On extended symmetries for the Galileon

TL;DR

This paper explores extended nonlinear symmetries of the Galileon field involving coordinate and field powers, identifying unique higher-order symmetry structures related to known kinetic term symmetries.

Contribution

It characterizes the structure of extended symmetries of the Galileon, revealing their uniqueness up to quadratic order and connections to kinetic term symmetries.

Findings

Unique symmetry structures up to quadratic order

Higher-order extensions relate to kinetic term symmetries

Identification of Galileon dual symmetries

Abstract

We investigate a large class of infinitesimal, but fully nonlinear in the field, transformations of the Galileon and search for extended symmetries. The transformations involve powers of the coordinates $x$ and the field $\pi$ up to any finite order $N$. Up to quadratic order the structure of these symmetry transformations is the unique generalisation of both the infinitesimal version of the standard Galileon shift symmetry as well as a recently discovered infinitesimal extension of this symmetry. The only higher-order extensions of this symmetry we recover are (`Galileon dual' versions of) symmetries of the standard kinetic term.