0 N ov 2 01 8 Spinor-Helicity Formalism for Massless Fields in AdS

In this letter we suggest a natural spinor-helicity formalism for massless fields in AdS4. It is based on the standard realization of the AdS4 isometry algebra so(3, 2) in terms of differential operators acting on sl(2,C) spinor variables. We start by deriving the AdS counterpart of plane waves in flat space and then use them to evaluate simple scattering amplitudes. Finally, based on symmetry arguments we classify all three-point amplitudes involving massless spinning fields. As in flat space, we find that the spinor-helicity formalism allows to construct additional consistent interactions compared to approaches employing Lorentz tensors.


I. INTRODUCTION
The spinor-helicity formalism by now has established itself as the most efficient framework for representing onshell scattering amplitudes of massless particles in the 4d Minkowski space, see e. g. [1,2] for reviews.Success of this formalism in the original setup has motivated various extensions -to other dimensions [3][4][5][6][7] and to massive particles [8][9][10].At the same time, literature on the spinor-helicity formalism in curved space remains very limited.In [11] a version of the spinor-helicity formalism for massless fields in dS 4 was proposed.Despite its virtues, in some aspects it departs from the spinorhelicity formalism in flat space, e.g. it loses manifest Lorentz covariance.
In this letter we make an alternative proposal for the spinor-helicity formalism in (A)dS 4 , which has all features of the flat space formalism and, in particular, reduces to the latter for conformal theories.In this respect, it is worth mentioning the twistor approach [12], which is naturally adapted to describing fields in conformally flat spaces.Upon specializing to AdS backgrounds, the twistor approach can be used to obtain certain representations of massless scattering amplitudes in AdS 4 [13][14][15][16][17][18][19].
Our approach provides a different perspective on these results, as it allows to compute amplitudes directly from the space-time action and does not rely on a twistor-space description of the massless theories in question.
Besides having obvious motivations -e.g. development of tools that could facilitate computations of holographic/inflationary correlators and simplify their analytic structure -we are also interested in gaining a better understanding of higher-spin interactions in flat and curved backgrounds and clarifying their relation.In particular, as was emphasized recently [20][21][22], in flat space the spinor-helicity formalism and the light-cone approach admit additional cubic higher-spin vertices compared to those built of Lorentz tensors.Moreover, these additional vertices are crucial for consistency of higher-spin interactions [23,24] and are present in chiral higher-spin theories [25][26][27], see also [28] for a related earlier result.Until recently, the fate of additional interactions in AdS was not clear.In [29] the expectation that they also exist in AdS 4 was confirmed in the light-cone approach.Below we classify all consistent 3-point amplitudes for massless particles in AdS 4 using the spinor-helicity formalism and find agreement with [29].

II. SPINOR-HELICITY AND FLAT SPACE
The basic fact about massless representations in the 4d Minkowski space is that they are labelled by two quantum numbers -helicity h and momentum p.Using the isomorphism so(3, 1) ∼ sl(2, C) we have p µ = − 1 2 (σ µ ) αα λ α λ α.For h ≥ 0 the associated state can be represented by a potential where ε + ν is a polarization vector defined by Here µ is an auxiliary spinor and the ambiguity of its choice reflects gauge ambiguity.Alternatively, states can be represented by gauge invariant field strengths.For (1) the field strength reads Extension to h < 0 and to fermions is straightforward.Once plane wave solutions (1) are available, one can evaluate amplitudes using the Feynman rules in any theory of massless fields.
The amplitudes are strongly constrained by Poincare covariance.These constraints allow to fix 3-point amplitudes up to a coupling constant [30] to be Here and p ≡ i p i is the total momentum.To make (4) non-trivial one assumes that momenta are complex, hence λ and λ are not complex conjugate to each other.Then, A I (A II ) is singular for i h i < 0 ( i h i > 0) in the limit of real momenta and should be dropped as unphysical.

III. ADS4 AND PLANE WAVES
Massless representations of the AdS 4 isometry algebra so(3, 2) can be obtained by deforming the flat space translation generator as follows [59] where R is the AdS radius.This realization of massless representations is often referred to as the twisted adjoint representation [31].Similarly to what happens in flat space, all algebra generators commute with the helicity operator 2H ≡ λ α ∂ α − λ α ∂ α , which allows to split the representation space into representations of definite helicity.
For our further purposes it will be convenient to choose coordinates in AdS that make Lorentz symmetry manifest.Starting from the ambient space description of AdS as a hyperboloid X M X M = −R 2 , M = 0, 1, . . ., 4 and making the stereographic projection from X M = (0, . . ., 0, −R), followed by the appropriate rescaling, we arrive at intrinsic coordinates x µ , µ = 0, 1, 2, 3 with the metric The AdS boundary in these coordinates is given by ) in ambient coordinates.We will refer to these patches as the inner and the outer patches, while their union will be referred to as the global AdS.Finally, we note that the inversion x µ ↔ x µ 4R 2 x 2 acts as the reflection with respect to the origin in ambient space.
The AdS isometries act on bulk fields by Lie derivatives along Killing vectors.In our analysis Lorentz symmetry will be manifest, so we only specify Killing vectors associated with deformed translations.They act on scalar fields by To deal with spinning fields in terms of spinors we introduce a local Lorentz frame by means of the frame field It can be used to convert tensor fields from the coordinate basis to the local Lorentz basis, e.g.
A a = e a µ A µ .To maintain invariance of e a µ with respect to the AdS isometries, the associated diffeomorphisms should be compensated by the appropriate Lorentz transformations.For the deformed translations the compensating local Lorentz transformations read Now we will find the AdS counterpart of flat plane wave solutions [60].These will be derived based on a consideration that plane waves should serve as intertwining kernels between two representations -the spinor-helicity representation and the space-time representation.We will focus on plane waves for field strengths, as these are gauge invariant and do not require any auxiliary objects, such as reference spinors.Then, Lorentz invariance requires that indices of field strengths can only be carried by λ α , λ α, (xλ All the remaining spinor indices should be covariantly contracted, which implies that plane waves may also depend on two scalars x α αλ α λ α and x α αx αα .Finally, we require that the action of the deformed translations on the plane wave in representation (5) agrees with that in space-time (6), supplemented with compensating Lorentz transformations (7).This results in a differential equation that fixes the functional dependence of plane waves on x α αλ α λ α and x α αx αα .For h ≥ 0 it has four linearly independent solutions where x + ≡ xθ(x) and x − ≡ −xθ(−x).Analogously, solutions can be constructed for h < 0. These solutions have the following properties.Plane wave F p|i (F p|o ) is supported on the inner (outer) patch and the inversion maps F p|i ↔ F s|o and F s|i ↔ F p|o .Note that the support of F s|i (F s|o ) is not the same as of F p|i (F p|o ) as x 2 can be negative.One can also consider the following linear combinations [61] which are supported on the global AdS patch.Note that both F p|g and F p|i reduce to familiar flat plane waves in the flat space limit R → ∞.Accordingly, we will call F p|g , F p|i as well as F p|o -their counterpart with the support in the outer patch -physical solutions, while the remaining solutions will be referred to as shadow ones [62].
Finally, we would like to comment on the role of conformal symmetry in this discussion.Massless fields in 4d are conformally invariant [32], however, their description in terms of potentials breaks conformal invariance except for the spin one case.Given that AdS and flat spaces are conformally equivalent, this means that at least the physical solution in (8) could have been obtained by applying the appropriate conformal transformation on a flat space plane wave solution.Putting differently, our labelling of AdS plane waves is consistent with the flat space one modulo conformal transformations.Conformal invariance also allows to conclude that the spin-1 potential is given by flat formula (1).A thorough investigation of potentials will be given elsewhere.

IV. ADS4 SCATTERING AMPLITUDES
In AdS one can define tree-level scattering amplitudes as the classical action evaluated on the solutions to the linearized equations of motion.Below we will evaluate some simple amplitudes using plane wave solutions we have just obtained.We will focus on the scattering of physical plane waves, as they have smooth flat limit and a clearer connection to the familiar flat space amplitudes.
In the following we will encounter integrals [33] which will be denoted as I which can be regarded as a result of a formal evaluation of the Fourier transform according to the rule x 2 → − p .Representation ( 10) makes the distributional nature of amplitudes manifest and the flat space limit more intuitive.Note that for non-negative n the right hand side for I p|g n in ( 10) is a well-defined distribution.It can be shown that this result is consistent with representation (9), see [33].
In these terms the n-point amplitudes for a scalar selfinteraction vertex L = 1 n! √ −gϕ n are given by depending on the AdS patch we are using.For n = 3 the amplitude is divergent, which is consistent with the standard AdS/CFT analysis [34], where the 3-point Witten diagram for ∆ = 1 scalars also gives a divergent result.Similarly, we can evaluate more general vertices involving field strengths of spinning fields.For example, for L = 1 2 √ −gϕF α1 α2 F α1 α2 for different patches we find Amplitudes of the form A p|g 3 have been previously derived in the twistor literature [13][14][15][16][17][18][19].
Finally, considering the Yang-Mills vertex, as a consequence of conformal invariance, we find exactly the same amplitude as in flat space, except that now we also have its variants associated with different patches.In fact, conformal invariance of the Yang-Mills action implies that the same conclusion holds for all tree-level spinor-helicity amplitudes in AdS.
Having studied some simple examples, we will now move to the case of general spinning 3-point amplitudes.These will be derived based on the symmetry considerations in complete analogy with the flat space case.As in flat space, Lorentz covariance is manifest and is achieved by combining spinors into spinor products.Moreover, once helicities on external lines are fixed, this imposes constraints on homogeneity degrees of spinors.For amplitudes being genuine functions of spinor products this leads to an ansatz (13) where x ≡ [12] 12 , y ≡ [23] 23 and z ≡ [31] 31 .It only remains to impose correct transformation properties with respect to deformed translations This gives a system of differential equations on f (x, y, z).
It can be shown that when at least one helicity is nonzero, one has four linearly independent solutions [63] A where I's are given by (9).When all helicities are vanishing, f I coincides with f III and f II coincides with f IV .
Classification (15) is different from (4) only in two respects.First is that so(3, 2) covariance turns out to be consistent with splitting the global AdS into two patches, each being associated with its own amplitude.This explains why we get four solutions in (15) instead of two solutions in flat space.The second difference is that the flat space momentum-conserving delta functions in AdS are replaced with one of I's (9) depending on the patch one is interested in.Based on the flat limit, we argue that A I and A II (A III and A IV ) for i h i < 0 ( i h i > 0) are unphysical.It is worth to note that these amplitudes are divergent, see discussion below (9).The same refers to all amplitudes with i h i = 0.
Amplitudes with three shadow fields using the inversion reduce to amplitudes where all fields are physical.Amplitudes where physical and shadow fields are mixed require separate analysis.If these are genuine functions, they should be given by linear combinations of ( 15).Another potential possibility is that they are given by distributions.In this respect it is worth to note that by considering an ansatz for a distribution supported on p = 0 and requiring (14), we again end up with (15), where I's appear in representation (10).

V. CONCLUSIONS
In the present letter we suggested a natural generalization of the spinor-helicity formalism to (A)dS 4 .We started by generalizing the familiar flat space plane wave solutions to AdS and then used them to evaluate some simple 3-point amplitudes.We also classified all consistent spinning 3-point amplitudes by requiring correct transformation properties.We found that, as in flat space, for three generic spins, by picking different signs of helicities, one can construct four different parityinvariant amplitudes.At the same time, approaches that involve Lorentz tensors result only in two consistent parity-invariant structures both on the bulk [35][36][37][38][39][40] and boundary [41,42] sides.This phenomenon directly generalizes an analogous one in flat space and is consistent with a recent analysis in the light-cone gauge [29].
The amplitudes that we computed were defined as the classical action evaluated on the particular basis of solutions to the linearized equations of motion.This definition is related to the holographic one -where amplitudes are identified with boundary correlators and computed in the bulk by Witten diagrams [43][44][45] -by a mere change of a basis for the states appearing on external lines.Unlike bulk-to-boundary propagators, the plane wave solutions that we employed do not have a boundary limit that would allow to associate them with boundary sources.Instead, they have a transparent flat limit, which also makes the flat limit of the spinor-helicity amplitudes more intuitive.In this respect, our plane waves serve as the properly focused scattering states necessary to access flat space physics from holography, see e. g. [46][47][48].An explicit transformation relating the two bases will be given elsewhere.
An obvious future direction is to extend these results to higher-point amplitudes and see how various bulk scattering processes manifests themselves in amplitudes' analytic structure, see e. g. [49][50][51] for related work.Optimistically, clear understanding of the analytic structure of AdS spinor-helicity amplitudes may lead to the development of the on-shell methods, which are as efficient as in flat space.
Finally, our construction may be useful in shedding the light on how higher-spin no-go theorems can be circumvented in flat space.Thus far it is known how to construct higher-spin theories in flat space only in the chiral sector [25][26][27], while their parity-invariant completions are obstructed by non-localities.At the same time, higher-spin theories in AdS have solid support from holography [52,53].We believe that the connection between higher spin theories in flat and in AdS spaces does exist and both sides will benefit from its clarification.

p|i λ and I o|i λ
respectively and K is the modified Bessel function of the second kind.These formulas should be understood in the sense of distributions and are valid for real λ except negative integers, where I p|i λ and I o|i λ diverge.In the following we will only need I p|i λ and I o|i λ for integer values of λ = n.We find it convenient to use the notation