Abstract
In this paper we study a relation between two positive geometries: the momen- tum amplituhedron, relevant for tree-level scattering amplitudes in \( \mathcal{N} \) = 4 super Yang-Mills theory, and the kinematic associahedron, encoding tree-level amplitudes in bi-adjoint scalar φ3 theory. We study the implications of restricting the latter to four spacetime dimensions and give a direct link between its canonical form and the canonical form for the momentum amplituhedron. After removing the little group scaling dependence of the gauge theory, we find that we can compare the resulting reduced forms with the pull-back of the associahedron form. In particular, the associahedron form is the sum over all helicity sectors of the reduced momentum amplituhedron forms. This relation highlights the common sin- gularity structure of the respective amplitudes; in particular, the factorization channels, corresponding to vanishing planar Mandelstam variables, are the same. Additionally, we also find a relation between these canonical forms directly on the kinematic space of the scalar theory when reduced to four spacetime dimensions by Gram determinant constraints. As a by-product of our work we provide a detailed analysis of the kinematic spaces relevant for the four-dimensional gauge and scalar theories, and provide direct links between them.
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References
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].
Y. Geyer, A.E. Lipstein and L.J. Mason, Ambitwistor strings in four dimensions, Phys. Rev. Lett. 113 (2014) 081602 [arXiv:1404.6219] [INSPIRE].
F. Cachazo and G. Zhang, Minimal basis in four dimensions and scalar blocks, arXiv:1601.06305 [INSPIRE].
N. Arkani-Hamed and J. Trnka, The amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed, Y. Bai and T. Lam, Positive geometries and canonical forms, JHEP 11 (2017) 039 [arXiv:1703.04541] [INSPIRE].
D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, The momentum amplituhedron, JHEP 08 (2019) 042 [arXiv:1905.04216] [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].
S. He and C. Zhang, Notes on scattering amplitudes as differential forms, JHEP 10 (2018) 054 [arXiv:1807.11051] [INSPIRE].
L. Ferro and T. Lukowski, Amplituhedra, and beyond, J. Phys. A 54 (2021) 033001 [arXiv:2007.04342] [INSPIRE].
T. Lukowski, M. Parisi and L.K. Williams, The positive tropical Grassmannian, the hypersimplex, and the m = 2 amplituhedron, arXiv:2002.06164 [INSPIRE].
N. Arkani-Hamed, T. Lam and M. Spradlin, Non-perturbative geometries for planar \( \mathcal{N} \) = 4 SYM amplitudes, arXiv:1912.08222 [INSPIRE].
V. Fock and A. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. I.H.E.S. 103 (2006) 1 [math/0311149].
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].
J.L. Bourjaily, Efficient tree-amplitudes in N = 4: automatic BCFW recursion in Mathematica, arXiv:1011.2447 [INSPIRE].
N. Arkani-Hamed et al., Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge U.K. (2016).
L. Ferro, T. Łukowski and R. Moerman, From momentum amplituhedron boundaries toamplitude singularities and back, JHEP 07 (2020) 201 [arXiv:2003.13704] [INSPIRE].
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Damgaard, D., Ferro, L., Łukowski, T. et al. Momentum amplituhedron meets kinematic associahedron. J. High Energ. Phys. 2021, 41 (2021). https://doi.org/10.1007/JHEP02(2021)041
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DOI: https://doi.org/10.1007/JHEP02(2021)041