New Soft Theorems for Goldstone Boson Amplitudes

TL;DR

This paper derives new soft theorems for Goldstone boson amplitudes that do not vanish in the soft limit, generalizing the Adler zero, and applies them to reconstruct tree-level amplitudes.

Contribution

It introduces a generalized soft theorem for theories where standard assumptions do not hold, expanding the understanding of Goldstone boson scattering amplitudes.

Findings

Derived a new soft theorem involving linear combinations of lower point amplitudes.

Applied the theorem to an $SU(N)/SU(N-1)$ sigma model.

Enabled reconstruction of tree-level amplitudes using modified soft recursion relations.

Abstract

In this letter we discuss new soft theorems for the Goldstone boson amplitudes with non-vanishing soft limits. The standard argument is that the non-linearly realized shift symmetry leads to the vanishing of scattering amplitudes in the soft limit, known as the Alder zero. This statement involves certain assumptions of the absence of cubic vertices and the absence of linear terms in the transformations of fields. For theories which fail to satisfy these conditions, we derive a new soft theorem which involves certain linear combinations of lower point amplitudes, generalizing the Adler zero statement. We provide an explicit example of $SU(N)/SU(N-1)$ sigma model which was also recently studied in the context of $U(1)$ fibrated models. The soft theorem can be then used as an input into the modified soft recursion relations for the reconstruction of all tree-level amplitudes.