Exploring the Landscape for Soft Theorems of Nonlinear Sigma Models

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.

46 pages + appendices, 15 figures; matched to the published version in v2

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• The SAGEX Review on Scattering Amplitudes, Chapter 11: Soft Theorems and Celestial Amplitudes
• Geometric Soft Theorems
• Off-Shell Color-Kinematics Duality for Chern-Simons
• Composing Effective Prediction at Five Points
• Nonrelativistic effective field theories with enhanced symmetries and soft behavior
• The ChPT: top-down and bottom-up
• Extended DBI and its generalizations from graded soft theorems