Abstract
Recently, the existence of an Amplituhedron for tree level amplitudes in the bi-adjoint scalar field theory has been proved by Arkani-Hamed et al. We argue that hyperbolic geometry constitutes a natural framework to address the study of positive geometries in moduli spaces of Riemann surfaces, and thus to try to extend this achievement beyond tree level. In this paper we begin an exploration of these ideas starting from the simplest example of hyperbolic geometry, the hyperbolic plane. The hyperboloid model naturally guides us to re-discover the moduli space Associahedron, and a new version of its kinematical avatar. As a by-product we obtain a solution to the scattering equations which can be interpreted as a special case of the two well known solutions in terms of spinor-helicity formalism. The construction is done in 1 + 2 dimensions and this makes harder to understand how to extract the amplitude from the dlog of the space time Associahedron. Nevertheless, we continue the investigation accommodating a loop momentum in the picture. By doing this we are led to another polytope called Halohedron, which was already known to mathematicians. We argue that the Halohedron fulfils many criteria that make it plausible to be understood as a 1-loop Amplituhedron for the cubic theory. Furthermore, the hyperboloid model again allows to understand that a kinematical version of the Halohedron exists and is related to the one living in moduli space by a simple generalisation of the tree level map.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Arkani-Hamed et al., Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge U.K. (2016).
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, 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].
S. Mizera, Combinatorics and topology of Kawai-Lewellen-Tye relations, JHEP 08 (2017) 097 [arXiv:1706.08527] [INSPIRE].
S. Mizera, Scattering amplitudes from intersection theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [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].
F. Cachazo, S. Mizera and G. Zhang, Scattering equations: real solutions and particles on a line, JHEP 03 (2017) 151 [arXiv:1609.00008] [INSPIRE].
S. He and E.Y. Yuan, One-loop scattering equations and amplitudes from forward limit, Phys. Rev. D 92 (2015) 105004 [arXiv:1508.06027] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, One-loop amplitudes on the Riemann sphere, JHEP 03 (2016) 114 [arXiv:1511.06315] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Loop integrands for scattering amplitudes from the Riemann sphere, Phys. Rev. Lett. 115 (2015) 121603 [arXiv:1507.00321] [INSPIRE].
Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, Two-loop scattering amplitudes from the Riemann sphere, Phys. Rev. D 94 (2016) 125029 [arXiv:1607.08887] [INSPIRE].
C. Cardona, B. Feng, H. Gomez and R. Huang, Cross-ratio identities and higher-order poles of CHY-integrand, JHEP 09 (2016) 133 [arXiv:1606.00670] [INSPIRE].
H. Gomez, S. Mizera and G. Zhang, CHY loop integrands from holomorphic forms, JHEP 03 (2017) 092 [arXiv:1612.06854] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, Properties of scattering forms and their relation to associahedra, JHEP 03 (2018) 064 [arXiv:1711.07942] [INSPIRE].
S. L Devadoss, T. Heath and C. Vipismakul, Deformations of bordered Riemann surfaces and associahedral polytopes, Not. Amer. Math. Soc. 58 (2011) 530 [arXiv:1002.1676].
W. Abikoff, The real analytic theory of Teichmüller space, Lecture note in mathematics, Springer, Germany (1980).
J.H. Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller Theory, Matrix Editions Ithaca, U.S.A. (2006).
D.B. Fairlie and D.E. Roberts, Dual models without tachyons — A new approach, unpublished Durham preprint PRINT-72-2440 (1972).
G. Salvatori, 1-loop amplitudes from the halohedron, arXiv:1806.01842 [INSPIRE].
S. Fomin, M. Shapiro and D. Thurston, Cluster algebras and triangulated surfaces. Part I: Cluster complexes, Acta Math. 201 (2008) 83.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1803.05809
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Salvatori, G., Cacciatori, S.L. Hyperbolic geometry and amplituhedra in 1+2 dimensions. J. High Energ. Phys. 2018, 167 (2018). https://doi.org/10.1007/JHEP08(2018)167
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)167