Abstract
In recent years, it has been understood that color-ordered scattering amplitudes can be encoded as logarithmic differential forms on positive geometries. In particular, amplitudes in maximally supersymmetric Yang-Mills theory in spinor helicity space are governed by the momentum amplituhedron. Due to the group-theoretic structure underlying color decompositions, color-ordered amplitudes enjoy various identities which relate different orderings. In this paper, we show how the Kleiss-Kuijf relations arise from the geometry of the momentum amplituhedron. We also show how similar relations can be realised for the kinematic associahedron, which is the positive geometry of bi-adjoint scalar cubic theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L. J. Dixon, Scattering amplitudes: the most perfect microscopic structures in the universe, J. Phys. A 44 (2011) 454001 [arXiv:1105.0771] [INSPIRE].
R. Kleiss and H. Kuijf, Multi - Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
V. Del Duca, L. J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
N. Arkani-Hamed, Y. Bai and T. Lam, Positive Geometries and Canonical Forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, The Momentum Amplituhedron, JHEP 08 (2019) 042 [arXiv:1905.04216] [INSPIRE].
L. Ferro, T. Łukowski and R. Moerman, From momentum amplituhedron boundaries toamplitude singularities and back, JHEP 07 (2020) 201 [arXiv:2003.13704] [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, J. L. Bourjaily, F. Cachazo, A. Postnikov and J. Trnka, On-Shell Structures of MHV Amplitudes Beyond the Planar Limit, JHEP 06 (2015) 179 [arXiv:1412.8475] [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].
S. He and C. Zhang, Notes on Scattering Amplitudes as Differential Forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].
L. Ferro and T. Lukowski, Amplituhedra, and beyond, J. Phys. A 54 (2021) 033001 [arXiv:2007.04342] [INSPIRE].
T. Lukowski, M. Parisi and L. K. Williams, The positive tropical Grassmannian, the hypersimplex, and the m = 2 amplituhedron, arXiv:2002.06164 [INSPIRE].
T. Łukowski and R. Moerman, Boundaries of the amplituhedron with amplituhedronBoundaries, Comput. Phys. Commun. 259 (2021) 107653, [arXiv:2002.07146].
J. L. Bourjaily, Positroids, Plabic Graphs, and Scattering Amplitudes in Mathematica, arXiv:1212.6974 [INSPIRE].
F. Cachazo, S. He and E. Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
T. Łukowski, M. Parisi, M. Spradlin and A. Volovich, Cluster Adjacency for m = 2 Yangian Invariants, JHEP 10 (2019) 158 [arXiv:1908.07618] [INSPIRE].
D. Damgaard, L. Ferro, T. Lukowski and R. Moerman, Momentum amplituhedron meets kinematic associahedron, JHEP 02 (2021) 041 [arXiv:2010.15858] [INSPIRE].
Z. Bern, J. J. M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
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: 2103.13908
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
Damgaard, D., Ferro, L., Łukowski, T. et al. Kleiss-Kuijf relations from momentum amplituhedron geometry. J. High Energ. Phys. 2021, 111 (2021). https://doi.org/10.1007/JHEP07(2021)111
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2021)111