Healthy scalar-tensor theories with third-order derivatives: Generalized disformal Horndeski and beyond

We systematically construct ghost-free scalar-tensor theories whose Lagrangian includes up to third-order derivatives of the scalar field. Using a spatially covariant action written in terms of the ADM variables, we impose degeneracy and consistency conditions that ensure the propagation of only one scalar and two tensor degrees of freedom. The resultant theories extend the generalized disformal Horndeski and U-DHOST theories. We discuss the transformation properties of these theories under generalized disformal transformations.