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

Tomoki Katayama

TL;DR

This paper reframes Horndeski theory through two constructive axioms—closure under invertible pure disformal transformations and inclusion of a minimal seed $\alpha X+R+\beta G^{\mu\nu}\phi_{\mu\nu}$—to enable systematic multi-field extensions from a well-defined single-field core. It shows how the standard single-field action is recovered up to boundary terms and introduces a practical, iterative disformal-mapping approach to build multi-field actions while preserving second-order structure. The work demonstrates that antisymmetric Allys-Akama-Kobayashi (AAK) terms arise naturally within the new framework and presents a conjecture that the resulting multi-Horndeski theory yields the most general equations of motion for multiple fields (at least for $\mathcal{N}=2$). By providing a constructive path and clarifying the role of AA terms, the paper offers a promising route for advancing multi-field scalar-tensor theories and their cosmological applications.

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.

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

TL;DR

This paper reframes Horndeski theory through two constructive axioms—closure under invertible pure disformal transformations and inclusion of a minimal seed —to enable systematic multi-field extensions from a well-defined single-field core. It shows how the standard single-field action is recovered up to boundary terms and introduces a practical, iterative disformal-mapping approach to build multi-field actions while preserving second-order structure. The work demonstrates that antisymmetric Allys-Akama-Kobayashi (AAK) terms arise naturally within the new framework and presents a conjecture that the resulting multi-Horndeski theory yields the most general equations of motion for multiple fields (at least for ). By providing a constructive path and clarifying the role of AA terms, the paper offers a promising route for advancing multi-field scalar-tensor theories and their cosmological applications.

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.

Paper Structure

This paper contains 22 sections, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Single-Field Case
  3. Horndeski theory
  4. disformal transformation
  5. Multi-Field Case
  6. Generalized Multi-Galileon Theory
  7. Extended multi-Galileon theory
  8. Challenges in Constructing the Multi-Horndeski Theory
  9. New definition of the Horndeski theory
  10. Introduction of a new definition
  11. Why the new definition?
  12. Application of the New Definition of Horndeski Theory to Multiple-Field
  13. A New Definition of the Multi-Horndeski Theory
  14. Conjecture on the New Definition of the multi-Horndeski Theory
  15. Allys-Akama-Kobayashi terms from the New Definition
  16. ...and 7 more sections

Figures (1)

  • Figure 1: Conceptual figure to construct $S_{n+1}$ from $S_n$

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Conjecture 1
  • Definition 3