Abstract
We present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar \( \mathcal{N} \) = 4 super-Yang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of hexagon functions which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematic limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi-Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann ζ valued constants that could not be fixed at the level of the symbol. The near-collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to −7.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge U.K. (1966).
G. Veneziano, Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories, Nuovo Cim. A 57 (1968) 190 [INSPIRE].
S. Ferrara, A. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
A. Polyakov, Nonhamiltonian approach to conformal quantum field theory, Zh. Eksp. Teor. Fiz. 66 (1974) 23 [INSPIRE].
A. Belavin, A.M. Polyakov and A. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
S. El-Showk et al., Solving the 3D Ising model with the conformal bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, The N = 4 superconformal bootstrap, Phys. Rev. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].
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].
Z. Bern, J.J. Carrasco, H. Johansson and D. Kosower, Maximally supersymmetric planar Yang-Mills amplitudes at five loops, Phys. Rev. D 76 (2007) 125020 [arXiv:0705.1864] [INSPIRE].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [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].
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].
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].
A. Sever, P. Vieira and T. Wang, OPE for super loops, JHEP 11 (2011) 051 [arXiv:1108.1575] [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].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [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].
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].
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].
J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].
K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977) 831.
F.C. Brown, Multiple zeta values and periods of moduli spaces \( {{\mathfrak{M}}_0}_{,n } \) Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
A. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE].
A. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].
F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].
F. Brown, Mixed Tate motives over \( \mathbb{Z} \), arXiv:1102.1312.
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].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].
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].
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] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
S. Caron-Huot and S. He, Jumpstarting the all-loop S-matrix of planar N = 4 super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [INSPIRE].
M. Bullimore and D. Skinner, Descent equations for superamplitudes, arXiv:1112.1056 [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux-tube S-matrix at finite coupling, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and flux tube S-matrix II. Extracting and matching data, arXiv:1306.2058 [INSPIRE].
B. Basso, A. Sever and P. Vieira, private communication.
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, to appear.
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].
J. Bartels, L. 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] [INSPIRE].
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] [INSPIRE].
L. Lipatov and A. Prygarin, Mandelstam cuts and light-like Wilson loops in N = 4 SUSY, Phys. Rev. D 83 (2011) 045020 [arXiv:1008.1016] [INSPIRE].
L. Lipatov and A. Prygarin, BFKL approach and six-particle MHV amplitude in N = 4 super Yang-Mills, Phys. Rev. D 83 (2011) 125001 [arXiv:1011.2673] [INSPIRE].
J. Bartels, L. Lipatov and A. Prygarin, MHV amplitude for 3 → 3 gluon scattering in Regge limit, Phys. Lett. B 705 (2011) 507 [arXiv:1012.3178] [INSPIRE].
V. Fadin and L. Lipatov, BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at N = 4 SUSY, Phys. Lett. B 706 (2012) 470 [arXiv:1111.0782] [INSPIRE].
A. Prygarin, M. Spradlin, C. Vergu and A. Volovich, All two-loop MHV amplitudes in multi-Regge kinematics from applied symbology, Phys. Rev. D 85 (2012) 085019 [arXiv:1112.6365] [INSPIRE].
J. Bartels, A. Kormilitzin, L. Lipatov and A. Prygarin, BFKL approach and 2 → 5 maximally helicity violating amplitude in \( \mathcal{N} \) = 4 super-Yang-Mills theory, Phys. Rev. D 86 (2012) 065026 [arXiv:1112.6366] [INSPIRE].
L. Lipatov, A. Prygarin and H.J. Schnitzer, The multi-Regge limit of NMHV amplitudes in N =4 SYM theory, JHEP 01 (2013) 068 [arXiv:1205.0186] [INSPIRE].
L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP 10 (2012) 074 [arXiv:1207.0186] [INSPIRE].
F.C.S. Brown, Single-valued multiple polylogarithms in one variable, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 527.
J. Pennington, The six-point remainder function to all loop orders in the multi-Regge limit, JHEP 01 (2013) 059 [arXiv:1209.5357] [INSPIRE].
S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].
S. Müller-Stach, S. Weinzierl and R. Zayadeh, A second-order differential equation for the two-loop sunrise graph with arbitrary masses, Commun. Num. Theor. Phys. 6 (2012) 203 [arXiv:1112.4360] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
N. Arkani-Hamed et al., Scattering amplitudes and the positive Grassmannian, arXiv:1212.5605 [INSPIRE].
A.E. Lipstein and L. Mason, From the holomorphic Wilson loop to ‘dlog’ loop-integrands for super-Yang-Mills amplitudes, JHEP 05 (2013) 106 [arXiv:1212.6228] [INSPIRE].
A.E. Lipstein and L. Mason, From dlogs to dilogs; the super Yang-Mills MHV amplitude revisited, arXiv:1307.1443 [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
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] [INSPIRE].
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].
C. Anastasiou et al., Two-loop polygon Wilson loops in N = 4 SYM, JHEP 05 (2009) 115 [arXiv:0902.2245] [INSPIRE].
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].
Y. Hatsuda, K. Ito, K. Sakai and Y. Satoh, g-functions and gluon scattering amplitudes at strong coupling, JHEP 04 (2011) 100 [arXiv:1102.2477] [INSPIRE].
Y. Hatsuda, K. Ito and Y. Satoh, T-functions and multi-gluon scattering amplitudes, JHEP 02 (2012) 003 [arXiv:1109.5564] [INSPIRE].
Y. Hatsuda, K. Ito, K. Sakai and Y. Satoh, Six-point gluon scattering amplitudes from Z 4 -symmetric integrable model, JHEP 09 (2010) 064 [arXiv:1005.4487] [INSPIRE].
Y. Hatsuda, K. Ito and Y. Satoh, Null-polygonal minimal surfaces in AdS 4 from perturbed W minimal models, JHEP 02 (2013) 067 [arXiv:1211.6225] [INSPIRE].
S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar 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 N = 4 SYM, JHEP 06 (2011) 100 [arXiv:1104.2787] [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].
E. Remiddi and J. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
D.E. Radford, A natural ring basis for the shuffle algebra and an application to group schemes, J. Alg. 58 (1979) 432.
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059.
C. Bogner and F. Brown, Symbolic integration and multiple polylogarithms, PoS(LL2012)053 [arXiv:1209.6524] [INSPIRE].
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
F. Chavez and C. Duhr, Three-mass triangle integrals and single-valued polylogarithms, JHEP 11 (2012) 114 [arXiv:1209.2722] [INSPIRE].
J. Drummond et al., Leading singularities and off-shell conformal integrals, JHEP 08 (2013) 133 [arXiv:1303.6909] [INSPIRE].
A. von Manteuffel and C. Studerus, Massive planar and non-planar double box integrals for light N f contributions to \( gg\to t\overline{t} \), JHEP 10 (2013) 037 [arXiv:1306.3504] [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic multiple zeta values and superstring amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
H.R.P. Ferguson and D.H. Bailey, A polynomial time, numerically stable integer relation algorithm, RNR Technical Report RNR-91-032, (1991).
H.R.P. Ferguson, D.H. Bailey and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comput. 68 (1999) 351.
N. Beisert, B. Eden and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. 01 (2007) P01021 [hep-th/0610251] [INSPIRE].
P. Vieira, private communication.
N. Arkani-Hamed, S. Caron-Huot and J. Trnka, private communication.
J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, arXiv:1305.1617 [INSPIRE].
J. Golden and M. Spradlin, The differential of all two-loop MHV amplitudes in \( \mathcal{N} \) = 4 Yang-Mills theory, JHEP 09 (2013) 111 [arXiv:1306.1833] [INSPIRE].
J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisoněk, Special values of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001) 907 [math/9910045] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1308.2276
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Dixon, L.J., Drummond, J.M., von Hippel, M. et al. Hexagon functions and the three-loop remainder function. J. High Energ. Phys. 2013, 49 (2013). https://doi.org/10.1007/JHEP12(2013)049
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2013)049