A duality for the S matrix

  • N. Arkani-Hamed
  • F. Cachazo
  • C. Cheung
  • J. Kaplan
Open Access


We propose a dual formulation for the S Matrix of \( \mathcal N \) = 4 SYM. The dual provides a basis for the “leading singularities” of scattering amplitudes to all orders in perturbation theory, which are sharply defined, IR safe data that uniquely determine the full amplitudes at tree level and 1-loop, and are conjectured to do so at all loop orders. The scattering amplitude for n particles in the sector with k negative helicity gluons is associated with a simple integral over the space of k planes in n dimensions, with the action of parity and cyclic symmetries manifest. The residues of the integrand compute a basis for the leading singularities. A given leading singularity is associated with a particular choice of integration contour, which we explicitly identify at tree level and 1-loop for all NMHV amplitudes as well as the 8 particle N2MHV amplitude. We also identify a number of 2-loop leading singularities for up to 8 particles. There are a large number of relations among residues which follow from the multi-variable generalization of Cauchy’s theorem known as the “global residue theorem”. These relations imply highly non-trivial identities guaranteeing the equivalence of many different representations of the same amplitude. They also enforce the cancellation of non-local poles as well as consistent infrared structure at loop level. Our conjecture connects the physics of scattering amplitudes to a particular subvariety in a Grassmannian; space-time locality is reflected in the topological properties of this space.


Supersymmetric gauge theory Duality in Gauge Field Theories 


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© The Author(s) 2010

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • N. Arkani-Hamed
    • 1
  • F. Cachazo
    • 2
  • C. Cheung
    • 3
    • 4
  • J. Kaplan
    • 5
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  2. 2.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  3. 3.Berkeley Center for Theoretical PhysicsUniversity of CaliforniaBerkeleyU.S.A.
  4. 4.Theoretical Physics GroupLawrence Berkeley National LaboratoryBerkeleyU.S.A.
  5. 5.Theory GroupSLAC National Accelerator LaboratoryMenlo ParkU.S.A.

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