Abstract
Inspired by the idea of viewing amplitudes in \( \mathcal{N}=4 \) SYM as differential forms on momentum twistor space, we introduce differential forms on the space of spinor variables, which combine helicity amplitudes in any four-dimensional gauge theory as a single object. In this note we focus on such differential forms in \( \mathcal{N}=4 \) SYM, which can also be thought of as “bosonizing” superamplitudes in non-chiral superspace. Remarkably all tree-level amplitudes in \( \mathcal{N}=4 \) SYM combine to a d log form in spinor variables, which is given by pushforward of canonical forms of Grassmannian cells. The tree forms can also be obtained using BCFW or inverse-soft construction, and we present all-multiplicity expression for MHV and NMHV forms to illustrate their simplicity. Similarly all-loop planar integrands can be naturally written as d log forms in the Grassmannian/on-shell-diagram picture, and we expect the same to hold beyond the planar limit. Just as the form in momentum twistor space reveals underlying positive geometry of the amplituhedron, the form in terms of spinor variables strongly suggests an “amplituhedron in momentum space”. We initiate the study of its geometry by connecting it to the moduli space of Witten’s twistor-string theory, which provides a pushforward formula for tree forms in \( \mathcal{N}=4 \) SYM.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the amplituhedron in binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [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, 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, P. Benincasa and A. Postnikov, Cosmological polytopes and the wavefunction of the universe, arXiv:1709.02813 [INSPIRE].
X. Gao, S. He and Y. Zhang, Labelled tree graphs, Feynman diagrams and disk integrals, JHEP 11 (2017) 144 [arXiv:1708.08701] [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].
G. Salvatori, 1-loop amplitudes from the halohedron, arXiv:1806.01842 [INSPIRE].
S.J. Parke and T.R. Taylor, An amplitude for n gluon scattering, Phys. Rev. Lett. 56 (1986) 2459 [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the simplest quantum field theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
Y.-T. Huang, Non-chiral S-matrix of N = 4 super Yang-Mills, arXiv:1104.2021 [INSPIRE].
J. Plefka, T. Schuster and V. Verschinin, From six to four and more: massless and massive maximal super Yang-Mills amplitudes in 6d and 4d and their hidden symmetries, JHEP 01 (2015) 098 [arXiv:1405.7248] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The all-loop integrand for scattering amplitudes in planar N = 4 SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [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. 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].
H. Elvang, Y.-T. Huang and C. Peng, On-shell superamplitudes in N < 4 SYM, JHEP 09 (2011) 031 [arXiv:1102.4843] [INSPIRE].
N. Arkani-Hamed et al., Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, U.K., (2016) [arXiv:1212.5605] [INSPIRE].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of residues and Grassmannian dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].
J.L. Bourjaily, Efficient tree-amplitudes in N = 4: automatic BCFW recursion in Mathematica, arXiv:1011.2447 [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
H. Elvang and Y.-T. Huang, Scattering amplitudes in gauge theory and gravity, Cambridge University Press, Cambridge, U.K., (2015) [INSPIRE].
F. Cachazo, Resultants and gravity amplitudes, arXiv:1301.3970 [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering in three dimensions from rational maps, JHEP 10 (2013) 141 [arXiv:1306.2962] [INSPIRE].
S. He, Z. Liu and J.-B. Wu, Scattering equations, twistor-string formulas and double-soft limits in four dimensions, JHEP 07 (2016) 060 [arXiv:1604.02834] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A. Hodges and J. Trnka, A note on polytopes for scattering amplitudes, JHEP 04 (2012) 081 [arXiv:1012.6030] [INSPIRE].
Y. Bai and S. He, The amplituhedron from momentum twistor diagrams, JHEP 02 (2015) 065 [arXiv:1408.2459] [INSPIRE].
S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the amplituhedron, JHEP 03 (2015) 128 [arXiv:1408.3410] [INSPIRE].
Y. Bai, S. He and T. Lam, The amplituhedron and the one-loop Grassmannian measure, JHEP 01 (2016) 112 [arXiv:1510.03553] [INSPIRE].
L. Ferro, T. Lukowski, A. Orta and M. Parisi, Towards the amplituhedron volume, JHEP 03 (2016) 014 [arXiv:1512.04954] [INSPIRE].
S.N. Karp and L.K. Williams, The m = 1 amplituhedron and cyclic hyperplane arrangements, arXiv:1608.08288 [INSPIRE].
S.N. Karp, L.K. Williams and Y.X. Zhang, Decompositions of amplituhedra, arXiv:1708.09525 [INSPIRE].
P. Galashin and T. Lam, Parity duality for the amplituhedron, arXiv:1805.00600 [INSPIRE].
L. Ferro, T. Lukowski and M. Parisi, Amplituhedron meets Jeffrey-Kirwan residue, arXiv:1805.01301 [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Singularity structure of maximally supersymmetric scattering amplitudes, Phys. Rev. Lett. 113 (2014) 261603 [arXiv:1410.0354] [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].
Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a nonplanar amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].
B. Feng, J. Wang, Y. Wang and Z. Zhang, BCFW recursion relation with nonzero boundary contribution, JHEP 01 (2010) 019 [arXiv:0911.0301] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, Scattering amplitudes for all masses and spins, arXiv:1709.04891 [INSPIRE].
Y.-T. Huang and C. Wen, ABJM amplitudes and the positive orthogonal Grassmannian, JHEP 02 (2014) 104 [arXiv:1309.3252] [INSPIRE].
Y.-T. Huang, C. Wen and D. Xie, The positive orthogonal Grassmannian and loop amplitudes of ABJM, J. Phys. A 47 (2014) 474008 [arXiv:1402.1479] [INSPIRE].
F. Cachazo, A. Guevara, M. Heydeman, S. Mizera, J.H. Schwarz and C. Wen, The S matrix of 6D super Yang-Mills and maximal supergravity from rational maps, JHEP 09 (2018) 125 [arXiv:1805.11111] [INSPIRE].
J.L. Bourjaily, Positroids, plabic graphs and scattering amplitudes in Mathematica, arXiv:1212.6974 [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: 1807.11051
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
He, S., Zhang, C. Notes on scattering amplitudes as differential forms. J. High Energ. Phys. 2018, 54 (2018). https://doi.org/10.1007/JHEP10(2018)054
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2018)054