On spin 3 interacting with gravity

TL;DR

This paper constructs a gauge-invariant cubic interaction vertex for spin 3 and spin 2 particles, shows its form in terms of the linearized Riemann tensor, and extends it to (A)dS space and massive cases, aligning with known theories.

Contribution

It explicitly constructs a gauge-invariant cubic vertex for spin 3 and 2 particles and extends it to (A)dS and arbitrary gravitational backgrounds.

Findings

Vertex expressed as R∂Φ∂Φ in linearized form

Deformation yields standard gravitational interaction for spin 3 in (A)dS

Higher derivative terms extend massive spin 3 description to arbitrary backgrounds

Abstract

Recently Boulanger and Leclercq have constructed cubic four derivative $3-3-2$ vertex for interaction of spin 3 and spin 2 particles. This vertex is trivially invariant under the gauge transformations of spin 2 field, so it seemed that it could be expressed in terms of (linearized) Riemann tensor. And indeed in this paper we managed to reproduce this vertex in the form $R \partial \Phi \partial \Phi$, where $R$ -- linearized Riemann tensor and $\Phi$ -- completely symmetric third rank tensor. Then we consider deformation of this vertex to $(A)dS$ space and show that such deformation produce "standard" gravitational interaction for spin 3 particles (in 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 allows one to extend gauge invariant description of massive spin 3 particle from constant curvature spaces to arbitrary gravitational backgrounds satisfying $R_{\mu\nu} = 0$.