Connecting scalar amplitudes using the positive tropical Grassmannian
Abstract The biadjoint scalar partial amplitude, m n I I $$ {m}_n\left(\mathbbm{I},\mathbbm{I}\right) $$ , can be expressed as a single integral over the positive tropical Grassmannian thus producing a Global Schwinger Parameterization. The first result in this work is an extension to all partial am...
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SpringerOpen
2024-12-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP12(2024)088 |
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| author | Freddy Cachazo Bruno Giménez Umbert |
| author_facet | Freddy Cachazo Bruno Giménez Umbert |
| author_sort | Freddy Cachazo |
| collection | DOAJ |
| description | Abstract The biadjoint scalar partial amplitude, m n I I $$ {m}_n\left(\mathbbm{I},\mathbbm{I}\right) $$ , can be expressed as a single integral over the positive tropical Grassmannian thus producing a Global Schwinger Parameterization. The first result in this work is an extension to all partial amplitudes m n (α, β) using a limiting procedure on kinematic invariants that produces indicator functions in the integrand. The same limiting procedure leads to an integral representation of ϕ 4 amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into C n/2−1 regions, with C q the q th-Catalan number. The contribution from each region is identified with a m n/2+1(α, I $$ \mathbbm{I} $$ ) amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for ϕ 4 amplitudes using the Lagrange inversion construction. We start the exploration of ϕ p theories, finding that their regions are encoded in non-crossing (p – 2)-chord diagrams. The structure of the expansion of ϕ p amplitudes in terms of ϕ 3 amplitudes is the same as that of Green functions in terms of connected Green functions in the planar limit of Φ p−1 matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra. |
| format | Article |
| id | doaj-art-a088a77eb2aa4f2d8f5f04a00fadaa1a |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-a088a77eb2aa4f2d8f5f04a00fadaa1a2024-12-22T12:09:12ZengSpringerOpenJournal of High Energy Physics1029-84792024-12-0120241213910.1007/JHEP12(2024)088Connecting scalar amplitudes using the positive tropical GrassmannianFreddy Cachazo0Bruno Giménez Umbert1Perimeter Institute for Theoretical PhysicsPerimeter Institute for Theoretical PhysicsAbstract The biadjoint scalar partial amplitude, m n I I $$ {m}_n\left(\mathbbm{I},\mathbbm{I}\right) $$ , can be expressed as a single integral over the positive tropical Grassmannian thus producing a Global Schwinger Parameterization. The first result in this work is an extension to all partial amplitudes m n (α, β) using a limiting procedure on kinematic invariants that produces indicator functions in the integrand. The same limiting procedure leads to an integral representation of ϕ 4 amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into C n/2−1 regions, with C q the q th-Catalan number. The contribution from each region is identified with a m n/2+1(α, I $$ \mathbbm{I} $$ ) amplitude. We provide a combinatorial description of the regions in terms of non-crossing chord diagrams and propose a general formula for ϕ 4 amplitudes using the Lagrange inversion construction. We start the exploration of ϕ p theories, finding that their regions are encoded in non-crossing (p – 2)-chord diagrams. The structure of the expansion of ϕ p amplitudes in terms of ϕ 3 amplitudes is the same as that of Green functions in terms of connected Green functions in the planar limit of Φ p−1 matrix models. We also discuss possible connections to recent constructions based on Stokes polytopes and accordiohedra.https://doi.org/10.1007/JHEP12(2024)088Scattering AmplitudesDifferential and Algebraic Geometry |
| spellingShingle | Freddy Cachazo Bruno Giménez Umbert Connecting scalar amplitudes using the positive tropical Grassmannian Journal of High Energy Physics Scattering Amplitudes Differential and Algebraic Geometry |
| title | Connecting scalar amplitudes using the positive tropical Grassmannian |
| title_full | Connecting scalar amplitudes using the positive tropical Grassmannian |
| title_fullStr | Connecting scalar amplitudes using the positive tropical Grassmannian |
| title_full_unstemmed | Connecting scalar amplitudes using the positive tropical Grassmannian |
| title_short | Connecting scalar amplitudes using the positive tropical Grassmannian |
| title_sort | connecting scalar amplitudes using the positive tropical grassmannian |
| topic | Scattering Amplitudes Differential and Algebraic Geometry |
| url | https://doi.org/10.1007/JHEP12(2024)088 |
| work_keys_str_mv | AT freddycachazo connectingscalaramplitudesusingthepositivetropicalgrassmannian AT brunogimenezumbert connectingscalaramplitudesusingthepositivetropicalgrassmannian |