Singular solutions in soft limits
Freddy Cachazo, Bruno Giménez Umbert, Yong Zhang
Abstract
A bstract A generalization of the scattering equations on X (2 , n ), the configuration space of n points on ℂℙ 1 , 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.