A New Definition of Horndeski Theory and the Possibility of Multiple Scalar Field Extensions

Abstract: In the single-field case, Horndeski provides the most general scalar-tensor theory with second-order field equations. By contrast, systematic multi-field extensions remain incomplete: while the general field equations for the bi-Horndeski case are known, a general action has not been established, and for cases with three or more fields, neither a general action nor general equations are available. We characterize Horndeski by two mild axioms: closure under invertible pure disformal transformations and the requirement that the theory includes the minimal Horndeski theory. Under this characterization, we recover the standard single-field action up to boundary terms and obtain a practical path to multi-field constructions. In particular, we show that antisymmetric structures, such as those identified by Allys, Akama, and Kobayashi, appear within this framework, and indicate that this viewpoint has the potential to account for features captured by known bi-Horndeski equations.