Abstract
Multi-loop scattering amplitudes/null polygonal Wilson loops in \( \mathcal{N} \) = 4 super-Yang-Mills are known to simplify significantly in reduced kinematics, where external legs/edges lie in an 1 + 1 dimensional subspace of Minkowski spacetime (or boundary of the AdS3 subspace). Since the edges of a 2n-gon with even and odd labels go along two different null directions, the kinematics is reduced to two copies of G(2, n)/T ∼ An−3. In the simplest octagon case, we conjecture that all loop amplitudes and Feynman integrals are given in terms of two overlapping A2 functions (a special case of two-dimensional harmonic polylogarithms): in addition to the letters v, 1 + v, w, 1 + w of A1 × A1, there are two letters v − w, 1 − vw mixing the two sectors but they never appear together in the same term; these are the reduced version of four-mass-box algebraic letters. Evidence supporting our conjecture includes all known octagon amplitudes as well as new computations of multi-loop integrals in reduced kinematics. By leveraging this alphabet and conditions on first and last entries, we initiate a bootstrap program in reduced kinematics: within the remarkably simple space of overlapping A2 functions, we easily obtain octagon amplitudes up to two-loop NMHV and three-loop MHV. We also briefly comment on the generalization to 2n-gons in terms of A2 functions and beyond.
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15 June 2022
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP06(2022)079
References
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].
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].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [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, C. Duhr, M. von Hippel and J. Pennington, Bootstrapping six-gluon scattering in planar N = 4 super-Yang-Mills theory, PoS LL2014 (2014) 077 [arXiv:1407.4724] [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].
D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [arXiv:1712.09610] [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. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic Amplitudes and Cluster Coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [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].
C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].
S. Fomin and A. Zelevinsky, Cluster algebras I: foundations, J. Am. Math. Soc. 15 (2002) 497.
S. Fomin and A. Zelevinsky, Cluster algebras II: Finite type classification, Invent. Math. 154 (2003) 63.
D. Speyer and L. Williams, The tropical totally positive grassmannian, J. Algeb. Comb. 22 (2005) 189.
Z. Li and C. Zhang, The Three-loop MHV Octagon from \( \overline{Q} \) equations, arXiv:2110.00350 [INSPIRE].
S. He, Z. Li and C. Zhang, Two-loop octagons, algebraic letters and \( \overline{Q} \) equations, Phys. Rev. D 101 (2020) 061701 [arXiv:1911.01290] [INSPIRE].
S. He, Z. Li and C. Zhang, The symbol and alphabet of two-loop NMHV amplitudes from \( \overline{Q} \) equations, JHEP 03 (2021) 278 [arXiv:2009.11471] [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].
J. Drummond, J. Foster, O. Gürdogan and C. Kalousios, Algebraic singularities of scattering amplitudes from tropical geometry, JHEP 04 (2021) 002 [arXiv:1912.08217] [INSPIRE].
N. Henke and G. Papathanasiou, How tropical are seven- and eight-particle amplitudes?, JHEP 08 (2020) 005 [arXiv:1912.08254] [INSPIRE].
N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar \( \mathcal{N} \) = 4 SYM amplitudes, JHEP 03 (2021) 065 [arXiv:1912.08222] [INSPIRE].
A. Herderschee, Algebraic branch points at all loop orders from positive kinematics and wall crossing, JHEP 07 (2021) 049 [arXiv:2102.03611] [INSPIRE].
J. Mago, A. Schreiber, M. Spradlin and A. Volovich, Symbol alphabets from plabic graphs, JHEP 10 (2020) 128 [arXiv:2007.00646] [INSPIRE].
S. He and Z. Li, A note on letters of Yangian invariants, JHEP 02 (2021) 155 [arXiv:2007.01574] [INSPIRE].
J. Mago, A. Schreiber, M. Spradlin, A.Y. Srikant and A. Volovich, Symbol alphabets from plabic graphs II: rational letters, JHEP 04 (2021) 056 [arXiv:2012.15812] [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. He, Z. Li and Q. Yang, Notes on cluster algebras and some all-loop Feynman integrals, JHEP 06 (2021) 119 [arXiv:2103.02796] [INSPIRE].
D. Chicherin, J.M. Henn and G. Papathanasiou, Cluster algebras for Feynman integrals, Phys. Rev. Lett. 126 (2021) 091603 [arXiv:2012.12285] [INSPIRE].
S. He, Z. Li, Q. Yang and C. Zhang, Feynman Integrals and Scattering Amplitudes from Wilson Loops, Phys. Rev. Lett. 126 (2021) 231601 [arXiv:2012.15042] [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].
L.F. Alday and J. Maldacena, Minimal surfaces in AdS and the eight-gluon scattering amplitude at strong coupling, arXiv:0903.4707 [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].
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].
P. Heslop and V.V. Khoze, Analytic Results for MHV Wilson Loops, JHEP 11 (2010) 035 [arXiv:1007.1805] [INSPIRE].
S. Caron-Huot and S. He, Three-loop octagons and n-gons in maximally supersymmetric Yang-Mills theory, JHEP 08 (2013) 101 [arXiv:1305.2781] [INSPIRE].
T. Goddard, P. Heslop and V.V. Khoze, Uplifting Amplitudes in Special Kinematics, JHEP 10 (2012) 041 [arXiv:1205.3448] [INSPIRE].
T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [INSPIRE].
M.A.C. Torres, Cluster algebras in scattering amplitudes with special 2D kinematics, Eur. Phys. J. C 74 (2014) 2757 [arXiv:1310.6906] [INSPIRE].
S. He, Z. Li, Y. Tang and Q. Yang, The Wilson-loop d log representation for Feynman integrals, JHEP 05 (2021) 052 [arXiv:2012.13094] [INSPIRE].
L. Ferro, Differential equations for multi-loop integrals and two-dimensional kinematics, JHEP 04 (2013) 160 [arXiv:1204.1031] [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].
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].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
F.C.S. Brown, Multiple zeta values and periods of moduli spaces \( \mathfrak{M} \)0,n, Annales Sci. Ecole Norm. Sup. 42 (2009) 371 [math/0606419] [INSPIRE].
N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet, JHEP 05 (2018) 096 [arXiv:1711.09102] [INSPIRE].
N. Arkani-Hamed, S. He, T. Lam and H. Thomas, Binary Geometries, Generalized Particles and Strings, and Cluster Algebras, arXiv:1912.11764 [INSPIRE].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
S. He and C. Zhang, Notes on Scattering Amplitudes as Differential Forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
J.L. Bourjaily, A.J. McLeod, M. von Hippel and M. Wilhelm, Rationalizing Loop Integration, JHEP 08 (2018) 184 [arXiv:1805.10281] [INSPIRE].
F. Brown, Iterated integrals in quantum field theory, in 6th Summer School on Geometric and Topological Methods for Quantum Field Theory, pp. 188–240 (2013) [DOI] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
J.L. Bourjaily, A.J. McLeod, M. Spradlin, M. von Hippel and M. Wilhelm, Elliptic Double-Box Integrals: Massless Scattering Amplitudes beyond Polylogarithms, Phys. Rev. Lett. 120 (2018) 121603 [arXiv:1712.02785] [INSPIRE].
J.L. Bourjaily, Y.-H. He, A.J. Mcleod, M. Von Hippel and M. Wilhelm, Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms, Phys. Rev. Lett. 121 (2018) 071603 [arXiv:1805.09326] [INSPIRE].
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He, S., Li, Z., Tang, Y. et al. Bootstrapping octagons in reduced kinematics from A2 cluster algebras. J. High Energ. Phys. 2021, 84 (2021). https://doi.org/10.1007/JHEP10(2021)084
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DOI: https://doi.org/10.1007/JHEP10(2021)084