Classification of all 3 particle S-matrices quadratic in photons or gravitons

bstract 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.