to connect the more general positivityproperties (C1) of the UV theory with the low-energy EFT amplitude. As discussed in the main text, this re-quires specifying which variables will be held fixed as oneanalytically continues into the complex s plane. Ratherthan the boosted Breit coordinates { γ, M } from equation(7), in this Appendix we will use the variables { E s , E u } ,defined by, ω s = s + 4 M M + E s , ω u = u + 4 M M + E u (C2)where M is a fixed mass scale. Once the amplitude iswritten in terms of these variables, which we denote byˆ A s ( s, t, E s , ω t , E u ) (= A s ( s, t, ω s , ω t , ω u ) with ω s and ω u replaced by (C2)), crossing (3) becomes,ˆ A s ( s, t, E s , ω t , E u ) = ˆ A u ( u, t, E u , ω t , E u ) , (C3)and so the kernel which appears in the dispersion relation(13) is now given by, P n ( µ, s ) = Im ˆ A s ( µ, t, E s , ω t , E u )( µ − s ) n +1 − Im ˆ A u ( µ, t, E u , ω t , E s )( u − µ ) n +1 . (C4)Despite not being manifestly crossing symmetric, it ispossible to keep { E s , E u } fixed so that both branch cutshave physical kinematics—this corresponds to satisfyingthe condition [128], s E s + E u | E s − E u | > M (2 M + E s + E u ) . (C5)This condition is always satisfied for sufficiently large s , and in particular when E s = E u it is always satis-fied. The reason for the change of notation is purelypedagogical—since { E s , E u } are not manifestly crossingsymmetric, they will better highlight some of the chal-lenges in going beyond the forward limit (and how toovercome them). ∂ t Bound.
In the Lorentz-invariant case, taking t -derivatives of the dispersion relation was shown in [15]to produce a second infinite series of bounds. In thatcase, although ∂ t Im A s is positive, the ∂ t also acts onthe 1 / ( µ − u ) n +1 of the crossed branch cut, and thisgives a negative contribution to the dispersion relationfor ∂ t ∂ ns A s . The remedy is to compensate for this neg-ative term by adding a sufficiently positive lower deriva-tive, which produces bounds such as, (cid:18) ∂ t + 2 n + 1 s b (cid:19) ∂ ns A s ≥ , (C6)in the Lorentz-invariant case [129].In our case, we no longer have Lorentz boosts and thedispersion relation is now given by (13) and (C4). Once ω s and ω u have been replaced by explicit functions of t , a3simple t derivative of Disc ˆ A s is no longer positive, since ∂ t ˆ A s = (cid:18) ∂ t − M ∂ ω u (cid:19) A s (C7)and ∂ ω u of Disc A s is not sign definite. Since the disper-sion relation involves both ˆ A s and ˆ A u , it is not possibleto compensate for this ∂ ω u simultaneously on both cuts.However, the integral of this term with respect to E u ispositive. In fact, if we define, I [ ˆ A s ] = (cid:90) E s dE (cid:48) s (cid:90) E u dE (cid:48) u ˆ A s ( s, t, E (cid:48) s , ω t , E (cid:48) u ) (C8)then we have that, ∂ E s Disc I [ ˆ A s ] ≥ ∂ E u Disc I [ ˆ A s ] ≥ ω t = t = 0, which follows from (C1) with i = j = 0,and also that, (cid:18) ∂ t + 14 M ∂ E u (cid:19) Disc I [ ˆ A s ] ≥ , (C10)when ω t = t = 0, thanks to (C1) with i = 1, j = 0. Inte-grating the dispersion relation with respect to E s and E u therefore provides the analogue of the Lorentz-invariantbound (C6) when boosts are broken, (cid:16) ∂ t + n +1 s b + M ∂ E s + M ∂ E u (cid:17) I [ ˆ A (2 N ) s ] (cid:12)(cid:12) t =0 ω t =0 ≥ N ≥ E s , E u must be physical on bothcuts (C5). To recap, I [ ˜ A (2 n ) s ] is the EFT scattering am-plitude A s ( s, t, ω s , ω t , ω u ) with ω s and ω u replaced by E s and E u as in (C2) and then a portion of the branch cut(up to a scale s b at which the EFT remains reliable) sub-tracted as in (12) and finally integrated over E s and E u as in (C8). This object is straightforward to compute inpractice, and we have demonstrated that the new posi-tivity bound (C11) may be used to diagnose whether aLorentz invariant, unitarity, causal, local UV completionof the EFT is possible.While it is straightforward to substitute the dispersionrelation for ˆ A s into (C11) and use (C9) and (C10) toconfirm that it is indeed positive, let us now re-derive(C11) using a more systematic approach which can bereadily generalised to higher derivatives. From Positivity to Monotonicity.
To systematicallygo to higher derivatives/integrals of ˆ A s , it is helpful to in-troduce the indefinite integrals, ˆ A ab ( E s , E u ), which obey, ∂ aE s ∂ bE u ˆ A ab ( E s , E u ) = ˆ A s ( s, t, E s , ω t , E u ) . (C12)Since the original Disc ˆ A s was positive for all E s and E u , then the first integrals Disc ˆ A and Disc ˆ A are monotonic in E s and E u respectively. This means that the combinations, I ( E s , δ s ; E u ) = ˆ A ( E s + δ s , E u ) − ˆ A ( E s + δ s , E u ) I ( E s ; E u , δ u ) = ˆ A ( E s , E u + δ u ) − ˆ A ( E s , E u + δ u )(C13)have Disc I ≥ I ≥ δ s ≥
0. Thedouble integral Disc ˆ A ( E s , E u ) is “doubly monotonic”,in that the following quantity, I ( E s , δ s ; E u , δ u ) = ˆ A ( E s + δ s , E u + δ u ) + ˆ A ( E s , E u ) − ˆ A ( E s + δ s , E u ) − ˆ A ( E s , E u + δ u )(C14)has Disc I ≥ δ s ≥ δ u ≥
0. Note that I (0 , E s ; 0 , E u ) is nothing more than the I [ A s ] definedearlier. This is useful because it makes transparent theaction of each derivative, ∂ δ s I ( E s , δ s ; E u , δ u ) = I ( E s + δ s ; E u , δ u ) ∂ E s I ( E s , δ s ; E u , δ u )= I ( E s + δ s ; E u , δ u ) − I ( E s ; E u , δ u ) . (C15)Similarly, since (cid:0) ∂ t + M ∂ E u (cid:1) Disc ˆ A s ( s, t, E s , ω t , E u )is positive for all E s and E u , then when we take twointegrals the resulting (cid:0) ∂ t + M ∂ E u (cid:1) Disc ˆ A ( E s , E u ) is“doubly monotonic”, so we have that, (cid:18) ∂ t + 14 M ∂ E u (cid:19) Disc I ( E s , δ s ; E u , δ u ) ≥ N ! ∂ Ns I ( E s , δ s ; E u , δ u )= (cid:90) dµπ (cid:20) Im I ( E s , δ s ; E u , δ u )( µ − s ) N +1 + Im I ( E u , δ u ; E s , δ s )( µ − u ) N +1 (cid:21) (C17)is that the E u derivative does not affect both branchcuts in the same way. In particular, from (C15) we seethat ∂ E u acting on the Disc I ( E u , δ (cid:48) u E s , δ s ) is not signdefinite. In order to have positive t derivatives on bothcuts, we need to use a crossing symmetric derivative ∂ E s + ∂ E u , and in order for the ∂ E s ( ∂ E u ) derivative on the right(left) cut to be positive we need to compensate for thenegative term appearing in (C15). That is to say, it isonly the combination, (cid:18) ∂ t + 14 M ∂ E s + 14 M ∂ E u (cid:19) I + I + I ≥ , (C18)which has positive Disc on both cuts (i.e. for both argu-ments ( E s , δ s ; E u , δ u ) and ( E u , δ u ; E s , δ s )). This can be4written more succinctly as, (cid:18) ∂ t + 14 M ∂ δ (cid:19) Disc I ≥ . (C19)where ∂ δ = ∂ δ s + ∂ δ u .Finally, defining D t = ∂ t + n +1 s b just like in theLorentz-invariant case (C6), so that, D t (cid:34) Im ˆ A s ( µ )( µ − u ) n +1 (cid:35) ≥ ∂ t Im ˆ A s ( µ )( µ − u ) n +1 (C20)for all µ ≥ s b when u = 0, then we arrive at the first t derivative bound (C11), (cid:18) D t + 14 M ∂ δ (cid:19) I (2 N )11 ( E s , δ s ; E u , δ u ) ≥ . (C21)where we have evaluated the amplitude at s = t = ω t =0. This bound constrains ∂ t ∂ Ns of the EFT amplitude.The positivity bounds (14) and (C11) follow from uni-tarity (C1) in the UV with j = 0 and i = 0 or i = 1.The construction we have described above generalisesstraightforwardly to any i or j , and in general requirestaking further energy integrals of the amplitude. We willnow illustrate this by deriving positivity bounds from(C1) with i = 2 , j = 0 and i = 0 , j = 1. From Monotonicity to Convexity.
Taking two E s derivatives, the indefinite integral ˜ A ( E s , E u ) is now a convex function, and so Disc ˆ A is only positive in thecombination, I ( E s , δ s , δ s ; E u ) = ˆ A ( E s + δ s , E u + δ u ) + ˆ A ( E s , E u ) − ˆ A ( E s + δ s , E u ) − ˆ A ( E s , E u + δ u ) , (C22)and similarly I ( E s ; E u , δ u , δ u ) is the analogous convexcombination of ˆ A ( E s , E u ). Taking two E s and one E u integral, ˆ A ( E s , E u ) is “convex-monotonic”, and so wemust use the combination, I ( E s , δ s , δ s ; E u , δ u ) = (cid:90) E s + δ s E s dE (cid:48) s (cid:90) E (cid:48) s + δ s E (cid:48) s dE s (cid:90) E u + δ u E u dE (cid:48) u ˜ A s ( s, t, E (cid:48)(cid:48) s , ω t , E (cid:48) u )(C23)which is a linear combination of ˆ A with 8 different ar-guments, in order to use Disc I ≥
0. Finally, taking twointegrals of E s and E u , ˆ A ( E s , E u ) is “doubly convex”,and the appropriate combination to take is, I ( E s , δ s , δ s ; E u , δ u , δ u )= (cid:90) ¯ E s + δ s ¯ E s d ¯ E (cid:48) s (cid:90) ¯ E (cid:48) s + δ s ¯ E (cid:48) s dE s (cid:90) ¯ E u + δ u ¯ E s d ¯ E (cid:48) u (cid:90) ¯ E (cid:48) u + δ u ¯ E (cid:48) u dE u ˜ A s . (C24) which can be written as a sum over ˆ A with 16 differ-ent arguments. Again, these I ab combinations are conve-nient because they have positive discontinuities and theirderivatives are related straightforwardly, e.g.( ∂ δ u − ∂ E u ) ∂ δ s I ( E s , δ s , δ s ; E u , δ u , δ u )= I ( E s + δ s , δ s ; E u , δ u ) (C25)These are the natural building blocks with which to con-struct the ∂ t and ∂ /∂ω ∂ω positivity bounds. ∂ t Bound.
The positivity property (C1) with i = 2 and j = 0 corresponds to, (cid:18) ∂ t + 14 M ∂ E u (cid:19) Disc I ≥ , (C26)but as with the first t -derivative this second derivativecannot be applied directly to a dispersion relation like(C17) for I since it is not symmetric in E s ↔ E u .One can instead begin with the na¨ıve symmetrisation of(C26), ( ∂ t + M ∂ E s + M ∂ E u ) I , and then add lower or-der I , I , etc. in order to compensate for the negativeterms which are generated on the left- and right-handcuts. One arrives at the following combination, (cid:18) ∂ t + 14 M ∂ δ (cid:19) (cid:18) ∂ t + 14 M ∂ δ (cid:19) Disc I ≥ . (C27)which is positive on both cuts (i.e. for both arguments( E s , δ s , δ s ; E u , δ u , δ u ) and ( E u , δ u , δ u ; E s , δ s , δ s )),where ∂ δ = ∂ δ s + ∂ δ u .To convert this into a constraint on the EFT ampli-tude, one must again promote the ∂ t derivatives as in(C20). This produces the positivity bound, (cid:18) D t + 14 M ∂ δ (cid:19) (cid:18) D t + 14 M ∂ δ (cid:19) I (2 N )22 ≥ . (C28)which constrains ∂ t ∂ Ns of the EFT amplitude. ∂ /∂ω ω Bound.
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Our assumption that the disper-sion relation for π (i.e. the free theory) has an acciden-tal Lorentz invariance (transformations which preserve Z µν ) is what allows the use of relativistic Mandelstam-like variables—in cases where the dispersion relationis far from linear, there would be little motivation forforming s and t combinations and in that case the non-relativistic problem is best treated in terms of the ener- gies directly (c.f. the Kramers-Kronig relations).[82] Note that in the Lorentz-invariant case it is not neces-sary to distinguish between u and ¯ u , so crossing π ↔ π or π ↔ π lead to the same relation for the ampli-tude. In contrast, (3) corresponds to crossing π ↔ π ,and had we instead performed π ↔ π we would havefound, A s ( s, t, ω s , ω t , ω u ) = A ¯ u ( u, t, − ω u , − ω t , − ω s ),which is related to A u by a time reversal.[83] R. J. Eden, P. V. Landshoff, D. I. Olive, and J. C. Polk-inghorne, S -matrix theory of strong interactions’ (Cam-bridge University Press, 1966).[84] Explicitly, any interaction with an odd number of n µ ∂ µ derivatives may contain ω n +1 s = s n √ s which has asquare-root branch cut. Without an additional symme-try (e.g. time-reversal invariance) to enforce that ω s appears only in even powers, in general the choice ofkinematics (5) does not produce an analytic A ( s ) whenboosts are broken.[85] When ω t = t = 0, the physical s -channel region nowcorresponds to s > m , γ ≥ M > ω s = f s ( s, t ) and ω u = f u ( s, t ), and these would havethe desired analyticity and crossing properties provid-ing that f s and f u are analytic in s at fixed t and obey f s ( s, t ) ≥ (cid:112) s + f u ( s, t ) and f u ( u, t ) ≥ (cid:112) s + f s ( u, t )for all real s greater than s b (where the branch cutbegins). However, we have found that it is the linearchoice (7) which allows the closest connection betweenanalyticity/causality and boundedness/locality (see Ap-pendices A and B).[87] A. Martin, Phys. Rev. , 1432 (1963).[88] M. Froissart, Phys. Rev. , 1053 (1961).[89] V. Gribov, The theory of complex angular momenta:Gribov lectures on theoretical physics , Cambridge Mono-graphs on Mathematical Physics (Cambridge UniversityPress, 2007).[90] Note that since we are neglecting the mass, the differ-ent channels appear to collide when t → s -channel, s > ω s > √ s + ω u , becomes the com-plement of the u -channel, s < ω s < √ s + ω u ). Inprinciple these are always regulated by the small mass(which is necessary to establish the Froissart bound (9)),and in practice once part of the branch cut up to s b hasbeen subtracted there is always a gap through which todeform the integration contour.[91] Note that the forward limit should be taken with ω t → first , and then t →
0. This ensures that there areno IR issues with the t -channel pole, which ∼ ω t /t inderivatively coupled theories.[92] Physically, the presence of a ˙ π interaction allows for aloop-level exchange process which dominates the tree-level amplitude at large centre-of-mass velocities—thismay also be related to the recent observation that suchan interaction cannot be consistently coupled to gravityon a Minkowski background [74].[93] In this context π also transforms non-linearly undertime translations, in which case any small breaking ofits shift symmetry also corresponds to a small breakingof time translation invariance (see e.g. [4]).[94] In this context Λ is often referred to as f π by analogywith the pion decay constant.[95] Y. Akrami et al. (Planck), (2018), arXiv:1807.06205[astro-ph.CO].[96] This corresponds to setting γ = 1 in (23). This bound can be improved by using a larger value for γ , but onemust ensure that it corresponds to kinematics which areboth subhorizon (25) and within the resolving power ofthe EFT—see e.g. the cutoffs identified in [73].[97] For instance, the EFT could be UV completed beforeLorentz invariance is restored, e.g. with a change in thepropagation (1) [77] or with additional non-Lorentz in-variant degrees of freedom [100].[98] A. Nicolis and F. Piazza, JHEP , 025 (2012),arXiv:1112.5174 [hep-th].[99] A. Nicolis, R. Penco, F. Piazza, and R. A. Rosen, JHEP , 055 (2013), arXiv:1306.1240 [hep-th].[100] S. Endlich, A. Nicolis, and R. Penco, Phys. Rev. D ,065006 (2014), arXiv:1311.6491 [hep-th].[101] A. Nicolis, R. Penco, F. Piazza, and R. Rattazzi, JHEP , 155 (2015), arXiv:1501.03845 [hep-th].[102] L. V. Delacretaz, T. Noumi, and L. Senatore, JCAP , 034 (2017), arXiv:1512.04100 [hep-th].[103] E. Pajer and D. Stefanyszyn, JHEP , 008 (2019),arXiv:1812.05133 [hep-th].[104] L. Alberte and A. Nicolis, JHEP , 076 (2020),arXiv:2001.06024 [hep-th].[105] T. Grall, S. Jazayeri, and D. Stefanyszyn, JHEP ,097 (2020), arXiv:2005.12937 [hep-th].[106] A good case study would be galileid theories [101]—the broken phase of Galileon EFTs—since around theMinkowksi background the Galileon (and Weakly Bro-ken Galileon) is known not to satisfy positivity bounds[19].[107] N. Arkani-Hamed, D. Baumann, H. Lee, and G. L.Pimentel, (2018), arXiv:1811.00024 [hep-th].[108] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, andG. L. Pimentel, JHEP , 204 (2020), arXiv:1910.14051[hep-th].[109] D. Baumann, C. Duaso Pueyo, and A. Joyce, AAPPSBull. , 2 (2020).[110] D. Baumann, C. Duaso Pueyo, A. Joyce, H. Lee, andG. L. Pimentel, (2020), arXiv:2005.04234 [hep-th].[111] S. C´espedes, A.-C. Davis, and S. Melville, JHEP ,012 (2021), arXiv:2009.07874 [hep-th].[112] H. Goodhew, S. Jazayeri, and E. Pajer, (2020),arXiv:2009.02898 [hep-th].[113] E. Pajer, (2020), arXiv:2010.12818 [hep-th].[114] H. J. Bremermann, R. Oehme, and J. G. Taylor, Phys.Rev. , 2178 (1958).[115] N. N. Bogoliubov, D. V. Shirkov, and S. Chomet, In-troduction to the theory of quantized fields , Vol. 59 (In-terscience New York, 1959).[116] A. Martin, Phys. Rev. , 1432 (1963).[117] K. Hepp, Helvetica Physica Acta (Switzerland)
Vol: 37 (1964).[118] J. Bros, H. Epstein, and V. J. Glaser, Nuovo Cim. , 1265 (1964).[119] Y. S. Jin and A. Martin, Phys. Rev. , B1375 (1964).[120] A. Martin, Nuovo Cim. A42 , 930 (1965).[121] G. Mahoux and A. Martin, Phys. Rev. , 2140 (1968).[122] When c π < c , then we are allowing for UV interac-tions which expand the π -cone ( x = | x | /c π ) to theMinkowski lightcone ( x = | x | /c ). Analyticity is moredifficult to prove non-perturbatively in that case sincethe usual arguments rely on the assumption that inter-actions do not change the causal support of the freetheory.[123] At energies which are comparable to the mass of π , (B5)becomes, ω j − ρ s k j − ρ s = ω s − ρ s cos ψ j ) , (C30)where k j = (cid:113) ω j − m j . This can be inverted forcos ψ j straightforwardly, although now cos ψ + cos ψ no longer vanishes (it is instead proportional to themass) and so the Bose symmetry condition (B8) is moreinvolved—it related a m m (cid:96) (cid:96) to an sum over other otherspherical waves, which is analogous to how crossing af-fects massive spinning particles (see [130, 131] and morerecently [16]).[124] Although note that [73] uses the true angles θ j , whereashere we define the spherical waves (B6) using the eccen-tric angles ϑ j .[125] H. Lehmann, Nuovo Cim. , 579 (1958).[126] Note that we are focussing on the large (cid:96) behaviour ofthe spherical wave expansion and neglecting the depen-dence on m . It is straightforward to retain the m depen-dence (and use the large s expansion of cos( ϕ − ϕ ))and this does not affect our final conclusion.[127] In contrast, cubic interactions—which generate ex-change contributions to 2 → t - and u -channel poles. For further details on how tocompute these we refer the reader to [73].[128] When E s or E u is negative, (C5) must be supplementedby √ s > M + 2 √− E s M and √ s > M + 2 √− E u M ,otherwise there is a interval of s near 4 M for which | k s | = √ ω s − s becomes imaginary.[129] Note we have not set s = 2 m − t/ s - u crossing to combine the Im ˜ A s and Im ˜ A u appearing in(11), and hence there is a factor of 2 difference between(C6) and the bound of [15].[130] T. L. Trueman and G. C. Wick, Annals Phys. , 322(1964).[131] G. Cohen-Tannoudji, A. Morel, and H. Navelet, Annalsof Physics46