Leading multi-soft limits from scattering equations

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.