Abstract
In the Grassmannian formulation of the S-matrix for planar \( \mathcal{N} \) = 4 Super Yang-Mills, Nk−2MHV scattering amplitudes for k negative and n − k positive helicity gluons can be expressed, by an application of the global residue theorem, as a signed sum over a collection of (k − 2)(n − k − 2)-dimensional residues. These residues are supported on certain positroid subvarieties of the Grassmannian G(k, n). In this paper, we replace the Grassmannian G(3, n) with its torus quotient, the moduli space of n points in the projective plane in general position, and planar \( \mathcal{N} \) = 4 SYM with generalized biadjoint scalar amplitudes \( {m}_n^{(3)} \) as introduced by Cachazo-Early-Guevara-Mizera (CEGM) [1]. Whereas in the Grassmannian formulation residues of the Parke-Taylor form correspond to individual BCFW, or on-shell diagrams, we show that each such (n − 5)-dimensional residue of \( {m}_n^{(3)} \) an entire biadjoint scalar partial amplitude \( {m}_n^{(2)} \), that is a sum over all tree-level Feynman diagrams for a fixed planar order. We make a proposal which would do the same for k ≥ 4.
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F. Cachazo, N. Early, A. Guevara and S. Mizera, Scattering equations: from projective spaces to tropical Grassmannians, JHEP 06 (2019) 039 [arXiv:1903.08904] [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].
L. Dolan and P. Goddard, Proof of the formula of Cachazo, He and Yuan for Yang-Mills tree amplitudes in arbitrary dimension, JHEP 05 (2014) 010 [arXiv:1311.5200] [INSPIRE].
D. García Sepúlveda and A. Guevara, A soft theorem for the tropical Grassmannian, arXiv:1909.05291 [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].
N. Arkani-Hamed, S. He and T. Lam, Stringy canonical forms, JHEP 02 (2021) 069 [arXiv:1912.08707] [INSPIRE].
N. Arkani-Hamed, S. He, T. Lam and H. Thomas, Binary geometries, generalized particles and strings, and cluster algebras, Phys. Rev. D 107 (2023) 066015 [arXiv:1912.11764] [INSPIRE].
M. Abhishek, S. Hegde and A.P. Saha, One-loop integrand from generalised scattering equations, JHEP 05 (2021) 012 [arXiv:2012.10916] [INSPIRE].
J.-L. Loday, Realization of the Stasheff polytope, math/0212126 [https://doi.org/10.48550/arXiv.math/0212126].
N. Arkani-Hamed et al., Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, U.K. (2016) [https://doi.org/10.1017/CBO9781316091548] [INSPIRE].
OEIS Foundation Inc., The On-line Encyclopedia of Integer Sequences — sequence A001263, https://oeis.org/A001263 (2019).
N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Unification of residues and Grassmannian dualities, JHEP 01 (2011) 049 [arXiv:0912.4912] [INSPIRE].
F. Cachazo, P. Svrcek and E. Witten, MHV vertices and tree amplitudes in gauge theory, JHEP 09 (2004) 006 [hep-th/0403047] [INSPIRE].
N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, Local spacetime physics from the Grassmannian, JHEP 01 (2011) 108 [arXiv:0912.3249] [INSPIRE].
M. Abhishek, S. Hegde, D.P. Jatkar and A.P. Saha, Double soft theorem for generalised biadjoint scalar amplitudes, SciPost Phys. 10 (2021) 036 [arXiv:2008.07271] [INSPIRE].
N. Early, Planar kinematic invariants, matroid subdivisions and generalized Feynman diagrams, arXiv:1912.13513 [INSPIRE].
N. Early, Introduction to planar bases for generalized biadjoint scalar amplitudes, in preparation.
F. Cachazo and N. Early, Residual embeddings of CEGM amplitudes: all (k, n), in preparation.
F. Santos, C. Stump and V. Welker, Noncrossing sets and a Grassmann associahedron, Forum Math. Sigma 5 (2017) e5.
N. Early, Planarity in generalized scattering amplitudes: PK polytope, generalized root systems and worldsheet associahedra, arXiv:2106.07142 [INSPIRE].
OEIS Foundation Inc., The On-line Encyclopedia of Integer Sequences — sequence A060854, https://oeis.org/A060854 (2019).
A. Postnikov, Permutohedra, associahedra, and beyond, math/0507163 [https://doi.org/10.48550/arXiv.math/0507163].
C. Ceballos, F. Santos and G.M. Ziegler, Many non-equivalent realizations of the associahedron, arXiv:1109.5544 [https://doi.org/10.48550/arXiv.1109.5544].
F. Cachazo and N. Early, Planar kinematics: cyclic fixed points, mirror superpotential, k-dimensional Catalan numbers, and root polytopes, arXiv:2010.09708 [INSPIRE].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological polytopes and the wavefunction of the universe, arXiv:1709.02813 [INSPIRE].
D. Agostini et al., Likelihood degenerations, Adv. Math. 414 (2023) 108863 [arXiv:2107.10518] [INSPIRE].
Acknowledgments
The authors thank Dani Kaufman and Bruno Umbert for useful discussions. The second author thanks Nima Arkani-Hamed, Johannes Henn and Bernd Sturmfels for encouragement and support. This research was supported in part by a grant from the Gluskin Sheff/Onex Freeman Dyson Chair in Theoretical Physics and by Perimeter Institute. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Colleges and Universities. This research received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 725110), Novel structures in scattering amplitudes.
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Cachazo, F., Early, N. Biadjoint scalars and associahedra from residues of generalized amplitudes. J. High Energ. Phys. 2023, 15 (2023). https://doi.org/10.1007/JHEP10(2023)015
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DOI: https://doi.org/10.1007/JHEP10(2023)015