Ghost-free, gauge invariant SVT generalizations of Horndeski theory

We analyze the generalizations of Kaluza--Klein compactifications of 5D Horndeski theory. They are Scalar--Vector--Tensor (SVT) theories with higher derivatives in the action, but with second order equations of motion. The vector field is invariant under a U(1) gauge transformation and the Scalar--Tensor sector corresponds to Horndeski theory. A subclass of these SVT theories is such that the Horndeski functions $G_4(π,X)$ and $G_5(π)$ remain free, while the speed of the tensor and vector modes is exactly the same. We show a subclass where the vector sector retains freedom through new functions of $π,\, X$ while the speed of the vector modes still tracks the speed of the tensor modes.