Lie-algebraic classification of effective theories with enhanced soft limits

TL;DR

This paper classifies effective theories with enhanced soft limits of scattering amplitudes using Lie algebra methods, identifying known theories and their possible generalizations based on symmetry structures.

Contribution

It introduces a Lie-algebraic framework to classify scalar effective theories with enhanced soft limits, revealing constraints on non-Abelian extensions and generalizing known models.

Findings

Reproduces known results for single NG boson theories with enhanced soft limits.

Shows that non-Abelian internal symmetries restrict NG bosons with enhanced soft limits to the center or Abelian subgroups.

Identifies two infinite classes of solutions generalizing multi-Galileon and multi-DBI theories.

Abstract

A great deal of effort has recently been invested in developing methods of calculating scattering amplitudes that bypass the traditional construction based on Lagrangians and Feynman rules. Motivated by this progress, we investigate the long-wavelength behavior of scattering amplitudes of massless scalar particles: Nambu-Goldstone (NG) bosons. The low-energy dynamics of NG bosons is governed by the underlying spontaneously broken symmetry, which likewise allows one to bypass the Lagrangian and connect the scaling of the scattering amplitudes directly to the Lie algebra of the symmetry generators. We focus on theories with enhanced soft limits, where the scattering amplitudes scale with a higher power of momentum than expected based on the mere existence of Adler's zero. Our approach is complementary to that developed recently by Cheung et al., and in the first step we reproduce their result. That is, as far as Lorentz-invariant theories with a single physical NG boson are concerned, we find no other nontrivial theories featuring enhanced soft limits beyond the already well-known ones: the Galileon and the Dirac-Born-Infeld (DBI) scalar. Next, we show that in a certain sense, these theories do not admit a nontrivial generalization to non-Abelian internal symmetries. Namely, for compact internal symmetry groups, all NG bosons featuring enhanced soft limits necessarily belong to the center of the group. For noncompact symmetry groups such as the ISO($n$) group featured by some multi-Galileon theories, these NG bosons then necessarily belong to an Abelian normal subgroup. The Lie-algebraic consistency constraints admit two infinite classes of solutions, generalizing the known multi-Galileon and multi-flavor DBI theories.