Scattering Equations, Twistor-string Formulas and Double-soft Limits in Four Dimensions

TL;DR

The paper develops four-dimensional scattering equations in both polynomial and rational forms, clarifying their GL$(k)$ relation and providing new compact formulas for tree amplitudes in the NLSM, a special Galileon, and the ${ m N}=4$ supersymmetric completion of DBI. By leveraging these 4d formulas, it derives comprehensive double-soft theorems across multiple theories, showing that the degenerate solution governs leading behavior and revealing coset-structure and nonlinearly realized symmetry features, illuminated through supersymmetric and scalar sectors. The work also unifies soft-limit results across singlet and non-singlet soft pairs within the super-DBI framework, and highlights the special simplicity of 4d scalar theories in the middle sector, with broader implications for loops and symmetry-based structures in scattering amplitudes.

Abstract

We study scattering equations and formulas for tree amplitudes of various theories in four dimensions, in terms of spinor helicity variables and on-shell superspace for supersymmetric theories. As originally obtained in Witten's twistor string theory and other twistor-string models, the equations can take either polynomial or rational forms, and we clarify the simple relation between them. We present new, four-dimensional formulas for all tree amplitudes in the non-linear sigma model, a special Galileon theory and the maximally supersymmetric completion of the Dirac-Born-Infeld theory. Furthermore, we apply the formulas to study various double-soft theorems in these theories, including the emissions of a pair of soft photons, fermions and scalars for super-amplitudes in super-DBI theory.