On spin 3 interacting with gravity

Recently Boulanger and Leclercq have constructed a cubic four derivative 3 - 3 - 2 vertex for the interaction of spin 3 and spin 2 particles. This vertex is trivially invariant under the gauge transformations of the spin 2 field, so it seemed that it could be expressed in terms of the (linearized) Riemann tensor. And indeed in this paper we managed to reproduce this vertex in the form R∂Φ∂Φ, where R is the linearized Riemann tensor and Φ is the completely symmetric third rank tensor. Then we consider the deformation of this vertex to (A)dS space and show that such deformation produces a 'standard' gravitational interaction for spin 3 particles (in the linear approximation) in agreement with general construction of Fradkin and Vasiliev. Then we turn to the massive case and show that the same higher derivative terms allow one to extend the gauge invariant description of a massive spin 3 particle from constant curvature spaces to arbitrary gravitational backgrounds satisfying Rμν = 0.