The two-loop hexagon 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, the conformal symmetry constrains multi-loop n-edged Wilson loops to be given in terms of the one-loop n-edged Wilson loop, augmented, for n ≥ 6, by a function of conformally invariant cross ratios. That function is termed the remainder function. In a recent paper, we have displayed the first analytic computation of the two-loop six-edged Wilson loop, and thus of the corresponding remainder function, in terms of known mathematical functions. Although the calculation was performed in the quasi-multi-Regge kinematics of a pair along the ladder, the Regge exactness of the six-edged Wilson loop in those kinematics entails that the result is the same as in general kinematics. We show in detail how the most difficult of the integrals is computed, which contribute to the six-edged Wilson loop. Finally, the remainder function is given as a function of uniform transcendental weight four in terms of Goncharov polylogarithms. We consider also some asymptotic values of the remainder function, and the value when all the cross ratios are equal.


Supersymmetric gauge theory Gauge Symmetry 


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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 (Roma)Italy
  3. 3.Institute for Particle Physics PhenomenologyUniversity of DurhamDurhamU.K.
  4. 4.Nuclear Physics Institute of Moscow State UniversityMoscowRussia

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