From k-essence to generalised Galileons

TL;DR

The paper achieves a complete classification of flat-space scalar field theories whose actions depend on π and its derivatives up to second order and whose equations of motion remain second order. It shows that every such theory is a linear combination of L_n{f} with f(π,X) multiplying a Galileon-core Lagrangian, and that curved-space extensions preserving second-order EOM for both the scalar and the metric can be constructed via a finite set of non-minimal curvature terms with specific coefficients. The authors establish a uniqueness result using an analysis of higher-derivative terms, connect their construction to Euler hierarchies, and demonstrate covariantization, including the conformal Galileons in four dimensions. The framework unifies k-essence, Galileons, k-Mouflage, and kinetically braided scalars under a single second-order, derivative-limited action principle with explicit covariant extensions and broad applicability to scalar-tensor phenomenology.

Abstract

We determine the most general scalar field theories which have an action that depends on derivatives of order two or less, and have equations of motion that stay second order and lower on flat space-time. We show that those theories can all be obtained from linear combinations of Lagrangians made by multiplying a particular form of the Galileon Lagrangian by an arbitrary scalar function of the scalar field and its first derivatives. We also obtain curved space-time extensions of those theories which have second order field equations for both the metric and the scalar field. This provide the most general extension, under the condition that field equations stay second order, of k-essence, Galileons, k-Mouflage as well as of the kinetically braided scalars. It also gives the most general action for a scalar classicalizer, which has second order field equations. We discuss the relation between our construction and the Euler hierachies of Fairlie et al, showing in particular that Euler hierachies allow one to obtain the most general theory when the latter is shift symmetric. As a simple application of our formalism, we give the covariantized version of the conformal Galileon.