Abstract
In this paper, we take a major step towards the construction and applications of an all-loop, all-multiplicity amplituhedron for three-dimensional planar \( \mathcal{N} \) = 6 Chern-Simons matter theory, or the ABJM amplituhedron. We show that by simply changing the overall sign of the positive region of the original amplituhedron for four-dimensional planar \( \mathcal{N} \) = 4 super-Yang-Mills (sYM) and performing a symplectic reduction, only three-dimensional kinematics in the middle sector of even-multiplicity survive. The resulting form of the geometry, combined with its parity images, gives the full loop integrand. This simple modification geometrically enforces the vanishing of odd-multiplicity cuts, and manifests the correct soft cuts as well as two-particle unitarity cuts. Furthermore, the so-called “bipartite structures” of four-point all-loop negative geometries also directly generalize to all multiplicities. We introduce a novel approach for triangulating loop amplituhedra based on the kinematics of the tree region, resulting in local integrands tailored to “prescriptive unitarity”. This construction sheds fascinating new light on the interplay between loop and tree amplituhedra for both ABJM and \( \mathcal{N} \) = 4 sYM: the loop geometry demands that the tree region must be dissected into chambers, defined by the simultaneous positivity of maximal cuts. The loop geometry is then the “fibration” of the tree region. Using the new construction, we give explicit results of one-loop integrands up to ten points and two-loop integrands up to eight points by computing the canonical form of ABJM loop amplituhedron.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Change history
12 April 2024
An Erratum to this paper has been published: https://doi.org/10.1007/JHEP04(2024)064
References
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].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
N. Arkani-Hamed, Y. Bai and T. Lam, Positive Geometries and Canonical Forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].
S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the Amplituhedron, JHEP 03 (2015) 128 [arXiv:1408.3410] [INSPIRE].
L. Ferro, T. Łukowski, A. Orta and M. Parisi, Towards the Amplituhedron Volume, JHEP 03 (2016) 014 [arXiv:1512.04954] [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.J. Dixon, M. von Hippel, A.J. McLeod and J. Trnka, Multi-loop positivity of the planar \( \mathcal{N} \) = 4 SYM six-point amplitude, JHEP 02 (2017) 112 [arXiv:1611.08325] [INSPIRE].
S.N. Karp and L.K. Williams, The m = 1 amplituhedron and cyclic hyperplane arrangements, Int. Math. Res. Not. 5 (2019) 1401 [arXiv:1608.08288] [INSPIRE].
S.N. Karp, L.K. Williams and Y.X. Zhang, Decompositions of amplituhedra, Ann. Inst. H. Poincare D Comb. Phys. Interact. 7 (2020) 303 [arXiv:1708.09525] [INSPIRE].
L. Ferro, T. Łukowski and M. Parisi, Amplituhedron meets Jeffrey-Kirwan residue, J. Phys. A 52 (2019) 045201 [arXiv:1805.01301] [INSPIRE].
P. Galashin and T. Lam, Parity duality for the amplituhedron, Compos. Math. 156 (2020) 2207 [arXiv:1805.00600] [INSPIRE].
N. Arkani-Hamed, C. Langer, A. Yelleshpur Srikant and J. Trnka, Deep Into the Amplituhedron: Amplitude Singularities at All Loops and Legs, Phys. Rev. Lett. 122 (2019) 051601 [arXiv:1810.08208] [INSPIRE].
G. Salvatori and S.L. Cacciatori, Hyperbolic Geometry and Amplituhedra in 1 + 2 dimensions, JHEP 08 (2018) 167 [arXiv:1803.05809] [INSPIRE].
R. Kojima, Triangulation of 2-loop MHV Amplituhedron from Sign Flips, JHEP 04 (2019) 085 [arXiv:1812.01822] [INSPIRE].
J. Rao, 4-particle amplituhedronics for 3-5 loops, Nucl. Phys. B 943 (2019) 114625 [arXiv:1806.01765] [INSPIRE].
A. Yelleshpur Srikant, Emergent unitarity from the amplituhedron, JHEP 01 (2020) 069 [arXiv:1906.10700] [INSPIRE].
C. Langer and A. Yelleshpur Srikant, All-loop cuts from the Amplituhedron, JHEP 04 (2019) 105 [arXiv:1902.05951] [INSPIRE].
T. Łukowski, On the Boundaries of the m = 2 Amplituhedron, arXiv:1908.00386 [INSPIRE].
E. Herrmann, C. Langer, J. Trnka and M. Zheng, Positive geometry, local triangulations, and the dual of the Amplituhedron, JHEP 01 (2021) 035 [arXiv:2009.05607] [INSPIRE].
R. Kojima and J. Rao, Triangulation-free Trivialization of 2-loop MHV Amplituhedron, JHEP 10 (2020) 140 [arXiv:2007.15650] [INSPIRE].
T. Łukowski, M. Parisi and L.K. Williams, The positive tropical Grassmannian, the hypersimplex, and the m = 2 amplituhedron, arXiv:2002.06164 [INSPIRE].
M. Parisi, M. Sherman-Bennett and L. Williams, The m = 2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers, arXiv:2104.08254 [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].
B. Eden, P. Heslop and L. Mason, The Correlahedron, JHEP 09 (2017) 156 [arXiv:1701.00453] [INSPIRE].
N. Arkani-Hamed, S. He and T. Lam, Stringy canonical forms, JHEP 02 (2021) 069 [arXiv:1912.08707] [INSPIRE].
N. Arkani-Hamed, S. He, G. Salvatori and H. Thomas, Causal diamonds, cluster polytopes and scattering amplitudes, JHEP 11 (2022) 049 [arXiv:1912.12948] [INSPIRE].
N. Arkani-Hamed, S. He, T. Lam and H. Thomas, Binary geometries, generalized particles and strings, and cluster algebras, Phys. Rev. D 107 (2023) 066015 [arXiv:1912.11764] [INSPIRE].
S. He and C. Zhang, Notes on Scattering Amplitudes as Differential Forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].
D. Damgaard, L. Ferro, T. Łukowski and M. Parisi, The Momentum Amplituhedron, JHEP 08 (2019) 042 [arXiv:1905.04216] [INSPIRE].
L. Ferro and T. Łukowski, The Loop Momentum Amplituhedron, JHEP 05 (2023) 183 [arXiv:2210.01127] [INSPIRE].
Y.-T. Huang, R. Kojima, C. Wen and S.-Q. Zhang, The orthogonal momentum amplituhedron and ABJM amplitudes, JHEP 01 (2022) 141 [arXiv:2111.03037] [INSPIRE].
S. He, C.-K. Kuo and Y.-Q. Zhang, The momentum amplituhedron of SYM and ABJM from twistor-string maps, JHEP 02 (2022) 148 [arXiv:2111.02576] [INSPIRE].
Y.-T. Huang, C.-K. Kuo and C. Wen, Dualities for Ising networks, Phys. Rev. Lett. 121 (2018) 251604 [arXiv:1809.01231] [INSPIRE].
N. Arkani-Hamed, Y.-T. Huang and S.-H. Shao, On the Positive Geometry of Conformal Field Theory, JHEP 06 (2019) 124 [arXiv:1812.07739] [INSPIRE].
K. Hosomichi et al., N=5,6 Superconformal Chern-Simons Theories and M2-branes on Orbifolds, JHEP 09 (2008) 002 [arXiv:0806.4977] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
T. Bargheer, F. Loebbert and C. Meneghelli, Symmetries of Tree-level Scattering Amplitudes in N = 6 Superconformal Chern-Simons Theory, Phys. Rev. D 82 (2010) 045016 [arXiv:1003.6120] [INSPIRE].
Y.-T. Huang and A.E. Lipstein, Dual Superconformal Symmetry of N = 6 Chern-Simons Theory, JHEP 11 (2010) 076 [arXiv:1008.0041] [INSPIRE].
S. Lee, Yangian Invariant Scattering Amplitudes in Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 105 (2010) 151603 [arXiv:1007.4772] [INSPIRE].
Y.-T. Huang and C.K. Wen, ABJM amplitudes and the positive orthogonal grassmannian, JHEP 02 (2014) 104 [arXiv:1309.3252] [INSPIRE].
Y.-T. Huang, C. Wen and D. Xie, The Positive orthogonal Grassmannian and loop amplitudes of ABJM, J. Phys. A 47 (2014) 474008 [arXiv:1402.1479] [INSPIRE].
S. He, C.-K. Kuo, Z. Li and Y.-Q. Zhang, All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron, Phys. Rev. Lett. 129 (2022) 221604 [arXiv:2204.08297] [INSPIRE].
S. He, C.-K. Kuo, Z. Li and Y.-Q. Zhang, Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron, JHEP 07 (2023) 212 [arXiv:2303.03035] [INSPIRE].
J.M. Henn, M. Lagares and S.-Q. Zhang, Integrated negative geometries in ABJM, JHEP 05 (2023) 112 [arXiv:2303.02996] [INSPIRE].
N. Arkani-Hamed, J. Henn and J. Trnka, Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron, JHEP 03 (2022) 108 [arXiv:2112.06956] [INSPIRE].
D. Chicherin and J.M. Henn, Symmetry properties of Wilson loops with a Lagrangian insertion, JHEP 07 (2022) 057 [arXiv:2202.05596] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
H. Elvang et al., Grassmannians for scattering amplitudes in 4d \( \mathcal{N} \) = 4 SYM and 3d ABJM, JHEP 12 (2014) 181 [arXiv:1410.0621] [INSPIRE].
S. Caron-Huot and Y.-T. Huang, The two-loop six-point amplitude in ABJM theory, JHEP 03 (2013) 075 [arXiv:1210.4226] [INSPIRE].
S. He, Y.-T. Huang, C.-K. Kuo and Z. Li, The two-loop eight-point amplitude in ABJM theory, JHEP 02 (2023) 065 [arXiv:2211.01792] [INSPIRE].
J.L. Bourjaily, E. Herrmann and J. Trnka, Prescriptive Unitarity, JHEP 06 (2017) 059 [arXiv:1704.05460] [INSPIRE].
S. He, Y.-T. Huang, C.-K. Kuo and M. Parisi, Chambers Unlocked: From Elliptic Leading Singularities to Local Triangulations, in progress.
T. Łukowski, R. Moerman and J. Stalknecht, On the geometry of the orthogonal momentum amplituhedron, JHEP 12 (2022) 006 [arXiv:2112.03294] [INSPIRE].
L. Ferro, T. Łukowski and R. Moerman, From momentum amplituhedron boundaries toamplitude singularities and back, JHEP 07 (2020) 201 [arXiv:2003.13704] [INSPIRE].
A. Brandhuber, G. Travaglini and C. Wen, A note on amplitudes in N = 6 superconformal Chern-Simons theory, JHEP 07 (2012) 160 [arXiv:1205.6705] [INSPIRE].
N. Arkani-Hamed et al., Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) [https://doi.org/10.1017/CBO9781316091548] [INSPIRE].
W.-M. Chen and Y.-T. Huang, Dualities for Loop Amplitudes of N = 6 Chern-Simons Matter Theory, JHEP 11 (2011) 057 [arXiv:1107.2710] [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].
Acknowledgments
It is our great pleasure to thank Nima Arkani-Hamed, Jacob Bourjaily, Tomasz Lukowski, Matteo Parisi, Justinas Rumbutis, Jonah Stalknecht, Jaroslav Trnka, Congkao Wen, Akshay Yelleshpur, Shun-Qing Zhang, and Yaoqi Zhang for stimulating discussions. The authors would also like to thank DIAS for hosting the conference “Amplituhedron at 10” where the results of this paper were discussed. SH thanks IAS (Princeton) for hospitality during the completion of the work; his research is supported in part by National Natural Science Foundation of China under Grant No. 11935013, 12047502, 12047503, 12247103, 12225510. Y-t H and C-k Kuo are funded by the National Science and Technology Council of Taiwan, Grant No. 111-2811-M-002 -125.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2306.00951
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
He, S., Huang, Yt. & Kuo, CK. The ABJM Amplituhedron. J. High Energ. Phys. 2023, 165 (2023). https://doi.org/10.1007/JHEP09(2023)165
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2023)165