Efficient Tree-Amplitudes in N=4: Automatic BCFW Recursion in Mathematica

TL;DR

This paper introduces a highly efficient Mathematica package 'bcfw' that automates the generation of tree-level super Yang-Mills amplitudes using BCFW recursion, offering flexible and faster analytic formulas for various helicity configurations.

Contribution

The paper presents an automated, flexible, and significantly faster implementation of BCFW recursion for N=4 super Yang-Mills amplitudes in Mathematica, including non-supersymmetric cases.

Findings

Automated generation of analytic amplitudes in Mathematica.

Significant speed improvements over previous methods.

Flexible recursive schemes using momentum-twistor Grassmannian integrals.

Abstract

We describe an efficient implementation of the BCFW recursion relations for tree-amplitudes in N=4 super Yang-Mills, which can generate analytic formulae for general N^kMHV colour-ordered helicity-amplitudes-which, in particular, includes all those of non-supersymmetric Yang-Mills. This note accompanies the public release of the Mathematica package "bcfw", which can quickly (and automatically) generate these amplitudes in a form that should be easy to export to any computational framework of interest, or which can be evaluated directly within Mathematica given external states specified by four-momenta, spinor-helicity variables or momentum-twistors. Moreover, bcfw is able to solve the BCFW recursion relations using any one of a three-parameter family of recursive `schemes,' leading to an extremely wide variety of distinct analytic representations of any particular amplitude. This flexibility is made possible by bcfw's use of the momentum-twistor Grassmannian integral to describe all tree amplitudes; and this flexibility is accompanied by a remarkable increase in efficiency, leading to formulae that can be evaluated much faster-often by several orders of magnitude-than those previously derived using BCFW.