Unification of Galileon Dualities

TL;DR

The work provides a unified framework for Galileon dualities by treating them as GL(2, R) coset-coordinate transformations acting on GAL(d,1)/SO(d−1,1. The authors derive the most general duality and show how it reparametrizes the couplings d_n via a linear representation on theory space, with invariants I_4 and I_5 (and I_4 in d=3) classifying duality classes. They demonstrate wide-ranging applications, including mapping classical solutions and observables, analyzing fluctuations and potential superluminality, and elucidating tree-level and one-loop S-matrix properties under duality. The results yield practical tools for simplifying calculations and for classifying Galileon theories through duality invariants, while clarifying how dual theories relate off-shell structures and counterterms in quantum corrections.

Abstract

We study dualities of the general Galileon theory in d dimensions in terms of coordinate transformations on the coset space corresponding to the spontaneously broken Galileon group. The most general duality transformation is found to be determined uniquely up to four free parameters and under compositions these transformations form a group which can be identified with GL(2,R). This group represents a unified framework for all the up to now known Galileon dualities. We discuss a representation of this group on the Galileon theory space and using concrete examples we illustrate its applicability both on the classical and quantum level.

Figures (5)

• Figure 1: The topologies of Feynman diagrams at the tree-level for the three-point, four-point (first line) and five-point (second line) Galileon scattering amplitudes.
• Figure 2: Graphical description of the definition of dual observable. To get the value of the observable ${\mathcal{O}}$ we can either use the phase space $\Sigma$ of the original theory or the dual phase space $\Sigma_{\alpha}$ and dual observable ${\mathcal{O}}_{\alpha}$
• Figure 3: The Feynman rules for the perturbative calculation of the one-particle amplitude ${\mathcal{M}}(\mathbf{k},\mathbf{k}^{^{\prime }})$ are given in the first two rows of this figure. This amplitude is given by the sum of Feynman graphs depicted in the last row.
• Figure 4: The surfaces $I_{4}=0$ (cylindrical one corresponding to duals of the quintic Galileon) and $I_{5}=0$ ($Z_{2}$ symmetric Galileons) in the theory space $D_{5}^{(1)}$ with $d_{2}=1/12$ fixed. The intersection of these surfaces corresponds to the duals of a free theory. Both surfaces are invariant with respect to the scaling.
• Figure 5: The Feynman graph for the Galileon self-energy and its counterterm.