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 in ref. [1], 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.
Highlights
Recent years have seen a surge of interest in novel computational methods for scattering amplitudes in particle physics [2]
In higher orders of the derivative expansion, WZ terms may be present in this non-Galileon sector as well, and are known to be classified by the de Rham cohomology of the coset space of the broken symmetry [67, 68]
In this paper and its companion [55], we have initiated a classification of effective theories featuring enhanced soft limits from the symmetry point of view
Summary
Recent years have seen a surge of interest in novel computational methods for scattering amplitudes in particle physics [2]. Our long-term goal is to clarify the general relationship between the presence of redundant symmetries and enhanced soft limits of scattering amplitudes of NG bosons, in both relativistic and nonrelativistic setting. Much of the work has already been done by Cheung et al [1, 19] They classified constructively all Lorentz-invariant theories for a single massless particle featuring enhanced soft limits, and noticed that a redundant symmetry is present in all cases. Our approach is to classify the extensions of the physical symmetry group by additional redundant generators, admitted by Lie-algebraic constraints This allows us to set rather stringent constraints on possible extensions of the Galileon and DBI theories to systems with multiple NG bosons. We provide most technical details of our work, and further extend the discussion, allowing for doubly enhanced soft limits
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