A Monte Carlo approach to the 4D scattering equations

Abstract

Abstract:The scattering equation formalism is a general framework for calculation of amplitudes in theories of massless particles. We provide a detailed introduction to the 4D scattering equation framework accessible to non-experts, outline current difficulties solving the equations numerically, and explain how to overcome them with a Monte Carlo algorithm. With this submission we include treeamps4dJAF, the first publicly available Mathematica package for calculating amplitudes by solving the scattering equations, supporting MHV analytical and N k−2 MHV numerical computations. The package provides a powerful and flexible computational tool for calculating tree-level amplitudes in super Yang Mills theories, Einstein supergravity and conformal supergravity. We tabulate sets of numerical solutions up to 9 points in all MHV sectors and 12 points in the NHMV sector which can be used for fast evaluation of amplitudes.

Scattering equations and a new factorization for amplitudes. Part I. Gauge theories

“…Like it was shown in [7], after integrating the y a coordinates around the solutions, C a = 0 ⇒ y a = ± σ 2 a − Λ 2 , ∀ a, we obtain a sum over all possible configurations (cuts), i.e. 2 n possibilities, where we call the sign, +/−, the upper/lower sheet 15 . For example, at six-point one has 2 6 = 2 × 32 possibilities given, schematically, by 16 A YM 6 (1, 2, 3, 4, 5, 6) = -Λ Λ…”

“…On the other hand, several methods have been developed to compute the CHY contour integral given in (1.1), most of them are applied to φ 3 or focused on solving the scattering equations [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In this work, from the double-cover representation, we have been able to achieve a graphic off-shell algorithm to carry out any color-ordered scattering of n-gluons and interactions with scalar fields, resulting in an expansion in terms of three-point amplitudes 4 .…”

New worldsheet formulae for conformal supergravity amplitudes

“…If more than two particles are in the left set, the worldsheet integrals can be evaluated numerically and we match the results against those obtained using Feynman diagrams and the double copy approach developed in [37] up to 8 points with any number of particles in the left set. An explicit algorithm for numerically solving the scattering equations and computing amplitudes with plane wave external states will be described in [39]. Moreover we generalize the scalar-graviton amplitudes to a supersymmetric formula using four types of vertex operators which describe states in the left or right set and the positive or negative helicity graviton multiplet.…”

“…For |L| > 2 we have verified (5) (and its extension to include fermions and spin one states) numerically by matching it against results obtained using Feynman diagrams and color-kinematics duality [37] up to eight points with any number of particles in the left set 2 . In order to do so, new techniques were developed for numerically solving the scattering equations which will be reported on in [39].…”

Bootstrapping solutions of scattering equations

“…Lastly, let us make a comparison between methods in our paper and in ref. [50]. In four dimensions in the spinor-helicity formalism, the scattering equations can be decomposed into 'helicity sectors' and written in terms of two-component spinors with additional variables involved (see e.g.…”