Ghost free Massive Gravity in the Stückelberg language

TL;DR

This work addresses the Boulware-Deser ghost problem in four-dimensional massive gravity by reconciling the Stückelberg and ADM formulations. It shows that beyond the decoupling limit, a nonperturbative constraint among the four Stückelberg fields persists, ensuring the BD ghost is eliminated; this is demonstrated via a vanishing kinetic determinant $\det\mathcal A=0$ and explicit primary and secondary constraints in both toy models and general settings, including nontrivial backgrounds. The results align with the ADM proofs of ghost-freedom for the dRGT/Hassan–Rosen class, thereby unifying the two cameras of analysis and clarifying the constraint structure responsible for ghost-freedom. The findings strengthen the theoretical foundation of ghost-free massive gravity and reinforce its consistency across perturbative orders and backgrounds, with implications for the viability of such theories in phenomenology.

Abstract

Massive Gravity in four dimensions has been shown to be free of the Boulware-Deser (BD) ghost in the ADM language for a specific choice of mass terms. We show here how this is consistent with the Stückelberg language beyond the decoupling limit, and how the constraint required to remove the BD ghost arises in this framework nonperturbatively, without the use of field redefinitions. We emphasize a subtlety in obtaining this constraint, that has been overlooked in previous literature. In both the ADM and Stückelberg formalisms the vanishing of the determinant of a Hessian guarantees the absence of the BD ghost.