Journal of High Energy Physics

, 2010:15 | Cite as

A two-loop octagon Wilson loop in \( \mathcal{N} = 4 \) SYM

  • Vittorio Del Duca
  • Claude DuhrEmail author
  • Vladimir A. Smirnov
Open Access


In the planar \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory at weak coupling, we perform the first analytic computation of a two-loop eight-edged Wilson loop embedded into the boundary of AdS 3. Its remainder function is given as a function of uniform transcendental weight four in terms of a constant plus a product of four logarithms. We compare to the strong-coupling result, and test a conjecture on the universality of the remainder function proposed in the literature.


Supersymmetric gauge theory Gauge Symmetry 


  1. [1]
    L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  2. [2]
    J.M. Drummond, G.P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [SPIRES]. MathSciNetADSCrossRefGoogle 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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  5. [5]
    J.M. Drummond, J. Henn, G.P. 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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude, Phys. Lett. B 662 (2008) 456 [arXiv:0712.4138] [SPIRES].MathSciNetADSGoogle Scholar
  7. [7]
    J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Hexagon Wilson loop = six-gluon MHV amplitude, Nucl. Phys. B 815 (2009) 142 [arXiv:0803.1466] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    Z. Bern, M. Czakon, D.A. Kosower, R. Roiban and V.A. Smirnov, Two-loop iteration of five-point N = 4 super-Yang-Mills amplitudes, Phys. Rev. Lett. 97 (2006) 181601 [hep-th/0604074] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  9. [9]
    Z. Bern et al., The two-loop six-gluon MHV amplitude in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 78 (2008) 045007 [arXiv:0803.1465] [SPIRES].MathSciNetADSGoogle Scholar
  10. [10]
    C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [SPIRES]. MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    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] [SPIRES].MathSciNetADSGoogle Scholar
  12. [12]
    C. Anastasiou et al., Two-loop polygon Wilson loops in N = 4 SYM, JHEP 05 (2009) 115 [arXiv:0902.2245] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    A. Brandhuber, P. Heslop, V.V. Khoze and G. Travaglini, Simplicity of polygon Wilson loops in N = 4 SYM, JHEP 01 (2010) 050 [arXiv:0910.4898] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  14. [14]
    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] [SPIRES].ADSCrossRefGoogle Scholar
  15. [15]
    V. Del Duca, C. Duhr and V.A. Smirnov, The two-loop hexagon Wilson loop in N = 4 SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [SPIRES].ADSCrossRefGoogle Scholar
  16. [16]
    J.-H. Zhang, On the two-loop hexagon Wilson loop remainder function in N = 4 SYM, arXiv:1004.1606 [SPIRES].
  17. [17]
    L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, arXiv:0911.4708 [SPIRES].
  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] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  19. [19]
    J.G.M. Gatheral, Exponentiation of eikonal cross-sections in nonabelian gauge theories, Phys. Lett. B 133 (1983) 90 [SPIRES].MathSciNetADSGoogle Scholar
  20. [20]
    J. Frenkel and J.C. Taylor, Nonabelian eikonal exponentiation, Nucl. Phys. B 246 (1984) 231 [SPIRES].ADSCrossRefGoogle Scholar
  21. [21]
    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One-loop n-point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  22. [22]
    I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [SPIRES].ADSGoogle Scholar
  23. [23]
    V. Del Duca, C. Duhr and E.W.N. Glover, Iterated amplitudes in the high-energy limit, JHEP 12 (2008) 097 [arXiv:0809.1822] [SPIRES].ADSCrossRefGoogle Scholar
  24. [24]
    V.S. Fadin and L.N. Lipatov, High-energy production of gluons in a quasimulti Regge kinematics, JETP Lett. 49 (1989) 352 [Yad. Fiz. 50 (1989) 1141] [SPIRES].ADSGoogle Scholar
  25. [25]
    V. Del Duca, Real next-to-leading corrections to the multigluon amplitudes in the helicity formalism, Phys. Rev. D 54 (1996) 989 [hep-ph/9601211] [SPIRES].ADSGoogle Scholar
  26. [26]
    C. Duhr, New techniques in QCD, Ph.D .thesis, Université Catholique de Louvain, Belgium (2009).Google Scholar
  27. [27]
    V.A. Smirnov, Analytical result for dimensionally regularized massless on-shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [SPIRES].ADSGoogle Scholar
  28. [28]
    J.B. Tausk, Non-planar massless two-loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [SPIRES].MathSciNetADSGoogle Scholar
  29. [29]
    V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1 [SPIRES].Google Scholar
  30. [30]
    V.A. Smirnov, Feynman integral calculus, Springer, Berlin Germany (2006), pag. 283 [SPIRES].Google Scholar
  31. [31]
    M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [SPIRES].ADSzbMATHCrossRefGoogle Scholar
  32. [32]
    A.V. Smirnov and V.A. Smirnov, On the resolution of singularities of multiple Mellin-Barnes integrals, Eur. Phys. J. C 62 (2009) 445 [arXiv:0901.0386] [SPIRES].ADSCrossRefGoogle Scholar
  33. [33]
    M. Czakon, MBasymptotics,
  34. [34]
    D.A. Kosower, barnesroutines,
  35. [35]
    A.V. Smirnov and M.N. Tentyukov, Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA), Comput. Phys. Commun. 180 (2009) 735 [arXiv:0807.4129] [SPIRES]. ADSzbMATHCrossRefGoogle Scholar
  36. [36]
    A.V. Smirnov, V.A. Smirnov and M. Tentyukov, FIESTA 2: parallelizeable multiloop numerical calculations, arXiv:0912.0158 [SPIRES].
  37. [37]
    S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multi-scale multi-loop integrals, J. Math. Phys. 43 (2002) 3363 [hep-ph/0110083] [SPIRES].MathSciNetADSzbMATHCrossRefGoogle Scholar
  38. [38]
    S. Moch and P. Uwer, XSummer: transcendental functions and symbolic summation in Form, Comput. Phys. Commun. 174 (2006) 759 [math-ph/0508008] [SPIRES].ADSzbMATHCrossRefGoogle Scholar
  39. [39]
    F. Jegerlehner, M.Y. Kalmykov and O. Veretin, MS-bar vs pole masses of gauge bosons. II: two-loop electroweak fermion corrections, Nucl. Phys. B 658 (2003) 49 [hep-ph/0212319] [SPIRES].ADSCrossRefGoogle Scholar
  40. [40]
    M.Y. Kalmykov, B.F.L. Ward and S.A. Yost, Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order ϵ-expansion of generalized hypergeometric functions with one half-integer value of parameter, JHEP 10 (2007) 048 [arXiv:0707.3654] [SPIRES].ADSCrossRefGoogle Scholar
  41. [41]
    L.F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, An operator product expansion for polygonal null Wilson loops, arXiv:1006.2788 [SPIRES].

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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

  • Vittorio Del Duca
    • 1
    • 2
  • Claude Duhr
    • 3
    Email author
  • Vladimir A. Smirnov
    • 4
  1. 1.PH Department, TH UnitCERNGeneva 23Switzerland
  2. 2.INFN, Laboratori Nazionali FrascatiFrascati, RomaItaly
  3. 3.Institute for Particle Physics PhenomenologyUniversity of DurhamDurhamU.K.
  4. 4.Nuclear Physics Institute of Moscow State UniversityMoscowRussia

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