Analytical amplitudes from numerical solutions of the scattering equations

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 (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 (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 (DF) 2 gauge theory are new to the best of our knowledge.

Cosmological Scattering Equations at Tree-level and One-loop

“…It may be possible to do this analytically using techniques from integrability [78,79], but a numerical approach may ultimately be needed. Indeed, even the scattering equations in flat space require numerical solution above five points for generic kinematics [99][100][101]. In summary, the study of the cosmological scattering equations is still in its infancy, and there are many exciting directions to explore.…”

Note on solutions of scattering equations

“…There are many works on studying scattering equations and their solutions [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. By these studies, now we have a very clear picture about the factorization property in the CHY-formalism.…”

Interpolation of Dense and Sparse Rational Functions and other Improvements in $\texttt{FireFly}$

“…The calculation of physical observables usually requires the insertion of the reduction table into a sum of Feynman integrals given by Eq. (42). For example, this has to be done to compute amplitudes as intermediate step of cross-section calculations.…”

“…Recently, a similar strategy to obtain amplitudes from floating-point probes has been presented [41,42].…”