Consistency Conditions on the S-Matrix of Massless Particles
aa r X i v : . [ h e p - t h ] J a n UWO-TH-07/09
Consistency Conditions On The S-Matrix Of Massless Particles
Paolo Benincasa ∗ and Freddy Cachazo † Department of Applied Mathematics,University of Western Ontario, London, Ontario N6A 5B7, Canada Perimeter Institute for Theoretical Physics,Waterloo, Ontario N2J 2W9, Canada
Abstract
We introduce a set of consistency conditions on the S-matrix of theories of massless particlesof arbitrary spin in four-dimensional Minkowski space-time. We find that in most cases the con-straints, derived from the conditions, can only be satisfied if the S-matrix is trivial. Our conditionsapply to theories where four-particle scattering amplitudes can be obtained from three-particleones via a recent technique called BCFW construction. We call theories in this class constructible.We propose a program for performing a systematic search of constructible theories that can havenon-trivial S-matrices. As illustrations, we provide simple proofs of already known facts like theimpossibility of spin s > ∗ Electronic address: [email protected] † Electronic address: [email protected] . INTRODUCTION The power of the constraints that Lorentz invariance imposes on the S-matrix of fourdimensional theories has been well known at least since the work of Weinberg [1, 2]. Impres-sive results like the impossibility of long-range forces mediated by massless particles withspin >
2, charge conservation in interactions mediated by a massless spin 1 particle, or theuniversality of the coupling to a massless spin 2 particle are examples beautifully obtainedby simply using the pole structure of the S-matrix governing soft limits in combination withLorentz invariance [2, 3].Weinberg’s argument does not rule out the possibility of non-trivial Lagrangians describ-ing self-interacting massless particles of higher spins. It rules out the possibility of thosefields producing macroscopic effects. Actually, the theory of massless particles of higher spinshas been an active research area for many years (see reviews [4, 5] and reference therein, alsosee [6, 7] for alternative approaches). Lagrangians for free theories have been well under-stood while interactions have been a stumbling block. Recent progress shows that in spaceswith negative cosmological constant it is possible to construct consistent Lagrangian theoriesbut no similar result exists for flat space-time [8, 9]. Despite the difficulties of constructingan interactive Lagrangian, several attempts have been made in studying the consistency ofspecific couplings among higher spin particles. For example, cubic interactions have beenstudied in [10, 11, 12, 13]. Also, very powerful techniques for constructing interaction ver-tices systematically have been developed using BRST-BV cohomological methods [14, 15, 16]and references therein.In this paper we introduce a technique for finding theories of massless particles that canhave non-trivial S-matrices within a special set of theories we call constructible. The startingpoint is always assuming a Poincar´e covariant theory where the S-matrix transformation isderived from that of one-particle states which are irreducible representations of the Poincar´egroup. There will also be implicit assumptions of locality and parity invariance.The next step is to show that for complex momenta, on-shell three-particle S-matricesof massless particles of any spin can be uniquely determined. As is well known, on-shellthree-particle amplitudes vanish in Minkowski space. That this need not be the case foramplitudes in signatures different from Minkowski or for complex momenta was explainedby Witten in [17]. 2e consider theories for which four-particle tree-level S-matrix elements can be com-pletely determined by three-particle ones. These theories are called constructible . This isdone by introducing a one parameter family of complex deformations of the amplitudesand using its pole structure to reconstruct it. The physical singularities are on-shell inter-mediate particles connecting physical on-shell three-particle amplitudes. This procedure isknown as the BCFW construction [18, 19]. One can also introduce the terminology fullyconstructible if this procedure can be extended to all n -particle amplitudes. Examples offully constructible theories are Yang-Mills [19] and General Relativity [20, 21, 22] (the factthat cubic couplings could play a key role in Yang-Mills theory and General Relativity wasalready understood in [23, 24]).The main observation is that by using the BCFW deformation, the four-particle am-plitude is obtained by summing over only a certain set of channels, say the s - and the u -channels. However, if the theory under consideration exists, then the answer should alsocontain the information about the t - channel. In particular, one could construct the four-particle amplitude using a different BCFW deformation that sums only over the t - and the u - channel.Choosing different deformations for constructing the same four-particle amplitude andrequiring the two answers to agree is what we call the four-particle test. This simple con-sistency condition turns out to be a powerful constraint that is very difficult to satisfy.It is important to mention that the constraints are only valid for constructible theories.Luckily, the set of constructible theories is large and we find many interesting results. Wealso discuss some strategies for circumventing this limitation.As illustrations of the simplicity and power of the four-particle test we present severalexamples. The first is a general analysis of theories of a single spin s particle. We find thatif s > s = 2 which passes the test.As a second example we allow for several particles of the same spin. We find that, againin the range s >
0, the only theories that can have a nontrivial S-matrix are those of spin1 with completely antisymmetric three-particle coupling constants which satisfy the Jacobiidentity and spin 2 particles with completely symmetric three-particle coupling constantswhich define a commutative and associative algebra. We also study the possible theoriesof particles of spin s , without self-couplings and with s >
1, that can couple non-triviallyto a spin 2 particle. In this case, we find that only s = 3 / N = 1 supergravity.The paper is organized as follows. In section II, we review the construction of the S-matrix and of scattering amplitudes for massless particles. In section III, we discuss howthree-particle amplitudes are non-zero and uniquely determined up the choice of the valuesof the coupling constant. In section IV, we apply the BCFW construction to show how,for certain theories, four-particle amplitudes can be computed from three-particle ones. Atheory for which this is possible is called constructible . We then introduce the four-particletest. In section V, we discuss sufficient conditions for a theory to be constructible. In sec-tion VI, we give examples of the use of the four-particle test. In section VII, we concludewith a discussion of possible future directions including how to relax the constructibility con-straint. Finally, in the appendix we illustrate one of the methods to relax the constructibilitycondition. II. PRELIMINARIESA. S-Matrix
In this section we define the S-matrix and scattering amplitudes. We do this in orderto set up the notation. Properties of the S-matrix, which we exploit in this paper likefactorization, have been well understood since at least the time of the S-matrix program[25, 26, 27].Recall that physically, one is interested in the probability for, say, two asymptotic statesto scatter and to produce n − out h p . . . p n − | p a p b i in = h p . . . p n − | S | p a p b i (1)where S is a unitary operator. As usual, it is convenient to write S = I + iT with h p . . . p n − | iT | p a p b i = δ (4) p a + p b − n − X i =1 p i ! M ( p a , p b → { p , p , . . . , p n − } ) . (2) M ( p a , p b → { p , p , . . . , p n − } ) is called the scattering amplitude (see for example chapter 4in [28]). 4ssuming crossing symmetry one can write p a = − p n − and p b = − p n and introduce ascattering amplitude where all particles are outgoing. Different processes are then obtainedby analytic continuation of M n = M n ( p , p , . . . , p n − , p n ) . (3) M n is our main object of study. Our goal is to determine when M n can be non-zero. Upto now we have exhibited only the dependence on momenta of external particles. However,if they have spin s > B. Massless Particles Of Spin s It turns out that all the information needed to describe the physical information of anon-shell massless spin s particle is contained in a pair of spinors { λ a , ˜ λ ˙ a } , left- and right-handed respectively, and the helicity of the particle [17, 29, 30, 31]. Recall that in a Poincar´einvariant theory, irreducible massless representations are classified by their helicity whichcan be h = ± s with s any integer or half-integer known as the spin of the particle.The spinors { λ a , ˜ λ ˙ a } transform in the representations (1 / ,
0) and (0 , /
2) of the universalcover of the Lorentz group, SL (2 , C ), respectively. Invariant tensors are ǫ ab , ǫ ˙ a ˙ b and ( σ µ ) a ˙ a where σ µ = ( I , ~σ ). The most basic Lorentz invariants, from which any other is made of, canbe constructed as follows: λ a λ ′ b ǫ ab ≡ h λ, λ ′ i , ˜ λ ˙ a ˜ λ ′ ˙ b ǫ ˙ a ˙ b ≡ [ λ, λ ′ ] . (4)Finally, using the third invariant tensor we can define the momentum of the particle by p µ = λ a ( σ µ ) a ˙ a ˜ λ ˙ a , where indices are raised using the first two tensors. A simple consequenceof this is that the scalar product of two vectors, p µ and q µ is given by 2 p · q = h λ p , λ q i [˜ λ p , ˜ λ q ]. III. THREE PARTICLE AMPLITUDES: A UNIQUENESS RESULT
In this section we prove that three-particle amplitudes of massless particles of any spincan be uniquely determined. 5he statement that on-shell scattering amplitudes of three massless particles can benon-zero might be somewhat surprising. However, as shown by Witten [17], three-particleamplitudes are naturally non-zero if we choose to work with the complexified Lorentz group SL (2 , C ) × SL (2 , C ), where (1 / ,
0) and (0 , /
2) are completely independent representationsand hence momenta are not longer real. In other words, if ˜ λ ˙ a = ± ¯ λ a then p µ is complex.Let us then consider a three-particle amplitude M ( { λ ( i ) , ˜ λ ( i ) , h i } ) where the spinors ofeach particle, λ ( i ) and ˜ λ ( i ) , are independent vectors in C .Momentum conservation ( p + p + p ) a ˙ a = 0 and the on-shell conditions, p i = 0, implythat p i · p j = 0 for any i and j . Therefore we have the following set of equations h , i [1 ,
2] = 0 , h , i [2 ,
3] = 0 , h , i [3 ,
1] = 0 . (5)Clearly, if [1 ,
2] = 0 and [2 ,
3] = 0 then [3 ,
1] must be zero. The reason is that the spinorslive in a two dimensional vector space and if ˜ λ (1) and ˜ λ (3) are proportional to ˜ λ (2) then theymust also be proportional.This means that the non-trivial solutions to (5) are either h , i = h , i = h , i = 0 or[1 ,
2] = [2 ,
3] = [3 ,
1] = 0.Take for example [1 ,
2] = [2 ,
3] = [3 ,
1] = 0 and set ˜ λ (2)˙ a = α ˜ λ (1)˙ a and ˜ λ (3)˙ a = α ˜ λ (1)˙ a . Thenmomentum conservation implies that λ (1) a + α λ (2) a + α λ (3) a = 0 which is easily seen to besatisfied if α = −h , i / h , i and α = −h , i / h , i .The conclusion of this discussion is that three-particle amplitudes, M ( { λ ( i ) , ˜ λ ( i ) , h i } ),which by Lorentz invariance are only restricted to be a generic function of h i, j i and [ i, j ]turn out to split into a “holomorphic” and an “anti-holomorphic” part . More explicitly M = M H ( h , i , h , i , h , i ) + M A ([1 , , [2 , , [3 , . (6)It is important to mention that we are considering the full three-particle amplitude andnot just the tree-level one. Therefore M H and M A are not restricted to be rational functions .In other words, we have purposefully avoided to talk about perturbation theory. We will beforced to do so later in section V but we believe that this discussion can be part of a moregeneral analysis. Using “holomorphic” and “anti-holomorphic” is an abuse of terminology since ˜ λ ˙ a = ± ¯ λ a . We hope thiswill not cause any confusion. We thank L. Freidel for discussions on this point. . Helicity Constraint and Uniqueness One of our basic assumptions about the S-matrix is that the Poincar´e group acts on thescattering amplitudes as it acts on individual one-particle states. This in particular meansthat the helicity operator must act as (cid:18) λ ai ∂∂λ ai − ˜ λ ai ∂∂ ˜ λ ai (cid:19) M (1 h , h , h ) = − h i M (1 h , h , h ) . (7)Equivalently, (cid:18) λ ai ∂∂λ ai + 2 h i (cid:19) M H ( h , i , h , i , h , i ) = 0 (8)on the holomorphic one and as (cid:18) ˜ λ ai ∂∂ ˜ λ ai − h i (cid:19) M A ([1 , , [2 , , [3 , d = h − h − h , d = h − h − h and d = h − h − h ,then F = h , i d h , i d h , i d , G = [1 , − d [2 , − d [3 , − d (10)are particular solutions of the equations (8) and (9) respectively.Therefore, M H /F and M A /G must be “scalar” functions, i.e., they have zero helicity.Let x be either h , i or [2 ,
3] depending on whether we are working with the holomorphicor the antiholomorphic pieces. Also let x be either h , i or [3 ,
1] and x be either h , i or[1 , M be either M H /F or M A /G . Then we find that x i ∂ M ( x , x , x ) ∂x i = 0 (11)for i = 1 , ,
3. Therefore, up to solutions with delta function support which we discard basedon analyticity, the only solution for M is a constant. Let such a constant be denoted by κ H or κ A respectively.We then find that the exact three-particle amplitude must be M ( { λ ( i ) , ˜ λ ( i ) , h i } ) = κ H h , i d h , i d h , i d + κ A [1 , − d [2 , − d [3 , − d . (12)Finally, we have to impose that M has the correct physical behavior in the limit ofreal momenta. In other words, we must require that M goes to zero when both h i, j i and7 i, j ] are taken to zero . Simple inspection shows that if d + d + d , which is equal to − h − h − h , is positive then we must set κ A = 0 in order to avoid an infinity while if − h − h − h is negative then κ H must be zero. The case when h + h + h = 0 is moresubtle since both pieces are allowed. In this paper we restrict our study to h + h + h = 0and leave the case h + h + h = 0 for future work. B. Examples
Let us consider few examples, which will appear in the next sections, as illustrations ofthe uniqueness of three-particle amplitudes.Consider a theory of several particles of a given integer spin s. Since all particles have thesame spin we can replace h = ± s by the corresponding sign. Let us use the middle lettersof the alphabet to denote the particle type.There are only four helicity configurations: M (1 − m , − r , + s ) = κ mrs (cid:18) h , i h , ih , i (cid:19) s , M (1 + m , + r , − s ) = κ mrs (cid:18) [1 , [2 , , (cid:19) s (13)and M (1 − m , − r , − s ) = κ ′ mrs ( h , ih , ih , i ) s , M (1 + m , + r , + s ) = κ ′ mrs ([1 , , , s . (14)The subscripts on the coupling constants κ and κ ′ mean that they can depend on the particletype . We will use the amplitudes in (13) in section VI.A simple but important observation is that if the spin is odd then the coupling constantmust be completely antisymmetric in its indices. This is because due to crossing symmetrythe amplitude must be invariant under the exchange of labels.This leads to our first result, a theory of less than three massless particles of odd spinmust have a trivial three-particle S-matrix. Under the conditions of constructibility, thiscan be extended to higher-particle sectors of the S-matrix and even to the full S-matrix. Taking to zero h i, j i means that λ ( i ) and λ ( j ) are proportional vectors. Therefore, all factors h i, j i can betaken to be proportional to the same small number ǫ which is then taken to zero. Note that here we have implicitly assumed parity invariance by equating the couplings of conjugateamplitudes. V. THE FOUR-PARTICLE TEST AND CONSTRUCTIBLE THEORIES
In this section we introduce what we call the four-particle test. Consider a four-particleamplitude M . Under the assumption that one-particle states are stable in the theory, M must have poles and multiple branch cuts emanating from them at locations where either s = ( p + p ) , t = ( p + p ) or u = ( p + p ) vanish .We choose to consider only the pole structure. Branch cuts will certainly lead to veryinteresting constraints but we leave this for future work. Restricting to the pole structurecorresponds to working at tree-level in field theory.As we will see, under certain conditions, one can construct physical on-shell tree-levelfour-particle amplitudes as the product of two on-shell three-particle amplitudes (evaluatedat complex momenta constructed out of the real momenta of the four external particles)times a Feynman propagator. In general this can be done in at least two ways. Roughlyspeaking, these correspond to summing over the s -channel and u -channel or summing overthe t -channel and u -channel. A necessary condition for the theory to exists is that the twofour-particle amplitudes constructed this way give the same answer. This is what we callthe four-particle test. It might be surprising at first that a sum over the s - and u -channelscontains information about the t -channel but as we will see this is a natural consequence ofthe BCFW construction which we now review. A. Review Of The BCFW Construction And Constructible Theories
The key ingredient for the four-particle test is the BCFW construction [19]. The con-struction can be applied to n -particle amplitudes, but for the purpose of this paper we onlyneed four-particle amplitudes.We want to study M ( { λ ( i ) a , ˜ λ ( i )˙ a , h i } ). Recall that momenta constructed from the spinorsof each particle are required to satisfy momentum conservation, i.e., ( p + p + p + p ) µ = 0.Choose two particles, one of positive and one of negative helicity , say i + s i and j − s j , We have introduced the notation s for the center of mass energy in order to avoid confusion with the spin s of the particles. Here we do not consider amplitudes with all equal helicities. s i and s j are the corresponding spins, and perform the following deformation λ ( i ) ( z ) = λ ( i ) + zλ ( j ) , ˜ λ ( j ) ( z ) = ˜ λ ( j ) − z ˜ λ ( i ) . (15)All other spinors remain the same.The deformation parameter z is a complex variable. It is easy to check that this deforma-tion preserves the on-shell conditions, i.e., p k ( z ) = 0 for any k and momentum conservationsince p i ( z ) + p j ( z ) = p i + p j .The main observation is that the scattering amplitude is a rational function of z whichwe denote by M ( z ). This fact follows from M (1 h , . . . , h ) being, at tree-level, a rationalfunction of spinor products. Being a rational function of z , M ( z ) can be determined ifcomplete knowledge of its poles, residues and behavior at infinity is found. Definition:
We call a theory constructible if M ( z ) vanishes at z = ∞ . As we will seethis means that M ( z ) can only be computed from M and hence the name.In the next section we study sufficient conditions for a theory to be constructible. Theproof of constructibility relies very strongly on the fact that on-shell amplitudes should onlyproduce the two physical helicity states of a massless particle . In this section we assumethat the theory under consideration is constructible.Any rational function that vanishes at infinity can be written as a sum over its poles withthe appropriate residues. In the case at hand, M ( z ) can only have poles of the form1( p i ( z ) + p k ) = 1 h λ ( i ) ( z ) , λ ( k ) i [ i, k ] = 1( h i, k i + z h j, k i )[ i, k ] (16)where k has to be different from i and j .As mentioned at the beginning of this section, M ( z ) can be constructed as a sum overonly two of the three channels. The reason is the following. For definiteness let us set i = 1 and j = 2, then the only propagators that can be z -dependent are 1 / ( p ( z ) + p ) and1 / ( p ( z ) + p ) . By construction 1 / ( p + p ) is z -independent.The rational function M ( z ) can thus be written as M (1 , ( z ) = c t z − z t + c u z − z u (17) This in turn is simply a consequence of imposing Lorentz invariance [2]. z t is such that t = ( p ( z ) + p ) vanishes, i.e., z t = −h , i / h , i while z u is where u = ( p ( z ) + p ) vanishes, i.e., z u = −h , i / h , i . Note that we have added the superscript(1 ,
2) to M ( z ) to indicate that it was obtained by deforming particles 1 and 2.Finally, we need to compute the residues. Close to the location of one of the poles, M ( z ) factorizes as the product of two on-shell three-particle amplitudes. Note that each ofthe three-particle amplitudes is on-shell since the intermediate particle is also on-shell. Seefigure 1 for a schematic representation. Therefore, we find that M (1 , ( z ) = X h M ( p h ( z t ) , p h , − P h , ( z t )) 1 P , ( z ) M ( p h ( z t ) , p h , P − h , ( z t ))+ X h M ( p h ( z u ) , p h , − P h , ( z u )) 1 P , ( z ) M ( p h ( z u ) , p h , P − h , ( z u )) . (18)where the sum over h runs over all possible helicities in the theory under consideration andalso over particle types if there is more than one.The scattering amplitude we are after is simply obtained by setting z = 0, i.e, M ( { λ ( i ) , ˜ λ ( i ) , h i } ) = M (1 , (0).Recall that we assumed h = s and h = − s . Let us further assume that h = − s .Therefore we could repeat the whole procedure but this time deforming particles 1 and 4.In this way we should find that M ( { λ ( i ) , ˜ λ ( i ) , h i } ) = M (1 , (0).We have finally arrived at the consistency condition we call the four-particle test. Onehas to require that M (1 , (0) = M (1 , (0) . (19)As we will see in examples, this is a very strong condition that very few constructibletheories satisfy non-trivially. In other words, most constructible theories satisfy (19) only ifall three-particle couplings are set to zero and hence four-particle amplitudes vanish. If thetheory is fully constructible, this implies that the whole S-matrix is trivial. B. Simple Examples
We illustrate the use of the four-particle test by first working out the general form of M (1 , (0) and M (1 , (0) for a theory containing only integer spin particles . We then spe-cialize to the case of a theory containing a single particle of integer spin s . It turns out that Including half-integer spins is straightforward and we give an example in section VI. (1 , = X h (cid:1) h − h ˆ1 h h ˆ2 h h P + X h (cid:1) h − h ˆ1 h h ˆ2 h h P FIG. 1: Factorization of a four-particle amplitude into two on-shell three-particle amplitudes. Inconstructible theories, four-particle amplitudes are given by a sum over simple poles of the 1-parameter family of amplitudes M ( z ) times the corresponding residues. At the location of thepoles the internal propagators go on-shell and the residues are the product of two on-shell three-particle amplitudes. the theory is constructible only when s >
0. For s >
0, we explicitly find the condition on s for the theory to pass the four-particle test.
1. General Formulas For Integer Spins
Consider first M (1 , (0). In order to keep the notation simple we will denote h λ (1) ( z ) , •i by h ˆ1 , •i and so on. The precise value of z depends on the deformation and channel beingconsidered. M (1 , (0) = X h (cid:16) κ H (1+ h + h + h ) h ˆ1 , i h − h − h h , ˆ P , i h − h − h h ˆ P , , ˆ1 i h − h − h + κ A (1 − h − h − h ) [1 , h + h − h [4 , ˆ P , ] h + h − h [ ˆ P , , h + h − h (cid:17) × P , × (cid:16) κ H (1+ h + h − h ) h , i − h − h − h h , ˆ P , i h − h + h h ˆ P , , i h − h + h + κ A (1 − h − h + h ) [3 , ˆ2] h + h + h [ˆ2 , ˆ P , ] − h + h − h [ ˆ P , , − h + h − h (cid:17) + X h (4 ↔ . (20)Here the subscripts on the three-particle couplings denote the dimension of the coupling.The range of values of the helicity of the internal particle depends on the details of thespecific theory under consideration. Even though (20) is completely general we choose to12xclude theories where h can take values such that h + h + h = 0 or − h + h + h = 0.The main reason is that formulas will simplify under this assumption.Note also that we have kept the two pieces of all three-particle amplitudes entering in(20). However, recall that we should set either the holomorphic or the anti-holomorphiccoupling to zero. As we will now see this condition is very important for the consistency of(20).Let us solve the condition P , ( z ) = 0. As mentioned above this leads to z t = −h , i / h , i . Since P , ( z t ), which we denoted by ˆ P , , is a null vector, it must be pos-sible to find spinors λ ( ˆ P ) and ˜ λ ( ˆ P ) such that ˆ P µ , = λ ( ˆ P ) a ( σ µ ) a ˙ a ˜ λ ( ˆ P ) ˙ a . Clearly, given ˆ P , itis not possible to uniquely determine the spinors since any pair of spinors { tλ ( ˆ P ) , t − ˜ λ ( ˆ P ) } gives rise to the same ˆ P , . This ambiguity drops out of (20) as we will see.After some algebra we find that P , ( z t ) = ˆ P , = [1 , , λ (4) ˜ λ (3) . (21)Therefore we can choose λ ( ˆ P ) = αλ , ˜ λ ( ˆ P ) = β ˜ λ , with αβ = [1 , , . (22)Moreover, it is also easy to getˆ λ = h , ih , i λ , ˆ˜ λ = [1 , ,
3] ˜ λ . (23)Using the explicit form of all the spinors one can check that the three-particle amplitudewith coupling constant κ H (1+ h + h + h ) in (20) possesses a factor of the form h , i = 0 to thepower − h − h − h . From our discussion in section III, if − h − h − h is less than zerothen the coupling κ H (1+ h + h + h ) = 0. In this way a possible infinity is avoided. Therefore weget a contribution from the term with coupling κ A (1 − h − h − h ) whenever h > − ( h + h ).Now, if − h − h − h is positive then κ H (1+ h + h + h ) need not vanish but the factor multiplyingit vanishes. In this case κ A (1 − h − h − h ) must be zero and we find no contributions.This meansthat the only non-zero contributions to the sum over h can only come from the region where h > − ( h + h ).Turning to the other three-particle amplitude, we find that the piece with coupling κ A (1 − h − h + h ) has a factor [3 ,
3] = 0 to the power − h + h + h . A similar analysis shows thatthe only nonzero contributions come from regions where h > ( h + h ).13utting the two conditions together we find that the first term gives a non-zero contri-bution only when h > max( − ( h + h ) , ( h + h )).Simplifying we find M (1 , (0) = X h> max( − ( h + h ) , ( h + h )) κ A − h − h − h κ H h + h − h ( − P , ) h P , (cid:18) [1 , , , (cid:19) h (cid:18) [1 , , , (cid:19) h (cid:18) h , ih , ih , i (cid:19) h (cid:18) h , ih , ih , i (cid:19) h ! + X h> max( − ( h + h ) , ( h + h )) (4 ↔ . (24)Finally, it is easy to obtain M (1 , (0) from (24) by simply exchanging the labels 2 and 4.Next we will write down all formulas explicitly for the case when | h i | = s for all i .
2. Theories Of A Single Spin s Particle
Consider now the case h = s , h = − s , h = s and h = − s . We also assume thatthe theory under consideration has a single particle of spin s. This restriction is again forsimplicity. If one decided to allow for more internal particles then the different terms wouldhave to satisfy the four-particle test independently since the dimensions of the couplingconstants would be different .Using (24) we find that the first sum contributing to M (1 , (0) allows only for h = s whilethe second one allows for h = − s and h = s . Using momentum conservation to simplifythe expressions we find M (1 , (0) = κ A − s κ H − s (cid:18) h , i [1 , h , ih , i (cid:19) s h , i [1 ,
4] + κ A − s κ H − s (cid:18) [1 , h , i [4 , , (cid:19) s h , i [1 , κ A − s κ H − s ([1 , h , i ) s (cid:0) − P , (cid:1) s P , . (25)We would like to set all couplings with the same dimension to the same value. In otherwords, we define κ = κ A − s = κ H − s . We also choose to study the case κ ′ = κ A − s = κ H − s = 0. There might be cases where the dimensions might agree by accident. Such cases might actually lead tonew interesting theories. We briefly elaborate in section VII but we leave the general analysis for futurework. One can easily show that momentum conservation for four particles implies that h a, b i / h a, c i = − [ d, c ] / [ d, b ]for any choice of { a, b, c, d } .
14t turns out that if we had chosen κ = 0 and κ ′ non-zero the resulting theories would nothave been constructible. In section VII we explore strategies for relaxing this condition.As mentioned above we can write M (1 , (0) by simply exchanging the labels 2 and 4. Wethen find M (1 , (0) = κ (cid:18) h , i [1 , h , ih , i (cid:19) s h , i [1 ,
4] + κ (cid:18) [1 , h , i [4 , , (cid:19) s h , i [1 , ,M (1 , (0) = κ (cid:18) h , i [1 , h , ih , i (cid:19) s h , i [1 ,
2] + κ (cid:18) [1 , h , i [2 , , (cid:19) s h , i [1 , . (26)Both amplitudes can be further simplified to M (1 , (0) = − ( − s κ ([1 , h , i ) s stu × s − s , M (1 , (0) = − ( − s κ ([1 , h , i ) s stu × t − s . (27)Finally, the four-particle test requires M (1 , (0) = M (1 , (0) or equivalently M (1 , (0) /M (1 , (0) = 1. The latter gives the condition ( s / t ) − s = 1 which can only besatisfied for generic choices of kinematical invariants if s = 2. If s = 2 the four-particle test M (1 , (0) = M (1 , (0) then requires κ = 0 and hence a trivial S-matrix. V. CONDITIONS FOR CONSTRUCTIBILITY
The example in the previous section showed that the only theory of a single massless spins particle that passes the four-particle test is that with s = 2. This theory turns out to belinearized General Relativity. For s = 1, the result is also familiar: a single photon should befree. However, if s = 0 one knows that a single scalar can have a non-trivial S-matrix. Thereason we did not find s = 0 as a possible solution in the previous example is that preciselyfor s = 0 the four-particle amplitude is not constructible. Therefore our calculation wasvalid only for s > s = 0 , , s > s . Polarizationtensors of particles of integer spin s can be expressed in terms of polarization vectors of spin1 particles as follows: ǫ + a ˙ a ,...,a s ˙ a s = s Y i =1 ǫ + a i ˙ a i , ǫ − a ˙ a ,...,a s ˙ a s = s Y i =1 ǫ − a i ˙ a i . (28)For half-integer spin s + 1 / ǫ + a ˙ a ,...,a s ˙ a s , ˙ b = ˜ λ ˙ b s Y i =1 ǫ + a i ˙ a i , ǫ − a ˙ a ,...,a s ˙ a s ,b = λ b s Y i =1 ǫ − a i ˙ a i , (29)and where polarization vectors of spin 1 particles are given by ǫ + a ˙ a = µ a ˜ λ ˙ a h µ, λ i , ǫ − a ˙ a = λ a ˜ µ ˙ a [˜ λ, ˜ µ ] (30)with µ a and ˜ µ ˙ a arbitrary reference spinors.This explains how all the physical data of a massless particle can be recovered from λ, ˜ λ and h . A comment is in order here. The presence of arbitrary reference spinors means thatpolarization tensors cannot be uniquely fixed once { λ, ˜ λ, h } is given. If a different referencespinor is chosen, say, µ ′ for ǫ + a ˙ a then ǫ + a ˙ a ( µ ′ ) = ǫ + a ˙ a ( µ ) + ωλ a ˜ λ ˙ a (31)where ω = h µ ′ , µ ih µ ′ , λ ih λ, µ i . If the particle has helicity h = 1 then it is easy to recognize (31) as a gauge transformationand the amplitude must be invariant.However, one does not have to invoke gauge invariance or assume any new principle. Asshown by Weinberg in [2] for any spin s , the only way to guarantee the correct Poincar´etransformations of the S-matrix of massless particles is by imposing invariance under (31). Inthat sense, there is no assumption in this section that has not already been made in sectionII. In other words, Poincar´e symmetry requires that M n gives the same answer independentlyof the choice of reference spinor µ . A. Behavior at Infinity
If a theory comes from a Lagrangian then the three-particle amplitudes derived in sectionIII can be computed as the product of three polarization tensors times a three-particle vertex16hat contains some power of momenta which we denote by L . Simple dimensional argumentsindicate that if all particles have integer spin then L = | h + h + h | . Let us denote thepower of momenta in the four-particle vertex by L .We are interested in the behavior of M , constructed using Feynman diagrams, under thedeformation of λ (1) and ˜ λ (2) defined in (15) as z is taken to infinity.Feynman diagrams fall into three different categories corresponding to different behav-iors at infinity. Representatives of each type are shown in figure 2. The first kind corre-sponds to the (1 , s -channel). The second corresponds to either the (1 , u -channel) or the (1 , t -channel). Finally, the third kind is the four-particlecoupling.Under the deformation λ (1) ( z ) = λ (1) + zλ (2) and ˜ λ (2) ( z ) = ˜ λ (2) − z ˜ λ (1) , polarizationtensors give contributions that go as z − s and z − s respectively in the case of integer spinand like z − s +1 / and z − s +1 / in the case of half-integer spin. Recall that we chose particle1 to have positive helicity while particle 2 to have negative helicity. Had we chosen theopposite helicities, polarization tensors would have given positive powers of z at infinity. Forsimplicity, let us restrict the rest of the discussion in this section to integer spin particles.For the first kind of diagrams, only a single three-particle vertex is z dependent and gives z L . Combining the contributions we find z L − s − s . Therefore, we need s + s > L .For the second kind of diagrams, two three-particle vertices contribute giving z L + L ′ .This time a propagator also contributes with z − . Combining the contributions we get z L + L ′ − s − s − . Therefore we need s + s > L + L ′ − z L . Combining the contributions we find z L − s − s . Therefore we need s + s > L .Summarizing, a four-particle amplitude is constructible, i.e., M (1 , ( z ) vanishes as z → ∞ if s + s > L , s + s > L + L ′ − s + s > L . It is important to mention that theseare sufficient conditions but not necessary. Recall that we are interested in the behavior ofthe whole amplitude and not on that of individual diagrams. Sometimes it is possible thatthe sum of Feynman diagrams vanishes at infinity even though individual diagrams do not.Also possible is that since our analysis does not take into account the precise structure ofinteraction vertices, there might be cancellations within the same diagram. In other words,our Feynman diagram analysis only provides an upper bound on the behavior at infinity.Let us go back to the example in the previous section. There s = s = s , L = L ′ = s .17 h h h h + (cid:1) h h h (3 h )1 h h (4 h ) + (cid:1) h h h h FIG. 2: The three different kinds of Feynman diagrams which exhibit different behavior as z → ∞ .They correspond to the s -channel, t ( u )-channel and the four-particle coupling respectively. Note that s + s > L implies s >
0, as mentioned at the beginning of this section. Thesecond condition is empty and the third implies that L < s . Thus, our conclusions in theexample are valid only if s > s − s = 1 this excludes ( F ) terms and for s = 2 this excludes R terms. We willcomment on possible ways to make these theories constructible in section VII. B. Physical vs. Spurious Poles
There is an apparent contradiction when in section IV we used that the only poles of M ( z ) come from propagators and when earlier in this section we used that polarizationtensors behave as z − s .The resolution to this puzzle is very simple yet amusing. Recall that polarization tensorsare defined only up to the choice of a reference spinor µ or ˜ µ of positive or negative chiralitydepending on the helicity of the particle. The z -dependence in polarization tensors comesfrom the factors in the denominator of the form h λ ( z ) , µ i s or [˜ λ ( z ) , ˜ µ ] s . The deformedspinors are given by λ ( z ) = λ + zλ ′ (or ˜ λ ( z ) = ˜ λ + z ˜ λ ′ ) where λ ′ (or ˜ λ ′ ) are the spinors ofa different particle. Now we see that if µ is not proportional to λ ′ then individual Feynmandiagrams go to zero as z becomes large due to the z dependence in the polarization tensors.In the same way, individual Feynman diagrams possess more poles than just those comingfrom propagators. Now let us choose µ proportional to λ ′ . Then the z dependence inpolarization tensors disappears. We then find that individual Feynman diagrams do notvanish as z becomes large but they show only poles at the propagators. Recall that we are18ot interested in individual Feynman diagrams, but rather in the full amplitude, which isindependent of the choice of reference spinor. Therefore, since M ( z ) vanishes for large z forsome choice of reference spinors it must also do so for any other choice. This means that thepole at infinity is spurious. Similarly, poles coming from polarization tensors are spuriousas well. VI. MORE EXAMPLES
In this section we give more examples of how the four-particle test can be used to constrainmany theories. In previous sections we studied theories of a single particle of integer spin s and found that only s = 2 admits self-interactions. Here we allow for several particles of thesame spin. In this section we consider the coupling of a particle of spin s and one of spin 2.The spin s can be integer or half-integer. A. Several Particles Of Same Integer Spin
Consider theories of several particles of the same integer spin s . The idea is to see whetherallowing for several particles relaxes the constraint found in section IV.B.2 that sets s = 2.We are interested in four-particle amplitudes where each particle carries an extra quantumnumber. We can call it a color label. The data for each particle is thus { λ ( i ) , ˜ λ ( i ) , h i , a i } .As discussed in section III.B, the most general three-particle amplitudes possess couplingconstants that can depend on the color of the particles. Here we drop the superscripts H and A in order to avoid cluttering the equations and define κ a a a = κ − s f a a a where f a a a are dimensionless factors. The subscript (1 − s ) is the dimension of the coupling constant.Repeating the calculation that led to (26) but this time keeping in mind that we have tosum not only over the helicity of the internal particle but also over all possible colors, wefind M (1 , (0) = κ − s X a I f a a a I f a I a a A + κ − s X a I f a a a I f a I a a B , (32)while M (1 , (0) = κ − s X a I f a a a I f a I a a C + κ − s X a I f a a a I f a I a a D (33)19ith A = h , i h , ih , ih , ih , i (cid:18) h , i [1 , h , ih , i (cid:19) s − , B = h , i h , ih , ih , i (cid:18) h , i [1 , h , ih , i (cid:19) s − , C = h , i h , ih , ih , ih , i (cid:18) h , i [1 , h , ih , i (cid:19) s − , D = h , i h , ih , ih , i (cid:18) h , i [1 , h , ih , i (cid:19) s − . (34)In order to understand why we have chosen to factor out the pieces that survive when s = 1 let us study this case in detail.
1. Spin 1
Before setting s = 1 it is important to recall that three-particle amplitudes for any oddinteger spin did not have the correct symmetry structure under the exchange of particlelabels. At the end of section III, we concluded that if no other labels were introduced thenthe three-particle couplings had to vanish. Now we have theories with a color label. In thiscase, it is easy to check that in order to ensure the correct symmetry properties we mustrequire f a a a to be completely antisymmetric in its indices.Let us now set s = 1. The four-particle test requires M (1 , (0) − M (1 , (0) = 0. Firstnote that the factor in front of B and D are equal up to a sign (due to the antisymmetricproperty of f ). Therefore they can be combined and simplified to give X a I f a a a I f a I a a ( B + D ) = − X a I f a a a I f a I a a (cid:18) h , i h , ih , ih , ih , i (cid:19) (35)where the right hand side was obtained by a simple application of the identity h , ih , i + h , ih , i = h , ih , i which follows from the fact that spinors are elements of a two-dimensional vector space .Note that the right hand side of (35) can nicely be combined with the other terms to giverise to the following condition X a I f a a a I f a I a a + X a I f a a a I f a I a a + X a I f a a a I f a I a a = 0 . (36) Readers familiar with color-ordered amplitudes possibly have recognized (35) as the U (1) decouplingidentity, i.e., A (1 , , ,
4) + A (2 , , ,
4) + A (2 , , ,
4) = 0. f a a a are the structure constants of a Lie algebra.
2. Spin 2
After the success with spin 1 particles, the natural question is to ask whether a similarstructure is possible for spin 2. Once again, before setting s = 2 let us mention that likein the case of odd integer spin particles, the requirement of having the correct symmetryproperties under the exchanges of labels implies that the dimensionless structure constants, f a a a , must be completely symmetric for even integer spin particles.Imposing the four-particle test using (32) and (33) we find that the most general solutionrequires X a I f a a a I f a I a a = X a I f a a a I f a I a a (37)which due to the symmetry properties of f abc implies that all the other products of structureconstants are equal and they factor out of (32) and (33) leaving behind the amplitudes fora single spin 2 particle which we know satisfy the four-particle test.Note that (37) implies that the algebra defined by E a ⋆ E b = f abc E c (38)must be commutative and associative. It turns out that those algebras are reducible andthe theory reduces to that of several non-interacting massless spin 2 particles. This provesthat it is not possible to define a non-abelian generalization of a theory of spin 2 particlesthat is constructible . The same conclusion was proven by using BRST methods in [35].Finally, let us mention that for s > B. Coupling Of A Spin s Particle To A Spin 2 Particle
Our final example of the use of the four-particle test is to theories of a single spin s particle (Ψ) and a spin 2 particle ( G ). Here we assume that the spin 2 particle only has We thank L. Freidel for useful discussions about this point. − ) and ( − − +). This means that we are dealing witha graviton. Let the coupling constant of three gravitons be κ while that of a graviton totwo Ψ’s be κ ′ . Assume that the graviton coupling preserves the helicity of the Ψ particle.This implies that κ and κ ′ have the same dimensions. Also assume that there no any cubiccoupling of Ψ’s .We need to analyze two different 4 particle amplitudes: M ( G , G , Ψ , Ψ ) and M (Ψ , Ψ , Ψ , Ψ ).Consider first M (Ψ − , Ψ +2 , Ψ − , Ψ +4 ) under a BCFW deformation. A Feynman diagramanalysis shows that the theory is constructible, i.e., the deformed amplitude vanishes atinfinity, for s >
1. This implies that the following discussion applies only to particles Ψ’swith spin higher than 1.Let us consider the four-particle test. We choose to deform (1 − , + ) and (1 − , + ): M (1 , = ( κ ′ ) h , i [1 ,
4] [2 , s [1 , s − [2 , [3 , s − M (1 , = ( κ ′ ) h , i [1 ,
2] [2 , s [1 , s − [3 , [2 , s − . (39)Notice that M (1 , is obtained from M (1 , by exchanging 2 and 4. Taking the ratio of thequantities in (39) leads to: M (1 , M (1 , = (cid:16) ts (cid:17) s − , (40)where s = P and t = P . This ratio is equal to one only if s = 3 /
2. Thus, the only particlewith spin higher than 1 which can couple to a graviton, giving a constructible theory, hasthe same spin as a gravitino in N = 1 supergravity.At this point the couplings κ and κ ′ are independent and it is not possible to concludethat the theory is linearized supergravity. Quite nicely, the next amplitude constrains thecouplings.Consider the four-particle test on the amplitude M ( G , G , Ψ , Ψ ). Again we choose to This last condition is not essential since such a coupling would have dimension different from that of κ and κ ′ and hence it would have to satisfy the four-particle test independently. ,
2) and (1 , M (1 , = − ( κ ′ ) h , i [2 , s +2 [1 , [3 , [2 , s − stu M (1 , = κ ′ h , i [2 , s +2 [1 , [2 , s − (cid:18) κ s + κ ′ u (cid:19) (41)where u = P .Taking their ratio and setting s = 3 /
2, we get M (1 , M (1 , = 1 − ut (cid:16) κκ ′ − (cid:17) . (42)Requiring the right hand side to be equal to one implies that κ ′ = κ . This means thatthis theory is unique and turns out to agree with linearized N = 1 supergravity.An interesting observation is that the local supersymmetry of this theory arises as anaccidental symmetry. The only symmetry we used in our derivation was under the Poincar´egroup; not even global supersymmetry was assumed. It has been known for a long time [36]that if one imposes global supersymmetry, then N = 1 supergravity is the unique theory ofspin 2 and spin 3 / N = 1 supergravity was succes-sively [37] derived from the non-interactive form by using gauge invariances. More recentlyand by using cohomological BRST methods, the assumption of global supersymmetry wasdropped [38].Finally, let us stress that this analysis does not apply to the coupling of particles withspin s ≤ VII. CONCLUSIONS AND FUTURE DIRECTIONS
Starting from the very basic assumptions of Poincar´e invariance and factorization of theS-matrix, we have derived powerful consistency requirements that constructible theoriesmust satisfy. We also found that many constructible theories satisfy the conditions only ifthe S-matrix is trivial. Non-trivial S-matrices seem to be rare.The consistency conditions we found came from studying theories where four-particlescattering amplitudes can be constructed out of three-particle ones via the BCFW con-struction. While failing to satisfy the four-particle constraint non-trivially means that the23heory should have a trivial S-matrix, passing the test does not necessarily imply that theinteracting theory exists. Once the four-particle test is satisfied one should check the five-and higher-particle amplitudes. A theory where all n -particle amplitudes can be determinedfrom the three-particle ones is called fully constructible.It is interesting to note that Yang-Mills [19] and General Relativity [22] are fully con-structible. This means that the theories are unique in that once the three-particle amplitudesare chosen (where the only ambiguity is in the value of the coupling constants) then thewhole tree-level S-matrix is determined. In the case of General Relativity it turns out thatgeneral covariance emerges from Poincar´e symmetry. In the case of Yang-Mills, the struc-ture of Lie algebras, i.e., antisymmetric structure constants that satisfy the Jacobi identity,also emerges from Poincar´e symmetry. In both cases, the only non-zero coupling constantsof three-particle amplitudes were chosen to be those of M (+ + − ) and M ( − − +). Itis important to mention that our analysis does not discard the possibility of theories withthree-particle amplitudes of the form M ( − − − ) and M (+ + +). Dimensional analysisshows that these theories are non-constructible due to the high power of momenta in thecubic vertex. For example, if s = 2 one finds six derivatives. Indeed, for spin 2, Wald [40]found consistent classical field theories that propagate only massless spin 2 fields and whichare not linearized General Relativity. Those theories do not possess general covariance andthe simplest of them possesses cubic couplings with six derivative interactions. In this classof theories might be the spin 3 self-interaction, which seems to be possible from [41], as wellas the recent proposal for spin 2 and spin 3 interaction of [42].There are some natural questions for the future. One of them is to ask what the cor-responding statements are if one replaces Poincar´e symmetry by some other group. Inparticular, it is known that interactions of higher spins are possible in anti-de Sitter space(see [39] and references therein). It would be interesting to reproduce such results from anS-matrix viewpoint.The constraints we obtained in this paper only concern the pole structure of the S-matrix.It is natural to expect that branch cuts might lead to more constraints. In field theory one isvery familiar with this phenomenon; some theories that are classically well defined becomeanomalous at loop level. It would be very interesting to find out whether the approachpresented in this paper can lead to constraints analogous to anomalies. Speculating evenmore, one could imagine that since three-particle amplitudes are determined exactly, even24on-perturbatively, then it might be possible to find constraints that are only visible outsideperturbation theory.A well known way to handle quantum corrections is supersymmetry. A natural general-ization of the results of this paper is to replace Poincar´e symmetry by super Poincar´e andthen explore consistency conditions for theories involving different supermultiplets.All of these generalizations, if possible, will only be valid for the set of constructibletheories. In order to increase the power of these constraints one has to find ways of relaxingthe condition of constructibility. Two possibilities are worth mentioning.The first approach is to compose several BCFW deformations [43] so that more polar-ization tensors vanish at infinity and make the amplitude constructible. This procedureworks in many cases but it is not very useful for four particles since deforming three parti-cles means that one has to sum over all channels at once and the four-particle constraint isguaranteed to be satisfied. One can however go to five and more particles and then therewill be non-trivial constraints.Some peculiar cases can arise because, as it was stressed in section V, the behavior atinfinity obtained by a Feynman diagram analysis is only an upper bound. It turns outin many examples that a Feynman diagram analysis shows a non-zero behavior at infinityunder a single BCFW deformation and a vanishing behavior under a composition of BCFWdeformations. Using the composition, one computes the amplitude which naturally comesout in a very compact form. When one takes this new compact, but equivalent, form ofthe amplitude and looks again at the behavior under a single BCFW deformation, one findsthat it does go to zero at infinity! This shows that there are cancellations that are notmanifest from Feynman diagrams. It would be very interesting if there was a simple andsystematic way of improving the Feynman diagram analysis so that it will produce tighterupper bounds. It would be even more interesting to find a way of carrying out the analysisonly in terms of the S-matrix.The second possibility is to introduce auxiliary massive fields such that quartic verticeswith too many derivatives arise as effective couplings once the auxiliary field is integratedout. Propagators of the auxiliary field create poles in z whose location is proportional to themass of the auxiliary field. The theory is then constructible, in the sense that no poles arelocated at infinity. Once the amplitudes are obtained one can take the mass of the auxiliaryfield to infinity and then recover the original theory. This gives a nice interpretation to the25hysics at infinity of some non-constructible theories: the presence of poles at infinity impliesthat the theory is an effective theory where some massive particles have been integrated out .The simplest example is a theory of a massless scalar s = 0. Recall that one condition fora theory to be constructible is that the quartic interaction has to have l < s derivatives.In the case at hand, with s = 0, this means that the quartic interaction must be absent.Therefore, a scalar theory with a λφ interaction is not constructible. In the appendix,we show that this theory can be made constructible by introducing an auxiliary field (anddeforming three particles).A necessary ingredient to carry out the program of auxiliary fields is to find three-particleamplitudes where one or more particles are massive. More generally, it will be interesting toextend our methods for general massive representations of the Poincar´e group. A good reasonto believe that this might be possible is the analysis of [44] where amplitudes of massivescalars and gluons were constructed using a suitable modification of BCFW deformations.In the case of massive particles of higher spins one might try to generate a mass term usingthe Higgs mechanism.Finally, there are two more directions that, in our view, deserve further study. The firstis the extension to theories in higher or less number of dimensions, including theories inten dimensions. The second is to carry out a systematic search for theories where severalthree-particle amplitudes might have coupling constants with different dimensions but thatwhen multiplied to produce four-particle amplitudes produce accidental degeneracies. Suchdegeneracies might lead to new consistent non-trivial theories which we might call exceptionaltheories . Acknowledgments
The authors would like to thank E. Buchbinder, B. Dittrich, L. Freidel, X. Liu and S.Speziale for useful discussion. PB would like to thank Perimeter Institute for hospitalityduring a visit where part of this research was done. The authors are also grateful to NatashaKirby for reading the manuscript. The research of FC at Perimeter Institute for TheoreticalPhysics is supported in part by the Government of Canada through NSERC and by theProvince of Ontario through MRI. 26
PPENDIX A: RELAXING CONSTRUCTIBILITY: AUXILIARY FIELDS
Our proposal for studying arbitrary spin theories is very general, but it suffers from thefact that some interesting theories are not constructible. In section VII, we mentioned severalways of trying to extend the range of applicability of our technique. One of them was theintroduction of auxiliary fields. In this appendix we illustrate the idea by showing how the λφ theory, which is not constructible (even under compositions of BCFW deformations),can be thought of as the effective theory of a constructible theory which contains a massivefield. The constructibility here is under a composition of two BCFW deformations.The failure to be constructible of the four-particle amplitude in the λφ theory is under-stood as a consequence of sending the mass of the heavy auxiliary field to infinity.Let us start with a massless scalar with a λφ interaction: L ( φ ) = 12 ( ∂ µ φ ) ( ∂ µ φ ) − λ φ . (A1)We can remove the quartic coupling by introducing a massive auxiliary field χ : L ( φ, χ ) = 12 ( ∂ µ φ ) ( ∂ µ φ ) + 12 ( ∂ µ χ ) ( ∂ µ χ ) − m χ χ − gχφ . (A2)It is straightforward to check that (A1) can be obtained from (A2) by integrating out thefield χ taking the limit of large g and large m χ , and by keeping g / m χ ≡ λ/
4! finite.The theory (A2) now has only cubic interactions. Since massless scalar fields do notpossess polarization tensors that can be made to vanish at infinity, the theory with onlycubic interactions is still not constructible under a BCFW deformation of two particles.This problem is resolved by applying a composition and deforming three particles.Another problem one has to deal with is that the new vertex in (A2) involves a massivescalar. This implies that the analysis of section III is not readily applicable. However, inthis specific case, the three particle amplitude is simply given by the coupling constant g .Since we are interested in the scattering of the massless scalars represented by the field φ , we consider only amplitudes where χ appears as an internal particle. This means that aninternal propagator takes the form 1 P − m χ . (A3)Let M ( φ , φ , φ , φ ) be the four particle amplitude of interest. From Feynman diagrams,27t is easy to see that it is given by M ( φ , φ , φ , φ ) = X j =2 g P j − m χ , (A4)where P j = p (1) + p ( j ) . Already from (A4), one can see that the correct limit leads to thefour point vertex of the original theory: g P j − m χ → − g m χ ∼ λ. (A5)Let us apply a three-particle deformation:˜ λ (1) ( z ) = ˜ λ (1) − z (cid:18) [1 , ,
3] ˜ λ (2) + [1 , ,
4] ˜ λ (4) (cid:19) λ (2) ( z ) = λ (2) + z [1 , , λ (1) λ (4) ( z ) = λ (4) + z [1 , , λ (1) . (A6)A Feynman diagram analysis shows that the deformed amplitude vanishes at infinity as z − .Taking the t -channel as an example, the deformed propagator in this channel is:1 P ( z ) − m χ , P ( z ) = P − z [1 , , λ (1) ˜ λ (2) , (A7)and its pole is given by z u = [2 , ,
3] ( P − m χ ) h , i [2 , . (A8)The momentum P on-shell becomes: P ( z u ) = P − ( P − m χ ) h , i [2 , λ (1) ˜ λ (2) . (A9)As stated at the beginning of the appendix, the three-particle amplitude is just thecoupling constant g , so it is easy to reconstruct the result (A4) and, as a consequence, (A5). [1] S. Weinberg, “Feynman Rules for Any Spin. 2. Massless Particles,” Phys. Rev. (1964)B882.[2] S. Weinberg, “Photons and Gravitons in S Matrix Theory: Derivation of Charge Conservationand Equality of Gravitational and Inertial Mass,” Phys. Rev. (1964) B1049.
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