Ghost-Free Derivative Interactions for a Massive Graviton

TL;DR

The paper extends ghost-free massive gravity by constructing pseudo-linear derivative interactions that preserve the correct degrees of freedom, beyond the standard dRGT potentials. Through Hamiltonian analysis and decoupling-limit reasoning, it demonstrates that a broad class of interactions L_{d,n} remains ghost-free and exhibits a scalar-tensor structure mirroring the fully non-linear theory in key sectors. It also predicts a possible missing two-derivative term in D=4 and argues these pseudo-linear terms imply the existence of additional fully non-linear derivative interactions, offering useful toy models and potential new parameters for massive gravity. Together, these results illuminate the landscape of ghost-free derivative interactions and guide exploration of the complete non-linear theory.

Abstract

If the kinetic interactions of a Lorentz-invariant massive graviton are Einstein-Hilbert, then the only possible potential terms free from the Boulware-Deser ghost are those of de Rham, Gabadadze and Tolley (dRGT). We point out there are other possibilities if the kinetic terms are not required to be Einstein-Hilbert. We construct pseudo-linear ghost-free potentials, terms that derive in a natural way from the linear theory. The simplicity of this approach allows us to construct diffeomorphism non-invariant higher-derivative interaction terms that do not introduce ghosts. We conjecture that these terms should have counterparts in the full dRGT theory. These terms would introduce new free parameters into the theory and may change some of the conclusions heretofore drawn.