Abstract
A generalization of the scattering equations on X (2, n), the configuration space of n points on ℂℙ1, to higher dimensional projective spaces was recently introduced by Early, Guevara, Mizera, and one of the authors. One of the new features in X (k, n) with k > 2 is the presence of both regular and singular solutions in a soft limit. In this work we study soft limits in X (3, 7), X (4, 7), X (3, 8) and X (5, 8), find all singular solutions, and show their geometrical configurations. More explicitly, for X (3, 7) and X (4, 7) we find 180 and 120 singular solutions which when added to the known number of regular solutions both give rise to 1 272 solutions as it is expected since X (3, 7) ∼ X (4, 7). Likewise, for X (3, 8) and X (5, 8) we find 59 640 and 58 800 singular solutions which when added to the regular solutions both give rise to 188 112 solutions. We also propose a classification of all configurations that can support singular solutions for general X (k, n) and comment on their contribution to soft expansions of generalized biadjoint amplitudes.
Highlights
These are the analogs of masslessness and momentum conservation conditions
For X(3, 7) and X(4, 7) we find 180 and 120 singular solutions which when added to the known number of regular solutions both give rise to 1 272 solutions as it is expected since X(3, 7) ∼ X(4, 7)
For X(3, 8) and X(5, 8) we find 59 640 and 58 800 singular solutions which when added to the regular solutions both give rise to 188 112 solutions
Summary
Let us write the scattering equations in a form that manifestly exhibits the dependence on particle n: Ea n−1 b=1 sab xab. Xab = 0 for all values of a and b Under this assumption it is easy to see from (2.1) that all n dependence can be dropped from the first n − 1 equations. This set of equations precisely corresponds to that of a system of n − 1 particles and can be solved to find Nn−1 solutions. The possible values of xn are not arbitrary since τ drops from the last equation in (2.1) to give n−1 snb b=1 xn − xIb. where xIb is any one of the Nn−1 solutions for the hard particles. Momentum conservation and it leads to n − 3 solutions for xn Since this is true for each xIb one finds Nnregular = (n − 3)Nn−1.
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