Transforming gravity: from derivative couplings to matter to
  second-order scalar-tensor theories beyond the Horndeski Lagrangian
  We study the structure of scalar-tensor theories of gravity based on
derivative couplings between the scalar and the matter degrees of freedom
introduced through an effective metric. Such interactions are classified by
their tensor structure into conformal (scalar), disformal (vector) and extended
disformal (traceless tensor), as well as by the derivative order of the scalar
field. Relations limited to first derivatives of the field ensure second order
equations of motion in the Einstein frame and hence the absence of Ostrogradski
ghost degrees of freedom. The existence of a mapping to the Jordan frame is not
trivial in the general case, and can be addressed using the Jacobian of the
frame transformation through its eigenvalues and eigentensors. These objects
also appear in the study of different aspects of such theories, including the
metric and field redefinition transformation of the path integral in the
quantum mechanical description. Although sane in the Einstein frame, generic
disformally coupled theories are described by higher order equations of motion
in the Jordan frame. This apparent contradiction is solved by the use of a
hidden constraint: the contraction of the metric equations with a Jacobian
eigentensor provides a constraint relation for the higher field derivatives,
which allows one to express the dynamical equations in a second order form.
This signals a loophole in Horndeski's theorem and allows one to enlarge the
set of scalar-tensor theories which are Ostrogradski-stable. The transformed
Gauss-Bonnet terms are also discussed for the simplest conformal and disformal
relations.
