Note on solutions of scattering equations

TL;DR

This paper investigates the relationship between different types of singularities in scattering equations within the CHY formalism, focusing on their mappings in various physical limits such as factorization, soft, and forward limits.

Contribution

It systematically studies the mapping between pole singularities and collapsed solutions in scattering equations, clarifying their relationship in key physical limits.

Findings

Identified the connection between pole singularities and solution collapse.

Demonstrated mapping patterns in factorization, soft, and forward limits.

Enhanced understanding of singularity behavior in scattering equations.

Abstract

In the CHY-frame for the amplitudes, there are two kinds of singularities we need to deal with. The first one is the pole singularities when the kinematics is not general, such that some of $S_A\to 0$. The second one is the collapse of locations of points after solving scattering equations (i.e., the singular solutions). These two types of singularities are tightly related to each other, but the exact mapping is not well understood. In this paper, we have initiated the systematic study of the mapping. We have demonstrated the different mapping patterns using three typical situations, i.e., the factorization limit, the soft limit and the forward limit.