Exploring the landscape for soft theorems of nonlinear sigma models

This is the authors' abstract. We don't add key points for this paper.

A bstract We generalize soft theorems of the nonlinear sigma model beyond the O O ( p 2 ) amplitudes and the coset of SU( N ) × SU( N ) / SU( N ). We first discuss the universal flavor ordering of the amplitudes for the Nambu-Goldstone bosons, so that we can reinterpret the known O O ( p 2 ) single soft theorem for SU( N ) × SU( N ) / SU( N ) in the context of a general symmetry group representation. We then investigate the special case of the fundamental representation of 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 O O ( p 4 ), where for at least two specific choices of the O 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 O 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 O O ( p 2 ) Lagrangian, while any possible corrections to the subleading part are determined by the O O ( p 4 ) Lagrangian alone. Higher order terms in the derivative expansion do not contribute any new corrections to the double soft theorem.