Lie-algebraic classification of effective theories with enhanced soft limits

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.

Symmetric superfluids

“…Importantly, in our analysis we do not assume the existence of a sensible theory of fluctuations around the Poincaré invariant vacuum φ(x) = const since ultimately we are interested in Lorentz breaking vacua. For this reason, our results extend those reported in a classification of scalar EFT's based on their amplitudes' soft scaling [15][16][17][18][19][20]. We find additional possibilities, namely Scaling (and Conformal) Superfluids, which do not admit a perturbative S-matrix when the Poincaré symmetries are unbroken.…”

“…Remarkably, we find that in D = 1 spacetime dimensions there are only two possibilities: the Dirac-Born-Infeld action and Scaling Superfluids where the Lagrangian is a monomial in X: P (X) = X α . As summarized 2 in Table I, for α = D/2 (Conformal Superfluid [4,5]) and α = 1/2 (Cuscuton [6,7]) the scaling symmetry is enhanced to the full conformal group and to additional vector and scalar generators (see (12), (20)), respectively. It is perhaps at first surprising to find scaling but not 2 In our notation for the semi-direct product ⋊, the normal subgroup N is on the left-hand side, G = N ⋊ H conformal symmetry as is the case for α = D/2, but this is compatible with all results in the literature (e.g.…”

The infrared structure of exceptional scalar theories

“…As for special Galileon, the shift symmetry that protects the soft behavior was discovered [31] after the enhanced soft behavior was identified, and the relation between the internal symmetry and enhanced soft behavior was further clarified in Refs. [27,33].…”

Gauged galileons

“…Motivated by the search of additional symmetries that could provide a rationale for the particular structure of dRGT theory, we also consider the gauging of the special galileon [27]. The special galileon theory is a one-parameter subset of the generic galileon that enjoys an extended shift symmetry that is responsible for an enhanced soft behavior of scattering amplitudes [28], among other interesting properties [29][30][31]. In our formalism, the fact that this symmetry is spontaneously broken will allow for the presence of an additional Goldstone mode in addition to the massive spin-2 degrees of freedom that we are interested in.…”