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Abstract
In the CHY-frame for the amplitudes, there are two kinds of singularities we need to deal with. The first one is the pole singularities when the kinematics is not general, such that some of $S_A\to 0$. The second one is the collapse of locations of points after solving scattering equations (i.e., the singular solutions). These two types of singularities are tightly related to each other, but the exact mapping is not well understood. In this paper, we have initiated the systematic study of the mapping. We have demonstrated the different mapping patterns using three typical situations, i.e., the factorization limit, the soft limit and the forward limit.
Highlights
Motivated by our curiosity of the singular solutions and potential application to other frontiers of researches, for example, the construction of two-loop CHY-integrands by double forward limits, in this paper, we have initiated the systematic study of the relation between the singular kinematic configurations and the singular solutions of scattering equations
We find that the singular solutions will always lead to singular kinematics
The layer structure of singular solutions gives a clear picture of the structure of singular kinematics
Summary
There are huge processes in the effective computation and deep understanding of scattering amplitudes using the so-called “on-shell program”.1 Among these on-shell methods, the CHY-formalism [6–10] provides a fantastic angle to study scattering amplitudes. There are many works on studying scattering equations, and their solutions [11–30] By these studies, we have an obvious picture about the factorization property in the CHY-formalism. The understanding of factorization in the CHY-formalism is crucial because, through its study, the “integration rule” for simple poles and its generalization for higher poles have been proposed [31–33] and we can read out the analytic expression for any CHY-integrand without solving the scattering equations. For forward limit, beautiful analysis has been done in [53] Motivated by these observations, in this paper, we will start a systematical discussion about the relation of the following two things: the kinematic singularities and the singular solutions of scattering equations. In the Appendix, how to implement the numerical kinematic configuration has been carefully discussed
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