Ward Identity and Scattering Amplitudes for Nonlinear Sigma Models

TL;DR

This paper derives a Ward identity for nonlinear sigma models using generalized shift symmetries, leading to new recursion relations and soft theorems that hold at the quantum level, enhancing understanding of scattering amplitudes.

Contribution

It introduces a novel Ward identity for nonlinear sigma models that does not rely on current algebra, providing new recursion relations and soft theorems at all orders.

Findings

Derivation of a Ward identity constraining correlation functions.

Establishment of a subleading single soft theorem valid quantum mechanically.

Development of a Berends-Giele recursion relation for tree amplitudes.

Abstract

We present a Ward identity for nonlinear sigma models using generalized nonlinear shift symmetries, without introducing current algebra or coset space. The Ward identity constrains correlation functions of the sigma model such that the Adler's zero is guaranteed for $S$-matrix elements, and gives rise to a subleading single soft theorem that is valid at the quantum level and to all orders in the Goldstone decay constant. For tree amplitudes, the Ward identity leads to a novel Berends-Giele recursion relation as well as an explicit form of the subleading single soft factor. Furthermore, interactions of the cubic biadjoint scalar theory associated with the single soft limit, which was previously discovered using the Cachazo-He-Yuan representation of tree amplitudes, can be seen to emerge from matrix elements of conserved currents corresponding to the generalized shift symmetry.