Stokes polytopes: the positive geometry for ϕ 4 interactions
Abstract In a remarkable recent work [1], the amplituhedron program was extended to the realm of non-supersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless ϕ 3 theory (and its close cousin, bi-adjoint ϕ 3 theory) a polytope known as the assoc...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-08-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1007/JHEP08(2019)067 |
| _version_ | 1818276391720845312 |
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| author | Pinaki Banerjee Alok Laddha Prashanth Raman |
| author_facet | Pinaki Banerjee Alok Laddha Prashanth Raman |
| author_sort | Pinaki Banerjee |
| collection | DOAJ |
| description | Abstract In a remarkable recent work [1], the amplituhedron program was extended to the realm of non-supersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless ϕ 3 theory (and its close cousin, bi-adjoint ϕ 3 theory) a polytope known as the associahedron sits inside the kinematic space and is the amplituhedron for the theory. Precisely as in the case of amplituhedron, it was shown that scattering amplitude can be obtained from the canonical form associated to the Associahedron. Combinatorial and geometric properties of associahedron naturally encode properties like locality and unitarity of (tree level) scattering amplitudes. In this paper we attempt to extend this program to planar amplitudes in massless ϕ 4 theory. We show that tree-level planar amplitudes in this theory can be obtained from geometry of objects known as the Stokes polytope which sits naturally inside the kinematic space. As in the case of associahedron we show that the canonical form on these Stokes polytopes can be used to compute scattering amplitudes for quartic interactions. However unlike associahedron, Stokes polytope of a given dimension is not unique and as we show, one must sum over all of them to obtain the complete scattering amplitude. Not all Stokes polytopes contribute equally and we argue that the corresponding weights depend on purely combinatorial properties of the Stokes polytopes. As in the case of φ3 theory, we show how factorization of Stokes polytope implies unitarity and locality of the amplitudes. |
| first_indexed | 2024-12-12T22:44:54Z |
| format | Article |
| id | doaj.art-bebe6ec53ac145b094bf8b6de2c59098 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-12T22:44:54Z |
| publishDate | 2019-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-bebe6ec53ac145b094bf8b6de2c590982022-12-22T00:09:14ZengSpringerOpenJournal of High Energy Physics1029-84792019-08-012019813510.1007/JHEP08(2019)067Stokes polytopes: the positive geometry for ϕ 4 interactionsPinaki Banerjee0Alok Laddha1Prashanth Raman2International Centre for Theoretical Sciences, Tata Institute of Fundamental ResearchChennai Mathematical InstituteInstitute of Mathematical SciencesAbstract In a remarkable recent work [1], the amplituhedron program was extended to the realm of non-supersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless ϕ 3 theory (and its close cousin, bi-adjoint ϕ 3 theory) a polytope known as the associahedron sits inside the kinematic space and is the amplituhedron for the theory. Precisely as in the case of amplituhedron, it was shown that scattering amplitude can be obtained from the canonical form associated to the Associahedron. Combinatorial and geometric properties of associahedron naturally encode properties like locality and unitarity of (tree level) scattering amplitudes. In this paper we attempt to extend this program to planar amplitudes in massless ϕ 4 theory. We show that tree-level planar amplitudes in this theory can be obtained from geometry of objects known as the Stokes polytope which sits naturally inside the kinematic space. As in the case of associahedron we show that the canonical form on these Stokes polytopes can be used to compute scattering amplitudes for quartic interactions. However unlike associahedron, Stokes polytope of a given dimension is not unique and as we show, one must sum over all of them to obtain the complete scattering amplitude. Not all Stokes polytopes contribute equally and we argue that the corresponding weights depend on purely combinatorial properties of the Stokes polytopes. As in the case of φ3 theory, we show how factorization of Stokes polytope implies unitarity and locality of the amplitudes.http://link.springer.com/article/10.1007/JHEP08(2019)067Scattering AmplitudesDifferential and Algebraic Geometry |
| spellingShingle | Pinaki Banerjee Alok Laddha Prashanth Raman Stokes polytopes: the positive geometry for ϕ 4 interactions Journal of High Energy Physics Scattering Amplitudes Differential and Algebraic Geometry |
| title | Stokes polytopes: the positive geometry for ϕ 4 interactions |
| title_full | Stokes polytopes: the positive geometry for ϕ 4 interactions |
| title_fullStr | Stokes polytopes: the positive geometry for ϕ 4 interactions |
| title_full_unstemmed | Stokes polytopes: the positive geometry for ϕ 4 interactions |
| title_short | Stokes polytopes: the positive geometry for ϕ 4 interactions |
| title_sort | stokes polytopes the positive geometry for ϕ 4 interactions |
| topic | Scattering Amplitudes Differential and Algebraic Geometry |
| url | http://link.springer.com/article/10.1007/JHEP08(2019)067 |
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