Abstract
We explicitly construct every kinematically allowed three particle graviton- graviton-P and photon-photon-P S-matrix in every dimension and for every choice of the little group representation of the massive particle P. We also explicitly construct the spacetime Lagrangian that generates each of these couplings. In the case of gravitons we demonstrate that this Lagrangian always involves (derivatives of ) two factors of the Riemann tensor, and so is always of fourth or higher order in derivatives. This result verifies one of the assumptions made in the recent preprint [1] while attempting to establish the rigidity of the Einstein tree level S-matrix within the space of local classical theories coupled to a collection of particles of bounded spin.
Highlights
The authors of the recent paper [1] have conjectured that four particle scattering amplitudes in ‘consistent’ classical theories can grow no faster that s2, as s is taken to infinity, at fixed t (s and t are Mandlestam variables)
They went on to use this ‘Classical Regge Growth’ (CRG) conjecture to classify consistent classical1 four graviton S-matrices. In particular they demonstrated that the classical Einstein S-matrix in D ≤ 6 spacetime dimensions admits no polynomial deformations2 consistent with the CRG conjecture
They argued that additions to the Einstein S-matrix with a finite number of additional poles3 always violates the CRG scaling bound in D ≤ 6
Summary
The authors of the recent paper [1] have conjectured that four particle scattering amplitudes in ‘consistent’ classical theories can grow no faster that s2, as s is taken to infinity, at fixed t (s and t are Mandlestam variables). In the special case of graviton-graviton-P scattering, it turns out (see below) that the dimensionality of this space does not grow as a function of the complexity of the Lorentz representation of the particle P or of the spacetime dimension in which we work, but is, instead, bounded to be less than or equal to eight For this reason it is not too difficult to enumerate all possible couplings for particles P transforming in any given representation of the massive little group SO(D − 1). It would be useful to perform such a study (if it has not already been done) and to establish the detailed map — induced by the AdS/CFT correspondence — from the S-matrix structures constructed in this paper to three point function CFT correlators.8 To end this introduction we note that three particle S-matrices play roughly the same role for the Lorentz group that Clebsh-Gordon coefficients play in compact groups. We hope that future investigations into diverse aspects of gravitational scattering will find several additional uses for the three point scattering amplitudes listed in this paper
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