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.
