Abstract
We continue the study of open associahedra associated with bi-color scattering amplitudes initiated in ref. [1]. We focus on the facet geometries of the open associahedra, uncovering many new phenomena such as fiber-product geometries. We then provide novel recursion procedures for calculating the canonical form of open associahedra, generalizing recursion relations for bounded polytopes to unbounded polytopes.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Herderschee, S. He, F. Teng and Y. Zhang, On positive geometry and scattering forms for matter particles, JHEP 06 (2020) 030 [arXiv:1912.08307] [INSPIRE].
N. Arkani-Hamed, Y. Bai and T. Lam, Positive geometries and canonical forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [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, 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. Banerjee, A. Laddha and P. Raman, Stokes polytopes: the positive geometry for ϕ4 interactions, JHEP 08 (2019) 067 [arXiv:1811.05904] [INSPIRE].
P.B. Aneesh, M. Jagadale and N. Kalyanapuram, Accordiohedra as positive geometries for generic scalar field theories, Phys. Rev. D 100 (2019) 106013 [arXiv:1906.12148] [INSPIRE].
P. Raman, The positive geometry for ϕp interactions, JHEP 10 (2019) 271 [arXiv:1906.02985] [INSPIRE].
N. Kalyanapuram and R.G. Jha, Positive geometries for all scalar theories from twisted intersection theory, Phys. Rev. Res. 2 (2020) 033119 [arXiv:2006.15359] [INSPIRE].
S. He and Q. Yang, An etude on recursion relations and triangulations, JHEP 05 (2019) 040 [arXiv:1810.08508] [INSPIRE].
G. Salvatori and S. Stanojevic, Scattering amplitudes and simple canonical forms for simple polytopes, arXiv:1912.06125 [INSPIRE].
Q. Yang, Triangulations for ABHY polytopes and recursions for tree and loop amplitudes, arXiv:1912.09163 [INSPIRE].
R. Kojima, Weights and recursion relations for ϕp tree amplitudes from the positive geometry, JHEP 08 (2020) 054 [arXiv:2005.11006] [INSPIRE].
R.R. John, R. Kojima and S. Mahato, Weights, recursion relations and projective triangulations for positive geometry of scalar theories, JHEP 10 (2020) 037 [arXiv:2007.10974] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
B. Eden, P. Heslop and L. Mason, The correlahedron, JHEP 09 (2017) 156 [arXiv:1701.00453] [INSPIRE].
G. Salvatori, 1-loop amplitudes from the halohedron, JHEP 12 (2019) 074 [arXiv:1806.01842] [INSPIRE].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological polytopes and the wavefunction of the universe, arXiv:1709.02813 [INSPIRE].
N. Arkani-Hamed, S. He, G. Salvatori and H. Thomas, Causal diamonds, cluster polytopes and scattering amplitudes, arXiv:1912.12948 [INSPIRE].
S.G. Naculich, Scattering equations and BCJ relations for gauge and gravitational amplitudes with massive scalar particles, JHEP 09 (2014) 029 [arXiv:1407.7836] [INSPIRE].
R.W. Brown and S.G. Naculich, Color-factor symmetry and BCJ relations for QCD amplitudes, JHEP 11 (2016) 060 [arXiv:1608.05291] [INSPIRE].
R.W. Brown and S.G. Naculich, KLT-type relations for QCD and bicolor amplitudes from color-factor symmetry, JHEP 03 (2018) 057 [arXiv:1802.01620] [INSPIRE].
H. Johansson and A. Ochirov, Double copy for massive quantum particles with spin, JHEP 09 (2019) 040 [arXiv:1906.12292] [INSPIRE].
S. He, G. Yan, C. Zhang and Y. Zhang, Scattering forms, worldsheet forms and amplitudes from subspaces, JHEP 08 (2018) 040 [arXiv:1803.11302] [INSPIRE].
S. Mizera, Combinatorics and topology of Kawai-Lewellen-Tye relations, JHEP 08 (2017) 097 [arXiv:1706.08527] [INSPIRE].
H. Frost and L. Mason, Lie polynomials and a twistorial correspondence for amplitudes, arXiv:1912.04198 [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative quantum gravity as a double copy of gauge theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
T. Melia, Dyck words and multiquark primitive amplitudes, Phys. Rev. D 88 (2013) 014020 [arXiv:1304.7809] [INSPIRE].
T. Melia, Getting more flavor out of one-flavor QCD, Phys. Rev. D 89 (2014) 074012 [arXiv:1312.0599] [INSPIRE].
H. Johansson and A. Ochirov, Color-kinematics duality for QCD amplitudes, JHEP 01 (2016) 170 [arXiv:1507.00332] [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].
A. Ochirov and B. Page, Multi-quark colour decompositions from unitarity, JHEP 10 (2019) 058 [arXiv:1908.02695] [INSPIRE].
N. Arkani-Hamed, S. He and T. Lam, Stringy canonical forms, arXiv:1912.08707 [INSPIRE].
N. Arkani-Hamed, S. He and T. Lam, Cluster configuration spaces of finite type, arXiv:2005.11419 [INSPIRE].
N. Arkani-Hamed, S. He, T. Lam and H. Thomas, Binary geometries, generalized particles and strings, and cluster algebras, arXiv:1912.11764 [INSPIRE].
V. Bazier-Matte, G. Douville, K. Mousavand, H. Thomas and E. Yıldırım, ABHY associahedra and Newton polytopes of F -polynomials for finite type cluster algebras, arXiv:1808.09986 [INSPIRE].
A. Padrol, Y. Palu, V. Pilaud and P.-G. Plamondon, Associahedra for finite type cluster algebras and minimal relations between g-vectors, arXiv:1906.06861 [INSPIRE].
D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, The momentum amplituhedron, JHEP 08 (2019) 042 [arXiv:1905.04216] [INSPIRE].
L. Ferro, T. Lukowski and R. Moerman, From momentum amplituhedron boundaries toamplitude singularities and back, JHEP 07 (2020) 201 [arXiv:2003.13704] [INSPIRE].
S. He and C. Zhang, Notes on scattering amplitudes as differential forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].
N. Craig, H. Elvang, M. Kiermaier and T. Slatyer, Massive amplitudes on the Coulomb branch of N = 4 SYM, JHEP 12 (2011) 097 [arXiv:1104.2050] [INSPIRE].
A. Herderschee, S. Koren and T. Trott, Constructing N = 4 Coulomb branch superamplitudes, JHEP 08 (2019) 107 [arXiv:1902.07205] [INSPIRE].
K. Kohn and K. Ranestad, Projective geometry of Wachspress coordinates, Found. Comput. Math. 20 (2019) 1135 [arXiv:1904.02123].
N. Arkani-Hamed, A. Hodges and J. Trnka, Positive amplitudes in the amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].
J. Warren, Barycentric coordinates for convex polytopes, Adv. Comput. Math. 6 (1996) 97.
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: 2008.06418
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
Herderschee, A., Teng, F. Open associahedra and scattering forms. J. High Energ. Phys. 2020, 134 (2020). https://doi.org/10.1007/JHEP12(2020)134
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)134