On massive higher spins and gravity. II. Spin 3

TL;DR

This work extends the gravitational coupling analysis from massive spin $5/2$ to bosonic spin $3$, using a gauge-invariant frame-like formalism to construct a minimal gravitational vertex that includes necessary non-minimal terms. It analyzes the massless, partially massless, and massive regimes, employing Fradkin–Vasiliev methods for massless cases and Skvortsov–Vasiliev techniques for partially massless theories, then applies a down–up strategy to obtain the massive vertex. Key results include a non-minimal, gauge-consistent vertex with a smooth $dS_4$ massless limit ($\Lambda\neq0$), a smooth flat limit ($m\neq0$), and a special unitary boundary at $m^2=6\Lambda$ where standard minimal couplings vanish and non-minimal terms prevail. The findings generalize the spin-$5/2$ program to the first bosonic case necessitating non-minimal corrections, clarifying the structure of higher-spin gravity couplings in curved backgrounds and their unitary regions. This provides a concrete framework for consistent interactions of massive higher-spin bosons with gravity and informs potential extensions to higher spins and cosmological settings.

Abstract

In this paper, we continue our investigation of gravitational interactions for massive higher spins, extending our recent work on massive spin 5/2 to massive spin 3, including its massless and partially massless limits. To construct the minimal gravitational interactions (i.e. vertexes containing both standard minimal interactions and non-minimal ones, which are necessary for any $s \ge 5/2$), we use a gauge invariant frame-like formalism. Similarly to the spin 5/2 case, there is a special point $m^2 = 6Λ$, which corresponds to a boundary of the unitary allowed region in $dS_4$, where minimal interactions disappear, leaving only the non-minimal ones.

Figures (1)

• Figure 1: Unitary forbidden region $m^2 < 6\Lambda$ for massive spin 3