Ghost-free, gauge invariant SVT generalizations of Horndeski theory

Abstract: We present a new type of Scalar-Vector-Tensor (SVT) theories with higher derivatives of all the fields in the action, but with second order equations of motion. The higher derivative vector field is invariant under a U(1) gauge transformation and the Scalar-Tensor sector corresponds to Horndeski theory. We also present a subclass of these SVT theories with 8 free functions of $\pi$ and $X$ where the speed of the tensor and vector modes is exactly the same. In particular, the Horndeski functions $G_4(\pi,X)$ and $G_5(\pi)$ remain free, while the speed of the vector modes tracks the speed of the tensor modes. Additionally, the vector sector retains freedom through the four new functions. All the theories here shown are a generalization of the Kaluza-Klein reduction of 5D Horndeski theory, sharing the main properties in cosmology, but including new free scalar functions in the Lagrangian.