Singular Solutions in Soft Limits

TL;DR

This paper investigates singular solutions in soft limits of generalized scattering equations on higher-dimensional projective spaces, providing explicit counts, geometric classifications, and implications for biadjoint amplitudes.

Contribution

It introduces a detailed analysis of singular solutions in specific higher-dimensional spaces and proposes a classification scheme for configurations supporting these solutions.

Findings

Identified all singular solutions for X(3,7), X(4,7), X(3,8), and X(5,8).

Counted total solutions including regular and singular, matching expected numbers.

Provided geometric interpretations and implications for soft expansions of amplitudes.

Abstract

A generalization of the scattering equations on $X(2,n)$, the configuration space of $n$ points on $\mathbb{CP}^1$, 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 $X(3,7)\sim X(4,7)$. 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.