A note on polytopes for scattering amplitudes

  • N. Arkani-Hamed
  • J. Bourjaily
  • F. Cachazo
  • A. Hodges
  • J. Trnka


In this note we continue the exploration of the polytope picture for scattering amplitudes, where amplitudes are associated with the volumes of polytopes in generalized momentum-twistor spaces. After a quick warm-up example illustrating the essential ideas with the elementary geometry of polygons in \( \mathbb{C}{\mathbb{P}^{{2}}} \), we interpret the 1-loop MHV integrand as the volume of a polytope in \( \mathbb{C}{\mathbb{P}^{{3}}} \) × \( \mathbb{C}{\mathbb{P}^{{3}}} \), which can be thought of as the space obtained by taking the geometric dual of the Wilson loop in each \( \mathbb{C}{\mathbb{P}^{{3}}} \) of the product. We then review the polytope picture for the NMHV tree amplitude and give it a more direct and intrinsic definition as the geometric dual of a canonical “square” of the Wilson-Loop polygon, living in a certain extension of momentum-twistor space into \( \mathbb{C}{\mathbb{P}^{{4}}} \). In both cases, one natural class of triangulations of the polytope produces the BCFW/CSW representations of the amplitudes; another class of triangulations leads to a striking new form, which is both remarkably simple as well as manifestly cyclic and local.


Scattering Amplitudes Duality in Gauge Field Theories 


  1. [1]
    R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The All-Loop Integrand For Scattering Amplitudes in Planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, arXiv:0905.1473 [INSPIRE].
  5. [5]
    J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  7. [7]
    L. Mason and D. Skinner, The Complete Planar S-matrix of N = 4 SYM as a Wilson Loop in Twistor Space, JHEP 12 (2010) 018 [arXiv:1009.2225] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, JHEP 07 (2011) 058 [arXiv:1010.1167] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  10. [10]
    J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    N. Berkovits and J. Maldacena, Fermionic T-duality, Dual Superconformal Symmetry and the Amplitude/Wilson Loop Connection, JHEP 09 (2008) 062 [arXiv:0807.3196] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    M. Bullimore, MHV Diagrams from an All-Line Recursion Relation, JHEP 08 (2011) 107 [arXiv:1010.5921] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    L. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Local Spacetime Physics from the Grassmannian, JHEP 01 (2011) 108 [arXiv:0912.3249] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  16. [16]
    M. Bullimore, L. Mason and D. Skinner, MHV Diagrams in Momentum Twistor Space, JHEP 12 (2010) 032 [arXiv:1009.1854] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, Multiloop Amplitudes in N = 4 SYM, in progress.Google Scholar
  18. [18]
    L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N = 4 super Yang-Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    A. Hodges, The Box Integrals in Momentum-Twistor Geometry, arXiv:1004.3323 [INSPIRE].
  20. [20]
    L. Mason and D. Skinner, Amplitudes at Weak Coupling as Polytopes in AdS_5, J. Phys. A A 44 (2011) 135401 [arXiv:1004.3498] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  21. [21]
    F.A. Berends and W. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Local Loop Integrals for Planar Scattering Amplitudes, submitted to JHEP (2012).Google Scholar

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  • N. Arkani-Hamed
    • 1
  • J. Bourjaily
    • 1
    • 2
  • F. Cachazo
    • 3
  • A. Hodges
    • 4
  • J. Trnka
    • 1
    • 2
  1. 1.School of Natural SciencesInstitute for Advanced StudyPrincetonU.S.A.
  2. 2.Department of PhysicsPrinceton UniersityPrincetonU.S.A.
  3. 3.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  4. 4.Wadham CollegeUniversity of OxfordOxfordU.K.

Personalised recommendations