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We study the structure of scalar-tensor theories of gravity based onderivative couplings between the scalar and the matter degrees of freedomintroduced through an effective metric. Such interactions are classified bytheir tensor structure into conformal (scalar), disformal (vector) and extendeddisformal (traceless tensor), as well as by the derivative order of the scalarfield. Relations limited to first derivatives of the field ensure second orderequations of motion in the Einstein frame and hence the absence of Ostrogradskighost degrees of freedom. The existence of a mapping to the Jordan frame is nottrivial in the general case, and can be addressed using the Jacobian of theframe transformation through its eigenvalues and eigentensors. These objectsalso appear in the study of different aspects of such theories, including themetric and field redefinition transformation of the path integral in thequantum mechanical description. Although sane in the Einstein frame, genericdisformally coupled theories are described by higher order equations of motionin the Jordan frame. This apparent contradiction is solved by the use of ahidden constraint: the contraction of the metric equations with a Jacobianeigentensor 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 theset of scalar-tensor theories which are Ostrogradski-stable. The transformedGauss-Bonnet terms are also discussed for the simplest conformal and disformalrelations.

Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian