Abstract
In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of G(2, n) as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the Δ-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the Δ-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to G(k, n).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press, Cambridge U.K. (2016) [arXiv:1212.5605] [INSPIRE].
A. Postnikov, Total positivity, Grassmannians, and networks, math/0609764.
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].
S. He and C. Zhang, Notes on Scattering Amplitudes as Differential Forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
E.I. Buchbinder and F. Cachazo, Two-loop amplitudes of gluons and octa-cuts in N = 4 super Yang-Mills, JHEP 11 (2005) 036 [hep-th/0506126] [INSPIRE].
F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [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].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
L. Ferro, T. Lukowski and M. Parisi, Amplituhedron meets Jeffrey-Kirwan Residue, J. Phys. A 52 (2019) 045201 [arXiv:1805.01301] [INSPIRE].
N. Early and V. Reiner, On configuration spaces and Whitehouse’s lifts of the Eulerian representations, arXiv:1808.04007.
D. Moseley, N. Proudfoot and B. Young, The Orlik-Terao algebra and the cohomology of configuration space, Exp. Math. 26 (2017) 373 [arXiv:1603.01189].
B. Knudsen, Configuration spaces in algebraic topology, arXiv:1803.11165.
V.I. Arnol’d, The cohomology ring of the colored braid group, Math. Notes Acad. Sci. USSR 5 (1969) 138.
W. Fulton and R. MacPherson, A compactification of configuration spaces, Annals Math. 139 (1994) 183 [INSPIRE].
B. Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996) 1057.
I. Kriz, On the rational homotopy type of configuration spaces, Ann. Math. 139 (1994) 227.
A. Ocneanu, Higher Representation Theory in Math and Physics, Harvard University course PHYSICS 267 (2017), https://youtu.be/9gHzFLfPFFU?t=380.
N. Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [INSPIRE].
N. Early, Generalized Permutohedra, Scattering Amplitudes and a Cubic Three-Fold, arXiv:1709.03686 [INSPIRE].
M. Enciso, Volumes of Polytopes Without Triangulations, JHEP 10 (2017) 071 [arXiv:1408.0932] [INSPIRE].
M. Enciso, Logarithms and Volumes of Polytopes, JHEP 04 (2018) 016 [arXiv:1612.07370] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
B. Feng and M. Luo, An Introduction to On-shell Recursion Relations, Front. Phys. (Beijing) 7 (2012) 533 [arXiv:1111.5759] [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [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].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes, JHEP 06 (2010) 003 [arXiv:1003.2403] [INSPIRE].
S. Franco, D. Galloni, B. Penante and C. Wen, Non-Planar On-Shell Diagrams, JHEP 06 (2015) 199 [arXiv:1502.02034] [INSPIRE].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a Nonplanar Amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].
R. Frassek and D. Meidinger, Yangian-type symmetries of non-planar leading singularities, JHEP 05 (2016) 110 [arXiv:1603.00088] [INSPIRE].
J.L. Bourjaily, S. Franco, D. Galloni and C. Wen, Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams, JHEP 10 (2016) 003 [arXiv:1607.01781] [INSPIRE].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
P. Deligne and J.W. Morgan, Notes on Supersymmetry (following Joseph Bernstein), in Quantum Fields and Strings: A Course for Mathematicians, AMS Press, New York U.S.A. (1999), pg. 41.
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].
E. Witten, Perturbative gauge theory as a string theory in twistor space, Commun. Math. Phys. 252 (2004) 189 [hep-th/0312171] [INSPIRE].
R. Roiban, M. Spradlin and A. Volovich, On the tree level S matrix of Yang-Mills theory, Phys. Rev. D 70 (2004) 026009 [hep-th/0403190] [INSPIRE].
S.J. Parke and T.R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
R. Kleiss and H. Kuijf, Multi-Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
L.J. Dixon, Calculating scattering amplitudes efficiently, in QCD and beyond. Proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics, TASI-95, Boulder U.S.A. (1995), pg. 539 [hep-ph/9601359] [INSPIRE].
F. Cachazo, Fundamental BCJ Relation in N = 4 SYM From The Connected Formulation, arXiv:1206.5970 [INSPIRE].
P. Orlik and H. Terao, Arrangements of Hyperplanes, Grundlehren der mathematischen Wissenschaften, Springer, Heidelberg Germany (2013).
H. Esnault, V. Schechtman and E. Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math. 109 (1992) 557.
S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].
S. He and Y. Zhang, New Formulas for Amplitudes from Higher-Dimensional Operators, JHEP 02 (2017) 019 [arXiv:1608.08448] [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].
U. Pachner, P.l. homeomorphic manifolds are equivalent by elementary shellings, Eur. J. Combin. 12 (1991) 129.
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].
S. He and Q. Yang, An Etude on Recursion Relations and Triangulations, arXiv:1810.08508 [INSPIRE].
P. Benincasa, On-shell diagrammatics and the perturbative structure of planar gauge theories, arXiv:1510.03642 [INSPIRE].
P. Heslop and A.E. Lipstein, On-shell diagrams for \( \mathcal{N}=8 \) supergravity amplitudes, JHEP 06 (2016) 069 [arXiv:1604.03046] [INSPIRE].
E. Herrmann and J. Trnka, Gravity On-shell Diagrams, JHEP 11 (2016) 136 [arXiv:1604.03479] [INSPIRE].
N. Arkani-Hamed, A. Hodges and J. Trnka, Positive Amplitudes In The Amplituhedron, JHEP 08 (2015) 030 [arXiv:1412.8478] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [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].
V. Reiner, Lectures on matroids and oriented matroids, http://www-users.math.umn.edu/~reiner/Talks/Vienna05/Lectures.pdf.
F. Cachazo and Y. Geyer, A ’Twistor String’ Inspired Formula For Tree-Level Scattering Amplitudes in N = 8 SUGRA, arXiv:1206.6511 [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].
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: 1812.01168
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
Cachazo, F., Early, N., Guevara, A. et al. Δ-algebra and scattering amplitudes. J. High Energ. Phys. 2019, 5 (2019). https://doi.org/10.1007/JHEP02(2019)005
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2019)005