Abstract
Seven-point amplitudes in planar \( \mathcal{N} \) = 4 super-Yang-Mills theory have previously been constructed through four loops using the Steinmann cluster bootstrap, but only at the level of the symbol. We promote these symbols to actual functions, by specifying their first derivatives and boundary conditions on a particular two-dimensional surface. To do this, we impose branch-cut conditions and construct the entire heptagon function space through weight six. We plot the amplitudes on a few lines in the bulk Euclidean region, and explore the properties of the heptagon function space under the coaction associated with multiple polylogarithms.
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R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The Analytic S-Matrix, Cambridge University Press (1966).
M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap II: two dimensional amplitudes, JHEP 11 (2017) 143 [arXiv:1607.06110] [INSPIRE].
J. Elias Miró, A.L. Guerrieri, A. Hebbar, J. Penedones and P. Vieira, Flux Tube S-matrix Bootstrap, Phys. Rev. Lett. 123 (2019) 221602 [arXiv:1906.08098] [INSPIRE].
L. Córdova, Y. He, M. Kruczenski and P. Vieira, The O(N ) S-matrix Monolith, JHEP 04 (2020) 142 [arXiv:1909.06495] [INSPIRE].
C. Bercini, M. Fabri, A. Homrich and P. Vieira, S-matrix bootstrap: Supersymmetry, Z2 , and Z4 symmetry, Phys. Rev. D 101 (2020) 045022 [arXiv:1909.06453] [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].
J.M. Drummond, J. Henn, V.A. 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.M. Carrasco, H. Johansson and D.A. 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].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].
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] [INSPIRE].
J.M. Drummond, J. Henn, G.P. 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].
S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, JHEP 12 (2011) 066 [arXiv:1105.5606] [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.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] [INSPIRE].
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] [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.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] [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, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [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].
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].
D.A. 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].
L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in \( \mathcal{N} \) = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].
L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [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].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016), [DOI] [arXiv:1212.5605] [INSPIRE].
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].
F. Brown and C. Duhr, A double integral of dlog forms which is not polylogarithmic, 6, 2020 [arXiv:2006.09413] [INSPIRE].
F.C.S. Brown, Multiple zeta values and periods of moduli spaces \( {\overline{\mathfrak{M}}}_{0,n}\left(\mathrm{\mathbb{R}}\right) \), Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [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].
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].
S. Fomin and A. Zelevinsky, Cluster algebras I: Foundations, math/0104151.
S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, math/0208229.
D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP 12 (2011) 011 [arXiv:1102.0062] [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.
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].
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].
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].
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].
J. Drummond, J. Foster and O. Gürdoğan, Cluster adjacency beyond MHV, JHEP 03 (2019) 086 [arXiv:1810.08149] [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].
J. Golden, A.J. McLeod, M. Spradlin and A. Volovich, The Sklyanin Bracket and Cluster Adjacency at All Multiplicity, JHEP 03 (2019) 195 [arXiv:1902.11286] [INSPIRE].
O. Gürdoğan and M. Parisi, Cluster patterns in Landau and Leading Singularities via the Amplituhedron, arXiv:2005.07154 [INSPIRE].
J. Mago, A. Schreiber, M. Spradlin and A. Volovich, A Note on One-loop Cluster Adjacency in N = 4 SYM, arXiv:2005.07177 [INSPIRE].
J. Drummond, J. Foster, O. Gürdogan and C. Kalousios, Tropical Grassmannians, cluster algebras and scattering amplitudes, JHEP 04 (2020) 146 [arXiv:1907.01053] [INSPIRE].
J. Drummond, J. Foster, O. Gürdogan and C. Kalousios, Algebraic singularities of scattering amplitudes from tropical geometry, arXiv:1912.08217 [INSPIRE].
N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar \( \mathcal{N} \) = 4 SYM amplitudes, arXiv:1912.08222 [INSPIRE].
N. Henke and G. Papathanasiou, How tropical are seven- and eight-particle amplitudes?, JHEP 08 (2020) 005 [arXiv:1912.08254] [INSPIRE].
J. Drummond, J. Foster, O. Gürdoğan and C. Kalousios, Tropical fans, scattering equations and amplitudes, arXiv:2002.04624 [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].
L. Dixon and F. Dulat, The Seven-Loop Six-Gluon NMHV Amplitude in Planar \( \mathcal{N} \) = 4 Super-Yang-Mills Theory, to appear.
S. Caron-Huot and S. He, Jumpstarting the All-Loop S-matrix of Planar \( \mathcal{N} \) = 4 Super Yang-Mills, JHEP 07 (2012) 174 [arXiv:1112.1060] [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. Golden and M. Spradlin, An analytic result for the two-loop seven-point MHV amplitude in \( \mathcal{N} \) = 4 SYM, JHEP 08 (2014) 154 [arXiv:1406.2055] [INSPIRE].
J. Golden and A.J. Mcleod, Cluster Algebras and the Subalgebra Constructibility of the Seven-Particle Remainder Function, JHEP 01 (2019) 017 [arXiv:1810.12181] [INSPIRE].
J.L. Bourjaily, M. Volk and M. Von Hippel, Conformally Regulated Direct Integration of the Two-Loop Heptagon Remainder, JHEP 02 (2020) 095 [arXiv:1912.05690] [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 \( \mathcal{N} \) = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
B. Basso, S. Caron-Huot and A. Sever, Adjoint BFKL at finite coupling: a short-cut from the collinear limit, JHEP 01 (2015) 027 [arXiv:1407.3766] [INSPIRE].
V. Del Duca et al., All-order amplitudes at any multiplicity in the multi-Regge limit, Phys. Rev. Lett. 124 (2020) 161602 [arXiv:1912.00188] [INSPIRE].
J. Bartels, Analytic structure of the 8-point scattering amplitude in multi-Regge kinematics in N = 4 SYM: conformal Regge pole and Regge cut contributions, arXiv:2005.08818 [INSPIRE].
B. Basso, L.J. Dixon and G. Papathanasiou, Origin of the Six-Gluon Amplitude in Planar \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 124 (2020) 161603 [arXiv:2001.05460] [INSPIRE].
N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].
E. Panzer and O. Schnetz, The Galois coaction on 𝜙4 periods, Commun. Num. Theor. Phys. 11 (2017) 657 [arXiv:1603.04289] [INSPIRE].
F. Brown, Feynman amplitudes, coaction principle, and cosmic Galois group, Commun. Num. Theor. Phys. 11 (2017) 453 [arXiv:1512.06409] [INSPIRE].
O. Schnetz, The Galois coaction on the electron anomalous magnetic moment, Commun. Num. Theor. Phys. 12 (2018) 335 [arXiv:1711.05118] [INSPIRE].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
M. Bullimore and D. Skinner, Descent Equations for Superamplitudes, arXiv:1112.1056 [INSPIRE].
B. Basso, L. Dixon, Y.-T. Liu and G. Papathanasiou, to appear.
L.J. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [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].
V. Del Duca, C. Duhr and V.A. Smirnov, The One-Loop One-Mass Hexagon Integral in D = 6 Dimensions, JHEP 07 (2011) 064 [arXiv:1105.1333] [INSPIRE].
B. Basso, private communication.
L.J. Dixon and I. Esterlis, All orders results for self-crossing Wilson loops mimicking double parton scattering, JHEP 07 (2016) 116 [Erratum ibid. 08 (2016) 131] [arXiv:1602.02107] [INSPIRE].
F. Brown, The Massless higher-loop two-point function, Commun. Math. Phys. 287 (2009) 925 [arXiv:0804.1660] [INSPIRE].
E. Panzer, Feynman integrals and hyperlogarithms, Ph.D. Thesis, Humboldt U. (2015) [DOI] [arXiv:1506.07243] [INSPIRE].
C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses, JHEP 08 (2019) 135 [arXiv:1904.07279] [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 III. The two-particle contributions, JHEP 08 (2014) 085 [arXiv:1402.3307] [INSPIRE].
V. Del Duca et al., The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy, JHEP 06 (2018) 116 [arXiv:1801.10605] [INSPIRE].
J. Bartels, A. Kormilitzin, L.N. 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].
J. Broedel, M. Sprenger and A. Torres Orjuela, Towards single-valued polylogarithms in two variables for the seven-point remainder function in multi-Regge-kinematics, Nucl. Phys. B 915 (2017) 394 [arXiv:1606.08411] [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].
V. Del Duca et al., Multi-Regge kinematics and the moduli space of Riemann spheres with marked points, JHEP 08 (2016) 152 [arXiv:1606.08807] [INSPIRE].
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Dixon, L.J., Liu, YT. Lifting heptagon symbols to functions. J. High Energ. Phys. 2020, 31 (2020). https://doi.org/10.1007/JHEP10(2020)031
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DOI: https://doi.org/10.1007/JHEP10(2020)031