Double Soft Theorems and Shift Symmetry in Nonlinear Sigma Models

TL;DR

The paper derives leading and subleading double soft theorems for nonlinear sigma models from shift symmetry enforcing Adler's zero, independent of the underlying coset G/H. It shows the double-soft behavior is controlled by infrared data, specifically a single four-point interaction and Adler's zeros, and expresses the full-amplitude double-soft limit in terms of distinct soft factors ${ m cal S}^{(0)}$, ${ m cal S}^{(1)}_{ m asym}$, and ${ m cal S}^{(1)}_{ m sym}$. The analysis uses Ward identities and a four-point vertex to fix the soft behavior, and demonstrates how color-ordered results follow from the full-amplitude expression. The work highlights infrared universality for Nambu-Goldstone bosons and suggests broader applicability to other massless theories and potential connections to other soft theorem approaches and symmetry structures.

Abstract

We show both the leading and subleading double soft theorems of the nonlinear sigma model follow from a shift symmetry enforcing Adler's zero condition in the presence of an unbroken global symmetry. They do not depend on the underlying coset G/H and are universal infrared behaviors of Nambu-Goldstone bosons. Although nonlinear sigma models contain an infinite number of interaction vertices, the double soft limit is determined entirely by a single four-point interaction, together with the existence of Adler's zeros.

Figures (1)

• Figure 1: An example of the pole diagram in double soft limit, where both soft legs are attached to the same external hard leg.