Scattering Equations: Real Solutions and Particles on a Line

TL;DR

This paper identifies regions in the space of kinematic invariants where all solutions to scattering equations are real, linking them to particles on a line and revealing combinatorial structures related to Eulerian numbers.

Contribution

It establishes a geometric interpretation of scattering equations as stationary points of particles on an interval, connecting solutions to particle orderings and combinatorial numbers.

Findings

Existence of regions with all real solutions for scattering equations.

One-to-one correspondence between solutions and particle orderings.

Connection between solution sectors and Eulerian numbers.

Abstract

We find $n(n-3)/2$-dimensional regions of the space of kinematic invariants, where all the solutions to the scattering equations (the core of the CHY formulation of amplitudes) for $n$ massless particles are real. On these regions, the scattering equations are equivalent to the problem of finding stationary points of $n-3$ mutually repelling particles on a finite real interval with appropriate boundary conditions. This identification directly implies that for each of the $(n-3)!$ possible orderings of the $n-3$ particles on the interval, there exists one stable stationary point. Furthermore, restricting to four dimensions, we find that the separation of the solutions into $k\in \{2,3,\ldots ,n-2\}$ sectors naturally matches that of permutations of $n-3$ labels into those with $k-2$ descents. This leads to a physical realization of the combinatorial meaning of the Eulerian numbers.