Covariant master theory for novel Galilean invariant models and massive gravity

Coupling the Galileons to a curved background has been a tradeoff between maintaining second order equations of motion and the Galilean shift symmetries, and allowing the background metric to be dynamical. We propose a construction which can achieve all three for a novel class of Galilean invariant models by coupling a scalar with the Galilean symmetry to a massive graviton. This generalizes the brane construction for Galileons by adding to the brane a dynamical metric (nonuniversally) interacting with the Galileon field. Alternatively, it can be thought of as an extension of the ghost-free massive gravity, or as a massive graviton-Galileon scalar-tensor theory. In the decoupling limit of these theories, new kinds of Galileon invariant interactions arise between the scalar and the longitudinal mode of the graviton. These have higher order equations of motion and infinite powers of the field, yet are ghost free.