Abstract
I present a conjecture that all two-loop MHV amplitudes in planar \( \mathcal{N} \) = 4 super-Yang-Mills theory possess an antipodal symmetry when evaluated on parity-even kinematics. The symmetry acts as a change of basis on the symbol letters, followed by the antipode operation associated with the Hopf algebra structure of multiple polylogarithms. At the symbol level, I provide the symmetry map explicitly for amplitudes with up to eight external particles, and also provide evidence at all multiplicities. Intriguingly, the map acts as an isomorphism on the normal fans of the Newton polytopes of the symbol letters. The conjectured symmetry is one of the rare known cases where the antipode map shows up in physically important examples.
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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].
L.F. Alday and R. Roiban, Scattering amplitudes, Wilson loops and the string/gauge theory correspondence, Phys. Rept. 468 (2008) 153 [arXiv:0807.1889] [INSPIRE].
L.J. Dixon, Ö. Gürdoğan, A.J. McLeod and M. Wilhelm, Folding amplitudes into form factors: an antipodal duality, Phys. Rev. Lett. 128 (2022) 111602 [arXiv:2112.06243] [INSPIRE].
S. Caron-Huot, L.J. Dixon, F. Dulat, M. von Hippel, A.J. McLeod and G. Papathanasiou, Six-gluon amplitudes in planar N = 4 super-Yang-Mills theory at six and seven loops, JHEP 08 (2019) 016 [arXiv:1903.10890] [INSPIRE].
L.J. Dixon, Ö. Gürdoğan, A.J. McLeod and M. Wilhelm, Bootstrapping a stress-tensor form factor through eight loops, JHEP 07 (2022) 153 [arXiv:2204.11901] [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].
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math.AG/0103059 [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) [arXiv:1212.5605] [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].
E. Gawrilow and M. Joswig, polymake: a framework for analyzing convex polytopes, in Polytopes — combinatorics and computation (Oberwolfach, 1997), DMV Sem. 29, Birkhäuser, Basel, Switzerland (2000), p. 43.
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].
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].
N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar N = 4 SYM amplitudes, JHEP 03 (2021) 065 [arXiv:1912.08222] [INSPIRE].
J. Drummond, J. Foster, Ö. Gürdoğan 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].
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 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. Golden and A.J. McLeod, The two-loop remainder function for eight and nine particles, JHEP 06 (2021) 142 [arXiv:2104.14194] [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic bubble ansatz, JHEP 09 (2011) 032 [arXiv:0911.4708] [INSPIRE].
G. Yang, A simple collinear limit of scattering amplitudes at strong coupling, JHEP 03 (2011) 087 [arXiv:1006.3306] [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].
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].
L.J. Dixon and Y.-T. Liu, Lifting heptagon symbols to functions, JHEP 10 (2020) 031 [arXiv:2007.12966] [INSPIRE].
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Liu, YT. Antipodal symmetry of two-loop MHV amplitudes. J. High Energ. Phys. 2022, 131 (2022). https://doi.org/10.1007/JHEP09(2022)131
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DOI: https://doi.org/10.1007/JHEP09(2022)131