Abstract
We provide a simple analytic formula for the two-loop six-point ratio function of planar \( \mathcal{N} = {4} \) super Yang-Mills theory. This result extends the analytic knowledge of multi-loop six-point amplitudes beyond those with maximal helicity violation. We make a natural ansatz for the symbols of the relevant functions appearing in the two-loop amplitude, and impose various consistency conditions, including symmetry, the absence of spurious poles, the correct collinear behaviour, and agreement with the operator product expansion for light-like (super) Wilson loops. This information reduces the ansatz to a small number of relatively simple functions. In order to fix these parameters uniquely, we utilize an explicit representation of the amplitude in terms of loop integrals that can be evaluated analytically in various kinematic limits. The final compact analytic result is expressed in terms of classical polylogarithms, whose arguments are rational functions of the dual conformal cross-ratios, plus precisely two functions that are not of this type. One of the functions, the loop integral Ω(2), also plays a key role in a new representa- tion of the remainder function \( {\text{R}}_6^{{(2)}} \) in the maximally helicity violating sector. Another interesting feature at two loops is the appearance of a new (parity odd) × (parity odd) sector of the amplitude, which is absent at one loop, and which is uniquely determined in a natural way in terms of the more familiar (parity even) × (parity even) part. The second non-polylogarithmic function, the loop integral \( \widetilde{\Omega } \) (2), characterizes this sector. Both Ω(2) and \( \widetilde{\Omega } \) (2) can be expressed as one-dimensional integrals over classical polylogarithms with rational arguments.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].
Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
J. Drummond, G. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].
J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].
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].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].
A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in \( \mathcal{N} = {4} \) super Yang-Mills and Wilson loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [INSPIRE].
J. Drummond, J. Henn, G. 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] [INSPIRE].
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] [INSPIRE].
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].
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].
J. Bartels, L. Lipatov and A. Sabio Vera, BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, An analytic result for the two-loop hexagon Wilson Loop in \( \mathcal{N} = {4} \) SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, The two-loop hexagon Wilson loop in \( \mathcal{N} = {4} \) SYM, JHEP 05 (2010) 084 [arXiv:1003.1702] [INSPIRE].
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].
K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831.
F. Brown, Multiple zeta values and periods of moduli spaces M 0,n , Ann. Sci. École Norm. Sup. (4) 42 (2009) 371 [math/0606419].
A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238.
J.M. Drummond and J.M. Henn, Simple loop integrals and amplitudes in \( \mathcal{N} = {4} \) SYM, JHEP 05 (2011) 105 [arXiv:1008.2965] [INSPIRE].
L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of \( \mathcal{N} = {4} \) super Yang-Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
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].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Bootstrapping null polygon Wilson loops, JHEP 03 (2011) 092 [arXiv:1010.5009] [INSPIRE].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [INSPIRE].
C. Anastasiou, A. Brandhuber, P. Heslop, V.V. Khoze, B. Spence and G. Travaglini, Two-loop polygon Wilson loops in \( \mathcal{N} = {4} \) SYM, JHEP 05 (2009) 115 [arXiv:0902.2245] [INSPIRE].
A. Brandhuber, P. Heslop, V.V. Khoze and G. Travaglini, Simplicity of polygon Wilson loops in \( \mathcal{N} = {4} \) SYM, JHEP 01 (2010) 050 [arXiv:0910.4898] [INSPIRE].
C. Vergu, Higher point MHV amplitudes in \( \mathcal{N} = {4} \) supersymmetric Yang-Mills theory, Phys. Rev. D 79 (2009) 125005 [arXiv:0903.3526] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, arXiv:1012.6032 [INSPIRE].
S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar \( \mathcal{N} = {4} \) super Yang-Mills, arXiv:1105.5606 [INSPIRE].
A. Sever and P. Vieira, Multichannel conformal blocks for polygon Wilson loops, arXiv:1105.5748 [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, A two-loop octagon Wilson loop in \( \mathcal{N} = {4} \) SYM, JHEP 09 (2010) 015 [arXiv:1006.4127] [INSPIRE].
P. Heslop and V.V. Khoze, Analytic results for MHV Wilson loops, JHEP 11 (2010) 035 [arXiv:1007.1805] [INSPIRE].
L.F. Alday, Some analytic results for two-loop scattering amplitudes, JHEP 07 (2011) 080 [arXiv:1009.1110] [INSPIRE].
P. Heslop and V.V. Khoze, Wilson loops @ 3-loops in special kinematics, JHEP 11 (2011) 152 [arXiv:1109.0058] [INSPIRE].
J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in \( \mathcal{N} = {4} \) super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
Z. Bern, V. Del Duca, L.J. Dixon and D.A. Kosower, All non-maximally-helicity-violating one-loop seven-gluon amplitudes in \( \mathcal{N} = {4} \) super-Yang-Mills theory, Phys. Rev. D 71 (2005) 045006 [hep-th/0410224] [INSPIRE].
Y.-t. Huang, \( \mathcal{N} = {4} \) SYM NMHV loop amplitude in superspace, Phys. Lett. B 631 (2005) 177 [hep-th/0507117] [INSPIRE].
K. Risager, S.J. Bidder and W.B. Perkins, One-loop nMHV amplitudes involving gluinos and scalars in \( \mathcal{N} = {4} \) gauge theory, JHEP 10 (2005) 003 [hep-th/0507170] [INSPIRE].
J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Generalized unitarity for \( \mathcal{N} = {4} \) super-amplitudes, arXiv:0808.0491 [INSPIRE].
D. Kosower, R. Roiban and C. Vergu, The six-point NMHV amplitude in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 83 (2011) 065018 [arXiv:1009.1376] [INSPIRE].
J. Drummond and J. Henn, All tree-level amplitudes in \( \mathcal{N} = {4} \) SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A duality for the S matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
L. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and C. Cheung, The Grassmannian origin of dual superconformal invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [INSPIRE].
G. Korchemsky and E. Sokatchev, Superconformal invariants for scattering amplitudes in \( \mathcal{N} = {4} \) SYM theory, Nucl. Phys. B 839 (2010) 377 [arXiv:1002.4625] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in \( \mathcal{N} = {4} \) super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
G. Korchemsky and E. Sokatchev, Symmetries and analytic properties of scattering amplitudes in \( \mathcal{N} = {4} \) SYM theory, Nucl. Phys. B 832 (2010) 1 [arXiv:0906.1737] [INSPIRE].
L. Mason and D. Skinner, The complete planar S-matrix of \( \mathcal{N} = {4} \) SYM as a Wilson loop in twistor space, JHEP 12 (2010) 018 [arXiv:1009.2225] [INSPIRE].
S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, JHEP 07 (2011) 058 [arXiv:1010.1167] [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: Part I, arXiv:1103.3714 [INSPIRE].
B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: Part II, arXiv:1103.4353 [INSPIRE].
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].
A. Sever, P. Vieira and T. Wang, OPE for super loops, JHEP 11 (2011) 051 [arXiv:1108.1575] [INSPIRE].
J.M. Henn, S. Moch and S.G. Naculich, Form factors and scattering amplitudes in \( \mathcal{N} = {4} \) SYM in dimensional and massive regularizations, arXiv:1109.5057 [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar \( \mathcal{N} = {4} \) SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Trnka, New differential equations for on-shell loop integrals, JHEP 04 (2011) 083 [arXiv:1010.3679] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in \( \mathcal{N} = {4} \) SYM, JHEP 06 (2011) 100 [arXiv:1104.2787] [INSPIRE].
V. Nair, A current algebra for some gauge theory amplitudes, Phys. Lett. B 214 (1988) 215 [INSPIRE].
G. Georgiou, E. Glover and V.V. Khoze, Non-MHV tree amplitudes in gauge theory, JHEP 07 (2004) 048 [hep-th/0407027] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, arXiv:0905.1473 [INSPIRE].
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].
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] [INSPIRE].
D.A. Kosower and P. Uwer, One loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, The massless hexagon integral in D = 6 dimensions, Phys. Lett. B 703 (2011) 363 [arXiv:1104.2781] [INSPIRE].
A. Kotikov, L. Lipatov, A. Onishchenko and V. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in \( \mathcal{N} = {4} \) SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [Erratum ibid. B 632 (2006) 754-756] [hep-th/0404092] [INSPIRE].
S. Caron-Huot, private communication.
E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].
J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, More loops and legs in Higgs-regulated \( \mathcal{N} = {4} \) SYM amplitudes, JHEP 08 (2010) 002 [arXiv:1004.5381] [INSPIRE].
I. Korchemskaya and G. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [INSPIRE].
J.M. Henn, S.G. Naculich, H.J. Schnitzer and M. Spradlin, Higgs-regularized three-loop four-gluon amplitude in \( \mathcal{N} = {4} \) SYM: exponentiation and Regge limits, JHEP 04 (2010) 038 [arXiv:1001.1358] [INSPIRE].
J.M. Henn, Scattering amplitudes on the Coulomb branch of \( \mathcal{N} = {4} \) super Yang-Mills, Nucl. Phys. Proc. Suppl. 205-206 (2010) 193 [arXiv:1005.2902] [INSPIRE].
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].
M. Czakon, MBasymptotics.m, http://projects.hepforge.org/mbtools/.
D. Kosower, barnesroutines.m, http://projects.hepforge.org/mbtools/.
T. Gehrmann and E. Remiddi, Two loop master integrals for γ* → 3 jets: the planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].
T. Gehrmann and E. Remiddi, Two loop master integrals for γ* → 3 jets: the nonplanar topologies, Nucl. Phys. B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].
D. Maître, HPL, a Mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222, package can be downloaded at http://krone.physik.unizh.ch/∼maitreda/HPL/ [hep-ph/0507152] [INSPIRE].
V. Smirnov, Feynman integral calculus, Springer Verlag, Heidelberg Germany (2006).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Dixon, L.J., Drummond, J.M. & Henn, J.M. Analytic result for the two-loop six-point NMHV amplitude in \( \mathcal{N} = {4} \) super Yang-Mills theory. J. High Energ. Phys. 2012, 24 (2012). https://doi.org/10.1007/JHEP01(2012)024
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2012)024