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Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations

We consider all degenerate scalar-tensor theories that depend quadratically on second order derivatives of a scalar field, which we have identified in a previous work. These theories, whose degeneracy in general ensures the absence of Ostrogradski instability, include the quartic Horndenski Lagrangian as well as its quartic extension beyond Horndeski, but also other families of Lagrangians. We study how all these theories transform under general conformal-disformal transformations and find that they can be separated into three main classes that are stable under these transformations. This leads to a complete classification modulo conformal-disformal transformations. Finally, we show that these higher order theories include mimetic gravity and some particular khronometric theories. They also contain theories that do not correspond, to our knowledge, to already studied theories, even up to field redefinitions.