Derivative interactions in de Rham-Gabadadze-Tolley massive gravity

TL;DR

This work investigates whether derivative interactions can be consistently added to ghost-free de Rham-Gabadadze-Tolley (dRGT) massive gravity without reintroducing BD ghosts. By constructing the most general derivative interactions from the Riemann tensor and organizing them by order (HR, H^2R) and dimensionality, the authors show these terms can be resummed into a two-parameter family expressed via the K-tensor, with all nonlinear pieces contributing at the Λ3 scale. However, analysis in the decoupling limit reveals that the scalar and tensor equations acquire fourth derivatives, signaling Boulware-Deser ghosts at Λ3, unless the derivative-interaction mass scale M is taken well below M_Pl to push ghosts to higher energies. Consequently, while a two-parameter derivative-interaction family can be written, ghost-free consistency at Λ3 requires suppressing these interactions (M ≪ M_Pl), effectively reducing the theory to pure dRGT below the cutoff, with important implications for model-building and cosmology in massive gravity.

Abstract

We investigate the possibility of a new massive gravity theory with derivative interactions as an extension of de Rham-Gabadadze-Tolley massive gravity. We find the most general Lagrangian of derivative interactions using Riemann tensor whose cutoff energy scale is $Λ_3$, which is consistent with de Rham-Gabadadze-Tolley massive gravity. Surprisingly, this infinite number of derivative interactions can be resummed with the same method in de Rham-Gabadadze-Tolley massive gravity, and remaining interactions contain only two parameters. We show that the equations of motion for scalar and tensor modes in the decoupling limit contain fourth derivatives with respect to spacetime, which implies the appearance of ghosts at $Λ_3$. We claim that consistent derivative interactions in de Rham-Gabadadze-Tolley massive gravity have a mass scale $M$, which is much smaller than the Planck mass $M_{\rm Pl}$.