Abstract
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.
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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].
Z. Bern, J.S. Rozowsky and B. Yan, Two-loop four-gluon amplitudes in N = 4 super-Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424] [SPIRES].
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].
F. Cachazo, M. Spradlin and A. Volovich, Iterative structure within the five-particle two-loop amplitude, Phys. Rev. D 74 (2006) 045020 [hep-th/0602228] [SPIRES].
V. Del Duca, C. Duhr, E.W. Nigel Glover and V.A. Smirnov, The one-loop pentagon to higher orders in epsilon, JHEP 01 (2010) 042 [arXiv:0905.0097] [SPIRES].
V. Del Duca, C. Duhr and E.W. Nigel Glover, The five-gluon amplitude in the high-energy limit, JHEP 12 (2009) 023 [arXiv:0905.0100] [SPIRES].
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] [SPIRES].
J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, Higgs-regularized three-loop four-gluon amplitude in N = 4 SYM: exponentiation andRegge limits, JHEP 04 (2010) 038 [arXiv:1001.1358] [SPIRES].
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].
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].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [SPIRES].
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].
J. Bartels, L.N. Lipatov and A. Sabio Vera, BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065] [SPIRES].
J. Bartels, L.N. Lipatov and A. Sabio Vera, N=4 supersymmetric Yang-Mills scattering amplitudes at high energies: the Regge cut contribution, Eur. Phys. J. C 65 (2010) 587 [arXiv:0807.0894] [SPIRES].
R.M. Schabinger, The Imaginary Part of the N = 4 super-Yang-Mills Two-Loop Six-Point MHV Amplitude in Multi-Regge Kinematics, JHEP 11 (2009) 108 [arXiv:0910.3933] [SPIRES].
F. Cachazo, M. Spradlin and A. Volovich, Leading Singularities of the Two-Loop Six-Particle MHV Amplitude, Phys. Rev. D 78 (2008) 105022 [arXiv:0805.4832] [SPIRES].
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].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [SPIRES].
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].
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].
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].
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].
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].
C. Anastasiou et al., Two-Loop Polygon Wilson Loops in N = 4 SYM, JHEP 05 (2009) 115 [arXiv:0902.2245] [SPIRES].
L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Antide-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [SPIRES].
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].
C. Vergu, The two-loop MHV amplitudes in N = 4 supersymmetric Yang- Mills theory, arXiv:0908.2394 [SPIRES].
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].
V.S. Fadin and L.N. Lipatov, High-Energy Production of Gluons in a QuasimultiRegge Kinematics, JETP Lett. 49 (1989) 352 [SPIRES].
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].
V. Del Duca, A. Frizzo and F. Maltoni, Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit, Nucl. Phys. B 568 (2000) 211 [hep-ph/9909464] [SPIRES].
C. Duhr, New techniques in QCD, PhD thesis, Université Catholique de Louvain (2009).
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, arXiv:0911.4708 [SPIRES].
J.G.M. Gatheral, Exponentiation of eikonal cross-sections in nonabelian gauge theories, Phys. Lett. B 133 (1983) 90 [SPIRES].
J. Frenkel and J.C. Taylor, Nonabelian eikonal exponentiation, Nucl. Phys. B 246 (1984) 231 [SPIRES].
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].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [SPIRES].
V.A. Smirnov, Analytical result for dimensionally regularized massless on-shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [SPIRES].
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].
V.A. Smirnov, Evaluating Feynman Integrals, Springer Tracts Mod. Phys. 211 (2004) 1 [SPIRES].
V.A. Smirnov, Feynman integral calculus, Springer, Berlin Germany (2006).
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [SPIRES].
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].
M. Czakon, MBasymptotics, http://projects.hepforge.org/mbtools/.
D.A. Kosower, barnesroutines, http://projects.hepforge.org/mbtools/.
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].
A.V. Smirnov, V.A. Smirnov and M. Tentyukov, FIESTA 2: parallelizeable multiloop numerical calculations, arXiv:0912.0158 [SPIRES].
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Research Lett. 5 (1998) 497.
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059.
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].
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].
L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for Scattering Amplitudes, arXiv:1002.2459 [SPIRES].
J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [SPIRES].
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].
S. Moch and P. Uwer, XSummer: Transcendental functions and symbolic summation in Form, Comput. Phys. Commun. 174 (2006) 759 [math-ph/0508008] [SPIRES].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [SPIRES].
T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [SPIRES].
U. Aglietti, V. Del Duca, C. Duhr, G. Somogyi and Z. Trócsányi, Analytic integration of real-virtual counterterms in NNLO jet cross sections I, JHEP 09 (2008) 107 [arXiv:0807.0514] [SPIRES].
R.C. Brower, H. Nastase, H.J. Schnitzer and C.-I. Tan, Implications of multi-Regge limits for the Bern-Dixon-Smirnov conjecture, Nucl. Phys. B 814 (2009) 293 [arXiv:0801.3891] [SPIRES].
R.C. Brower, H. Nastase, H.J. Schnitzer and C.-I. Tan, Analyticity for Multi-Regge Limits of the Bern-Dixon-Smirnov Amplitudes, Nucl. Phys. B 822 (2009) 301 [arXiv:0809.1632] [SPIRES].
H.R.P. Ferguson and D.H. Bailey, A Polynomial Time, Numerically Stable Integer Relation Algorithm, RNR Technical Report, RNR-91-032.
H.R.P. Ferguson, D.H. Bailey and S. Arno, Analysis of PSLQ, an Integer Relation Finding Algorithm, NASA Technical Report, NAS-96-005.
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [SPIRES].
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Del Duca, V., Duhr, C. & Smirnov, V.A. The two-loop hexagon Wilson loop in \( \mathcal{N} = 4 \) SYM. J. High Energ. Phys. 2010, 84 (2010). https://doi.org/10.1007/JHEP05(2010)084
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DOI: https://doi.org/10.1007/JHEP05(2010)084