Positivity of CurvatureSquared Corrections in Gravity
(5 pages)

Using a positivity constraint from tree level scattering, it is shown that a GaussBonnet term term must have nonnegative coefficient (in d>4) in order for a UV completion to be free of ghosts and tachyons. To do this, assume a weakly coupled UV completion of gravity  so that highenergy graviton scattering is made unitary by the treelevel exchange of massive states. These states can be described by a KallenLehmann spectral representation, where unitarity demands a positive spectral density. Integrating out the massive states, the corresponding low energy EFT has a GaussBonnet coefficient which depends on the integral of this spectral density, and is therefore nonnegative.

As a low energy effective field theory, gravity consists of a dominant EinsteinHilbert term (from usual General Relativity) plus subleading highercurvature corrections. Onshell and up to field redefinitions, the only higher order terms which appear are the socalled Lovelock invariants  the first of which is the GaussBonnet term R_{abcd}R^{abcd}, This is a total derivative in four dimensions, but contributes nontrivially in d>4.

Lagrange Multipliers and Third Order ScalarTensor Field Theories
(43 pages)

Using a scalar Lagrange multiplier, one can find EulerLagrange equations which are second order in the scalar field and third order in the metric. Applying a disformal transformation generates field equations which are at most third order. This allows the construction of second order scalartensor Lagrangians which yield third order EulerLagrange equations, and allows a discussion of all possible third order scalartensor field equations (which could have come from a d=4 Lagrangian).

Horndeski theories are all scalartensor field theories which have at most secondorder equations of motion (also known as `generalized Galileons'). `Beyond Horndeski' theories are scalartensor field theories with higherderivative equations of motion, but where there is a redundancy in the eom which ensures that the propagating degrees of freedom are healthy.
