The Boostless Bootstrap and BCFW Momentum Shifts
TThe Boostless Bootstrap and BCFW Momentum Shifts
David Stefanyszyn ∗ and Jakub Supe(cid:32)l † Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
In a recent paper [1], three-particle interactions without invariance under Lorentz boosts were con-strained by demanding that they yield tree-level four-particle scattering amplitudes with singularitiesas dictated by unitarity and locality. In this brief note, we show how to obtain an independent ver-ification and consistency check of these boostless bootstrap results using BCFW momentum shifts.We point out that the constructibility criterion, related to the behaviour of the deformed amplitudeat infinite BCFW parameter z , is not strictly necessary to obtain non-trivial constraints for thethree-particle interactions. I. INTRODUCTION
The on-shell approach to computing scattering am-plitudes had led to tremendous advances in ourunderstanding of gauge theory and gravity [2–4]. Wein-berg’s seminal papers from the 60’s [5, 6] showed thatthe combination of Poincar´e invariance and unitarityuniquely picks out Maxwell’s equations as the descrip-tion of a massless spin-1 particle (photon) and Einstein’sequations as the description of a massless spin-2 particle(graviton), at tree-level. In addition, charge conservationand the equivalence principle follow from consistency ofscattering amplitudes involving photons and gravitons,respectively.Since then, there have been many attempts to constrainor bootstrap interactions using a purely on-shell ap-proach, by invoking the principles of unitarity, localityand causality. One of these attempts is the techniqueknown as BCFW deformations [7], in which some of theexternal momenta are deformed by a complex param-eter z . Consideration of the analytic structure of theamplitude as a function of z often imposes non-trivialconstraints on interactions [8], as we will explain in detailin Section II. Constraints can also be derived in anotherway [9–11], without deforming the amplitude, but ratherby demanding consistent factorisation. Here one simplywrites down the most general form of the tree-levelfour-particle amplitude, and demands that poles onlyarise due to intermediate (exchanged) particles goingon-shell, with the corresponding residues given by aproduct of on-shell three-particle amplitudes.Notable results of this bootstrap programme include theobservations that long-range forces cannot be mediatedby particles with spin ≥
3, Yang-Mills (YM) is theunique theory of multiple massless spin-1 particles,General Relativity (GR) is the unique low-energy theoryof a massless spin-2 particle, and the existence of ∗ [email protected] † [email protected] massless spin-3 / [8, 10]. Thison-shell approach shows that the low-energy proper-ties of Poincar´e invariant theories are virtually inevitable.Given such spectacular success, attention has recentlyturned to applying similar methods to spatial correlationfunctions which are the fundamental observables incosmology, see [12–28] and references therein. Thesecosmological correlators live on the late-time boundaryof an approximate de Sitter spacetime and encode detailsabout the bulk spacetime through their dependenceon spatial momenta. A key ingredient in this cosmo-logical bootstrap is the fact that correlators containflat-space scattering amplitudes in the residues of theirtotal-energy poles [29, 30] (for a detailed proof see [28]).These amplitudes form part of the theoretical datarequired to bootstrap the corresponding correlators,and therefore the cosmological bootstrap programmerequires us to have a solid understanding of flat-spacescattering amplitudes.Much effort has so far focused on correlators fixed byde Sitter symmetries (or conformal symmetries on theboundary). Intuitively, taking the cosmological boot-strap programme beyond exact de Sitter symmetries inorder to construct inflationary correlators, requires tak-ing the S -matrix bootstrap beyond exact Poincar´e sym-metries. Motivated by this, a recent paper [1] derivedthe singular parts of tree-level four-particle amplitudeswhere the free propagators are assumed to be masslessand Poincar´e invariant, i.e. all dispersion relations arelinear with each particle propagating at the same speed,but with interactions that are allowed to break Lorentzboosts. Within this set-up, analytically continued three-particle amplitudes are fixed by the helicities of the ex-ternal particles up to an almost arbitrary function oftheir energies and these functions are then constrained bydemanding consistent factorisation, but without makinguse of BCFW shifts. The most interesting results from One can also show that there is an upper limit on the number ofspin-3 / a r X i v : . [ h e p - t h ] S e p this boostless bootstrap are [1]: • When there is a graviton in the spectrum, all three-particle interactions must reduce to their Poincar´einvariant form, even those that do not involve thegraviton itself . Universal coupling of gravity to allother particles is then recovered. • Low-energy self-interactions of a photon must van-ish. The leading allowed operator has six deriva-tives and is therefore mass dimension-9. • There exists at least one large class of boost-breaking theories involving scalars, spin-1 / boost-breaking masslessQED ). The corresponding Lagrangians containgeneralised, boost-breaking gauge redundancies.In the Poincar´e invariant cases, all the results derived byimposing consistent factorisation have also been reachedvia the BCFW formalism. Likewise, in this paper weshow that the results of [1] can be derived using BCFWshifts as a tool to automate consistent factorisation andtherefore provide a neat consistency check. In Section IIwe briefly introduce the spinor helicity formalism that wewill use throughout, present general three-particle am-plitudes for boost-breaking theories, and briefly reviewBCFW momentum shifts. In Section III we argue thatBCFW shifts remain a useful tool for non-constructibletheories by making a distinction between accessible and inaccessible singularities. In this case, while the four-particle amplitudes are not completely fixed by the three-particle ones, we can still use BCFW shifts to constrainthe latter. We illustrate this in Section IV for boost-breaking theories of self-interacting spin- S particles. Weend the paper with some concluding remarks. II. SPINOR HELICITY FORMALISM ANDBCFW DEFORMATIONS
We work in four spacetime dimensions and use the spinorhelicity formalism to present amplitudes in a compactform. In this formalism, a complex null four-momentum p µ is represented as a product of two-component spinorsas p α ˙ α = σ µα ˙ α p µ = λ α ˜ λ ˙ α , (1)where σ µα ˙ α are the Pauli matrices and the undotted anddotted indices transform in the spinor representations ofthe Lorentz group i.e. (1 / ,
0) and (0 , /
2) respectively. Related results based on field-theoretic methods have been de-rived in [31–33]. See also [34] where Lorentz boosts arise in thepresence of soft gravitons. The inverse of this equation is p µ = (¯ σ µ ) ˙ αα p α ˙ α and so theenergy of a particle in terms of the spinors is E = (¯ σ ) ˙ αα λ α ˜ λ ˙ α . Throughout we follow the conventions of [35] and we as-sume that each particle satisfies p µ p µ = E − p = 0on-shell. When boosts are broken, amplitudes are con-structed from SO (3) invariant quantities rather than SO (1 ,
3) invariant ones. In [1], it was shown that suchthree-particle amplitudes are functions of the followingobjects: • “angle” brackets: (cid:104) ij (cid:105) = (cid:15) αβ λ ( i ) α λ ( j ) β , • “square” brackets: [ ij ] = (cid:15) ˙ α ˙ β ˜ λ ( i )˙ α ˜ λ ( j )˙ β , • energies: E i .Here latin indices label the external particles: i, j =1 , ,
3. We remind the reader that the spinors are not grassmanian and therefore the angle and square brack-ets are anti-symmetric. By demanding that the ampli-tudes scale in the appropriate way under helicity trans-formations, on-shell, non-perturbative three-particle am-plitudes take the form [1] A = (cid:40) (cid:104) (cid:105) d (cid:104) (cid:105) d (cid:104) (cid:105) d F H ( E , E , E ) , h < , [12] − d [23] − d [31] − d F AH ( E , E , E ) , h > , (2)where d i = 2 h i − h , h i is the helicity of the i th particleand h is the sum of the helicities (if h = 0, the ampli-tude can be a sum of the two expressions.) The presenceof functions F H , F AH (which depend on the helicities)reflects the fact that Lorentz boosts are no longer an as-sumed symmetry. The Poincar´e invariant amplitudes arerecovered when these functions are constant. As energyis conserved, we will often consider the F ’s as functionsof two arguments only. As an example, the three-particleamplitude for three incoming gravitons, two with nega-tive helicity and one with positive helicity, is A (1 − , − , +2 ) = (cid:18) (cid:104) (cid:105) (cid:104) (cid:105)(cid:104) (cid:105) (cid:19) F H ( E , E ) . (3)In this case, Bose symmetry dictates that F H ( E , E ) = F H ( E , E ).Now, to be consistent with unitarity and locality, a tree-level four-particle amplitude must factorise into a prod-uct of on-shell three-particle amplitudes on each of itspoles. Poles correspond to exchanged particles going on-shell and consistent factorisation dictates that, for exam-ple, lim s → ( s A ) = A (1 , , − I ) × A (3 , , I ) , (4)where s = ( p + p ) is the propagator of the exchangedparticle which is labelled by I . Analogous relations hold The superscripts refer to holomorphic and anti-holomorphic kine-matic configurations [8]. when t = ( p + p ) → u = ( p + p ) →
0. Forfuture reference, in terms of the spinors these Mandel-stam variables are given by s = (cid:104) (cid:105) [12] = (cid:104) (cid:105) [34], t = (cid:104) (cid:105) [13] = (cid:104) (cid:105) [24] and u = (cid:104) (cid:105) [14] = (cid:104) (cid:105) [23].Requiring the amplitude to factorise correctly on eachpole is often highly non-trivial [1, 8–11] since, for exam-ple, the s -channel residue can contain poles in t and u which then need to be interpreted as propagation of aparticle in those channels. This is beautifully illustratedfor multiple spin-1 particles where consistency in eachchannel requires the coupling constants to satisfy theJacobi identity.In [8], Benincasa and Cachazo elegantly used BCFWshifts [7] to formally assess this consistency for a num-ber of tree-level four-particle amplitudes. The simplestBCFW shift takes two particles i and j and deforms theirenergies and momenta according to λ ( i ) ( z ) = λ ( i ) + zλ ( j ) , ˜ λ ( j ) ( z ) = ˜ λ ( j ) − z ˜ λ ( i ) , (5)with all other spinors kept fixed. This choice pre-serves the on-shell conditions for both particles as wellas energy-momentum conservation. The deformed am-plitude A ( i,j )4 ( z ) is a rational function of the complex pa-rameter z and thus can be fully deduced from knowledgeof its poles, residues and behaviour at infinity. It takesthe form A ( i,j )4 ( z ) = (cid:88) n res z = z n A ( i,j )4 ( z ) z − z n + B ( i,j ) ( z ) , (6)where the boundary term B ( i,j ) ( z ) is regular in the entirecomplex plane. For four-particle amplitudes, only twopoles can be reached by a given deformation (since p i + p j is independent of z ) and as we remarked above, the cor-responding residues are evaluated from the three-particleamplitudes alone. For example, summing over the possi-ble helicities of the exchanged particle, for A (1 , ( z ) thetwo poles that can be reached are z t and z u and we have A (1 , ( z ) = (cid:88) h I A (ˆ1 , , − ˆ I ) A (ˆ2 , , ˆ I ) t ( z )+ (cid:88) h I A (ˆ1 , , − ˆ I ) A (ˆ2 , , ˆ I ) u ( z )+ B (1 , ( z ) , (7)where a hat indicates that the particle has had one of itsspinors deformed and evaluated at the appropriate pole.We have t ( z ) = ( p ( z ) + p ) = (cid:104) (cid:105) [13] + z (cid:104) (cid:105) [13] and u ( z ) = ( p ( z ) + p ) = (cid:104) (cid:105) [14] + z (cid:104) (cid:105) [14] and thereforethe locations of the poles are z t = − (cid:104) (cid:105)(cid:104) (cid:105) , z u = − (cid:104) (cid:105)(cid:104) (cid:105) . (8) A ( i,j )4 (0) corresponds to the amplitude for unshifted mo-menta, and then constraints on three-particle couplings can be derived by demanding that distinct A ( i,j )4 (0) co-incide at z = 0 [8] i.e. A ( i,j )4 (0) = A ( k,l )4 (0) ∀ i, j, k, l. (9)This is the four-particle test . III. CONSTRUCTIBILITY CRITERION
In its original formulation, the above described methodis reserved for constructible theories for which B ( i,j ) ( z )vanishes. In this case the singular parts of undeformedamplitudes can be compared with one another and thefull four-particle amplitude is determined by the three-particle ones. Since B ( i,j ) ( z ) is regular, it is sufficient toprove that the amplitude tends to zero as z → ∞ . Thisis usually a non-trivial matter, necessitating a referenceto the Lagrangian and a detailed counting of powers ofmomenta. Fortunately, many theories describing natureare constructible including, most notably, YM [7, 36] andGR [37] (scalar field theories are not constructible in thesense described above. This has lead to new, interestingmomentum shifts and on-shell recursion relations beingderived for scalar theories with non-linearly realisedsymmetries [38, 39]).However, for the boost-breaking amplitudes of interesthere, it is unlikely that B ( i,j ) ( z ) would vanish, since theunknown functions of energies will in general contributepositive powers of z to the tree-level amplitude. Indeed,for both particles i and j , the deformation of their ener-gies is linear in z and so the divergence at large z getsworse as additional powers of energy are included: E i ( z ) = E i (0) + z σ ) ˙ αα λ ( j ) α ˜ λ ( i )˙ α , (10) E j ( z ) = E j (0) − z σ ) ˙ αα λ ( j ) α ˜ λ ( i )˙ α . (11)The BCFW method can still be useful for non-constructible theories, however. One possibility is to in-troduce a distinction between accessible and inaccessible singularities of A ( i,j )4 ( z ), along the lines of [40]. We saya singularity is accessible via a deformation of momenta i and j if this singularity is approached as z → z ∗ forsome z ∗ . Otherwise we say it is inaccessible . The regularterm B ( i,j ) ( z ), by definition, cannot have any singular-ities in the z − plane and therefore cannot contribute toany residues of the accessible singularities of A ( i,j )4 ( z ).But it may exhibit inaccessible singularities. As an illus-tration of this distinction, consider a single scalar theorywhich is famously non-constructible. In the absence ofadditional global charges, the three-particle amplitude isa non-zero constant, A = g , and so we have A (1 , (0) = g (cid:18) t + 1 u (cid:19) + B (1 , (0) , (12) A (1 , (0) = g (cid:18) s + 1 t (cid:19) + B (1 , (0) . (13)The consistency condition A (1 , (0) = A (1 , (0) can besatisfied by choosing B (1 , ( z ) = g s and B (1 , ( z ) = g u ,since these two functions do not have any accessiblesingularities with regards to their own deformations.In the following section we will constrain boost-breakingamplitudes using the fact that the regular term B ( i,j ) ( z )does not have any accessible singularities. We will seethat for spinning particles, we can derive the highly non-trivial constraints first found in [1]. IV. CONSTRAINING THREE-PARTICLEINTERACTIONS
In this section we will constrain three-particle inter-actions for theories of a single spin- S particle withinteger S . We will derive the constraints first presentedin [1]. We also checked that the BCFW techniquesallow us to recover other results in [1], namely those of(gravitational) Compton scattering and the full analysisfor a scalar or a photon coupled to gravity. Thosecalculations contain only minor differences comparedwith what is presented below so we omit the details infavour of brevity. We remind the reader that we do notimpose boost invariance, but only demand that the freetheory is Poincar´e invariant, with the on-shell condition E − p = 0 for each particle.Consider the amplitude A (1 + S − S + S − S ), where su-perscripts denote the helicities of incoming particles ofsome integer spin S . We will impose matching condi-tions between deformations (1 ,
2) and (1 , , A (1 + S , + S , − S )and A (1 − S , − S , + S ) given in (2), and p ( z t ) + p = [13][14] λ (3) ˜ λ (4) , (14) p ( z u ) + p = [14][13] λ (4) ˜ λ (3) , (15)to eliminate all copies of λ ( I ) and ˜ λ ( I ) , which are thespinors associated with the exchanged particle , we find A (1 , (0) = B (1 , (0) + (cid:18) t F ˆ1 , F ˆ2 , + 1 u F ˆ1 , − ˆ1 − F ˆ2 , − ˆ2 − (cid:19) (cid:18) [13] (cid:104) (cid:105) s (cid:19) S . (16) These amplitudes arise from the leading order couplings.Higher-dimension operators give rise to the A (1 + S , + S , + S ), A (1 − S , − S , − S ) amplitudes but we don’t consider these here.We refer the reader to [1] for a discussion on these amplitudes. Here we have assumed [13] and [14] are non-zero, and therefore t = 0 and u = 0 are approached as (cid:104) (cid:105) = 0 and (cid:104) (cid:105) = 0 respec-tively (or as [24] = 0 and [23] = 0 respectively, by momentumconservation). For example, in the t -channel we set λ ( I ) = αλ (3) and ˜ λ ( I ) = β ˜ λ with αβ = [13][14] . When computing the residue, α and β onlyappear in the product αβ . In the u -channel we have summed over the two possi-bilities for the helicity configuration of the exchangedparticle but given (15), only one of these is non-zero.To keep formulae compact, here we have introducedsubscripts to the F ’s to denote their arguments e.g. F ( E i , E j ) ≡ F i,j and F ( E i , E j + E k ) ≡ F i,j + k . Again,hats denote deformed objects evaluated at the appropri-ate points e.g. in the 1 /t coefficient, F ˆ1 , ≡ F ( ˆ E ( z t ) , E )where ˆ E ( z t ) is the deformed energy of particle 1 evalu-ated at z = z t . Likewise, in the 1 /u coefficient, hattedenergies are evaluated at z = z u . We have also removedthe H/AH superscripts since the functions are identical,due to parity, up to an inconsequential overall sign [1].Now, we can also write A (1 , (0) = ˜ B (1 , (0) + (cid:18) t F , F , + 1 u F , − − F , − − (cid:19) (cid:18) [13] (cid:104) (cid:105) s (cid:19) S , (17)where here we dropped the hat above all the energies,which indicates that the expression is evaluated at their undeformed values. This can be justified as follows. Weassume that the F ’s can be Taylor expanded around theundeformed energies. The deformed energies areˆ E ( z t ) = E − t (cid:104) (cid:105) [13] (¯ σ ) ˙ αα λ (2) α ˜ λ (1)˙ α , (18)ˆ E ( z t ) = E + t (cid:104) (cid:105) [13] (¯ σ ) ˙ αα λ (2) α ˜ λ (1)˙ α , (19)with similar expressions evaluated at z = z u . Herewe see that potential new singularities generated bythe deformed energies are all inaccessible , as theycorrespond to the vanishing of (cid:104) (cid:105) or [13], but these donot depend on z . Moreover, only the leading term inthe Taylor expansion will exhibit accessible singularities,since in all subleading terms t and u will be cancelledout. We can therefore simply absorb all subleadingterms into B (1 , , thus introducing ˜ B (1 , that still doesnot contain any terms singular in t or u . Although itcould become singular in some kinematic configurations,especially at s = 0, that is not a problem, becausethis singularity is inaccessible and we only demandthat ˜ B (1 , , for those configurations for which it can bedefined, does not have any singularities as a function of z .We now play the same game for the (1 ,
4) deformationwhich amounts to interchanging particles 2 and 4. Wehave A (1 , (0) = ˜ B (1 , (0) + (cid:18) t F , F , + 1 s F , − − F , − − (cid:19) (cid:18) [13] (cid:104) (cid:105) u (cid:19) S . (20)We discussed the S = 0 case earlier where we showedthat equating A (1 , (0) and A (1 , (0) requires us to makecertain choices for the boundary terms. Let us now con-sider S > S an integer. We see that A (1 , (0) in(17) contains terms proportional to 1 / ( ts S ) and 1 / ( us S )which are both singular in more than one Mandelstamvariable and thus cannot be accounted for or modifiedby ˜ B (1 , (0) nor ˜ B (1 , (0). A similar observation appliesto A (1 , (0) in (20). Thus, by matching the amplitudeswe find the necessary condition as S t + bs S u = cu S t + du S s , (21)where a = F , F , , (22) b = F , − − F , − − , (23) c = F , F , , (24) d = F , − − F , − − . (25)Recalling that s + t + u = 0, this constraint, given thatit must be valid for all kinematics, is equivalent to au S − b ( s + u ) u S − − cs S + d ( s + u ) s S − = 0 . (26)For S = 1 we therefore have a = ( b − d ) = − c , or equiv-alently, F , F , − F , − − F , − − + F , − − F , − − = 0 , (27)which is simply an alternative form of (4.25) from [1]. As-suming that the F ’s are polynomials, in [1] it was shownthat the only solution to this system is F ≡ S = 1 functions are alternating poly-nomials as dictated by Bose symmetry . We thereforesee that the leading order three-particle interactions forthree-photons must vanish, as is the case for Poincar´einvariant theories. For S = 2 we require a = b = c = d ,or equivalently, F , F , = F , − − F , − − = F , − − F , − − , (28)which gives rise to the constraints (4 . − (4 .
33) from [1]once we use the fact that the S = 2 functions are sym-metric in their arguments by Bose symmetry (this alsomakes the a = c constraint trival). In [1] it was shownthat the only solution to this system is F = constant and so again the three-particle interactions are reducedto their Poincar´e invariant form, but this time the am-plitudes are non-zero and are those of GR. Finally, for S >
2, it is simple to see that a = b = c = d = 0 is re-quired and therefore there are no consistent three-particleinteractions for these massless, higher-spin particles evenwhen boosts are broken, as was also concluded in [1]. The spinor helicity parts of the S = 1 three-particle amplitudesare odd under the exchange of identical particles so if the ampli-tudes are to be even by Bose symmetry, the F ’s must be alter-nating. For S = 2, the spinor parts of the three-particle amplitudes areeven under the exchange of identical particles and so the F ’s aresymmetric polynomials. V. SUMMARY
Very recently, the singular parts of four-particle am-plitudes were bootstrapped in [1] by demanding thatthey factorise into a product of on-shell three-particleamplitudes on simple poles. In that work, consistentfactorisation was implemented directly without makinguse of BCFW momentum shifts. In this short note,we have shown that the same results can be derivedby using BCFW shifts to automate consistent factori-sation. We presented full details for the illustrativecases of single spin- S particle amplitudes but havealso checked that the procedure produces the expectedresults for Compton scattering, and its gravitationalanalogue, as well as for scalars or photons coupledminimally to gravity. For single spin- S particles,the boostless bootstrap teaches us that the leadingthree-particle couplings for a photon must vanish, theleading three-particle couplings for a graviton must bethose of GR, while massless higher-spinning particlesdo not self-interact. For photon Compton scattering,boost-breaking interactions between the photon, scalarsand spin-1 / a priori constructible, in the sense that the boundary terms donot necessarily vanish at large z , we have still been ableto use BCFW shifts to constrain the three-particle cou-plings contributing to particle exchange. This does mean,of course, that the three-particle amplitudes themselvesdo not fully fix the four-particle ones. Indeed, all of thefour-particle amplitudes we have constructed are definedup to the presence of “contact” terms that are regular forall kinematic configurations. It would be very interestingto investigate the possibility of using generalised momen-tum shifts, possibly along the lines of [41], to recursivelyderive exact higher-point amplitudes even if only for asubset of boost-breaking theories. It would also be veryinteresting to investigate the generalised on-shell recur-sion relations introduced in [42], where boundary termsare fixed with additional knowledge of a subset of thezeros of the deformed amplitude, in our boost-breakingsetting. ACKNOWLEDGEMENTS
We would like to thank Paolo Benincasa, Tanguy Grall,Sadra Jazayeri and Enrico Pajer for useful discussionsand comments on a draft of this paper. D.S. has beensupported in part by the research program VIDI withProject No. 680-47-535, which is (partly) financed by the Netherlands Organisation for Scientific Research (NWO).J.S. has been supported by a grant from STFC. [1] E. Pajer, D. Stefanyszyn, and J. Supe(cid:32)l, (2020),arXiv:2007.00027 [hep-th].[2] H. Elvang and Y.-t. Huang, (2013), arXiv:1308.1697[hep-th].[3] C. Cheung, in
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