Gapped Goldstones at the cut-off scale: a non-relativistic EFT

TL;DR

This paper develops a nonrelativistic effective field theory for gapped Goldstone modes at finite density, demonstrating how to incorporate both gapless and gapped modes within a universal EFT framework, with applications to conformal field theories.

Contribution

It introduces a method to construct an EFT that includes gapped Goldstones at finite density, extending nonrelativistic EFT techniques to non-Abelian symmetry breaking scenarios.

Findings

EFT can describe both gapless and gapped Goldstones simultaneously.

The EFT preserves gapped Goldstone number but allows non-unitary processes.

Application to large charge sectors of non-Abelian conformal field theories.

Abstract

At finite density, the spontaneous breakdown of an internal non-Abelian symmetry dictates, along with gapless modes, modes whose gap is fixed by the algebra and proportional to the chemical potential: the gapped Goldstones. Generically the gap of these states is comparable to that of other non-universal excitations or to the energy scale where the dynamics is strongly coupled. This makes it non-straightforward to derive a universal ef effective field theory (EFT) description realizing all the symmetries. Focusing on the illustrative example of a fully broken SU (2) group, we demonstrate that such an EFT can be constructed by carving out around the Goldstones, gapless and gapped, at small 3-momentum. The rules governing the EFT, where the gapless Goldstones are soft while the gapped ones are slow, are those of standard nonrelativistic EFTs, like for instance nonrelativistic QED. In particular, the EFT Lagrangian formally preserves gapped Goldstone number, and processes where such number is not conserved are described inclusively by allowing for imaginary parts in the Wilson coefficients. Thus, while the symmetry is manifestly realized in the EFT, unitarity is not. We comment on the application of our construction to the study of the large charge sector of conformal field theories with non-Abelian symmetries.