Abstract
The amplituhedron provides a beautiful description of perturbative superamplitude integrands in \( \mathcal{N}=4 \) SYM in terms of purely geometric objects, generalisations of polytopes. On the other hand the Wilson loop in supertwistor space also gives an explicit description of these superamplitudes as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that these geometric images of the Feynman diagrams give a tessellation of the amplituhedron. This turns out to be the case for NMHV amplitudes. We argue however that beyond NMHV this is not true. Specifically, each Feynman diagram leads to an image with a physical boundary and spurious boundaries. The spurious ones should be “internal”, matching with neighbouring diagrams. We however show that there is no choice of geometric image of the Wilson loop Feynman diagrams which yields a geometric object without leaving unmatched spurious boundaries.
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References
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. Hodges and J. Trnka, A Note on Polytopes for Scattering Amplitudes, JHEP 04 (2012) 081 [arXiv:1012.6030] [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].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].
Y. Bai and S. He, The Amplituhedron from Momentum Twistor Diagrams, JHEP 02 (2015) 065 [arXiv:1408.2459] [INSPIRE].
S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the Amplituhedron, JHEP 03 (2015) 128 [arXiv:1408.3410] [INSPIRE].
T. Lam, Amplituhedron cells and Stanley symmetric functions, Commun. Math. Phys. 343 (2016) 1025 [arXiv:1408.5531] [INSPIRE].
N. Arkani-Hamed, A. Hodges and J. Trnka, Positive Amplitudes In The Amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].
S. Agarwala and E. Marin-Amat, Wilson Loop diagrams and Positroids, Commun. Math. Phys. 350 (2017) 569 [arXiv:1509.06150] [INSPIRE].
Y. Bai, S. He and T. Lam, The Amplituhedron and the One-loop Grassmannian Measure, JHEP 01 (2016) 112 [arXiv:1510.03553] [INSPIRE].
L. Ferro, T. Lukowski, A. Orta and M. Parisi, Towards the Amplituhedron Volume, JHEP 03 (2016) 014 [arXiv:1512.04954] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a Nonplanar Amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].
D. Galloni, Positivity Sectors and the Amplituhedron, arXiv:1601.02639 [INSPIRE].
S.N. Karp and L.K. Williams, The m=1 amplituhedron and cyclic hyperplane arrangements, arXiv:1608.08288 [INSPIRE].
T. Dennen, I. Prlina, M. Spradlin, S. Stanojevic and A. Volovich, Landau Singularities from the Amplituhedron, JHEP 06 (2017) 152 [arXiv:1612.02708] [INSPIRE].
L. Ferro, T. Lukowski, A. Orta and M. Parisi, Yangian symmetry for the tree amplituhedron, J. Phys. A 50 (2017) 294005 [arXiv:1612.04378] [INSPIRE].
L. Ferro, T. Lukowski, A. Orta and M. Parisi, Tree-level scattering amplitudes from the amplituhedron, J. Phys. Conf. Ser. 841 (2017) 012037 [arXiv:1612.06276] [INSPIRE].
B. Eden, P. Heslop and L. Mason, The Correlahedron, JHEP 09 (2017) 156 [arXiv:1701.00453] [INSPIRE].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
S.N. Karp, L.K. Williams and Y.X. Zhang, Decompositions of amplituhedra, arXiv:1708.09525 [INSPIRE].
J. Rao, 4-particle Amplituhedron at 3-loop and its Mondrian Diagrammatic Implication, JHEP 06 (2018) 038 [arXiv:1712.09990] [INSPIRE].
Y. An, Y. Li, Z. Li and J. Rao, All-loop Mondrian Diagrammatics and 4-particle Amplituhedron, JHEP 06 (2018) 023 [arXiv:1712.09994] [INSPIRE].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological Polytopes and the Wavefunction of the Universe, arXiv:1709.02813 [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].
P. Galashin and T. Lam, Parity duality for the amplituhedron, arXiv:1805.00600 [INSPIRE].
S. Agarwala and S. Fryer, A study in \( {\mathbb{G}}_{\mathbb{R},\ge 0} \) : from the geometric case book of Wilson loop diagrams and SYM N = 4, arXiv:1803.00958 [INSPIRE].
L. Ferro, T. Lukowski and M. Parisi, Amplituhedron meets Jeffrey-Kirwan Residue, arXiv:1805.01301 [INSPIRE].
J. Rao, 4-particle Amplituhedronics for 3-5 loops, arXiv:1806.01765 [INSPIRE].
J. Bourjaily and H. Thomas, What is the Amplituhedron?, Not. Amer. Math. Soc. 65 (2018) 167 [INSPIRE].
N. Arkani-Hamed, Y. Bai and T. Lam, Positive Geometries and Canonical Forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].
S. Agarwala and C. Marcott, Wilson loops in SYM N = 4 do not parametrize an orientable space, arXiv:1807.05397 [INSPIRE].
L.J. Mason and D. Skinner, The Complete Planar S-matrix of N = 4 SYM as a Wilson Loop in Twistor Space, JHEP 12 (2010) 018 [arXiv:1009.2225] [INSPIRE].
D. Chicherin and E. Sokatchev, \( \mathcal{N}=4 \) super-Yang-Mills in LHC superspace part I: classical and quantum theory, JHEP 02 (2017) 062 [arXiv:1601.06803] [INSPIRE].
D. Chicherin, P. Heslop, G.P. Korchemsky and E. Sokatchev, Wilson Loop Form Factors: A New Duality, JHEP 04 (2018) 029 [arXiv:1612.05197] [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].
D. Chicherin et al., Correlation functions of the chiral stress-tensor multiplet in \( \mathcal{N}=4 \) SYM, JHEP 06 (2015) 198 [arXiv:1412.8718] [INSPIRE].
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Heslop, P., Stewart, A. The twistor Wilson loop and the amplituhedron. J. High Energ. Phys. 2018, 142 (2018). https://doi.org/10.1007/JHEP10(2018)142
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DOI: https://doi.org/10.1007/JHEP10(2018)142