Analytical amplitudes from numerical solutions of the scattering equations

TL;DR

This paper introduces a Python package that numerically solves scattering equations within the CHY formalism, enabling computation of various tree-level amplitudes and analysis of their singularity structures, including new analytical expressions.

Contribution

A novel high-precision numerical implementation for solving scattering equations and computing amplitudes across multiple theories, facilitating analytical reconstruction and exploration of amplitude singularities.

Findings

Successfully computed tree amplitudes for multiple theories.

Reconstructed new analytical expressions for conformal gravity and $(DF)^2$ theories.

Analyzed the singularity structure of scattering amplitudes.

Abstract

The CHY formalism for massless scattering provides a cohesive framework for the computation of scattering amplitudes in a variety of theories. It is especially compelling because it elucidates existing relations among theories which are seemingly unrelated in a standard Lagrangian formulation. However, it entails operations that are highly non-trivial to perform analytically, most notably solving the scattering equations. We present a new Python package ( https://github.com/GDeLaurentis/seampy ) to solve the scattering equations and to compute scattering amplitudes. Both operations are done numerically with high-precision floating-point algebra. Elimination theory is used to obtain solutions to the scattering equations for arbitrary kinematics. These solutions are then applied to a variety of CHY integrands to obtain tree amplitudes for the following theories: Yang-Mills, Einstein gravity, biadjoint scalar, Born-Infeld, non-linear sigma model, Galileon, conformal gravity and $(\text{DF})^2$. Finally, we exploit this high-precision numerical implementation to explore the singularity structure of the amplitudes and to reconstruct analytical expressions which make manifest their pole structure. Some of the expressions for conformal gravity and the $(\text{DF})^2$ gauge theory are new to the best of our knowledge.