Abstract
In this note, we study the permutohedral geometry of the singularities of a certain differential form introduced in recent work of Arkani-Hamed, Bai, He and Yan. There it was observed that the poles of the form determine a family of polyhedra which have the same face lattice as that of the permutohedron. We realize that family explicitly, proving that it in fact fills out the configuration space of a particularly well-behaved family of generalized permutohedra, the zonotopal generalized permutohedra, that are obtained as the Minkowski sums of line segments parallel to the root directions ei − ej.
Finally we interpret Mizera’s formula for the biadjoint scalar amplitude m(𝕀n, 𝕀n), restricted to a certain dimension n − 2 subspace of the kinematic space, as a sum over the boundary components of the standard root cone, which is the conical hull of the roots e1 − e2, … , en−2 − en−1.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Aguiar and F. Ardila, Hopf monoids and generalized permutahedra, arXiv:1709.07504.
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 et al., Hidden zeros for particle/string amplitudes and the unity of colored scalars, pions and gluons, arXiv:2312.16282 [INSPIRE].
A. Barvinok and J. Pommersheim. An algorithmic theory of lattice points, in New perspectives in algebraic combinatorics 38 (1999), p. 91 [ISBN: 9780521770873].
L. Billera, N. Jia and V. Reiner. A quasisymmetric function for matroids, Eur. J. Combinatorics 30 (2009) 1727. math/0606646.
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
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 and N. Early. Minimal kinematics: an all k and n peek into Trop+G(k, n), SIGMA 17 (2021) 078.
S. Cho. Polytopes of roots of type AN, Bull. Aust. Math. Soc. 59 (1999) 391.
V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
N. Early, Canonical Bases for Permutohedral Plates, arXiv:1712.08520 [INSPIRE].
N. Early, Honeycomb Tessellations and Graded Permutohedral Blades, arXiv:1810.03246 [INSPIRE].
N. Early, From weakly separated collections to matroid subdivisions, Comb. Theory 2 (2022) 2.
N. Early, A. Pfister and B. Sturmfels, Minimal Kinematics on \( \mathcal{M} \)0,n, arXiv:2402.03065 [INSPIRE].
H. Frost, Biadjoint scalar tree amplitudes and intersecting dual associahedra, JHEP 06 (2018) 153 [arXiv:1802.03384] [INSPIRE].
I. Gelfand, M. Graev and A. Postnikov. Combinatorics of hypergeometric functions associated with positive roots, In The Arnold-Gelfand mathematical seminars. V.I. Arnold, I.M. Gelfand, V.S. Retakh and M. Smirnov eds., Birkhäuser Boston, (1997), p. 205–221 [https://doi.org/10.1007/978-1-4612-4122-5_10].
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].
J-L. Loday. The multiple facets of the associahedron, preprint (2005).
S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett. 120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].
S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, JHEP 08 (2017) 097 [arXiv:1706.08527] [INSPIRE].
J. Morton, L. Pachter, A. Shiu, B. Sturmfels and O. Wienand. Convex Rank Test and Semigraphoids, SIAM J. Discrete Math. 23 (2009) 1117.
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].
A. Postnikov, V. Reiner and L. Williams, Faces of Generalized Permutohedra, Doc. Math 13 (2008) 73 [math/0609184].
A. Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not. 6 (2009) 1026 [math/0507163].
G. Salvatori and S.L. Cacciatori, Hyperbolic Geometry and Amplituhedra in 1 + 2 dimensions, JHEP 08 (2018) 167 [arXiv:1803.05809] [INSPIRE].
D. Speyer and L.K. Williams, The tropical totally positive Grassmannian, J. Algebr. Comb. 22 (2005) 189 [math/0312297] [INSPIRE].
R. Stanley. A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. 111 (1995) 166.
J. Pitman and R. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom. 27 (2002) 603 [math/9908029].
Acknowledgments
The author was partially supported by RTG grant NSF/DMS-1148634, University of Minnesota.
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: 1804.05460
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
Early, N. Generalized permutohedra in the kinematic space. J. High Energ. Phys. 2024, 72 (2024). https://doi.org/10.1007/JHEP06(2024)072
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2024)072