Galileon Duality

TL;DR

Galileon theories possess a nontrivial duality realized by a field-dependent diffeomorphism $\tilde{x}^\mu = x^\mu + \partial^\mu \pi$, which maps a Galileon with coefficients $\{c_n\}$ to a dual Galileon with coefficients $\{p_n\}$ and an invertibility condition on $\Pi=\partial_\mu\partial_\nu \pi$. This duality is equivalent to a Legendre transform and is manifest in the decoupling limit of bigravity, while allowing a free scalar to be dual to a higher-order Galileon with all operators nonzero; importantly, classical superluminal propagation in one frame does not imply acausality since the S-matrix remains analytic and (expected) front velocity is luminal after quantum corrections. The work further shows that matter couplings map locally, enabling a novel realization of the Vainshtein mechanism in the dual frame, and that the S-matrix is invariant under duality, with concrete demonstrations in plane-wave backgrounds and tree-level amplitudes. It also discusses the strong-weak coupling interchange, dual coupling to point sources, and the consistent definition of a duality-preserving path-integral measure, suggesting robustness of the duality at the quantum level under diffeomorphism-like transformations.

Abstract

We show that every Galileon theory admits a dual formulation as a Galileon theory with new operator coefficients. In n dimensions a free scalar field in Minkowski spacetime is dual to a (n+1)-th order Galileon theory which exhibits the Vainshtein mechanism when coupled to sources and superluminal propagation even on-shell. This demonstrates that superluminal propagation is compatible with an analytic S-matrix and causality. For point sources, the duality interchanges the strongly coupled Vainshtein regime with the weakly coupled asymptotic regime. The duality is made manifest in the context of the decoupling limit of bigravity, but is independent of this.