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 ∈ {2, 3, . . . , 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.
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ArXiv ePrint: 1609.00008v2
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Cachazo, F., Mizera, S. & Zhang, G. Scattering equations: real solutions and particles on a line. J. High Energ. Phys. 2017, 151 (2017). https://doi.org/10.1007/JHEP03(2017)151
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DOI: https://doi.org/10.1007/JHEP03(2017)151