On spin 3 interacting with gravity

TL;DR

This work constructs and analyzes a four-derivative cubic vertex for the spin-3–spin-3–spin-2 system, showing it can be written in terms of the linearized Riemann tensor R_{mu nu, alpha beta} and the spin-3 field Phi_{mu nu alpha}. It then demonstrates how to deform this vertex to (A)dS space, restoring gauge invariance with lower-derivative and non-minimal terms to recover the standard linear gravitational interaction for spin-3. The authors further develop a gauge-invariant description of massive spin-3 in Minkowski space and show that, in backgrounds with R_{mu nu} = 0, the same higher-derivative construction can extend to arbitrary gravitational backgrounds, preserving gauge invariance at linear order. In four dimensions these results simplify and illustrate a consistent pathway connecting massless high-spin interactions in curved spaces with massive high-spin dynamics in general gravitational backgrounds, aligning with unitarity considerations and broader high-spin program.

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 $R \partial Φ\partial Φ$, where $R$ -- linearized Riemann tensor and $Φ$ -- 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 $R_{μν} = 0$.