Exploring the Landscape for Soft Theorems of Nonlinear Sigma Models

TL;DR

This paper extends soft theorems in nonlinear sigma models beyond leading order and specific cosets, providing new amplitude relations, formulas, and a comprehensive double soft theorem applicable to various representations and derivative orders.

Contribution

It generalizes soft theorems for nonlinear sigma models to higher orders and arbitrary groups, introducing new amplitude relations, formulas, and a universal double soft theorem.

Findings

Extended single soft theorems to $ ext{O}(p^4)$ for general groups.

Derived a Cachazo-He-Yuan formula for special flavor orderings.

Established a universal double soft theorem valid to all derivative orders.

Abstract

We generalize soft theorems of the nonlinear sigma model beyond the $\mathcal{O} (p^2)$ amplitudes and the coset of $\text{SU} (N) \times \text{SU} (N) / \text{SU} (N) $. We first discuss the universal flavor ordering of the amplitudes for the Nambu-Goldstone bosons, so that we can reinterpret the known $\mathcal{O} (p^2)$ single soft theorem for $\text{SU} (N) \times \text{SU} (N) / \text{SU} (N) $ in the context of a general symmetry group representation. We then investigate the special case of the fundamental representation of $\text{SO} (N)$, where a special flavor ordering of the "pair basis" is available. We provide novel amplitude relations and a Cachazo-He-Yuan formula for such a basis, and derive the corresponding single soft theorem. Next, we extend the single soft theorem for a general group representation to $\mathcal{O} (p^4)$, where for at least two specific choices of the $\mathcal{O} (p^4)$ operators, the leading non-vanishing pieces can be interpreted as new extended theory amplitudes involving bi-adjoint scalars, and the corresponding soft factors are the same as at $\mathcal{O} (p^2)$. Finally, we compute the general formula for the double soft theorem, valid to all derivative orders, where the leading part in the soft momenta is fixed by the $\mathcal{O}(p^2)$ Lagrangian, while any possible corrections to the subleading part are determined by the $\mathcal{O}(p^4)$ Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.