Abstract
We study the six-particle amplitude in planar \( \mathcal{N} \) = 4 super Yang-Mills theory in the double scaling (DS) limit, the only nontrivial codimension-one boundary of its positive kinematic region. We construct the relevant function space, which is significantly constrained due to the extended Steinmann relations, up to weight 13 in coproduct form, and up to weight 12 as an explicit polylogarithmic representation. Expanding the latter in the collinear boundary of the DS limit, and using the Pentagon Operator Product Expansion, we compute the non-divergent coefficient of a certain component of the Next-to-Maximally-Helicity-Violating amplitude through weight 12 and eight loops. We also specialize our results to the overlapping origin limit, observing a general pattern for its leading divergences.
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L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills Theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].
F. Gliozzi, J. Scherk and D.I. Olive, Supersymmetry, Supergravity Theories and the Dual Spinor Model, Nucl. Phys. B 122 (1977) 253 [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
F. Levkovich-Maslyuk, A review of the AdS/CFT Quantum Spectral Curve, J. Phys. A 53 (2020) 283004 [arXiv:1911.13065] [INSPIRE].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [INSPIRE].
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] [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].
B. Basso, A. Sever and P. Vieira, Spacetime and Flux Tube S-Matrices at Finite Coupling for N = 4 Supersymmetric Yang-Mills Theory, 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, JHEP 01 (2014) 008 [arXiv:1306.2058] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions, JHEP 08 (2014) 085 [arXiv:1402.3307] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux-tube S-matrix IV. Gluons and Fusion, JHEP 09 (2014) 149 [arXiv:1407.1736] [INSPIRE].
A.V. Belitsky, Nonsinglet pentagons and NMHV amplitudes, Nucl. Phys. B 896 (2015) 493 [arXiv:1407.2853] [INSPIRE].
A.V. Belitsky, Fermionic pentagons and NMHV hexagon, Nucl. Phys. B 894 (2015) 108 [arXiv:1410.2534] [INSPIRE].
B. Basso, J. Caetano, L. Cordova, A. Sever and P. Vieira, OPE for all Helicity Amplitudes, JHEP 08 (2015) 018 [arXiv:1412.1132] [INSPIRE].
B. Basso, J. Caetano, L. Cordova, A. Sever and P. Vieira, OPE for all Helicity Amplitudes II. Form Factors and Data Analysis, JHEP 12 (2015) 088 [arXiv:1508.02987] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Hexagonal Wilson loops in planar \( \mathcal{N} \) = 4 SYM theory at finite coupling, J. Phys. A 49 (2016) 41LT01 [arXiv:1508.03045] [INSPIRE].
A.V. Belitsky, Matrix pentagons, Nucl. Phys. B 923 (2017) 588 [arXiv:1607.06555] [INSPIRE].
J.M. Drummond, Review of AdS/CFT Integrability, Chapter V.2: Dual Superconformal Symmetry, Lett. Math. Phys. 99 (2012) 481 [arXiv:1012.4002] [INSPIRE].
D. Fioravanti, S. Piscaglia and M. Rossi, Asymptotic Bethe Ansatz on the GKP vacuum as a defect spin chain: scattering, particles and minimal area Wilson loops, Nucl. Phys. B 898 (2015) 301 [arXiv:1503.08795] [INSPIRE].
A. Bonini, D. Fioravanti, S. Piscaglia and M. Rossi, Strong Wilson polygons from the lodge of free and bound mesons, JHEP 04 (2016) 029 [arXiv:1511.05851] [INSPIRE].
A. Bonini, D. Fioravanti, S. Piscaglia and M. Rossi, The contribution of scalars to \( \mathcal{N} \) = 4 SYM amplitudes, Phys. Rev. D 95 (2017) 041902 [arXiv:1607.02084] [INSPIRE].
A. Bonini, D. Fioravanti, S. Piscaglia and M. Rossi, The contribution of scalars to \( \mathcal{N} \) = 4 SYM amplitudes II: Young tableaux, asymptotic factorisation and strong coupling, Nucl. Phys. B 931 (2018) 19 [arXiv:1707.05767] [INSPIRE].
A. Bonini, D. Fioravanti, S. Piscaglia and M. Rossi, Fermions and scalars in \( \mathcal{N} \) = 4 Wilson loops at strong coupling and beyond, Nucl. Phys. B 944 (2019) 114644 [arXiv:1807.09743] [INSPIRE].
S. Caron-Huot, L.J. Dixon, M. von Hippel, A.J. McLeod and G. Papathanasiou, The Double Pentaladder Integral to All Orders, JHEP 07 (2018) 170 [arXiv:1806.01361] [INSPIRE].
J.M. Drummond and G. Papathanasiou, Hexagon OPE Resummation and Multi-Regge Kinematics, JHEP 02 (2016) 185 [arXiv:1507.08982] [INSPIRE].
L. Córdova, Hexagon POPE: effective particles and tree level resummation, JHEP 01 (2017) 051 [arXiv:1606.00423] [INSPIRE].
H.T. Lam and M. von Hippel, Resumming the POPE at One Loop, JHEP 12 (2016) 011 [arXiv:1608.08116] [INSPIRE].
L.V. Bork and A.I. Onishchenko, Pentagon OPE resummation in N = 4 SYM: hexagons with one effective particle contribution, Phys. Rev. D 102 (2020) 026002 [arXiv:1909.13675] [INSPIRE].
L.V. Bork and A.I. Onishchenko, Pentagon OPE Resummation in N = 4 SYM: One Effective Particle and MHV Amplitude, Phys. Part. Nucl. 51 (2020) 531 [INSPIRE].
G. Papathanasiou, Hexagon Wilson Loop OPE and Harmonic Polylogarithms, JHEP 11 (2013) 150 [arXiv:1310.5735] [INSPIRE].
G. Papathanasiou, Evaluating the six-point remainder function near the collinear limit, Int. J. Mod. Phys. A 29 (2014) 1450154 [arXiv:1406.1123] [INSPIRE].
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [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].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov and A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) [DOI].
J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic Amplitudes and Cluster Coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].
J. Drummond, J. Foster and O. Gürdoğan, Cluster adjacency beyond MHV, JHEP 03 (2019) 086 [arXiv:1810.08149] [INSPIRE].
S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multiscale multiloop integrals, J. Math. Phys. 43 (2002) 3363 [hep-ph/0110083] [INSPIRE].
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].
C. Schneider, Symbolic summation assists combinatorics, Sem. Lothar. Combin. 56 (2007) 1.
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].
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].
L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].
J.M. Drummond, G. Papathanasiou and M. Spradlin, A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].
L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].
S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].
L.J. Dixon, J. Drummond, T. Harrington, A.J. McLeod, G. Papathanasiou and M. Spradlin, Heptagons from the Steinmann Cluster Bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].
J. Drummond, J. Foster, O. Gürdoğan and G. Papathanasiou, Cluster adjacency and the four-loop NMHV heptagon, JHEP 03 (2019) 087 [arXiv:1812.04640] [INSPIRE].
S. Caron-Huot, L.J. Dixon, F. Dulat, M. von Hippel, A.J. McLeod and G. Papathanasiou, Six-Gluon amplitudes in planar \( \mathcal{N} \) = 4 super-Yang-Mills theory at six and seven loops, JHEP 08 (2019) 016 [arXiv:1903.10890] [INSPIRE].
S. Caron-Huot, L.J. Dixon, F. Dulat, M. Von Hippel, A.J. McLeod and G. Papathanasiou, The Cosmic Galois Group and Extended Steinmann Relations for Planar \( \mathcal{N} \) = 4 SYM Amplitudes, JHEP 09 (2019) 061 [arXiv:1906.07116] [INSPIRE].
L.J. Dixon and Y.-T. Liu, Lifting Heptagon Symbols to Functions, JHEP 10 (2020) 031 [arXiv:2007.12966] [INSPIRE].
S. Caron-Huot et al., The Steinmann Cluster Bootstrap for N = 4 Super Yang-Mills Amplitudes, PoS CORFU2019 (2020) 003 [arXiv:2005.06735] [INSPIRE].
O. Steinmann, Über den Zusammenhang zwischen den Wightmanfunktionen und der retardierten Kommutatoren, Helv. Phys. Acta 33 (1960) 257.
O. Steinmann, Wightman-Funktionen und retardierten Kommutatoren. II, Helv. Phys. Acta 33 (1960) 347.
K.E. Cahill and H.P. Stapp, Optical theorems and steinmann relations, Annals Phys. 90 (1975) 438 [INSPIRE].
J. Drummond, J. Foster and O. Gürdoğan, Cluster Adjacency Properties of Scattering Amplitudes in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 120 (2018) 161601 [arXiv:1710.10953] [INSPIRE].
C. Duhr, Mathematical aspects of scattering amplitudes, in Theoretical Advanced Study Institute in Elementary Particle Physics: Journeys Through the Precision Frontier: Amplitudes for Colliders, (2014), DOI [arXiv:1411.7538] [INSPIRE].
B. Basso, L.J. Dixon and G. Papathanasiou, Origin of the Six-Gluon Amplitude in Planar N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 124 (2020) 161603 [arXiv:2001.05460] [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].
A.J. McLeod, H. Munch, G. Papathanasiou and M. von Hippel, A Novel Algorithm for Nested Summation and Hypergeometric Expansions, JHEP 11 (2020) 122 [arXiv:2005.05612] [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mil ls theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Generalized unitarity for N = 4 super-amplitudes, Nucl. Phys. B 869 (2013) 452 [arXiv:0808.0491] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov and A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, Cambridge University Press (2012) [DOI].
F. Brown and C. Duhr, A double integral of dlog forms which is not polylogarithmic, (2020) [arXiv:2006.09413] [INSPIRE].
K.-T. Chen, Algebras of Iterated Path Integrals and Fundamental Groups, Trans. Am. Math. Soc. 156 (1971) 359.
K.-T. Chen, Iterated path integrals, Bul l. Am. Math. Soc. 83 (1977) 831 [INSPIRE].
A. Goncharov, Geometry of Configurations, Polylogarithms, and Motivic Cohomology, Adv. Math. 114 (1995) 197,.
A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].
A.B. 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].
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].
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].
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].
SpaSM: a Sparse direct Solver Modulo p, The SpaSM group, v1.2 (2017) [http://github.com/cbouilla/spasm].
F.C.S. Brown, Multiple zeta values and periods of moduli spaces M 0 ,n ( R ), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. thesis, Humboldt U., 2015. arXiv:1506.07243. 10.18452/17157 [INSPIRE].
J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The Multiple Zeta Value Data Mine, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
L. Dixon and F. Dulat, to appear.
L. Dixon, private communication.
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Chestnov, V., Papathanasiou, G. Hexagon bootstrap in the double scaling limit. J. High Energ. Phys. 2021, 7 (2021). https://doi.org/10.1007/JHEP09(2021)007
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DOI: https://doi.org/10.1007/JHEP09(2021)007