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

Jacob L. Bourjaily

TL;DR

The paper presents an efficient, publicly available Mathematica implementation of tree-level BCFW recursion for $\mathcal{N}=4$ SYM using momentum-twistor Grassmannian contours. It provides a flexible three-parameter family of recursion schemes that generate many analytic representations of any $n$-point, $N^kMHV$ amplitude, while achieving large speedups through a fully supersymmetric Grassmannian description and momentum-twistor variables. The bc fw package offers analytic generation, symbolic projection of helicity components, and numeric evaluation with a variety of kinematic input options, plus practical demonstrations and comparisons to other tools. The work facilitates intuition-building, identity checks (e.g., SUSY Ward identities), and efficient amplitude computations, with broader implications for leveraging Grassmannian and twistor methods in collider phenomenology.

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.

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

TL;DR

The paper presents an efficient, publicly available Mathematica implementation of tree-level BCFW recursion for SYM using momentum-twistor Grassmannian contours. It provides a flexible three-parameter family of recursion schemes that generate many analytic representations of any -point, amplitude, while achieving large speedups through a fully supersymmetric Grassmannian description and momentum-twistor variables. The bc fw package offers analytic generation, symbolic projection of helicity components, and numeric evaluation with a variety of kinematic input options, plus practical demonstrations and comparisons to other tools. The work facilitates intuition-building, identity checks (e.g., SUSY Ward identities), and efficient amplitude computations, with broader implications for leveraging Grassmannian and twistor methods in collider phenomenology.

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.

Paper Structure

This paper contains 12 sections, 32 equations, 10 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Kinematics: Momenta to Momentum-Twistors (and Back)
  3. Tree-Amplitudes as Contour Integrals in the Grassmannian
  4. Generalized BCFW Recursion Schemes
  5. Extracting Helicity Component-Amplitudes from Tree-Contours
  6. The bcfw Mathematica Package
  7. Setup and Initialization
  8. Getting Started with Analytic Tree Amplitudes
  9. Referencing, Generating, or Specifying Kinematical Data
  10. Numerical Evaluation of Tree Amplitudes
  11. Example Applications
  12. Comparison of bcfw with Other Computational Tools

Figures (10)

  • Figure 1: $\mathcal{A}_{6}^{(3)}\left(-,-,-,+,+,+\right)$. The split-helicity $6$-point NMHV amplitude.
  • Figure 2: $\mathcal{A}_{6}^{(3)}\left(-,-,\psi_{-1/2}^{(123)},+,+,\psi_{+1/2}^{(4)}\right)$. A $6$-point NMHV amplitude involving two gluinos and four gluons.
  • Figure 3: $\mathcal{A}_{8}^{(4)}\left(\psi_{+1/2}^{(1)},\psi_{+1/2}^{(1)},\psi_{+1/2}^{(1)},\phi_0^{(13)},\psi_{-1/2}^{(234)},\psi_{-1/2}^{(234)},\psi_{-1/2}^{(234)},\phi_0^{(24)}\right).$$~$ An example 8-point N$^2$MHV helicity-amplitude involving 6 gluinos and 2 squarks.
  • Figure 4: $\mathcal{A}_{10}^{(5)}\left(\psi_{+1/2}^{(1)},\psi_{+1/2}^{(1)},\psi_{+1/2}^{(1)},\psi_{+1/2}^{(1)},\psi_{-1/2}^{(123)},\psi_{-1/2}^{(234)},\psi_{-1/2}^{(234)},\psi_{-1/2}^{(234)},\psi_{-1/2}^{(234)},\psi_{+1/2}^{(4)}\right).$$~$ An example 10-point N$^3$MHV helicity-amplitude involving only gluinos.
  • Figure 5: The map connecting momentum-twistor variables and dual-coordinates.
  • ...and 5 more figures