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Helicity Decomposition of Ghost-free Massive Gravity

Claudia de Rham, Gregory Gabadadze, Andrew J. Tolley

TL;DR

This work performs a detailed helicity Decomposition of the full Lagrangian for ghost-free massive gravity to track the five propagating degrees of freedom and identify the true strong-coupling scale. It shows that the first interactions of the helicity-0 mode arise at Λ3 = (M_{ m Pl} m^2)^{1/3} and that perturbation theory remains well-defined below this scale, with no ghosts appearing at Λ5 or Λ4 once all contributions are accounted for. The analysis demonstrates that the Stueckelberg and helicity pictures are equivalent below Λ3, and that matter couplings can be chosen to preserve ghost-freedom and perturbative control in realistic settings. These results are consistent with ADM and Stueckelberg analyses and clarify misconceptions about lower strong-coupling scales in massive gravity, reinforcing the viability of the ghost-free models.

Abstract

We perform a helicity decomposition in the full Lagrangian of the class of Massive Gravity theories previously proven to be free of the sixth (ghost) degree of freedom via a Hamiltonian analysis. We demonstrate, both with and without the use of nonlinear field redefinitions, that the scale at which the first interactions of the helicity-zero mode come in is $Λ_3=(M_Pl m^2)^{1/3}$, and that this is the same scale at which helicity-zero perturbation theory breaks down. We show that the number of propagating helicity modes remains five in the full nonlinear theory with sources. We clarify recent misconceptions in the literature advocating the existence of either a ghost or a breakdown of perturbation theory at the significantly lower energy scales, $Λ_5=(M_Pl m^4)^{1/5}$ or $Λ_4=(M_Pl m^3)^{1/4}$, which arose because relevant terms in those calculations were overlooked. As an interesting byproduct of our analysis, we show that it is possible to derive the Stueckelberg formalism from the helicity decomposition, without ever invoking diffeomorphism invariance, just from a simple requirement that the kinetic terms of the helicity-two, -one and -zero modes are diagonalized.

Helicity Decomposition of Ghost-free Massive Gravity

TL;DR

This work performs a detailed helicity Decomposition of the full Lagrangian for ghost-free massive gravity to track the five propagating degrees of freedom and identify the true strong-coupling scale. It shows that the first interactions of the helicity-0 mode arise at Λ3 = (M_{ m Pl} m^2)^{1/3} and that perturbation theory remains well-defined below this scale, with no ghosts appearing at Λ5 or Λ4 once all contributions are accounted for. The analysis demonstrates that the Stueckelberg and helicity pictures are equivalent below Λ3, and that matter couplings can be chosen to preserve ghost-freedom and perturbative control in realistic settings. These results are consistent with ADM and Stueckelberg analyses and clarify misconceptions about lower strong-coupling scales in massive gravity, reinforcing the viability of the ghost-free models.

Abstract

We perform a helicity decomposition in the full Lagrangian of the class of Massive Gravity theories previously proven to be free of the sixth (ghost) degree of freedom via a Hamiltonian analysis. We demonstrate, both with and without the use of nonlinear field redefinitions, that the scale at which the first interactions of the helicity-zero mode come in is , and that this is the same scale at which helicity-zero perturbation theory breaks down. We show that the number of propagating helicity modes remains five in the full nonlinear theory with sources. We clarify recent misconceptions in the literature advocating the existence of either a ghost or a breakdown of perturbation theory at the significantly lower energy scales, or , which arose because relevant terms in those calculations were overlooked. As an interesting byproduct of our analysis, we show that it is possible to derive the Stueckelberg formalism from the helicity decomposition, without ever invoking diffeomorphism invariance, just from a simple requirement that the kinetic terms of the helicity-two, -one and -zero modes are diagonalized.

Paper Structure

This paper contains 28 sections, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Relationship between the Helicity and Stückelberg Decompositions
  3. A Two Scalar Field Toy-Model
  4. How to have higher derivatives and evade ghosts
  5. Coupling to sources
  6. Perturbation theory is well-defined
  7. Ghost-free Massive Gravity
  8. The model
  9. Helicity decomposition
  10. Relevant interactions at the scale $\Lambda_5$
  11. Triviality of the S-matrix in the $\Lambda_5$ theory
  12. Absence of ghosts nonlinearly or around arbitrary backgrounds
  13. Coupling to matter
  14. A first look at external sources
  15. Consistency Relation for matter
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Tree-level diagrams contributing to the 4-point function $\langle \phi^4 \rangle$ at the scale $\Lambda$.
  • Figure 2: Tree-level diagrams contributing to the 3-point function $\langle A \pi \pi \rangle$ at the scale $\Lambda_4$.