Complete Decoupling Limit of Ghost-free Massive Gravity

TL;DR

This work delivers the complete decoupling-limit (DL) of ghost-free massive gravity with a Minkowski reference metric, including full interactions among helicity-2, helicity-1, and helicity-0 modes. By employing a vierbein formulation with diffeomorphism and local Lorentz Stückelberg fields, the authors derive a resummed, all-orders expression for helicity-1 interactions and identify the Lorentz-Stückelberg equations that enforce the symmetric vierbein condition, ensuring equivalence with the metric dRGT action. They provide explicit solutions for the Lorentz Stückelberg field, both in a general formal form and via a nonlinear, covariantized diagonalization approach, highlighting the universal nature of the symmetric vierbein constraint. The DL action is assembled to show a manifestly gauge-invariant vector sector with an infinite tower of vector-scalar interactions, while vector self-interactions are absent, offering a complete and practical framework for analyzing Vainshtein-screening and related phenomena in massive gravity. The results pave the way for applications to multi-vierbein theories and further explorations of the theory’s phenomenology in different backgrounds.

Abstract

We present the complete form of the decoupling limit of ghost-free massive gravity with a Minkowski reference metric, including the full interactions of the helicity-1 and helicity-0 modes of the massive spin-2 field. While in the metric language the square root structure of the mass terms makes it difficult to find a simple way to write down the interactions, we show that using the vierbein formulation of massive gravity, including Stueckelberg fields for both diffeomorphism and local Lorentz symmetries, we can find an explicitly resummed expression for the helicity-1 field interactions. We clarify the equations of motion for the Lorentz Stueckelberg fields and how these generate the symmetric vierbein condition which guarantees equivalence between the vierbein and metric formulations of massive gravity.