Leading multi-soft limits from scattering equations

TL;DR

This paper derives a universal formula for leading soft factors in various theories using scattering equations, providing explicit examples and recursive procedures, advancing understanding of soft limits in quantum field theory.

Contribution

It introduces a CHY-based formula for leading soft factors applicable in multiple theories and develops recursive methods for their calculation, including in four dimensions.

Findings

Explicit formulas for up to three soft legs analytically.

Numerical evaluation for four soft legs confirms the formula.

Recursive procedures for gluon soft factors in four dimensions.

Abstract

A Cachazo-He-Yuan (CHY) type formula is derived for the leading gluon, bi-adjoint scalar $\phi^3$, Yang-Mills-scalar and non-linear sigma model $m$-soft factors $S_m$ in arbitrary dimension. The general formula is used to evaluate explicit examples for up to three soft legs analytically and up to four soft legs numerically via comparison with amplitude ratios under soft kinematics. A structural pattern for gluon $m$-soft factor is inferred and a simpler formula for its calculation is conjectured. In four dimensions, a Cachazo-Svr\v{c}ek-Witten (CSW) recursive procedure producing the leading $m$-soft gluon factor in spinor helicity formalism is developed as an alternative, and Britto-Cachazo-Feng-Witten (BCFW) recursion is used to obtain the leading four-soft gluon factor for all analytically distinct helicity configurations.