Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space

Lionel Mason, David Skinner

TL;DR

This work posits that the complete planar S-matrix of $\mathcal{N}=4$ SYM is encoded in the correlation function of a supersymmetric Wilson loop in momentum twistor space. By employing the twistor action in axial gauge, the authors show that tree-level NMHV/N^2MHV amplitudes and loop integrands (1-loop and 2-loop MHV/NMHV) arise naturally as contractions in this Wilson-loop framework, yielding expressions in terms of dual superconformal $R$-invariants that match momentum-twistor MHV rules. The paper also develops a space-time interpretation via an off-shell Penrose-Ward transform, constructing a space-time superconnection whose Wilson loop counterpart should reproduce the same planar S-matrix, thereby linking twistor-space and space-time formulations of supersymmetric Wilson loops. Collectively, these results illuminate a coherent, gauge-friendly route to all-order planar amplitudes in $\mathcal{N}=4$ SYM and reinforce the deep connection between twistor geometry, Wilson loops, and scattering amplitudes.

Abstract

We propose that the complete planar S-matrix of N=4 super Yang-Mills - including all N^kMHV partial amplitudes to all loops - is equivalent to the correlation function of a supersymmetric Wilson loop in twistor space. Remarkably, the entire classical S-matrix arises from evaluating the correlation function in the self-dual sector, while the expansion of the correlation function in powers of the Yang-Mills coupling constant provides the loop expansion of the amplitudes. We support our proposal with explicit computations of the n particle NMHV and NNMHV trees, the integrands of the 1-loop MHV and NMHV amplitudes, and the n particle 2-loop MHV amplitude. These calculations are performed using the twistor action in axial gauge. In this gauge, the Feynman diagrams of the correlation function are the planar duals of the usual MHV diagrams for the scattering amplitude. The results are presented in the form of a sum of products of dual superconformal invariants in (momentum) twistor space, and agree with the expressions derived in arXiv:1009.1854 directly from the MHV rules. The twistor space Wilson loop is a natural supersymmetric generalization of the standard Wilson loop used to compute MHV amplitudes. We show how the Penrose-Ward transform can be used to determine a corresponding supersymmetrization on space-time.

The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space

TL;DR

This work posits that the complete planar S-matrix of SYM is encoded in the correlation function of a supersymmetric Wilson loop in momentum twistor space. By employing the twistor action in axial gauge, the authors show that tree-level NMHV/N^2MHV amplitudes and loop integrands (1-loop and 2-loop MHV/NMHV) arise naturally as contractions in this Wilson-loop framework, yielding expressions in terms of dual superconformal -invariants that match momentum-twistor MHV rules. The paper also develops a space-time interpretation via an off-shell Penrose-Ward transform, constructing a space-time superconnection whose Wilson loop counterpart should reproduce the same planar S-matrix, thereby linking twistor-space and space-time formulations of supersymmetric Wilson loops. Collectively, these results illuminate a coherent, gauge-friendly route to all-order planar amplitudes in SYM and reinforce the deep connection between twistor geometry, Wilson loops, and scattering amplitudes.

Abstract

We propose that the complete planar S-matrix of N=4 super Yang-Mills - including all N^kMHV partial amplitudes to all loops - is equivalent to the correlation function of a supersymmetric Wilson loop in twistor space. Remarkably, the entire classical S-matrix arises from evaluating the correlation function in the self-dual sector, while the expansion of the correlation function in powers of the Yang-Mills coupling constant provides the loop expansion of the amplitudes. We support our proposal with explicit computations of the n particle NMHV and NNMHV trees, the integrands of the 1-loop MHV and NMHV amplitudes, and the n particle 2-loop MHV amplitude. These calculations are performed using the twistor action in axial gauge. In this gauge, the Feynman diagrams of the correlation function are the planar duals of the usual MHV diagrams for the scattering amplitude. The results are presented in the form of a sum of products of dual superconformal invariants in (momentum) twistor space, and agree with the expressions derived in arXiv:1009.1854 directly from the MHV rules. The twistor space Wilson loop is a natural supersymmetric generalization of the standard Wilson loop used to compute MHV amplitudes. We show how the Penrose-Ward transform can be used to determine a corresponding supersymmetrization on space-time.

Paper Structure

This paper contains 19 sections, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Twistor space and Wilson loops
  3. The off-shell Penrose-Ward correspondence
  4. Parallel propagation and Wilson loops
  5. A supersymmetric Wilson Loop
  6. The Twistor Action for $\mathcal{N}=4$ SYM
  7. Correlation functions from the twistor action in an axial gauge
  8. The propagator in an axial gauge
  9. Tree amplitudes
  10. NMHV
  11. N$^2$MHV
  12. Dual MHV diagrams and N$^k$MHV amplitudes
  13. Loop amplitudes
  14. MHV
  15. NMHV
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Correlation function diagrams with two propagators (and no vertices) contribute to the N$^2$MHV tree amplitude. Only those diagrams whose contractions are compatible with the ordering of the trace in the Wilson loop survive in the planar limit.