Characterizing the solutions to scattering equations that support tree-level $\text{N}^{k}\text{MHV}$ gauge/gravity amplitudes

TL;DR

This paper introduces a method to classify solutions to scattering equations in four dimensions based on discriminant matrix ranks, linking solutions to specific N^kMHV amplitudes and proving amplitude vanishing in certain Einstein-Yang-Mills configurations.

Contribution

It defines discriminant matrices to categorize solutions and connects these to the support of specific N^kMHV amplitudes within the CHY formalism, providing new insights into solution structure and amplitude properties.

Findings

Solutions are classified by discriminant matrix ranks.

Each solution subset supports only a specific N^kMHV amplitude.

Single-helicity gluon amplitudes in Einstein-Yang-Mills vanish at tree level.

Abstract

In this paper we define, independent of theories, two discriminant matrices involving a solution to the scattering equations in four dimensions, the ranks of which are used to divide the solution set into a disjoint union of subsets. We further demonstrate, {entirely within the Cachazo-He-Yuan formalism,} that each subset of solutions gives nonzero contribution to tree-level $\text{N}^{k}\text{MHV}$ gauge/gravity amplitudes only for a specific value of $k$. Thus the solutions can be characterized by the rank of their discriminant matrices, which in turn determines the value of $k$ of the $\text{N}^{k} \text{MHV}$ amplitudes a solution can support. As another application of the technique developed, we show analytically that in Einstein-Yang-Mills theory, if all gluons have the same helicity, the tree-level single-trace amplitudes must vanish.