Causal diamonds, cluster polytopes and scattering amplitudes
Abstract
The “amplituhedron” for tree-level scattering amplitudes in the bi-adjoint φ3 theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1 + 1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic “spacetime” with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain “walk”, associated with any acyclic quiver, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The An–3, Bn–1/Cn–1 and Dn polytopes are the amplituhedra for n-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope D¯n, which chops the Dn polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.
- Authors:
-
- Inst. for Advanced Study, Princeton, NJ (United States); Harvard Univ., Cambridge, MA (United States)
- Chinese Academy of Sciences (CAS), Beijing (China); International Centre for Theoretical Physics Asia-Pacific, Beijing/Hangzhou (China); Peng Huanwu Center for Fundamental Theory, Anhui (China)
- Brown Univ., Providence, RI (United States)
- Univ. du Québec à Montréal, QC (Canada)
- Publication Date:
- Research Org.:
- Institute for Advanced Study, Princeton, NJ (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1898392
- Grant/Contract Number:
- SC0009988
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of High Energy Physics (Online)
- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2022; Journal Issue: 11; Journal ID: ISSN 1029-8479
- Publisher:
- Springer Nature
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Differential and Algebraic Geometry; Scattering Amplitudes
Citation Formats
Arkani-Hamed, N., He, S., Salvatori, G., and Thomas, H. Causal diamonds, cluster polytopes and scattering amplitudes. United States: N. p., 2022.
Web. doi:10.1007/jhep11(2022)049.
Arkani-Hamed, N., He, S., Salvatori, G., & Thomas, H. Causal diamonds, cluster polytopes and scattering amplitudes. United States. https://doi.org/10.1007/jhep11(2022)049
Arkani-Hamed, N., He, S., Salvatori, G., and Thomas, H. Thu .
"Causal diamonds, cluster polytopes and scattering amplitudes". United States. https://doi.org/10.1007/jhep11(2022)049. https://www.osti.gov/servlets/purl/1898392.
@article{osti_1898392,
title = {Causal diamonds, cluster polytopes and scattering amplitudes},
author = {Arkani-Hamed, N. and He, S. and Salvatori, G. and Thomas, H.},
abstractNote = {The “amplituhedron” for tree-level scattering amplitudes in the bi-adjoint φ3 theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1 + 1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic “spacetime” with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain “walk”, associated with any acyclic quiver, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The An–3, Bn–1/Cn–1 and Dn polytopes are the amplituhedra for n-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope D¯n, which chops the Dn polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.},
doi = {10.1007/jhep11(2022)049},
journal = {Journal of High Energy Physics (Online)},
number = 11,
volume = 2022,
place = {United States},
year = {Thu Nov 10 00:00:00 EST 2022},
month = {Thu Nov 10 00:00:00 EST 2022}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Y -systems and generalized associahedra
journal, January 2003
- Fomin, Sergey; Zelevinsky, Andre
- Annals of Mathematics, Vol. 158, Issue 3
1-loop amplitudes from the Halohedron
journal, December 2019
- Salvatori, Giulio
- Journal of High Energy Physics, Vol. 2019, Issue 12
Dual superconformal symmetry of scattering amplitudes in super-Yang–Mills theory
journal, March 2010
- Drummond, J. M.; Henn, J.; Korchemsky, G. P.
- Nuclear Physics B, Vol. 828, Issue 1-2
Magic identities for conformal four-point integrals
journal, January 2007
- Drummond, James M.; Henn, Johannes; Smirnov, Vladimir A.
- Journal of High Energy Physics, Vol. 2007, Issue 01
An etude on recursion relations and triangulations
journal, May 2019
- He, Song; Yang, Qinglin
- Journal of High Energy Physics, Vol. 2019, Issue 5
A Hopf Operad of Forests of Binary Trees and Related Finite-Dimensional Algebras
journal, November 2004
- Chapoton, Frédéric
- Journal of Algebraic Combinatorics, Vol. 20, Issue 3
Powers of Coxeter elements in infinite groups are reduced
journal, October 2008
- Speyer, David
- Proceedings of the American Mathematical Society, Vol. 137, Issue 4
Unwinding the amplituhedron in binary
journal, January 2018
- Arkani-Hamed, Nima; Thomas, Hugh; Trnka, Jaroslav
- Journal of High Energy Physics, Vol. 2018, Issue 1
Polytopal Realizations of Generalized Associahedra
journal, December 2002
- Chapoton, Frédéric; Fomin, Sergey; Zelevinsky, Andrei
- Canadian Mathematical Bulletin, Vol. 45, Issue 4
Scattering forms and the positive geometry of kinematics, color and the worldsheet
journal, May 2018
- Arkani-Hamed, Nima; Bai, Yuntao; He, Song
- Journal of High Energy Physics, Vol. 2018, Issue 5
The Amplituhedron
journal, October 2014
- Arkani-Hamed, Nima; Trnka, Jaroslav
- Journal of High Energy Physics, Vol. 2014, Issue 10
Scattering amplitudes and simple canonical forms for simple polytopes
journal, March 2021
- Salvatori, Giulio; Stanojevic, Stefan
- Journal of High Energy Physics, Vol. 2021, Issue 3
Positive geometries and canonical forms
journal, November 2017
- Arkani-Hamed, Nima; Bai, Yuntao; Lam, Thomas
- Journal of High Energy Physics, Vol. 2017, Issue 11
Sortable elements in infinite Coxeter groups
journal, February 2011
- Reading, Nathan; Speyer, David E.
- Transactions of the American Mathematical Society, Vol. 363, Issue 02
Homotopy Associativity of H-Spaces. II
journal, August 1963
- Stasheff, James Dillon
- Transactions of the American Mathematical Society, Vol. 108, Issue 2
Eliminating spurious poles from gauge-theoretic amplitudes
journal, May 2013
- Hodges, Andrew
- Journal of High Energy Physics, Vol. 2013, Issue 5
One-loop scattering equations and amplitudes from forward limit
journal, November 2015
- He, Song; Yuan, Ellis Ye
- Physical Review D, Vol. 92, Issue 10
Moduli space of paired punctures, cyclohedra and particle pairs on a circle
journal, May 2019
- Li, Zhenjie; Zhang, Chi
- Journal of High Energy Physics, Vol. 2019, Issue 5
Grassmannian Geometry of Scattering Amplitudes
book, May 2016
- Arkani-Hamed, Nima; Bourjaily, Jacob; Cachazo, Freddy
Stringy canonical forms
journal, February 2021
- Arkani-Hamed, Nima; He, Song; Lam, Thomas
- Journal of High Energy Physics, Vol. 2021, Issue 2