On the Theory of Continuous-Spin Particles: Helicity Correspondence in Radiation and Forces
OOn the Theory of Continuous-Spin Particles:Helicity Correspondence in Radiation and Forces
Philip Schuster ∗ and Natalia Toro † Perimeter Institute for Theoretical Physics, Ontario, Canada, N2L 2Y5 (Dated: November 5, 2013)We have recently shown that continuous-spin particles (CSPs) have covariantsingle-emission amplitudes with the requisite properties to mediate long-range forces.CSPs, the most general massless particle type consistent with Lorentz symmetry, arecharacterized by a scale ρ . Here, we demonstrate a helicity correspondence at CSPenergies larger than ρ , in which these amplitudes are well approximated by thefamiliar ones for particles of helicity 0, ±
1, or ±
2. These properties follow fromLorentz invariance. We also construct tree-level multi-emission and CSP-exchangeamplitudes that are unitary, appropriately analytic, and consistent with helicity-0correspondence. We propose sewing rules from which these amplitudes and otherscan be obtained. We also exhibit a candidate CSP-graviton matrix element, whichshows that the Weinberg-Witten theorem does not apply to CSPs. These resultsraise the surprising possibility that the known long-range forces might be mediatedby CSPs with very small ρ rather than by helicity 1 and 2 particles. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] N ov CONTENTS
I. Introduction 3II. Review of Continuous-Spin Particles 5A. Poincar´e Transformations on One-Particle States and the Little Group 5B. Wavefunctions 8C. Soft Factors for CSP Emission 10III. Helicity Correspondence in Soft Emission 11A. Preliminaries: Soft Factors and Amplitudes 13B. Scalar Correspondence 14C. Photon Correspondence 17D. Graviton Correspondence 20E. General CSP Soft Factors Reconsidered 22IV. Infinite-Spin Limit vs. Correspondence Limit 24V. Helicity Correspondence and Unitarity 26A. Correspondence in Multi-CSP Scattering Amplitudes 27B. A Unitary Ansatz for Intermediate-CSP Amplitudes 29C. Scalar-Like CSP Sewing Rules 34VI. CSP Interactions with Gravity 36VII. Conclusions and Future Directions 38Acknowledgments 41References 41
I. INTRODUCTION
The known long-range forces in Nature are consistently modelled by exchange of helicity 1and 2 particles. The absence of forces mediated by higher or lower helicities can be simplyexplained: Lorentz-invariance requires higher-helicity interactions to fall too fast in theinfrared to mediate long-range forces [1], while scalars’ masses are unstable to radiativecorrections. But another type of massless particle is allowed by Lorentz symmetry — theso-called “continuous-spin” particles (CSPs) [2]. New CSP emission amplitudes [3] satisfyconstraints analogous to Weinberg’s “soft theorems”[1, 4, 5], suggesting that CSPs can alsoconsistently mediate long-range forces.A CSP in 3+1 dimensions is labeled by a spin-scale ρ with units of mass. Its single-particle basis states take on arbitrary integer spins (a second type of CSP, not consideredhere, takes on all half-integer spins). Like massive particle polarizations, these spin statesmix under Lorentz boosts by an amount proportional to ρ . In the ρ = 0 limit the CSPfactorizes into a tower of states labeled by Lorentz-invariant integer helicities (see Figure 1).In this paper we show, firstly, that the soft CSP emission amplitudes found in [3] smoothlyapproach helicity-0, 1, and 2 amplitudes in the high-energy helicity-correspondence limit (energy E (cid:29) ρv for emitters at velocity v ). Secondly, we provide evidence that helicity-correspondence amplitudes are consistent with tree-level unitarity and can couple to gravity,focusing on the scalar-like case. These findings suggest a twist on Nature’s apparent prefer-ence for low helicities: in theories with CSPs, polarization modes of spin 2 or less dominateinteractions at energies larger than ρv . A world with macroscopic ρ − would appear, to agood approximation, to be governed at short distances by fixed-helicity gauge theories andgeneral relativity. If a full CSP theory of this form exists, it is conceivable that known long-range forces could be mediated by CSPs with very small (perhaps Hubble-scale) ρ rather thanhelicity 1 and 2 particles. In a forthcoming paper [6], we elaborate on the thermodynamicsof such a world, and estimate constraints from stellar production and radio transmission.Further related studies will appear in [7]. More sharply testing this speculation requires afull classical theory of interacting CSPs, a topic to be pursued in [8].In Section II we review the kinematics of CSPs and the main results of [3]: new wave-functions and “soft factors” (41) into which single CSP-emission amplitudes should factorizewhen the CSP becomes soft compared to other interacting particles.Section III exhibits the helicity correspondence of these soft factors, which arises in thehigh-energy limit E (cid:29) ρ or, for non-relativistic emitters with velocity v , E (cid:29) ρv (a moreprecise definition of the correspondence limit including angular dependence is ρ | z | (cid:28)
1, with z given by (50)). We define helicity- h correspondence to mean that spin- h emission ampli-tudes differ from standard helicity- h amplitudes by ( ρ | z | ) -suppressed corrections, whileamplitudes for other spins are suppressed by increasing powers of ρ | z | . We present scalar-,photon-, and graviton-like soft factors in the family (42) that exhibit helicity 0, ±
1, and ± Spin Basis | i| i| i| i| i| i| i ...... D nn ( ⇢~b ) | n i e in✓ | n i | n i rotationeigenstatetranslationsmix states rotationsmix states | i | + ✓ i e i~b.~t | i translationeigenstate Angle Basis ~t | ~t | = ⇢ FIG. 1. The figure summarizes the Little Group (LG) transformation of massless particle states (see § II A). Particle types are characterized by a scale ρ . Basis states may be labeled by a tower of integeror half-integer spins, or equivalently by angles on a circle. The two bases are related by Fouriertransform. The LG has the structure of the isometries of the Euclidean 2-plane, or ISO (2). Thespin basis diagonalizes LG rotations, while the angle basis diagonalizes LG translations. Lorentzboosts induce LG translations (and rotations), which mix states in the spin basis. The scale ρ controls the amount of mixing under boosts, much like the combination m × S for a spin- S massiveparticle. When ρ = 0, spin labels become Lorentz-invariant helicities. ple, required to obtain a helicity correspondence, follow from perturbative unitarity of theCSP interactions. This hierarchical structure permits approximate thermal equilibrium forthe correspondence state and any matter it couples to without thermalizing the full CSPtower [6], settling Wigner’s concern about CSP thermodynamics [9]. We exclude higher-helicity correspondence in theories with a cutoff scale much larger than ρ , while raising thepossibility that spins may interact democratically in theories with an ultraviolet cutoff near ρ . A related conjecture for double-valued CSPs was made in [10].In Section IV, we explore the connection between continuous-spin particles and the si-multaneous high-spin limit of a massive particle, with ( mass ) × ( spin ) = ρ held fixed. Thegroup-theory of CSPs is closely related to this high-spin limit [11]; the CSP soft factors allowus to also compare their dynamics. The relation of soft factors to this infinite-spin limit pro-vides some intuition for how the hierarchical coupling structure required by correspondencecan be consistent with Lorentz-invariance. But the connection is purely formal, not physical— finite-spin matrix elements do not approach CSP matrix elements in the infinite-spinlimit. This result underscores that the connection we highlight between helicity- h particlesand CSPs in the small- ρ limit is much closer and more physical than that between CSPsand the massive high-spin limit.Section V examines the consistency of scalar-like CSP soft factors with unitarity. We con-struct unitary tree-level multi-emission and exchange amplitudes mediated by an off-shellCSP that factorize into scalar-like soft factors, and propose candidate sewing rules for a fullperturbative S -matrix. Because the amplitudes and sewing rules are not rational functionsof momenta, they are less constrained by factorization than familiar tree-level amplitudes.Studying the loop-level unitarity of these sewing rules is an important problem that maysharpen either the structure of a CSP theory or potential physical obstructions. The inter-play of unitarity and covariance for photon-like or graviton-like CSPs is more subtle than inscalar-like theories, just as it is for photons and gravitons compared to scalars. Analogousconstructions in these cases are an important direction for future work; we highlight severalsubtleties particular to the construction of photon- and graviton-like CSP amplitudes.No simple analogue of the high-helicity Weinberg-Witten theorem [12] forbids CSP cou-plings to gravity, as discussed in VI. We exhibit a covariant, symmetric, and conservedrank-two tensor matrix element between single-CSP states, in contrast to the non-existenceof such matrix elements for high-helicity particles. These matrix elements also have a corre-spondence limit in which they approach familiar stress-energy tensors for a scalar or gaugeboson, though this correspondence is governed by the graviton momentum and possessessome unusual characteristics. It is still not clear whether these matrix elements arise froma conserved tensor operator or whether they are compatible with CSP-matter interactions.Of course, a helicity-2-like CSP might yield consistent gravitational theories even if suchobstructions are found, and merits further study. II. REVIEW OF CONTINUOUS-SPIN PARTICLES
This section summarizes properties of continuous-spin particles. The classification ofparticle degrees of freedom that transform as unitary irreducible representations (irreps)of the Poincar´e group has proved a valuable principle for understanding theories of na-ture. Continuous-spin particles are naturally defined from this perspective. We reviewthe definition, state-space, and kinematics for continuous-spin particles (SSII A), covariant“wavefunctions” on which Lorentz transformations induce transformations appropriate toa single-particle state (SSII B), and the form of soft emission factors required by Lorentz-invariance (SSII C). All of the material in this section is expanded upon in [3].
A. Poincar´e Transformations on One-Particle States and the Little Group
Because the translation generators P µ mutually commute, one-particle states can belabeled by a c -number momentum eigenvalue k µ and by internal labels a whose detailedform we will constrain shortly. We may write the action of a Lorentz transformation Λ oneach state as U (Λ) | k, a (cid:105) = (cid:88) a (cid:48) D (Λ , k ) aa (cid:48) | Λ k, a (cid:48) (cid:105) , (1)where the transformation matrix D must be unitary with respect to the norm (cid:104) k, a | k (cid:48) , a (cid:48) (cid:105) = 2 k δ (3) ( k − k (cid:48) ) δ aa (cid:48) . (2)For (1) to be satisfied, the a labels must furnish an irreducible representation of the LittleGroup LG k , the subgroup of Lorentz transformations such that Λ µν k ν = k µ . We exhibitsome properties of this group (focusing on the massless case), then we will return to (1) inthe case of general Lorentz transformations.The Little Group LG k is generated by the components of w µ ≡ (cid:15) µνρσ k ν J ρσ , (3)closely related to the Pauli-Lubanski pseudo-vector. Because w.k = 0, w µ has three indepen-dent components and the Little Group in 3+1 dimensions is always 3-dimensional. Thesecomponents obey the commutation relation[ w µ , w ν ] = − i (cid:126) (cid:15) µνρσ w ρ k σ . (4)The operator w = w µ w µ , with dimensions of mass , is a Casimir of the Poincar´e group,and therefore constant on all irreps of the Little Group. In the case of a massive particles,we see that for k µ = ( m, , , w µ = (0 , m(cid:126) J ); the components have group structure SO (3)and in the spin- S representation w = − m (cid:126) J = − m S ( S + 1) . (5)Indeed, it is reassuring that the left- and right-hand sides of this equation are manifestlycovariant, unlike the three-vector expression in the middle.For null k µ , we introduce a light-cone frame with k as one of its axes (see [13] for a similartreatment). We choose space-like (cid:15) , ( k ) with (cid:15) i ( k ) = − (cid:15) ( k ) .(cid:15) ( k ) = (cid:15) i ( k ) .k = 0, andthe unique vector q µ ( k ) satisfying q ( k ) = 0 , q ( k ) .k = 1, and q ( k ) .(cid:15) i ( k ) = 0. It will also beuseful to work with (cid:15) µ ± ( k ) ≡ ( (cid:15) µ ( k ) ± i(cid:15) µ ( k )) / √
2. In terms of this frame, we can identify thecomponents of w µ and their commutation relations as R ≡ q.w T , ≡ (cid:15) , .w, ( T ± ≡ √ (cid:15) ± .w ) , (6)[ R, T , ] = ± iT , , [ T , T ] = 0 , ([ R, T ± ] = ± T ± , [ T ± , T ∓ ] = 0) . (7)For example, taking ¯ k µ = ( ω, , , ω ) , (cid:15) = (0 , , , , (cid:15) = (0 , , , , (8)we find w µ = − ¯ k µ R + ˆ (cid:15) µ T + ˆ (cid:15) µ T (9)with R = J , T = ω ( J + J ), and T = − ω ( J + J ).These are the commutation relations of the rotation and translation generators of ISO (2).The Casimir in this case is w = W = − (cid:126)T = − T + T − , (10)which is completely independent of the action of the rotation generator! Any Little Groupelement can be labeled by an angle θ and a two-vector (cid:126)b (or a complex number β = ( b + ib ) / √
2) with units of length through the decomposition W ( θ, β ) ≡ e i(cid:126)b. (cid:126)T e − iθR = e i √ ( βT − + β ∗ T + ) e − iθR . (11)This group permits two types of single-valued unitary representation (we will not discussdouble-valued representations, though these also exist). The helicity representations consistof a single state | k, h (cid:105) transforming as W ( θ,(cid:126)b ) | k, h (cid:105) = e ihθ | k, h (cid:105) . (12)Since these states are annihilated by the translation generators, they have vanishing w .The so-called “continuous-spin” representations of ISO (2) are characterized by w = − ρ for any ρ with dimensions of momentum. These have infinitely many states, which can beparametrized in two particularly useful bases: simultaneous eigenstates of T , (the anglebasis, with states labeled by an angle φ ) or eigenstates of R (spin basis, with states labeledby an arbitrary integer n ). These are simply related by Fourier transform. We begin withthe former basis which has much simpler Little Group transformation rules, then transformto the latter, which recovers a tower of all integer-helicity particles in the ρ → T , , subject to W = − ρ , have the interpretation of plane-waves in the ISO (2) of fixed “momentum” norm (cid:126)T = ρ . They can be labeled by an angle φ such that (cid:126)T | k, φ (cid:105) = ( ρ cos φ, ρ sin φ ) | k, φ (cid:105) , R | k, φ (cid:105) = i∂ φ | k, φ (cid:105) . (13)Generic Little Group elements transform the states as W ( θ,(cid:126)b ) | k, φ (cid:105) = e i(cid:126)b.(cid:126)t φ + θ | k, φ + θ (cid:105) (14)= e iρ Re[ √ βe − i ( φ + θ ) ] | k, φ + θ (cid:105) (15)= (cid:90) dφ (cid:48) π D φφ (cid:48) [ θ, β ] | k, φ (cid:48) (cid:105) , with D φφ (cid:48) [ θ, β ] = (2 π ) δ ( φ (cid:48) − φ − θ ) e iρ Re[ √ βe − iφ (cid:48) ] , (16)which is unitary with respect to the inner product (cid:104) k, φ | k (cid:48) , φ (cid:48) (cid:105) = 2 π k δ ( k − k (cid:48) ) δ ( φ − φ (cid:48) ) . (17)The Lorentz-invariant sum over states is (cid:90) d (cid:126)kk dφ π . (18)We may transform to the spin basis by defining | k, n (cid:105) ≡ (cid:90) dφ π e inφ | k, φ (cid:105) , (19)for integer n , which have inner product (cid:104) k (cid:48) , n (cid:48) | k, n (cid:105) = δ nn (cid:48) k δ ( k − k (cid:48) ) . (20)Little Group transformations act as W [ θ, β ] | k, n (cid:105) = D nn (cid:48) [ θ, β ] | k, n (cid:48) (cid:105) , (21) D nn (cid:48) [ θ, β ] = e − inθ ( ie iα ) ( n − n (cid:48) ) J n − n (cid:48) ( ρ √ | β | ) , (22)where β = | β | e iα . The appearance of Bessel functions is to be expected, as they are repre-sentation functions for the Euclidean group in two dimensions (see for example [14] or [15]).We call this a “spin” rather than “helicity” basis because they mix under general LittleGroup actions, in contrast with helicity states. In the limit ρ →
0, however, J n − n (cid:48) ( ρ | β | )approaches zero for n (cid:54) = n (cid:48) and 1 for n = n (cid:48) , so we recover a direct sum of all integer-helicitystates with the transformation rule D nn (cid:48) [ θ, β ] → e − inθ δ nn (cid:48) . (23)So far we have considered only the action of little-group elements that keep a momentum k µ invariant. The action of other Lorentz transformations on single-particle states is dictatedpartly by convention. It is standard to construct all states in a given Poincar´e irrep fromthe states at fixed reference momentum ¯ k µ , as follows. For each k we choose a “standardLorentz transformation” B k such that ( B k ) µν ¯ k ν = k µ , for which we define D ( B k , k ) aa (cid:48) ≡ δ aa (cid:48) . (24)The action of any Lorentz transformation Λ on one-particle states is then determined bygroup composition of (15) and (24) to be U (Λ) | k, a (cid:105) = D ( W Λ ,k , k ) aa (cid:48) | Λ k, a (cid:48) (cid:105) with W Λ ,k ≡ B − k Λ B k ∈ LG ¯ k , . (25)We will find it useful to think of states at given momentum k as having their “own” LittleGroup. This is equivalent to the B k construction, provided that we choose, for each k µ ,a light-cone frame with (cid:15) µ ± ( k ) ≡ B kµν (cid:15) ν ± (¯ k ) . Since the action of B k on ¯ k µ and (cid:15) µ ± (¯ k ) fullyspecifies a generator B k , we can view any choice of (cid:15) ± ( k ) as implicily defining the “standardLorentz transformation” B k .It will be desirable to work in a basis where, for every k , (cid:15) ± ( k ) = 0. Among other things,in this basis R is the usual “helicity operator”, ˆk .(cid:126) J (the non-covariance of this operatorcould have been our first hint that no symmetry guarantees Lorentz-invariance of masslessparticles’ helicities). One choice of B k that guarantees this, starting from (8), is B k = e iφJ z e iθJ y e i log( | k | /ω ) K z for k = | k | (1 , sin θ cos φ, sin θ sin φ, cos θ ) . (26)Generalizations of continuous-spin representations in higher dimensions, with supersym-metry, and in other contexts are discussed in [16–21] B. Wavefunctions
Although Wigner long ago formulated covariant wave equations for continuous-spin par-ticles [2, 9, 22–24], they are not the most general wave equations whose solutions transformas CSPs. As Wigner himself noted several times, wave equations and Poincar´e representa-tions are not associated uniquely, and we can expect multiple alternative wave equations todescribe the same type of particle excitation. Nonetheless, most of the literature on CSPsfor the last seven decades [25–28] relies almost exclusively on Wigner’s equations of motion(one exception is [29]).In [3], we sought families of wavefunctions possessing both Lorentz and Little Grouplabels, on which the actions of the two were related by the covariance equation (cid:88) a (cid:48) D aa (cid:48) [ W Λ ,k ] ψ ( Λk , a (cid:48) , l ) = (cid:88) ¯ l D − l ¯ l [Λ] ψ (cid:0) k , a, ¯ l (cid:1) , (27)where l , ¯ l and a , a (cid:48) are Lorentz and little-group indices respectively. In the case of infinite-dimensional single-valued representations of the Lorentz group, one is led to seek out so-lutions whose Lorentz transformation is carried not by a tensor index, but by an auxiliaryvector η µ , with D [Λ] ψ ( η, . . . ) ≡ ψ (Λ η, . . . ) , (28)or similarly for an auxiliary Weyl spinor field ξ α [30]. While the latter approach was pursuedby [29], we have found the auxiliary vector form to be and at least equally useful, even thoughthe resulting Lorentz representations are generically not irreducible. In this form, and usingthe angle basis for the little group, the covariance equation (27) becomes (cid:90) dφ (cid:48) π ψ ( { Λk , φ (cid:48) } , η µ ) D φ (cid:48) φ [ W (Λ , k )] = ψ (cid:0) { k , φ } , Λ − η (cid:1) . (29)It is sufficient to solve this equation for Λ ∈ LG k , which is readily done by linearizing theLorentz and Little Group actions of the three generators: − i ( η.(cid:15) − (cid:15) + .∂ η − η.(cid:15) + (cid:15) − .∂ η ) ψ = ∂ φ ψ (30) − ( η.(cid:15) − k.∂ η − η.k(cid:15) − .∂ η ) ψ = ρ √ e − iφ ψ (31)( η.(cid:15) + k.∂ η − η.k(cid:15) + .∂ η ) ψ = ρ √ e iφ ψ (32)These equations are homogeneous in η , are Fourier-conjugate to themselves, and imply(using k = 0) W ψ = (cid:0) k.η k.∂ η η.∂ η − ( k.η ) ∂ η − η ( k.∂ η ) (cid:1) ψ = − ρ ψ. (33)The covariance equations above have two classes of solutions, depending on whether theyhave singular or smooth support near η.p = 0. The first, singular class can be written as ψ ( { k, φ, f } , η ) = (cid:90) drf ( r ) (cid:90) dτ δ ( η − r(cid:15) ( kφ ) − rτ k ) e − iτρ , (34)with k = 0, f ( r ) arbitrary, and (cid:15) ( kφ ) ≡ i √ (cid:15) + ( k ) e − iφ − (cid:15) − ( k ) e iφ ) = −√ (cid:2) (cid:15) + e − iφ (cid:3) . (35)0These solutions are supported on a plane in η -space, with a profile under η -rescaling de-termined by f ( r ). Because they satisfy p ψ = p.ηψ = 0, the W equation simplifies to( − η ( p · ∂ η ) + ρ ) ψ = 0. These, together with the equation ( η + 1) ψ = 0, are the Wignerequations, which are satisfied by f ( r ) = δ ( r −
1) (in which case ψ is only supported on aline in η ). However, alternate η -space equations such as η.∂ η ψ = 0 single out different f ( r )but yield equally covariant wavefunctions.Smooth solutions of the covariance equations (31)–(32) take the form ψ ( { k, φ } , η ) = f ( η.k, η ) e iρ η.(cid:15) ( kφ ) η.k = f ( η.k, η ) e − iρ √ (cid:18) η.(cid:15) +( k ) e − iφη.k (cid:19) (36)in the angle basis, with f an arbitrary function. Two further wave equations besides k ψ = 0and ( W + ρ ) ψ = 0 must be imposed to fully fix f . One simple choice is k.∂ η ψ = 0, η.∂ η ψ = mψ , which fixes f ( η.k, η ) = ( η.k ) m . Equivalently, the smooth wavefunctions inthe spin-basis are ψ ( { k, n } , η ) = f ( η.k, η ) e in arg (cid:104) − η.(cid:15) +( k ) η.k (cid:105) J n (cid:18) ρ √ (cid:12)(cid:12)(cid:12)(cid:12) η.(cid:15) + ( k ) η.k (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) . (37)The subclass of wavefunctions supported on f ( η.k, η ) = δ ( η )( η.k ) c for arbitrary complex c are closely related to the wavefunctions found by [29]. For a discussion of the relationbetween the above wavefunctions and the equations of motion for high-helicity particles(appropriate when ρ = 0) [31, 32], see [3]. C. Soft Factors for CSP Emission
Together, Lorentz-invariance, locality, and unitarity impose significant constraints onhow different massless particles can interact. These constraints are particularly simple inthe case of amplitudes involving n particles of momentum p , . . . , p n , plus a single massless“soft particle” whose momentum k satisfies k.p i (cid:28) p i .p j for all i, j . Famously, Weinbergfound these limits to be so constraining as to exclude any coupling of helicity h > k.p i →
0, unitarity implies that these amplitudes are dominated by “externalemission” terms where the soft particle is emitted by one of the in- or out-going hardparticles. The sum of these contributions takes the form A ( p , . . . , p n , { k, a } ) = A ( p , . . . , p n ) × (cid:34) n (cid:88) i =1 ± p i .k + i(cid:15) × s i ( { k, a } , p i ) (cid:35) + O ( | k | ) , (38)where a denotes the Little Group state of the soft particle, and the upper (lower) sign in thepropagator corresponds to outgoing (incoming) momenta. In this expression, the “parent1amplitude” A , which involves the n hard particles but not the soft particle, is universal,while each external emission off the i ’th leg is proportional to a distinct “soft factor” s i .Simultaneous Lorentz-covariance of the parent n -point amplitude and the n + 1-point softamplitude imply a very simple covariance relation on the object in square brackets: f ( { k, a } , p , . . . , p n ) = (cid:88) a (cid:48) D ∗ aa (cid:48) [ W Λ ,k ] f ( { Λ k, a (cid:48) } , p (cid:48) , . . . , p (cid:48) n ) + O ( | k | ) , (39)where f ( { k, a } , p , . . . , p n ) ≡ n (cid:88) i =1 ± p i .k + i(cid:15) × s i ( { k, a } , p i ) . (40)This condition is most simply satisfied if the soft factors s i are separately covariant, thoughcancellations between terms are possible (and indeed required in the case of helicity ampli-tudes, which is the origin of the constraints in [1]).In [3], we exhibited a simple soft-factor ansatz that satisfies (39) term by term, obtainedsimply by evaluating the smooth covariant wavefunctions (36) at η i = p i : for CSP emission this gives s i ( { k, φ } , p i ) = f i ( k.p i ) e − iρ (cid:15) ( kφ ) .pik.pi = f i ( k.p i ) e + i √ ρ Im (cid:20) e − iφ (cid:15) + .pik.pi (cid:21) , and (41) s i ( { k, n } , p i ) = f i ( k.p i ) ˜ J n (cid:18) ρ √ (cid:15) + .p i k.p i (cid:19) (42)in the angle and spin bases, respectively, with˜ J n ( w ) ≡ ( − n e − in arg( w ) J n ( | w | ) . (43)CSP absorption soft factors are obtained by complex conjugation of the above. To highlightthe helicity correspondence, we will use the spin basis in most of this paper. But we remindthe reader that most non-trivial manipulations of the soft factors are much simpler if onefirst Fourier transforms them back to the angle basis.We have allowed for the possibility that each species i has a different f i , and omitted theexplicit dependence on η → p i = m i , since this is independent of kinematics for on-shell p . Whatever their form, the f i must have mass dimension 1. Since f must be smooth as k.p i →
0, a natural decomposition of f i is into terms f ( m ) ( k.p ) = g ( m ) ( k.p ) m . (44)where g ( m ) has mass-dimension 1 − m . III. HELICITY CORRESPONDENCE IN SOFT EMISSION
This section and the next explore helicity correspondence in a variety of CSP scatteringamplitudes. We have already seen from (23) a kinematic connection between CSPs and2fixed-helicity particles: in the ρ → interactions havea well-defined (and non-trivial) ρ → ρ is controlled by ρ/ ( energy ) ondimensional grounds. Thus for any finite ρ , the physical high-energy limit is also helicity-like. The precise scaling of the correspondence parameter z i ≡ −√ (cid:15) ∗ + ( k ) .p i /k.p i in differentkinematic regimes is discussed in III B.In this section, we explore helicity correspondence in the CSP soft factors (42). Becausesoft factors are almost fully constrained by Lorentz-invariance, we will be able to classifytheir correspondence systematically. At the same time, because generic amplitudes in aunitary theory of CSPs (if one exists) must factorize into soft factors, we expect this cor-respondence to persist in more general amplitudes of any full theory. As we will see in thenext section, the construction of more general CSP amplitudes is not always straightfor-ward. Nonetheless, soft factor correspondence will be a useful guide to exploring physicalconsequences of correspondence in a universe with CSPs [6].The pattern of correspondence exhibited by CSP soft factors, summarized in Table I, hasthree striking features: • Although the ρ → spectrum contains all integer helicities, only helicities 0, ±
1, and ± f ( p i .k ) in (42) forwhich only helicity 0, helicity ±
1, or helicity ± ρ → h correspondence soft factors, or as scalar-, photon-,or graviton-like soft factors respectively. • The soft factors recovered in the ρ → F µν ,rather than the soft factors associated with A µ J µ couplings in gauge theories). Thisis especially surprising because unlike the CSP soft factors (42), helicity-1 and 2 softfactors are not Lorentz-covariant. • Though we define correspondence through a ρ → ρ ! The criterion that selects helicity-0 and 1 correspondence is the absence ofan ultraviolet cutoff imposed by the tree-level interactions. Helicity-2-like interactionsmust of course have a cutoff (the Planck scale) but these are the interactions with the weakest cutoff-dependence, after the helicity-0 and 1 cases.Each of the features emphasized above will be made precise and justified in this section.We recall in § III A the relationship of soft factors to scattering amplitudes and the form of3
Scalar-like Photon-like Graviton-likeHelicity Soft Factor 1 q i (cid:15) ± ( k ) .p i M ∗ ( (cid:15) ± ( k ) .p i ) CSP Soft Factor(covariant form) a i ˜ J n ( ρz i ) − q i √ ρ p i .k ˜ J n ( ρz i ) ρ M ∗ [( p i .k ) − ρ p i ] ˜ J n ( ρz i )Constraint — (cid:80) in q i = (cid:80) out q i universal M ∗ Subtraction — − q i √ ρ p i .kδ n ρ M ∗ p i .k ( p i .kδ n + ρ(cid:15) + .kδ n − ρ(cid:15) − .kδ n, − )TABLE I. Soft factors for helicity ± h and CSP soft factors that exhibit correspondence; soft factoramplitudes are written in terms of z i ≡ √ (cid:15) ∗ + ( k ) .p i /k.p i and ˜ J n ( w ) ≡ ( − n e − in arg( w ) J n ( | w | ). Theconstraint in the third row for helicities ± M ∗ ). When the constraint is satisfied, the “subtraction” in the fourth row sums to zeroover all legs of any amplitude, ensuring both perturbative unitarity and correspondence of the CSPemission amplitudes. The combination (covariant CSP soft factor) – (subtraction) also defines acompletely equivalent (albeit non-covariant) subtracted CSP soft factor, which in the limit of small z approaches the helicity soft factor. gauge and gravity soft factors — these preliminaries form a necessary baseline for the dis-cussion of helicity correspondence. In § III B we demonstrate the helicity-0 correspondenceof the simplest CSP soft factors, and elaborate on the kinematic parameter that controlsdeviations of CSP soft emission amplitudes from the helicity-like form. Sections III C andIII D introduce soft factors with helicity-1 (photon-like) and helicity-2 (graviton-like) cor-respondence, and show how charge conservation conditions are related to the perturbativeunitarity of high-energy scattering. This discussion resolves the puzzle that CSP soft factorsare Lorentz-covariant while gauge/gravity soft factors are not, and motivates the introduc-tion of equivalent but non-covariant CSP soft factors that tighten the connection to gaugeand gravity soft-factors. We step back in Section III E to see why the most general CSP softfactor has only scalar-like, photon-like, and graviton-like correspondence.
A. Preliminaries: Soft Factors and Amplitudes
Before exploring correspondence in CSP soft factors, we review the standard soft factorsfor helicities 0, 1, and 2 that we will recover from CSPs in the ρ → § III E.4Soft factors are defined through the limit k.p i (cid:28) p i .p j of a single-emission amplitude A ( p , . . . , p n , { k, a } ). In this limit, the ( n +1)-point amplitude A must factorize as A ( p , . . . , p n , { k, a } ) = A ( p , . . . , p n ) × (cid:34) n (cid:88) i =1 ± p i .k + i(cid:15) × s i ( { k, a } , p i ) (cid:35) + O ( | k | ) , (45)where A is a parent amplitude of n “matter” legs, and the s i are soft factors that dependonly on the soft momentum k and the i ’th matter leg. In the case of helicity 0, ±
1, and ± s i ( { k, h = 0 } , p i ) = a i , s i ( { k, h = ± } , p i ) = q i (cid:15) ± ( k ) .p i , s i ( { k, h = ± } , p i ) = g ( (cid:15) ± ( k ) .p i ) (46)where a i , q i , and g have mass-dimension 1, 0, and -1 respectively. The latter two soft factorsare not individually Lorentz-covariant. Covariance of (45) therefore requires that the sum ofincoming charges q i equals the sum of outgoing charges (helicity ±
1) and that g is universal(helicity ± k → k , poles in A must be shifted and the sum over matter spins becomesnon-trivial. However, for scalar matter with momentum-independent parent amplitude A (i.e. matter legs interacting only through a non-derivative contact interaction), (45) is avalid amplitude for all k . For example, if p . . . p correspond to distinct particle types with A ( p , p → p , p ) = λ , A ( p , p → p , p , { k, a } ) = λ (cid:18) s ( { k, a } , p )( k − p ) + i(cid:15) + s ( { k, a } , p )( k − p ) + i(cid:15) (47)+ s ( { k, a } , p )( p + k ) + i(cid:15) + s ( { k, a } , p )( p + k ) + i(cid:15) (cid:19) . (48)This should be interpreted as the lowest O( λ ) contribution, at tree level. For the specialcase that only the outgoing particles couple to the radiated particle of momentum k , thisis illustrated diagrammatically in Figure 2. This formula implicitly defines s ( { k, a } , p i ) atarbitrary k (not just in the soft limit), which we will continue to refer to as a “soft factor”in the following discussion. We will return to the original definition of soft factors, as the dominant contributions to the k → § III E.
B. Scalar Correspondence
The general soft factor (42) involved an arbitrary pre-factor f i ( k.p i ) of mass-dimension1. The simplest example of correspondence arises for constant f i ( k.p i ) = a i , yielding s (0) i ( { k, n } , p i ) = a i ˜ J n ( ρz i ) (49)with z i ≡ √ (cid:15) ∗ + ( k ) .p i /k.p i , ˜ J n ( w ) ≡ ( − n e − in arg( w ) J n ( | w | ) . (50)5 FIG. 2. The construction of a candidate on-shell CSP amplitude is illustrated above. CSPs areattached to a parent amplitude A ( p , p ; p , p ) using the CSP soft factors, with appropriate matterpropagators included. This example presumes that only the outgoing matter legs couple to theCSP. The final result is the 5-point amplitude A ( p , p ; p , p , { k, φ } ) used below as an example toinvestigate certain aspects of CSP interactions. Importantly, differential emission cross-sections obtained from (48) using this soft factor arefinite, because (cid:88) n | s (0) i ( { k, n } , p i ) | = | a i | (51) (cid:88) n s (0) i ( { k, n } , p i ) s (0) j ( { k, n } , p j ) ∗ = a i a ∗ j J ( ρ | z i − z j | ) . (52)These results follow from Bessel identities, or even more simply by working in the anglebasis (in this case, the sums become integrals of pure phases over a finite interval, which areclearly bounded).We turn now to the limit ρ | z i | (cid:28)
1, where helicity-0 correspondence is recovered. Taylor-expanding (49) at small z i , we find s (0) i ( { k, n } , p i ) ≈ a i (cid:16) ρz i (cid:17) n (cid:18) − ρ | z i | n + 1) + . . . (cid:19) /n ! . (53)for n ≥ n <
0. Thus for ρ | z i | (cid:28)
1, the dominant matter-CSPcoupling is to the n = 0 spin-mode of the CSP, and is well approximated by the helicity-0soft factor s i ( { k, n } , p i ) = a i . The emission amplitudes for other modes are suppressed by | ρz i | n . This is illustrated in Figure 3, which shows the squared soft factors for the spin- n modes for a range of small and large values of ρz .This is our first example of correspondence. We will shortly elaborate on the kinematicdependence of the dimensionful parameter | z i | that controls the correspondence, but com-ment first on its frame-dependence. Our z i depends explicitly on the choice of frame vectors6 Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) (cid:45) (cid:45) (cid:45) n (cid:200) s (cid:72) n (cid:76) (cid:200) Squared Scalar (cid:45)
Like Soft Factors for Spin n Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) (cid:45) (cid:45) (cid:45) n (cid:200) s (cid:72) n (cid:76) (cid:200) Squared Scalar (cid:45)
Like Soft Factors for Spin n FIG. 3. Both plots illustrate on a log scale the scaling of squared scalar-like soft factors | s (0) ( { k, n } , p ) | with ρ | z | ≡ √ ρ (cid:12)(cid:12)(cid:12) (cid:15) + ( k ) .pk.p (cid:12)(cid:12)(cid:12) . The left plot focuses on low n modes, while the right plotzooms out to show the large- n scaling of the soft factors for large z . For small ρ | z | , the amplitudesare sharply peaked at n = 0. For large ρ | z | , spin- n soft factors for n (cid:46) ρ | z | are all O (1 / √ ρz ), while n (cid:38) ρ | z | modes are further suppressed. Negative n modes scale the same way as positive n . (cid:15) ± introduced in § II A, or equivalently on the choice of a standard boost B p . This is expected,because the CSP spin-states are not distinguished in a Lorentz-invariant way, and mix un-der boosts. It is similar to the frame-dependence encountered in polarization amplitudes formassive particles. Here, we can gain intuition for the frame-dependence by recalling thatthe spin label n dictates how the state re-phases under the little group rotation defined by(6). For generic (cid:15) + ( k ), the LG rotation is a linear combination of a Lorentz rotation and aLorentz boost. Only when (cid:15) ± = 0 in a particular frame does the LG rotation coincide witha pure Lorentz rotation about the CSP 3-momentum direction ˆ k , R = (cid:126) k · (cid:126) J | k | , making contactwith the standard definition of helicity and with SO (3) rotation properties of amplitudes.The case z = 0 is realized by an emitter at rest for the choice of B p where R is a purerotation . This is why the standard boost (26) with (cid:15) ( k ) = 0 for all k is particularly useful.But there must be a Lorentz-invariant notion of correspondence limits that applies toany Lorentz-invariant quantity. In general, these will take the form of ρ | z i − z j | , which isclearly invariant under both shifts of (cid:15) + → (cid:15) + + k (which cancel between the two z ’s) andre-phasings of (cid:15) + , and therefore under changes of frame. For example, though the n = 0emission amplitude off a particular scattering reaction is not Lorentz-invariant, the totalemission probability is. For the CSP-emission process shown in Figure 2 this is (cid:88) n | A ( p , p → p , p , { k, n } ) | = | λ | (cid:12)(cid:12)(cid:12)(cid:12) s (0) ( { k, n } , p )( p + k ) + i(cid:15) + s (0) ( { k, n } , p )( p + k ) + i(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (54)= | λ | (cid:18) | a | (( p + k ) ) + | a | (( p + k ) ) + 2 Re [ a a ∗ ] J ( ρ | z i − z j | )( p + k ) ( p + k ) (cid:19) , (55)7where a , are coupling constants and we have used (52) to obtain the second line. When ρ | z i − z j | (cid:28)
1, the Bessel function is approximately 1 and we recover the total emissionprobability of a scalar theory; when ρ | z i − z j | is large, the interference terms drop outand the emission pattern (and total emission probability) changes significantly. A relatedexample, which we will discuss further in [6], is the emission and absorption of a CSPby two different radiators. There too, deviations from a helicity theory are controlled by ρ | z emitter − z absorber | . It will always be simplest, however, to work in either the rest frameof one particle of interest or a center-of-mass frame for high-energy processes, so that evenframe-dependent quantities such as soft factors exhibit correspondence.We now consider the physical meaning of small ρz i , for both relativistic and non-relativistic p . Indeed, z i is a rather familiar expression that appears in the Klein-Nishinaformula and other photon emission problems. As was discussed previously, a convenientchoice of B k is (26), for which (cid:15) ( k ) = 0 for all k . Then if p = ( p , p ) and ˜k is at a relativeangle of θ , | z | ≈ | p | sin θ | k | ( p − | p | cos θ ) . (56)For non-relativistic p with velocity v p , this approaches v p sin θ/ ( | k | ).In the relativistic, non-collinear limit | z | ≈ cot( θ/ / ( | k | ). The latter expression has a collinear singularity, whichfor massive p is regulated by the matter particle’s mass. For fixed | k | , | z | is maximized atemission angles sin θ = m/E p and bounded above by | p || k | m .Having considered the case of a constant f i ( k.p i ) = a i in (42), a natural next step is toconsider simple powers f i ( k.p i ) = ( k.p i ) m c ( m ) i , where the c i have mass-dimension 1 − m .We would not have considered these forms at all for helicity-0 particles, because they vanishin the k → ρ and the z -dependent Bessel function/phase — allow non-trivial soft limits evenwith m >
0. As we will see below, m = 1 and 2 will, under suitable conditions, lead tohelicity-1-like and helicity-2-like CSP interactions. C. Photon Correspondence
On a first look, it seems that all of the soft factors would also be scalar-like in thecorrespondence limit, albeit with momentum-dependent interactions. As an example, thesoft factor with f linear in p i .k is s (1) i ( { k, n } , p i ) = − q i µ k.p i ˜ J n ( ρz i ) , (57)where we have written the coefficients in terms of a uniform mass scale µ and dimensionlesscoefficients q i . In analogy to (53), we find s (1) i ( { k, n } , p i ) ≈ − q i µ k.p i (cid:16) ρz i (cid:17) n /n ! (cid:18) − ρ | z i | n + 1) + . . . (cid:19) (58)8for n ≥
0, and the conjugate for n <
0. The leading interaction is still that of the spin-0state, but now its coefficient grows with the momenta p i and k . The n = ± (cid:15) ∗± .p (since z ∝ (cid:15) ∗ + .p i /k.p i cancels the p i .k pre-factors),but they are still ρz -suppressed relative to spin-0 interactions.If we were to multiply this soft factor into a four-scalar amplitude as in (48) or Figure 2,the spin-0 emission amplitude would be simply A ( p , p → p , p , { k, n = 0 } ) ≈ − λ µ ( q + q − q − q ) + O ( ρ | z i | ) , (59)where the explicit pre-factors in each soft factor have been cancelled against the propagators(dropping i(cid:15) ’s). The absence of any propagator suppression to this amplitude implies thatthe resulting 2-to-3 scattering cross-section grows at large center-of-mass energy, becomingstrongly coupled at a scale Λ ≈ ( µ/λ )( q + q − q − q ) − . Explicitly, integrating (59) overthe phase-space of the 2 original final-state particles and the final-state phase-space of theCSP, σ soft ( s ) ∼ λ s (cid:90) ρ< | k |
1, this approaches the soft-emission amplitude for a gauge boson coupling toeach leg with strength e q i where e = ρ/ ( √ µ ). The next largest interactions are for n = 0and ± A ( { k, n = 0 } , p . . . ) = + λ (cid:88) i eq i (cid:15) ∗− .p i ( ± p i + k ) × ρz i (cid:0) − ρ | z i | /
16 + . . . (cid:1) . (62) A ( { k, n = 2 } , p . . . ) = − λ (cid:88) i eq i (cid:15) ∗ + .p i ( ± p i + k ) × ρz i (cid:0) − ρ | z i | /
12 + . . . (cid:1) . (63)More generally, the couplings for | n | (cid:54) = 1 scale as e ( ρz/ || n |− | / | n | !, and hence vanish in thesmall- z limit.We have already noted the apparent inconsistency of a covariant CSP soft factor ap-proaching the (non-covariant) gauge theory soft factor as ρ →
0. However, recall that had9we considered a single soft factor (57) rather than a full amplitude, with µ/ρ fixed as ρ → e fixed, the n = 0 soft factor actually diverges!Although this is a perfectly consistent soft factor for finite ρ , and the resulting amplitudeshave a smooth ρ → ρ → ρ → s (1) i ( { k, n } , p i ) = −√ eq i ρ p i .k (cid:16) ˜ J n ( ρz i ) − δ n (cid:17) . (64)The δ n term does not transform like a single-CSP state under Lorentz-transformations, butlike a Lorentz scalar, so this soft factor is non-covariant. It is well-behaved in the ρ → /ρ pre-factor, because the factor in parentheses scales as a positive powerof ρ for all n ( O ( ρ | n | ) for n (cid:54) = 0 and O ( ρ ) for n = 0). The scaling of these soft factors isillustrated in Figure 4. This ρ → ρ → ˜ s (1) i ( { k, n } , p i ) = eq i (cid:0) δ n, − (cid:15) ∗− .p i + δ n, (cid:15) ∗ + .p i (cid:1) . (65)To make the correspondence even more manifest, and remove the somewhat misleading 1 /ρ pre-factor, we can introduce the function F n ( w ) ≡ − n ) ˜ J n ( w ) − δ n ( w ) sgn( n ) , (66)where ( w ) sgn( n ) = w for n ≥ w ∗ for n < F n ,˜ s (1) i ( { k, n } , p i ) = eq i (cid:15) ∗ sgn( n ) .p i F n ( ρz i ) . (67)The limiting values F ± (0) = 1, F n (0) = 0 for all other n establish the correspondence. Moreprecisely, the first few F n ’s has the small-argument behavior F ( w ) ≈ w/ − | w | /
16 + . . . ) (68) F ( w ) = F ∗− ( w ) ≈ (1 − | w | / . . . ) (69) F ( w ) = F ∗− ( w ) ≈ − w/ − | w | /
12 + . . . ) . (70)The small- and large-argument behavior of | F n ( w ) | is easily derived from that of the Besselfunctions: | F n ( w ) | ≈ ( w/ || n |− | | n | ! for | w | (cid:28) max( n, , (71)while at large arguments it is contained in the envelope | F ( w ) | < / | w | ( w = 0) , | F n ( w ) | < / | w | / ( w (cid:54) = 0) . (72)0 Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) (cid:45) (cid:45) (cid:45) n (cid:200) s (cid:142) (cid:72) n (cid:76) (cid:200) Squared Photon (cid:45)
Like Soft Factors for Spin n Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) Ρ z (cid:61) (cid:45) (cid:45) (cid:45) n (cid:200) s (cid:142) (cid:72) n (cid:76) (cid:200) Squared Photon (cid:45)
Like Soft Factors for Spin n FIG. 4. Both plots illustrate on a log scale the scaling of squared photon-like soft factors | ˜ s ( { k, n } , p ) | with ρ | z | ≡ √ ρ (cid:12)(cid:12)(cid:12) (cid:15) + ( k ) .pk.p (cid:12)(cid:12)(cid:12) . The plots are normalized to the standard QED softfactor for a photon with the same coupling strength. The left plot focuses on low n modes, whilethe right plot zooms out to show the large- n scaling of soft factors at large z . For small ρ | z | , theamplitudes are sharply peaked at n = 1 (and -1, not shown). For large ρz , spin- n soft factorsfor n (cid:46) ρ | z | are O ( ρ | z | ) − / ( O ( ρ | z | ) − for n = 0), while n (cid:38) ρ | z | modes are further suppressed.Negative n modes scale the same way as positive n . The scales are the same as in Figure 3. The small-argument approximation is also a bound on F n ( w ), which intersects the large-argument envelope at w ∼ n for n (cid:54) = 0 and w ∼ n = 0.We have seen that the covariant soft factor (57) with generic q i has correspondence witha momentum-dependent scalar interaction, which violates perturbative unitarity above thecutoff scale Λ ∼ µ . However, in the special case that all interactions conserve q i (i.e. thesum of incoming q i ’s equals the sum of outgoing q i ’s in any process), the theory has nocutoff and has small- ρ and high-energy correspondence with gauge theory soft factors withgauge coupling e = ρ/ ( √ µ ). In this case, the non-covariant soft factor (67) is physicallyequivalent to (57), and makes the correspondence more manifest. The perturbative unitarityargument suggests the intriguing possibility that in CSP theories with a cutoff Λ, chargeconservation (where “charge” is defined as the coupling constant of the n = ± O ( ρ/ Λ), though the genericity of this effect can only bestudied in a full theory.
D. Graviton Correspondence
Analogous arguments apply to the soft factors with pre-factor f ( k.p i ) = g (2) i ( k.p i ) . Thecoefficients g (2) i have mass-dimension −
3, and we write them as g (2) i = g i /µ . However, itwill be convenient to consider a more general f ( k.p i ) quadratic in p i : s (2) i ( { k, n } , p i ) = 1 µ (cid:18) g i ( p.k ) + g (cid:48) i ρ p (cid:19) ˜ J n ( ρz i ) . (73)1Again, the leading interaction at high energies — the O ( ρ ) term in the emission amplitudefor the n = 0 state — violates unitarity at a cutoff scale Λ ∼ µ for generic O (1) g i .This fastest-growing part of the amplitude is proportional to 1 /µ (cid:80) i ± g i k.p µi (+/ − signfor outgoing/incoming legs). The cutoff can only be raised parametrically higher than µ if g i = g for all i . In this case, the above leading term vanishes by momentum conservation.Moreover, the O ( ρ ) terms of the n = ± gρ/µ (cid:80) i ± (cid:15) ± ( k ) .p µi ,also vanish by momentum conservation.After these cancellations, the largest amplitudes are for spins 0 and ±
2; these begin atorder 1 /M ∗ ≡ ρ /µ and their leading behavior at small z is given by A ( { k, n = 0 } , p . . . ) = A ( p . . . ) (cid:88) i M ∗ − g | (cid:15) ∗ + .p i | + g (cid:48) i p i ± k.p i (1 + O ( | ρz i | )) (74) A ( { k, n = ± } , p . . . ) = A ( p . . . ) (cid:88) i M ∗ g ( (cid:15) ∗± .p i ) ± k.p i (1 + O ( | ρz i | )) . (75)The n = ± ρz i , a gravitational amplitude with Planckscale M ∗ ; the first term of the numerator for the n = 0 mode can be simplified using (cid:15) ( µ + (cid:15) ν ) − = η µν − k ( µ q ν ) to the form − / p i + k.p i q.p i ). The second term vanishes in the fullamplitude by again using momentum conservation. Thus A ( { k, n = 0 } , p . . . ) = A ( p . . . ) (cid:88) i M ∗ ± k.p i
14 ( g (cid:48) i − g ) p i (1 + O ( | ρz i | )) . (76)For the special choice g (cid:48) i = g , the n = 0 amplitude is governed by the next-leading term, of O ( ρ z /M ∗ ). In the limit ρzρ (cid:28)
1, this theory approximately reproduces general relativity,with emission amplitudes for the n ’th mode scaling as M ∗ ( ρz/ (cid:12)(cid:12) | n |− (cid:12)(cid:12) / | n | !. For generic g (cid:48) i ,the theory also has a scalar with Brans-Dicke-like couplings. It is not clear, without a fulltheory, whether the condition g (cid:48) i = g arises from minimal theories, or is non-generic in animportant way.In analogy with (67), we may again construct a modified non-covariant soft factor thatproduces the same amplitudes as (73) and recovers precisely the helicity-2 soft factor in the z → s (2) i ( { k, n } , p i ) = 1 M ∗ ρ (cid:0) ( p.k ) + ρ p / (cid:1) ˜ J n ( ρz i ) − ( p.k ) [ δ n − δ n ρz i / δ n, − ρz ∗ i / . (77)As in the photon-like case, it is straightforward to define “correspondence functions” analo-gous to F n ( w ) that smoothly approach δ | n | as w → w as n getsfarther from ±
2. In terms of these functions, (77) will look manifestly like a correction togravitational soft factors.Of course, the vanishing in full amplitudes of the non-covariant terms in (67) and (77)involves precisely the same cancellation of terms that appeared in Weinberg’s soft-emissionamplitudes. Our subtractions are proportional to the “pure gauge” components of a tensor2field that are generated by Lorentz transformation. But the justification for insisting thatthese terms cancel is entirely different. In Weinberg’s construction, the constraint wasLorentz invariance; here, covariant soft factors (57) and (73) are easily constructed, butin order to raise or remove a high-energy cutoff, the interaction coefficients must be eitherconserved (for the p.k pre-factor) or universal (for the ( p.k ) pre-factor), and only in thiscase do the correspondence limits become photon-like or graviton-like. E. General CSP Soft Factors Reconsidered
One might ask whether this trend continues: can we continue exploiting cancellations tobuild CSP interactions that couple predominantly to the n = 3 state in the correspondencelimit? We can certainly construct “soft factors” with increasingly high powers g ( m ) i ( p i .k ) m in their pre-factors, but the leading term in an n = 0 amplitude will always take theform (cid:80) i g ( m ) i ( p i .k ) m − . If the amplitude is to have non-singular support in momentumspace (aside from the single momentum-conserving delta-function arising from translationinvariance), then for m > g ( m ) i ’s will lead to the vanishing of this leading term.Therefore the “generic” CSP soft factor couples dominantly to the spin-0 state; the forms(57) and (73) are very special cases.This brings us back to a question about which we have so far been rather glib: whetherthe “soft factor” with generic pre-factor f ( k.p i ) is really worthy of the name. That is, doesLorentz-invariance of the S -matrix require covariance of the sum over these momentum-dependent soft factors? A true soft factor is guaranteed to dominate some class of amplitudesat small | k | — only in that case is covariance of the sum of soft factors necessary to ensurecovariance of the amplitude. Let us consider the growth of the amplitude shown in Figure2 in the soft limit for two cases: a scalar-like CSP with arbitrary couplings a and a anda vector-like CSP interaction with q = − q = q . In the scalar-like case, the emissionamplitude is A (0) ( p , . . . , p , { k, n } ) = λ (cid:18) a p .k ˜ J n ( ρz ) + a p .k ˜ J n ( ρz ) (cid:19) + O (( p i .k ) ) , (78)while in the vector-like case it is A (1) ( p , . . . , p , { k, n } ) = λ (cid:18) − q √ ρ ˜ J n ( ρz ) + q √ ρ ˜ J n ( ρz ) (cid:19) + O (( p i .k ) ) . (79)The behavior of both amplitudes changes crucially between the “soft correspondence”regime ρ | p i | (cid:28) k.p i (cid:28) p i .p j , where the ˜ J n is evaluated at small arguments (and is approxi-mately polynomial) and the “ultra-soft” regime k.p i (cid:28) ρ | p i | (cid:28) p i .p j , where the falling large-argument behavior of ˜ J n takes over. In the ultra-soft regime, ˜ J n ( ρz ) ∼ / (cid:112) ρ | z | ∝ √ k.p i actually falls as k gets increasingly soft. The scalar-like amplitude A (0) continues to growwithout bound because the propagators grow as 1 /k.p i while the Bessel functions fall only3as √ k.p i . But the vector-like amplitude A (1) indeed falls in this ultra-soft limit. “Internal”emission contributions to the amplitude, which are not enhanced by 1 /k.p i poles, could stillcontribute comparably to the amplitude (and to the total cross-section, which also falls as √ k.p i ). Therefore, Lorentz-covariance of the external emission components in (79) is neithernecessary to guarantee Lorentz-covariance of the amplitude, nor even sufficient to guaranteeLorentz-covariance of a leading part. The simplification usually obtained by going to thesoft limit has been lost.What about the “soft correspondence” regime ρ | p i | (cid:28) k.p i (cid:28) p i .p j < Λ , which onlyexists in a theory with cutoff Λ (cid:29) ρ ? Here A (1) ( p , . . . , p , { k, ± } ) is well-approximatedby helicity 1 soft factors, which do grow in the soft limit as 1 /k.p i . Indeed, the “soft” partof the amplitude (79) reaches a maximum strength of order 1 /ρ for k.p i ∼ ρ | p i | ( ρz i ∼ k . Although we cannot cleanly extract the “soft part” of theamplitude as a residue at k.p i = 0, we can still expect that it dominates over internal emis-sions in the soft correspondence regime. Internal emission contributions to the amplitudeare suppressed by propagators 1 /p i .p j , and should on dimensional grounds be bounded byΛ /p i .p j . For momenta near Λ, this is parametrically smaller than the “soft” contributions tothe amplitude. Thus, the scaling of single-emission amplitudes in the “soft correspondence”regime provides the justification for demanding covariance of the part of the amplitudeobtained from soft factors.With the above remarks in mind, we summarize the general soft factor that respects per-turbative unitary below a cutoff scale Λ (cid:29) ρ and is truly constrained by Lorentz-invarianceas s i ( { k, n } , p i ) = (cid:20) a i − eq i √ ρ p i .k + 1 M ∗ (cid:0) ( p i .k ) − ρ p i (cid:1)(cid:21) ˜ J n ( ρz i ) , (80)where the first term is scalar-like with arbitrary dimension-1 a i , the second term is photon-like and the dimensionless coupling q i is conserved, and the third term is graviton-like witha universal coupling 1 /M ∗ with dimensions of inverse mass. The latter two terms couldequally well be replaced by the non-covariant soft factors (67) and (77). It is consistent, asfar as we can tell from soft factors, for a single CSP with small ρ to mediate helicity 0, 1,and 2-like interactions! These helicities are not singled out by the spectrum (which containsa proliferation of high-spin modes), but by interactions.A separate but important question is whether some new structure emerges in the “ultra-soft” regime. The correspondence soft factors we have considered, with monomial or bino-mial f ( p i .k ), lead to anarchic ultra-soft spin-basis amplitudes; the “angle” basis amplitudesare anarchic for all k . It remains possible that, in some other basis or with another choiceof f ( p i .k ), the ultra-soft amplitudes display some interesting structure. This could be avery promising way of untangling the deep infrared physics of CSPs, which is clearly verydifferent from that of either definite-helicity or massive particles.4 IV. INFINITE-SPIN LIMIT VS. CORRESPONDENCE LIMIT
The discussion above motivates a physical picture of CSP physics as a generalizationof fixed-helicity physics, with a physical correspondence in the limit ρz →
0. Anotherconnection often encountered in the literature is of the CSP as a simultaneous high-spin andlow-mass limit of a massive particle. It’s very easy to see that the limit S → ∞ of a massiveparticle, with ρ = M S held fixed at some finite value, will have W = − M S ( S + 1) → − ρ in the infinite-spin limit (we use M for mass here, to distinguish it from the quantumnumber m ). The counting of states in the spin-basis is also indicative of a connectionbetween the continuous-spin representations of the Poincar´e group and high-spin limits ofmassive particles. Inonu and Wigner formalized this notion in their description of contractinggroups and their representations [11]. Up until now, however, it was unclear whether thisrelationship was physical or merely group-theoretic.Our soft factors allow us to address this question quantitatively: how are emission ampli-tudes for a mass- M , spin- S limit (in the limit of large S , and M small compared to particlemomenta) related to those for continuous-spin particles? For simplicity, we perform theanalysis on soft factors rather than full amplitudes, though the conclusions would extendnaturally to more general amplitudes. We are reassured to recover some relationship be-tween the massive spin- S soft factors and their CSP counterparts, since much of the basicstructure of these amplitudes is dictated by Little group symmetry. However, it appearsto us that the “infinite-spin” connection is just group-theoretic – CSP soft factors have amuch better behaved analytic structure than the high-spin limit of massive particle softfactors. In particular, to obtain a sensible infinite-spin limit, the spin- S soft factors must bemultiplied by an S -fold pole that, for any finite S , grossly violates unitarity. Only when wefinally take S to infinity do we recover a CSP soft factor where the S -fold pole is replacedby a Bessel function, which smoothly cuts off the low-momentum limit of the amplitude. Itwould appear unlikely that the physics of an interacting CSP theory is usefully related tothe physics of interacting massive particles in the infinite-spin limit.It is well-known [33] that massive spin- S fields can be described by traceless and sym-metric rank- S tensor wavefunctions A µ ...µ S with k µ A µ ...µ S = 0. The natural soft factor foremission of a spin- S particle of momentum k µ off an external momentum p is s S ( p, k, m ) = 1Λ S − A µ ...µ S m p µ . . . p µ S , (81)where Λ is a cutoff scale for the interaction, and A µ ...µ S m the wavefunction for the m ’th spinstate.We must now specify the basis wavefunctions A m for the spin- S particle of mass M . Tomake contact with the massless limit, we will want to consider highly relativistic k , andwork in a basis where m is the eigenvalue of J = W /M , where W is given by equation5(B6) of [3]. We note first that (cid:15) + , √ J − µν (cid:15) ν + = − (cid:15) µ , and 12 ( J − ) µν (cid:15) ν + = − (cid:15) µ − (82)diagonalize J and form a canonical basis for the spin-1 A µm with m = 1, 0, and − S wavefunctions out of a symmetric product of S spin-1wavefunctions, using the definitions A µ ...µ S S = (cid:15) µ + . . . (cid:15) µ S + , A µ ...µ S m = (cid:115) ( S + m )!(2 j )!( S − m )! ( J − ) S − m A µ ...µ s m (83)where each copy of J − is a direct sum of lowering operators ( J − ,i ) µ i ν i acting on the i ’th index.It is now easy to compute s ( p, k, m ) from the above, accounting for all combinatoric factors.In particular, each term in A m can be classified according to the number k of indices actedon by ( J − ) (for given k , it follows that S − m − k indices are acted on by a single J − , and m + k are unaffected), which can range from max(0 , − m ) ≤ k ≤ ( S − m ). Then s S ( p, k, m ) = 1Λ S − (cid:115) ( S + m )!(2 j )!( S − m )! (cid:88) k (cid:18) ( S − m )!2 k (cid:19) (cid:18) S ! k !( m + k )!( S − m − k )! (cid:19) (84) × (cid:104) ( − (cid:15) − .p ) k ( −√ (cid:15) .p ) S − m − k ( (cid:15) + .p ) m + k (cid:105) , (85)where the first factor in parentheses counts the number of ways a given permutation, e.g.the one proportional to (cid:15) µ − . . . (cid:15) µ k − (cid:15) µ k +1 + . . . (cid:15) µ k + m + (cid:15) µ k + m +1 . . . (cid:15) µ S can arise from action of J − m lowering operators, the second factor in parentheses counting the number of such permuta-tions, and the square brackets encoding the resulting contraction with p µ . . . p µ S . This softfactor can be reorganized as s S ( p, k, m ) = 1Λ S − (cid:115) S ( S + m )!( S − m )!(2 S )! ( − (cid:15) .p ) S ( − e iϕ ) m (86) × k ≤ ( S − m ) / (cid:88) max(0 , − m ) ≤ k ( − k k !( m + k )! (cid:20) S !( S − m − k )! S k + m (cid:21) (cid:18) M S | z | (cid:19) k + m , (87)where e iϕ ≡ (cid:15) + .p(cid:15) − .p and | z | ≡ √ | (cid:15) + .p | M(cid:15) .p .Several simplifications occur in the limit of large S and small mass M . For m (cid:28) √ S ,the overall square-root pre-factor becomes an m -independent factor C S (its detailed form isunimportant). In the small-mass limit, (cid:15) .p → M k.p and | z | → √ | (cid:15) + .p | k.p . The sum over k hasbeen written in a form closely resembling the Bessel function series expansion. Indeed, for2 k + m (cid:28) √ S (corresponding to terms built mainly out of (cid:15) ), the factor in square bracketsapproaches 1, so that these terms approach the first few terms of the Taylor expansion for J m ( ρ | z | ) where ρ = M S (for m < k to k + m is required tobring these terms into the canonical form). The series is dominated for any | z | by terms6with k < | z | /
2, so the whole series can be said to converge to J m ( | z | ) at large S . Then inthe large- S and small- M limit, s S ( p, k, m ) ∼ C s p.k S Λ S − M S ( − e iϕ ) m J m ( ρ | z | ) . (88)This makes it clear that formally, s CSP ( p, k, m ) = lim S →∞ s S ( p, k ∗ , m ) × Λ S − M S C s p.k S , (89)where k ∗ is a momentum close to the null k µ , but with k ∗ = ( ρ/S ) .The fact that the limiting dependence on the spin quantum number m resembles a CSPsoft factor is expected, on the basis of group theory. Nonetheless, we see that the large- S limit of the massive particle soft factor does not exist in any clear sense, because the overallmomentum-dependent (but m -independent) pre-factor is singular. The right-hand side ofequation (89) is clearly not a quantity that, at any finite S , could be considered a soft factor– it has an unphysical order- S pole in p.k . To us, this suggest that the physical interpretationof a CSP as the infinite-spin limit of a massive particle is at best useful for understandingkinematics, but likely not a useful approach for interpreting interactions. This conclusioncan and will be made sharper in [8]. V. HELICITY CORRESPONDENCE AND UNITARITY
We have seen that the soft factors of Table I display correspondence with helicity 0, ±
1, and ± g µν is not equal7to the sum over physical states (cid:80) λ = ± (cid:15) µλ ( k ) (cid:15) ∗ νλ ( k ); Feynman rules are only unitarity becausethe difference between these two expressions, when contracted into the two lower-pointfactors obtained by Feynman rule construction, vanishes by Ward identities when k = 0.Although powerful methods exist to compute gauge and gravity amplitudes via unitarity,the simplest and most efficient methods like BCFW recursion rely on analytic continuationto complex momenta and on the introduction of a three-point “amplitude” that has noreal-momentum support. The complex-momentum technique might be useful for CSPs,but there is no obvious complex-momentum continuation of our soft factor that we havebeen able to interpret as a three-point amplitude! Thus, we must confine ourselves to real-momentum arguments – a more robust starting point, but one that makes gauge theoryunitarity arguments fairly involved. Extending the line of inquiry pursued here to photon-and graviton-like CSP amplitudes or finding obstacles to doing so is of great interest, boththeoretically and as a means to build CSP generalizations of known gauge theories andgravity. We will indicate several subtleties unique to vector- or graviton-like CSPs as theyarise in this section. A. Correspondence in Multi-CSP Scattering Amplitudes
Constructing a limited class of multi-CSP scattering amplitudes that are Lorentz-invariant and unitary (or at least not conspicuously non-unitary!) is rather straightforwardfor the m = 0 (scalar-like) soft factors. Consider for example the amplitude for radiationof two CSPs labeled by { k, n } and { k (cid:48) , n (cid:48) } off a parent amplitude A ( p , p ; p , p ) = λ (illustrated in Figure 5). As before, we will start by assuming that only the legs p and p couple to the CSP. To maintain unitarity, we expect that when a CSP is radiated off eitherleg, there must be a propagator factor in the amplitude, which produces a pole when theintermediate line goes off-shell, as well as a soft factor for each emission off an external line.There is a bit more ambiguity in the internal vertex, when both emissions are off the sameline (large dot vertices in Figure 5) — these vertices could deviate from the soft factor byterms that vanish when either leg is on-shell, but we ignore this possibility as there is nonatural deformation of this form. Neglecting for now the terms in parentheses in Figure 5,we obtain an ansatz six-point amplitude A ( p , p ; p , p , { k, n } , { k (cid:48) , n (cid:48) } ) = λ a s ( { k, n } , p )(( p + k ) + i(cid:15) ) a s ( { k (cid:48) , n (cid:48) } , p + k )(( p + k + k (cid:48) ) + i(cid:15) ) (90)+ a s ( { k, n } , p )(( p + k ) + i(cid:15) ) a s ( { k (cid:48) , n (cid:48) } , p + k )(( p + k + k (cid:48) ) + i(cid:15) ) (91)+ a s ( { k, n } , p )(( p + k ) + i(cid:15) ) a s ( { k (cid:48) , n } , p )(( p + k (cid:48) ) + i(cid:15) ) + ( k, n ↔ k (cid:48) , n (cid:48) ) , (92)where s ( { k, n } , p ) is the scalar-like soft factor (49). It is straightforward to verify that thisamplitude continues to yield finite scattering cross-sections, aside from the standard IR8 FIG. 5. The construction of a candidate on-shell two-CSP amplitude is illustrated above. CSPsare attached to a parent amplitude A ( p , p ; p , p ) using the CSP soft factors, with appropriatematter propagators included. This example presumes that only the outgoing matter legs coupleto the CSP. The internal vertex (with black dots) could deviate from the naive soft-factor form,but we ignore this possibility. The terms in parentheses (with grey dots) describe “contact”-likestructure that could arise. The final result is the 6-point amplitude A ( p , p ; p , p , { k, n } , { k (cid:48) , n (cid:48) } )used below as an example to investigate certain aspects of CSP interactions. divergences. This finiteness is most easily verified in the angle-basis amplitude, obtainedeither by Fourier transform or by replacing each s ( { k, n } , p ) by s ( { k, φ } , p ) from (41) with f i = 1. Because both s ( { k, n } , p ) and s ( { k (cid:48) , n (cid:48) } , p ) have helicity-0 correspondence, the n = n (cid:48) = 0 amplitude is well approximated by a product of scalar amplitudes whenever the ρz are small (in particular, in the high-energy limits). Amplitudes for emitting one non-zero n state receive the usual suppression, while amplitudes for n (cid:54) = 0 and n (cid:48) (cid:54) = 0 are suppressedby O ( ρ | z | | n | + | n (cid:48) | ). Multi-CSP amplitudes constructed in the same way would follow the samepattern.There could also be new terms in the amplitude, involving multiple CSPs at a singlevertex, or a CSP self-interaction (the terms in parentheses in Figure 5). Any covariantfunction without spurious poles would be consistent with the factorization of general multi-emission amplitudes. Indeed there is considerable flexibility in constructing such a covariantvertex — for example, the four-point vertex can be obtained from ψ ( η, k, φ ) ψ ( η (cid:48) , k (cid:48) , n (cid:48) ) byreplacing η by k , the symmetric sum ( p + k + k (cid:48) ) + p of the two matter momenta at thevertex, or any function thereof. There is, however, no apparent need to add such terms toour soft factor. With or without such vertices, there seems to be no obstacle to building aunitary collection of amplitudes for multi-CSP emission.Each of the ambiguities mentioned above become more important in the case of photon-9like (or graviton-like) CSP interactions. Here, we may approach the problem starting froman expression similar to (92), using the “subtracted” (non-covariant but manifestly corre-spondent) soft factors (67)). Just as in QED, the non-covariance in a sum of soft factorsvanishes, but the non-covariance in the sum of products of soft factors doesn’t vanish .In QED, this non-cancellation is cured by the addition of the two-scalar, two-photonvertex; in non-Abelian theories, it further requires a self-interaction vertex. These vertexrules could have been guessed without reference to an action, simply from unitarity andLorentz-covariance of amplitudes. It is clear that CSPs need some similar correction, butthe precise form is more difficult to guess. In contrast to gauge theories, where each vertexis a fixed-rank polynomial in momenta and polarizations (a basis for which can be enumer-ated), here we face a plethora of candidate vertices. First, the pre-factor of the soft factor˜ s (1) ( { k (cid:48) , n (cid:48) } , p + k ), for example, may be simply ( p + k ) .k (cid:48) , but could also receive correc-tions when both p + k and p + k + k (cid:48) are off-shell. Second, the argument of the four-pointvertex Bessel function could depend on several combinations of momenta — or it might becomposed of several terms, each multiplying a different Bessel. While this freedom makes iteasy to find rules consistent with unitarity, Lorentz-invariance, and helicity correspondencefor a special case (e.g. CSP pair production), we have not found simple rules that guaranteethese properties for all multi-emission amplitudes. It is unclear whether simple rules exist,or whether the theory requires an infinite tower of self-interactions (like gravity or the CSWconstruction of Yang-Mills theory). The gauge-theory-like structure for CSPs found in [8]suggests that similar machinations might be required here, while ultimately arising from asimple theory. B. A Unitary Ansatz for Intermediate-CSP Amplitudes
In this section, we use unitarity and crossing to construct a family of matter-matter scat-tering amplitudes mediated by off-shell
CSPs. One example, with distinct scalar particlesof momentum a → a (cid:48) and b → b (cid:48) , is M a + b → a (cid:48) + b (cid:48) = 1 k + i(cid:15) J (cid:32) ρ (cid:112) − ( (cid:15) µνρσ k ν p ρ q σ ) k.pk.q + p.qk (cid:33) , (93)where p = a + a (cid:48) , q = b + b (cid:48) , and k = a − a (cid:48) . Despite appearances, J ( √ w ) is analytic onthe complex plane ( J ( w ) has an even Taylor expansion), so this amplitude is analytic inmomenta, except at isolated points. As is now becoming familiar, unitarity arguments do notfully fix the form of the amplitude — since the amplitude is not a rational function, one caninsert terms proportional to t = k (the virtuality of the intermediate CSP) in several places One could also have attempted to build a version of (92) out of the un-subtracted photon-like soft factors(57) — in this case, covariance would be manifest but we would still need to make sure the leading termscancelled, to preserve unitary factorization of general amplitudes into soft factors. If there is a field theoryfor photon-like CSPs, it would likely favor one grouping of Feynman rules or the other, but we don’t knowwhich. without becoming collinear (i.e. when its virtuality k becomes parametricallysmaller than other momentum scales in the problem, including k.p ), we cannot directlyapply unitarity to a massless intermediate state in a 2 → → and Φ that couple to the CSP and four φ a,b,c,d that do not. For simplicity we take allparticles to be distinguishable, so that crossed diagrams need not contribute. Introducingthree-point scalar couplings y Φ φ c φ d and y φ a φ b Φ (94)allows a non-zero amplitude, of the form shown in the diagram. In the limit k = ( p c + p d − p ) →
0, this should factorize into the product of diagrams in the right of Figure6, each of which can be computed by our soft-factor ansatz to obtain an expression (upto terms proportional to k ) for the full six-point amplitude. Taking the limits ( p c + p d ) and ( p a + p b ) → k -dependence, weobtain constraints on two-to-two scattering amplitudes. In effect, this construction providesa physical regulator of the four-point amplitude by moving the CSP propagator pole awayfrom the collinear region.Of course, this construction presumes that the tri-scalar interactions (94) can consistentlyappear in the same theory. While there is no reason to doubt this assumption for scalar-like CSPs (and it gives a reasonable answer), it would certainly fail for a photon (or aphoton- or graviton-like CSP). In that case, at least one of φ a,b and one of φ c,d is charged.Otherwise, (94) would violate charge conservation! This in turn implies that at least fourdiagrams would contribute to the six-point master amplitude, with only their sum beinggauge-invariant — a significant complication that we will not tackle here. We focus here oninteractions where in- and out-going particles have equal masses; many of the complicationswe discuss would actually simplify for unequal masses.Let us consider the factorized amplitude on the right side of Figure 6. The four-pointamplitudes are derivable from the CSP soft factors of Section II C: M left = y P − m c exp (cid:20) − iρ (cid:15) φ .pk.p (cid:21) (95) M right = y ∗ P − m c ∗ exp (cid:20) + iρ (cid:15) φ .qk.q (cid:21) (96)where p = p + P , q = P + p , and c i denote couplings to the CSPs. We have written thephases in a symmetric manner (for k null, ( p + k ) .k = p.k so the above is equivalent to (41)),to assist us in checking crossing later.1 a b2*43*c d FIG. 6. The Master Amplitude (bottom-left) used to derive factorization constraints for two-scalaramplitudes mediated by a “scalar CSP”, and two factorization limits. The thick dot-dashed linerepresents an intermediate CSP, solid lines are scalars with a CSP coupling, dashed lines are scalarswith no CSP coupling. In the limit that the pairs c − d and a − b become collinear (3 ∗ and 2 ∗ goon-shell) the amplitude factorizes as a product of two splittings and the CSP-mediated amplitude12 →
34. In the limit that the CSP goes on-shell, it factorizes into CSP emission and absorptionamplitudes.
The factorization limit of the six-point amplitude is thereforelim k → k M ab → cd = S ab S ∗ cd a a ∗ × (cid:90) dφ π exp (cid:20) + iρ (cid:18) (cid:15) φ .qk.q − (cid:15) φ .pk.p (cid:19)(cid:21) (97)= S ab S ∗ cd a a ∗ × (cid:90) dφ π exp (cid:20) + i √ ρ Im (cid:18) e − iφ (cid:15) + .Ak.qk.p (cid:19)(cid:21) , (98)where S ab = y / ( P − m ), S cd = y / ( P − m ), and A µ = k.pq µ − k.qp µ . The φ -dependent2phase integrates to a Bessel function J (cid:32) ρ √ | (cid:15) + .A | k.pk.q (cid:33) . (99)The quantity √ | (cid:15) + .A | is Lorentz-invariant because k.A = 0. To express it in a manifestlycovariant form, one can introduce a light-cone vector Q such that k.Q = 1, k.(cid:15) ± = 0, interms of which g µν = − (cid:15) µ + (cid:15) ν − − (cid:15) µ − (cid:15) ν + + k µ Q ν + Q µ k ν . (100)Contracting this identity into A µ A ν , we find A = − (cid:15) + .A(cid:15) − .A = − | (cid:15) + .A | . (101)For the null k that we are considering, we can also write A = V with V µ = (cid:15) µνρσ k µ p ν q σ ,as these differ only by terms proportional to k . Thus, we havelim k → k M ab → cd = S ab S ∗ cd a a ∗ × J (cid:18) ρ √− V k.pk.q (cid:19) . (102)Although writing this in terms of V µ seems rather arbitrary, it will be justified later by arather convenient property of V µ : as long as p , q , and k are built from linear combinationsof time-like momenta, V µ will always be space-like, and √− V real. Despite appearances, J ( √ w ) is analytic on the complex plane ( J ( w ) has an even Taylor expansion), though ithas an essential singularity at infinity. However, J ( √ w ) is only bounded for positive w (fornegative w it grows exponentially in (cid:112) | w | ), so it is important that the argument of thesquare root be bounded from below.We note that when the intermediate momentum P goes on-shell, p.k = ( P + p ) . ( P − p ) = 0 so the argument of the Bessel function diverges, and the Bessel function itselfvanishes! This does not, however, imply that CSP-mediated scattering amplitudes vanish!We must not forget that (102) is only a k → M ab → cd = S ab S ∗ cd a a ∗ × k + i(cid:15) J (cid:32) ρ (cid:112) − V + F ( p, q, k ) k k.p k.q + F ( p, q, k ) k (cid:33) , (103)where F and F are Lorentz-invariant polynomials of mass-dimension 4 and 2 respectively(larger classes of deformations are also conceivable, but seem unnecessary). In this case, thefour-point amplitude would have to take the form M → = a a ∗ × k + i(cid:15) J (cid:32) ρ (cid:112) − V + F ( p, q, k ) k k.p k.q + F ( p, q, k ) k (cid:33) , (104)We further constrain the functions F by the following assumptions:31. Invariance under p , → − p , ( s ↔ u crossing), which flips p → − p leaving k and q unchanged (and similarly under q → − q with k and p unchanged,2. Invariance under relabeling the diagram to its reflection, i.e. exchanging p → p , p → p , which swaps p → q , q → p , and k → − k , and3. The amplitude and its crossing variants should all have − V + F ( p, q, k ) k positive-semidefinite.The first two requirements nearly fix F — since k.pk.q is odd under crossing and even underreflection, and k is even under both, the unique polynomial F is αp.q . There is no obviousconstraint on the coefficient α from unitarity, except that it should be non-zero so that theargument of the Bessel function does not diverge for generic on-shell momenta.As for the numerator, it is easy to verify that F = 0 satisfies the third requirement,because V µ is an epsilon-contraction of physical (null or time-like) momenta, so V µ willalways be either space-like or null. One can further check that any polynomial F satisfyingthe first two conditions would give rise to − V + F < t -channel amplitudewe consider here or one of its crossed variants. Subject to our assumption (103) for thegeneral amplitude, this fully fixes the numerator of the Bessel argument to be ρ √− V .Our simple ansatz amplitude, M → = a a ∗ × k + i(cid:15) J (cid:32) ρ (cid:112) − ( (cid:15) µνρσ k ν p ρ q σ ) k.p k.q + αp.qk (cid:33) , (105)satisfies all straightforward unitarity checks. Let’s go through this explicitly. The opticaltheorem T X → Y − ( T Y → X ) ∗ = − i (cid:88) Z T X → Z ( T Y → Z ) ∗ , (106)applied to the matter-matter amplitude above is non-trivial. The left hand side of (106) isproportional to iδ ( k ) (resulting from the i(cid:15) terms), T X → Y − ( T Y → X ) ∗ = − iδ ( k ) δ ( P X − P Y ) J ( ... ) . (107)The right hand side is obtained by integrating over the phase space (and little group labels)of a product of the soft-factor amplitudes T X →{ k,φ } + ... and ( T Y →{ k,φ } + ... ) ∗ above, − i (cid:88) Z T X → Z ( T Y → Z ) ∗ = − i (cid:90) d kδ + ( k ) δ ( P X − P Z ) δ ( P Z − P Y ) × (cid:90) dφ π M left ( X → { k, φ } + ... ) × M right ( { k, φ } + ... → Y ) , (108)where the ... refers to whatever else is left in the Z -state. In the case where X and Y eachcontain three matter legs, Z is a CSP + matter particle state. We can use (95) and (96) forthe above “left” and “right” factors, integrate over phase space and little group labels, and4obtain exactly the same as (107), as required by the optical theorem at this order. This is,after all, how the J amplitude ansatz was constructed.The more subtle case is when X and Y each contain two particles. The only relevantcontribution to the right-hand side at tree-level comes from Z consisting of a CSP state andnothing else. If either of the two scattered particles has non-zero mass, then the momentum-conserving δ -function has no overlap with δ ( k ) and both sides of (106) vanish trivially. Themore subtle case is when all particles are massless. In this case, the product of δ -functionshas non-vanishing support on the kinematic configuration where X and Y each consist ofmultiple massless particles collinear with the CSP momentum k . The argument of the Besselin (107) diverges on the support of δ ( k ), so that the Bessel function itself vanishes. Thus,the right-hand side of (106) must also vanish. But precisely in this limit, the argument ofthe Bessel functions in each spin-state’s contribution to M left and M right also diverges. Thenatural way to define the soft factors in this limit is by regulating them at slightly off-shellmatter momenta. In this case, the sum over n reduces precisely to the form we found in thesix-point amplitude, yielding a single J ( . . . ) that vanishes when the regulator is removedto take the matter legs on-shell. If one were to do the same calculation in the angle basis,the contribution from each φ would be a phase of unit norm, but this phase would vary sorapidly near the collinear momentum configuration that we can view it as averaging to zero.Of course, it would be preferable to frame this argument in a manner where we do not needto invoke a limiting procedure to obtain the right-hand side of (106) from off-shell objects,but we do not know of a natural way to do so.In summary, we can obtain 6-point tree-level pure matter amplitudes that factorize cor-rectly into 4-point amplitudes with on-shell CSPs. We expect this to persist for higher-pointamplitudes, built for example using the sewing rules described below. But for 4-point purematter amplitudes, factorization occurs most directly into 3-point soft factors . If 3-point on-shell CSP amplitudes are defined as the on-shell limit of these soft factors, then they vanishfor real momentum. With this interpretation, unitarity in the sense of (106) is maintainedat tree-level. C. Scalar-Like CSP Sewing Rules
The results of this section (with a non-unique modification to (105) to be describedshortly) can be taken as ansatz sewing rules for scalar-like CSPs that appear to be consistentwith unitarity at tree-level. These are summarized in this section and in Figure 7.As we showed in V A, the soft factor (49) can be sewn as easily to intermediate scalarmatter lines as to external ones. At tree-level, the resulting amplitudes factorize in a mannerconsistent with unitarity and preserve the correspondence. This sewing rule is s (0) ( { k, n } , p, p (cid:48) ) out = a ˜ J n ( ρz ) s (0) ( { k, φ } , p, p (cid:48) ) = a exp (cid:0) iρ Im [ e − iφ z ] (cid:1) , (109)5 FIG. 7. A graphical summary of the sewing rules defined in Section V C. The left sewing rule (109)applies to outgoing on-shell CSPs (its conjugate to incoming CSPs) and the right one (112) tointermediate CSPs. In all cases, the matter four-momenta are arbitrary, and the arrows on matterlegs serve merely to define conventions in the formulas. for outgoing CSPs, where z ≡ √ (cid:15) ∗ + ( k ) .p/k.p = √ (cid:15) ∗ + ( k ) .p (cid:48) /k.p (cid:48) ˜ J n ( w ) ≡ ( − n e − in arg( w ) J n ( | w | ) . (110)For incoming CSPs, the complex conjugate sewing rule s (0) ( { k, φ } , p, p (cid:48) ) in ≡ s (0) ( { k, φ } , p, p (cid:48) ) ∗ out (111)should be used (similarly in the spin-basis). This was shown in [3] (Section IVD) to yieldstandard crossing relations when analytically continued.The generalization of the amplitude (105) to off-shell external momenta (a correlationfunction) is a natural candidate for an intermediate-CSP sewing rule. However, for off-shellmatter momenta there is a region of phase space where all three of k , p , and q lie in aspace-like (0 ,
3) plane. In this case, V µ is time-like, and the argument of the Bessel functionin (105) is unbounded from below. Indeed, as all momenta become soft in this space-likeconfiguration, (105) diverges exponentially. Clearly, something must cut off the resultingexponential growth of the J in any physical theory. There are several candidates, and thisdifficulty is so far removed from the regime where our unitarity argument is valid that weonly mention them briefly. The sewing rule for attaching a CSP between two off-shell matterlegs may be different from the soft factor, though we have found no evidence that this mustbe the case. Alternately, (104) can be generalized in a way that keeps the Bessel functionbounded. One example (keeping terms appropriate for off-shell momenta) is G (0) ( p , p , q , q ) = a p a ∗ q × k + i(cid:15) J (cid:32)(cid:115) − ρ V A + β ( ρ k p q ) /A (cid:33) , A = k.p k.q − αk p.q (112)where p = ( p + p ) / q = ( q + q ) /
2, and any positive β . This leaves the form of(105) unchanged in both high-momentum and soft k → | V | < | k p q | ), theargument of the Bessel cannot become large. At the same time, the arbitrariness of (112),with its complicated form and two free coefficients, makes it clear that a better physicalinterpretation for these objects is badly needed. It is provided merely as a demonstrationthat a bounded sewing rule can be consistent with tree-level unitarity, and can be used forconstructing n-point CSP amplitudes for a simple theory with a single CSP interacting witha scalar. General amplitudes for a given process are constructed by summing over all allowedchannels (graphs) using the rules illustrated in Figure 7 to compute each graph.It would be very interesting to examine quantum properties of a theory built from thesesewing rules, both to see whether it is unitary and physically reasonable at a quantum leveland to see whether any physical arguments can further constrain the form of (112) – thisis an important open problem. It would also be very interesting to see whether analogous(but presumably more complicated) sewing rules exist for photon- and graviton-like CSPs. VI. CSP INTERACTIONS WITH GRAVITY
Another classic and powerful constraint on high-helicity theories is based on the difficultyof coupling high helicity particles to gravity. One may worry that similar obstructions arisein coupling CSPs to gravity. We have endeavoured to exclude CSP couplings to gravity byseveral standard arguments, and have so far been unsuccessful. All the same, we lack amodel of fully consistent gravitational interactions of CSPs. In this section, we show howsome of the widely known no-go arguments against high helicity don’t quite work for CSPs,and suggest some more subtle but potentially severe constraints on CSP interactions withgravity. We close with a roadmap of distinct possibilities for CSP gravitation. We hope theseremarks will inspire more conclusive work to constrain or develop each of these options.At the outset, we distinguish two very different classes of potential obstructions to gravita-tional CSP couplings: one formal, and one physical. The first question concerns the existenceof Lorentz-covariant graviton-CSP interactions. Though the simplest generalization of theWeinberg-Witten theorem [12] fails to exclude such interactions, further obstructions mayexist. The second concerns problematic physical effects of gravitational CSP interactions.The simplest such problem would be if cross-sections σ ( e + e − → CSP CSP ) through anoff-shell graviton formally diverge, as intuition motivated by the equivalence principle in thespin-basis would suggest. We will consider each potential problem in turn.The Weinberg-Witten theorem [12] is a remarkably simple argument forbidding a non-vanishing and covariant energy momentum tensor T µν for massless particles of helicity >
1. Essentially, the argument considers the action of rotations on matrix elements M µν ≡(cid:104) p (cid:48) , h | T µν | p, h (cid:105) in a brick-wall frame (cid:126)p = − (cid:126)p (cid:48) . Lorentz-invariance dictates that rotationsabout the (cid:126)p axis must leave M µν invariant, as they do not change p , p (cid:48) , or h . On the otherhand, these rotations re-phase both single-particle states by e ihθ (with the same sign) and7components of T µν by e imθ with | m | ≤
2. For h >
1, the net transformation e i (2 h + m ) θ (cid:54) = 1 isinconsistent with the invariance of M µν , unless M µν = 0.Unlike the high-helicity case, one can easily construct a non-zero, Lorentz-covariant, andconserved tensor matrix element for CSPs. For example: (cid:104) p (cid:48) , φ (cid:48) | T µν ( k ) | p, φ (cid:105) = ( p µ p (cid:48) ν + p (cid:48) µ p ν − p.p (cid:48) g µν ) e iρ (cid:18) (cid:15)φ (cid:48) ( p (cid:48) ) .kp (cid:48) .k − (cid:15)φ ( p ) .kp.k (cid:19) , (113)= ( p µ p (cid:48) ν + p (cid:48) µ p ν − p.p (cid:48) g µν ) e − iρ (cid:18) (cid:15)φ (cid:48) ( p (cid:48) ) .p + (cid:15)φ ( p ) .p (cid:48) p.p (cid:48) (cid:19) , (114)or in the spin basis (cid:104) p (cid:48) , n (cid:48) | T µν ( k ) | p, n (cid:105) = ( p µ p (cid:48) ν + p (cid:48) µ p ν − p.p (cid:48) g µν ) ˜ J n (cid:48) (cid:18) ρ (cid:15) + ( p (cid:48) ) .pp.p (cid:48) (cid:19) ˜ J ∗ n (cid:18) ρ (cid:15) + ( p ) .p (cid:48) p.p (cid:48) (cid:19) . (115)These matrix elements furnish appropriately leading-order covariant gravity scattering am-plitudes; their contraction into the graviton momentum k µ = p (cid:48) µ − p µ vanishes, ensuringgauge-invariance of this leading-order amplitude. While this is encouraging, it certainly doesnot guarantee that CSPs can couple consistently to gravity at all orders – it is not clearthat the above can be interpreted as matrix elements of an energy-momentum tensor thatis conserved as an operator.Though the relevance of (115) to gravity is unclear, we mention for completeness severalnotable physical properties. For k (cid:29) ρ , the matrix element (115) approaches δ n δ n (cid:48) T scalarµν ,so that it has helicity-0 correspondence, and there may be similar operators with helicity-1 correspondence. At the same time, it has a new and somewhat worrisome property:exchange of a single soft graviton with k (cid:28) ρ would maximally mix different spin states.This unphysical effect could potentially be ameliorated by a very small ρ , and we do not knowwhether it persists in a full eikonal calculation of CSP scattering in a weak gravitationalfield. In any case, total scattering cross-sections obtained from (115) are finite, as are theproduction cross-sections that would be obtained from the crossed matrix element.This brings us to the second question about (115), and about CSP coupling to gravitymore generally. Does an amplitude like (115) satisfy the equivalence principle? And ifnot, would equivalence-principle-respecting amplitudes necessarily lead to divergent ratesfor gravitational processes, from perturbative CSP pair production to Hawking emission?A sharp way of framing the equivalence principle for CSPs in terms of an S matrix isto seek Lorentz-covariant amplitudes with at least one graviton and one CSP among theirexternal states and study their properties (the simplest such amplitude would be for anni-hilation of two scalars into a graviton and a CSP). The equivalence principle in this contextusually means that each state should have unit coupling to the graviton [1]. It is not clearhow this requirement should be modified when dealing with CSP matter, which introducesa CSP propagator. Though this argument is not a direct test of (115) (the amplitude inquestion involves an off-shell CSP and on-shell graviton, while (115) is only defined for on-shell CSPs), it may shed some light on potential obstructions to the equivalence principlefor CSPs.8The form of (115) also underscores why the worry about divergent cross-sections is prema-ture. While we might expect s -channel graviton matrix elements M µν = (cid:104) T µν |{ k, n } , { k (cid:48) , n (cid:48) }(cid:105) ∼ k µ k (cid:48) ν δ nn (cid:48) based on the equivalence principle, leading to a divergent sum over n , this simpleguess is not Lorentz-covariant — modifying it by adding J n form factors as in (115) wouldlead to finite cross-sections, as already noted. And such form factors need not be at oddswith the equivalence principle. After all, the gravitational Hydrogen-Antihydrogen produc-tion cross-section is finite, despite the infinite multiplicity of hydrogen bound states below m e + m p . That puzzle is resolved by form factors with a simple physical interpretation:gravitons have exponentially suppressed couplings to the tower of increasingly de-localizedbound states. Though we do not have the same sharp physical picture in the case of CSPs,the very specialized non-local character of the J n ( ρz )’s on which CSP form factors dependdoes seem consistent with a similar effect. While this is still poorly understood, it is not auniquely gravitational problem.All of these questions can be framed far more precisely in a field-theory context [8]. Inthis case, we can identify several possible resolutions. The first and most concrete is basedon coupling of a helicity-2 h µν to the canonical Belinfante energy-momentum tensor Θ µν in aCSP gauge field theory. If this Θ µν is suitably invariant under the CSP gauge symmetry, thenit forms the basis for a formally consistent coupling to gravity, and allows a sharp evaluationof physical questions like production cross-sections. If no suitably invariant Θ µν exists, itrules out the standard geometric interpretation for CSP couplings to gravity. It might stillbe possible that helicity-2 gravity could couple to a conserved and covariant tensor T µν that is not the Belinfante stress-energy tensor. In the ρ = 0 limit, it is certainly consistentfor only the correspondence states to interact gravitationally — the accompanying towerof non-gravitating high helicity states would pose no problems, since they have no otherinteractions. A ρ (cid:54) = 0 generalization of this physics would be surprising, but we do not knowof general arguments against this possibility in systems with infinitely many species. Afinal possibility is that CSPs may more naturally couple to gravity mediated by a CSP withhelicity-2 correspondence, rather than by exact helicity-2 gravitons. This would at least bean economical use of CSPs. If such interactions can be realized (even in the simplest form ofa single self-interacting “CSP graviton”), their non-linear structure would likely generalizein an interesting way the curved space interpretation of general relativity! This possibilityheightens the motivation to better understand multi-CSP interactions. VII. CONCLUSIONS AND FUTURE DIRECTIONS
In [3], we presented new wave equations and wavefunctions for continuous-spin particles,and identified the unique family of Lorentz-covariant and smooth soft factors for the interac-tion of continuous-spin particles with matter. In this paper, we have demonstrated that thesoft factors exhibit a helicity correspondence at high energies, and that the simplest “scalar-9like” soft factors can be sewn together into consistent tree-level amplitudes for multi-CSPemission and for CSP-mediated scattering, and are consistent with gravity at leading order.We first showed that if a CSP theory has a high cutoff Λ (cid:29) ρ , three specific forms of softfactor yield cutoff un-suppressed (or in the last case, weakly suppressed) interactions. Thesethree soft factors exhibit a helicity correspondence : amplitudes for emission of a CSP whoseenergy E exceeds ρv are dominated by a single spin mode, this spin mode’s amplitudes areapproximated by helicity amplitudes up to corrections of order ( ρv/E ) , and production ofother spin modes is suppressed by increasing powers of ρv/E . Here v denotes the velocity ofthe emitter in the frame where the Little Group rotation coincides with a Lorentz rotation,i.e. where (cid:15) ± = 0.We demonstrated that multi-CSP-emission amplitudes that continue to exhibit scalar-likecorrespondence can be obtained by sewing together CSP soft factors. In a similar vein, weuse unitary factorization of CSP-mediated matter scattering to motivate a new sewing rulefor off-shell CSPs. Unitarity does not fully constrain the matter-matter scattering amplitudeor sewing rule, and appears to be consistent with several deformations, all of which maintainscalar-like correspondence. We also exhibited a consistent leading-order CSP coupling togravity that also exhibits scalar-like correspondence. Here, however, the correspondence iscontrolled by the soft graviton momentum rather than the CSP momentum. It is an openquestion whether and how these results generalize to photon- and graviton-like correspon-dence.Taken together, these results motivate speculation that theories of interacting CSPs mightexhibit all three types of correspondence, either simultaneously or on different “branches”of the theory; one theory might be approximately described by scalar interactions at highenergies, another by gauge bosons, and a third by gravity. This is of course an excitingpossibility — it suggests that CSPs might furnish natural generalizations of QED and gravity,in which the flat-space photon (graviton) remains massless but does not have a Lorentz-invariant helicity. Some constraints on this possibility and avenues for testing it will bediscussed in [6, 7].It is likely that many of the most basic questions we would like to answer about CSPs— including some with obvious phenomenological implications, like the modification offorce-laws at distances of O ( ρv ) − , the possible existence of a massive phase, and even CSP“electrostatics” — are more efficiently described in a field theory that elucidates the classicallimit of these theories. Nonetheless, many basic theoretical questions that can and should beapproached from an S-matrix viewpoint are left unanswered. Several of these are highlightedbelow, with an emphasis on those that can help to untangle the consistency and behaviorof photon-like and graviton-like CSP theories. • Scattering Mediated by Off-Shell CSPs:
A variety of CSP-mediated scatteringreactions warrant further investigation. In § V B we constructed a six-point amplitudethat factorizes into scalar-like soft factors at real momenta, then used it to infer two-0to-two scattering amplitudes consistent with real-momentum unitarity. It would bevery valuable to construct analogous amplitudes that factorize to photon- or graviton-like soft factors, or to prove that no such amplitude exists. Besides enabling the studyof CSP force laws, this would also provide clues for how to naturally describe CSPdegrees of freedom off the mass shell. The technique of § V B can be generalized tothese cases, but charge conservation implies that multiple channels must contributeto the six-point amplitude. This introduces subtleties to the photon- and graviton-like arguments that were absent from the scalar-like case. It would be very usefulto develop further tools for constraining these amplitudes, which do not require the“physical regulator” of the six-point amplitudes or simplify computation of this six-point amplitude. One possible strategy is using matter-leg shifted BCFW [34, 35]or Risager [36] deformations to expose poles at complex momentum. This requires aprescription for handling the complexified soft factor, and a generalization of BCFWrecursion relations to amplitudes with isolated essential singularities. • Multi-CSP Amplitudes:
The single-emission soft factors for scalar-like CSPs couldbe readily interpreted as sewing rules to build multi-CSP emission amplitudes ( § V C)with a product structure much like that of ordinary scalar amplitudes, which automat-ically factorize in a manner consistent with both unitarity and helicity correspondence.The analogous task for photon-like amplitudes is harder. Here, the soft factors can-not be interpreted as a complete set of sewing rules, but appear to require correlatedcontact interactions to simultaneously preserve the Lorentz-covariance of multi-CSPamplitudes or their high-energy boundedness. This is not surprising, as scalar QED(to which the CSP theory would have to reduce) has four-point vertices required topreserve Lorentz-invariance and unitarity. It would be interesting and very useful toseek a systematic set of rules for higher-point photon-like CSP amplitudes (and like-wise for the graviton-like amplitudes). The structure of these rules may also shed lighton the extent to which CSP theories are less local than ordinary gauge theories. • Self-Interacting CSPs:
Motivated by the possibility of a general helicity correspon-dence, it would be interesting to construct self-interaction amplitudes for CSPs, and inparticular to seek amplitudes that approach non-abelian gauge theory amplitudes inthe ρ → • Fermionic and Supersymmetric Generalizations:
This paper and [3] have fo-cused principally on CSP-scalar amplitudes where the CSP has integer spins. It wouldbe valuable to generalize the single and multi-CSP amplitudes constructed in this paper1to include coupling to fermionic matter. Fermionic matter may also have new typesof interactions with “double-valued” continuous-spin representations along the linesconsidered in [10]. Continuous-spin representations also have supersymmetric gener-alizations, discussed in [16], which may permit non-trivial interactions that generalizethe ones found in this paper. The additional symmetry constraints on supersymmet-ric CSP amplitudes might facilitate progress in understanding CSP interactions moregenerally. • Low-Energy Behavior:
The correspondence highlights a remarkable simplificationof CSP interactions at energies E (cid:29) ρ . In the spin-basis, CSP amplitudes in thishigh-energy limit approach helicity amplitudes. It is not clear what structure, if any,CSP soft factors have in the ultra-low-energy limit E (cid:28) ρ . Spin- and angle-basismatrix elements are both fairly anarchic in this regime, making the physical pictureof processes with characteristic energies E < ρ very unclear. The uniform large-argument behavior of J n with different n suggests that perhaps there is some otherbasis in which the dynamics of this regime is hierarchical. Finding or excluding sucha structure would shed light on the puzzling long-distance physics of CSPs. ACKNOWLEDGMENTS
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