Healthy scalar-tensor theories with third-order derivatives: Generalized disformal Horndeski and beyond

TL;DR

This work develops a broad class of ghost-free scalar-tensor theories whose Lagrangians contain up to third-order derivatives of the scalar field by formulating a spatially covariant action in ADM variables and enforcing degeneracy and consistency conditions. The action is cast in unitary gauge with a central kinetic structure $Q_{ij}=K_{ij}+\mathcal{U}_{ij}V+\frac{1}{2}\mathcal{V}_{ij}^kV_k$, and the degeneracy conditions render the kinetic sector as a completed square, while the consistency conditions restrict the allowed couplings to depend on three functions $W_0$, $W_1$, and $W_2$ of $(N,Z)$. Generalized disformal transformations preserve the theory class and enable moving to frames where $\mathcal{U}_{ij}=\mathcal{V}_{ij}^k=0$, connecting these theories to GDH and extending U-DHOST. The framework thus provides a systematic, stable extension of higher-derivative scalar-tensor theories with potential cosmological and gravitational-wave phenomenology, and outlines future work on perturbations, matter couplings, and screening mechanisms.

Abstract

We systematically construct ghost-free scalar-tensor theories whose Lagrangian includes up to third-order derivatives of the scalar field. Using a spatially covariant action written in terms of the ADM variables, we impose degeneracy and consistency conditions that ensure the propagation of only one scalar and two tensor degrees of freedom. The resultant theories extend the generalized disformal Horndeski and U-DHOST theories. We discuss the transformation properties of these theories under generalized disformal transformations.