Multichannel conformal blocks for polygon Wilson loops



We introduce the notion of Multichannel Conformal Blocks relevant for the Operator Product Expansion for Null Polygon Wilson loops with more than six edges. As an application of these, we decompose the one loop heptagon Wilson loop and predict the value of its two loop OPE discontinuities. At the functional level, the OPE discontinuities are roughly half of the full result. Using symbols they suffice to predict the full two loop result. We also present several new predictions for the heptagon result at any loop order.


Exact S-Matrix Integrable Field Theories Supersymmetric gauge theory AdS-CFT Correspondence 


  1. [1]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007)064 [arXiv:0705.0303] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    G. Korchemsky, J. Drummond and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].MathSciNetADSGoogle Scholar
  3. [3]
    A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in N = 4 super Yang-Mills and Wilson loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    Z. Bern, L. Dixon, D. Kosower, R. Roiban, M. Spradlin, et al., The two-loop six-gluon MHV amplitude in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [INSPIRE].MathSciNetADSGoogle Scholar
  5. [5]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    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
  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]
    M. Bullimore and D. Skinner, Holomorphic linking, loop equations and scattering amplitudes in twistor space, arXiv:1101.1329 [INSPIRE].
  10. [10]
    A. Belitsky, G. Korchemsky and E. Sokatchev, Are scattering amplitudes dual to super Wilson loops?, Nucl. Phys. B 855 (2012) 333 [arXiv:1103.3008] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    L.F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, An operator product expansion for polygonal null Wilson loops, JHEP 04 (2011) 088 [arXiv:1006.2788] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    F. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    F. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011)011 [arXiv:1102.0062] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    V. Del Duca, C. Duhr and V.A. Smirnov, An analytic result for the two-loop hexagon Wilson loop in N = 4 SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  17. [17]
    D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Bootstrapping null polygon Wilson loops, JHEP 03 (2011) 092 [arXiv:1010.5009] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  18. [18]
    L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, Nucl. Phys. B 826 (2010) 337 [arXiv:0712.1223] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  20. [20]
    B. Basso, Exciting the GKP string at any coupling, arXiv:1010.5237 [INSPIRE].
  21. [21]
    V. Del Duca, C. Duhr and V.A. Smirnov, A two-loop octagon Wilson loop in N = 4 SYM, JHEP 09 (2010) 015 [arXiv:1006.4127] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    P. Heslop and V.V. Khoze, Analytic results for MHV Wilson loops, JHEP 11 (2010) 035 [arXiv:1007.1805] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  23. [23]
    S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, arXiv:1105.5606 [INSPIRE].
  24. [24]
    Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].MathSciNetADSGoogle Scholar
  25. [25]
    L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for scattering amplitudes, J. Phys. A 43 (2010) 485401 [arXiv:1002.2459] [INSPIRE].MathSciNetGoogle Scholar
  26. [26]
    J. Murley and N. Saad, Tables of the appell hypergeometric functions F 2, arXiv:0809.5203.
  27. [27]
    V.V. Bytev, M.Y. Kalmykov and B.A. Kniehl, HYPERDIRE: HYPERgeometric functions differential reduction Mathematica based packages for differential reduction of generalized hypergeometric functions: now with p F p−1 , F 1 ,F 2 ,F 3 ,F 4, arXiv:1105.3565 [INSPIRE].
  28. [28]
    L. Lipatov, Reggeization of the vector meson and the vacuum singularity in nonabelian gauge theories, Sov. J. Nucl. Phys. 23 (1976) 338 [INSPIRE].Google Scholar
  29. [29]
    V.S. Fadin, E. Kuraev and L. Lipatov, On the Pomeranchuk singularity in asymptotically free theories, Phys. Lett. B 60 (1975) 50 [INSPIRE].ADSGoogle Scholar
  30. [30]
    E. Kuraev, L. Lipatov and V.S. Fadin, Multi - Reggeon processes in the Yang-Mills theory, Sov. Phys. JETP 44 (1976) 443 [INSPIRE].ADSGoogle Scholar
  31. [31]
    I. Balitsky and L. Lipatov, The Pomeranchuk singularity in quantum chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [INSPIRE].Google Scholar
  32. [32]
    J. Bartels, L. Lipatov and A. Prygarin, Collinear and Regge behavior of 2 → 4 MHV amplitude in N = 4 super Yang-Mills theory, arXiv:1104.4709 [INSPIRE].

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical Physics WaterlooOntarioCanada

Personalised recommendations