^{*}

^{†}

^{3}.

In this Letter, we suggest a natural spinor-helicity formalism for massless fields in four-dimensional anti–de Sitter space (

The spinor-helicity formalism by now has established itself as the most efficient framework for representing on-shell scattering amplitudes of massless particles in the 4D Minkowski space; see, e.g.,

In this Letter, we make an alternative proposal for the spinor-helicity formalism in four-dimensional anti–de Sitter space (

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

The basic fact about massless representations in the 4D Minkowski space is that they are labelled by two quantum numbers—helicity

The amplitudes are strongly constrained by the Poincaré covariance. These constraints allow us to fix three-point amplitudes up to a coupling constant

Massless representations of the

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

The AdS isometries act on bulk fields by Lie derivatives along Killing vectors. In our analysis, the Lorentz symmetry will be manifest, and so we only specify Killing vectors associated with deformed translations. They act on scalar fields by

Now, we will find the AdS counterpart of the flat plane wave solutions

These solutions have the following properties: Plane waves

At this point, one may wonder whether splitting of the plane wave solutions into patches as in Eq.

Finally, we would like to comment on the role of conformal symmetry in this discussion. Massless fields in 4D are conformally invariant

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 regular plane waves because they have a smooth flat limit and a clearer connection to the familiar flat space amplitudes

In the following, we will encounter integrals

In these terms, the

Similarly, we can evaluate more general vertices involving the field strengths of spinning fields. For example, for

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 three-point amplitudes. In contrast to the previous analysis, in which we computed amplitudes using their field-theoretic definition, in the following, the amplitudes will be found by requiring correct transformation properties—that is, solely from representation theory considerations. As in flat space, the 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 the homogeneity degrees of spinors. For amplitudes being genuine functions of spinor products, this leads to an ansatz

Classification

Amplitudes with three singular plane waves using the inversion reduce to amplitudes in which all plane waves are regular. Amplitudes in which regular and singular plane waves are mixed require a separate analysis. If these are genuine functions, they should be given by linear combinations of Eq.

In the present Letter, we suggested a natural generalization of the spinor-helicity formalism to

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—in which amplitudes are identified with boundary correlators and computed in the bulk by Witten diagrams

An obvious future direction is to extend these results to higher-point amplitudes and see how various bulk scattering processes manifest themselves in an amplitude’s analytic structure; see, e.g.,

Finally, our construction may be useful in shedding light on how higher-spin no-go theorems (see

We are grateful to V. Didenko, R. Metsaev, K. Mkrtchyan, T. McLoughlin, and E. Skvortsov for fruitful discussions and to T. Adamo, E. Skvortsov, and A. Tseytlin for comments on the draft. We would also like to thank E. Joung for pointing out

In the following, we will be primarily concerned with the AdS space case, but our analysis can be straightforwardly extended to de Sitter space.

A related approach that makes all higher-spin symmetry manifest was developed in Refs.

In fact, by appropriately deforming the translation generator, one can realize all symmetric representations of

This discussion parallels parts of Ref.

Note that the dual representation to Eq.

For earlier discussions on the flat limit of higher-spin vertices, see Ref.

What we actually derive is the solutions for