Scattering Equations: Real Solutions and Particles on a Line

TL;DR

This work identifies a $n(n-3)/2$-dimensional subregion ${\cal K}_n^+$ of the kinematic space ${\cal K}_n$ in which all scattering-equation solutions are real and maps these solutions to the equilibrium configurations of $n-3$ mutually repelling particles on a finite interval, yielding exactly $(n-3)!$ stable equilibria. In four dimensions, the authors show a sector decomposition of the solutions into $k$-sectors, counted by Eulerian numbers ${\genfrac{<>{0pt}}{}{n-3}{k-2}}$, with a precise physical realization connecting desent statistics to sector assignment. They further develop the CHY framework on ${\cal K}_n^+$, demonstrating real-valued integrals over the interval and proposing numerical methods, a vector-space structure on the kinematic space, and potential links to the positive Grassmannian and Jacobi polynomials. The results provide a concrete, real-geometry realization of scattering equations, enable real-analytic CHY computations, and open avenues for efficient amplitudes calculations and new physical kinematic regimes. The work thus bridges scattering amplitudes, combinatorics, and real-geometry, with broad implications for both theory and computation.

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.

Figures (4)

• Figure 1: From $\mathbb{CP}^1$ to $\mathbb{RP}^1$. Left: Generic kinematics, $\mathcal{K}_{n}$ leads to configurations of punctures living on a Riemann sphere. Right: Positive kinematics, $\mathcal{K}^+_{n}$ makes the punctures arrange themselves on a real projective line. The black punctures $\sigma_1,\sigma_2,\ldots,\sigma_{n-3}$ can be thought of as a system of mutually repelling particles on the interval $\mathtt{I}=[\sigma_A,\sigma_B]$. Punctures $\sigma_A, \sigma_B, \sigma_C$ can be fixed using ${\rm SL}(2,\mathbb{R})$ invariance. The first two then provide repelling boundary conditions for the black particles.
• Figure 2: (a) Configuration of particles for physical kinematics from section \ref{['sec:physical kinematics']}. The vertical forces exerted on particles $1,2,\ldots,n-3$ due to $A$ and $B$ are equal and opposite, making them stay on the real axis. (b) Corresponding $2 \to n-2$ scattering process. Two massless particles of equal and opposite spatial momenta collide to produce $n-2$ particles confined to $d-2$ dimensions of space.
• Figure A.3: Example of the potential, $V(\sigma_7)$, for the soft particle with label $7$. The minima of the potential mark the positions of the puncture $\sigma_7$ in different solutions.
• Figure C.4: Determining which solution becomes degenerate. Top: Equilibrium positions that the particles try to reach. Middle: In the case $\sigma_a < \sigma_b$ the particles converge to distinct locations. Bottom: In the case $\sigma_b < \sigma_a$ the particles pinch at a common location creating a degenerate solution.