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The complete planar S-matrix of \( \mathcal{N} = 4 \) SYM as a Wilson loop in twistor space

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Abstract

We show that the complete planar S-matrix of \( \mathcal{N} = 4 \) super Yang-Mills — including all NkMHV 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 N2MHV 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 the companion paper [1] 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 and give the corresponding superconnection in the abelian case.

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Authors and Affiliations

  1. The Mathematical Institute, 24-29 St. Giles’, Oxford, OX1 3LB, United Kingdom

    Lionel Mason

  2. Perimeter Institute for Theoretical Physics, 31 Caroline St., Waterloo, ON, N2L 2Y5, Canada

    David Skinner

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  1. Lionel Mason
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  2. David Skinner
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Correspondence to David Skinner.

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ArXiv ePrint: 1009.2225

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Mason, L., Skinner, D. The complete planar S-matrix of \( \mathcal{N} = 4 \) SYM as a Wilson loop in twistor space. J. High Energ. Phys. 2010, 18 (2010). https://doi.org/10.1007/JHEP12(2010)018

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