All-loop Mondrian diagrammatics and 4-particle amplituhedron
Abstract Based on 1712.09990 which handles the 4-particle amplituhedron at 3-loop, we have found an extremely simple pattern, yet far more non-trivial than one might naturally expect: the all-loop Mondrian diagrammatics. By further simplifying and rephrasing the key relation of positivity in the amp...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-06-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1007/JHEP06(2018)023 |
| _version_ | 1819103425788379136 |
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| author | Yang An Yi Li Zhinan Li Junjie Rao |
| author_facet | Yang An Yi Li Zhinan Li Junjie Rao |
| author_sort | Yang An |
| collection | DOAJ |
| description | Abstract Based on 1712.09990 which handles the 4-particle amplituhedron at 3-loop, we have found an extremely simple pattern, yet far more non-trivial than one might naturally expect: the all-loop Mondrian diagrammatics. By further simplifying and rephrasing the key relation of positivity in the amplituhedron setting, remarkably, we find a completeness relation unifying all diagrams of the Mondrian types for the 4-particle integrand of planar N $$ \mathcal{N} $$ = 4 SYM to all loop orders, each of which can be mapped to a simple product following a few plain rules designed for this relation. The explicit examples we investigate span from 3-loop to 7-loop order, and based on them, we classify the basic patterns of Mondrian diagrams into four types: the ladder, cross, brick-wall and spiral patterns. Interestingly, for some special combinations of ordered subspaces (a concept defined in the previous work), we find failed exceptions of the completeness relation which are called “anomalies”, nevertheless, they substantially give hints on the all-loop recursive proof of this relation. These investigations are closely related to the combinatoric knowledge of separable permutations and Schröder numbers, and go even further from a diagrammatic perspective. For physical relevance, we need to further consider dual conformal invariance for two basic diagrammatic patterns to correct the numerator for a local integrand involving one or both of such patterns, while the denominator encoding its pole structure and also the sign factor, are already fixed by rules of the completeness relation. With this extra treatment to ensure the integrals are dual conformally invariant, each Mondrian diagram can be exactly translated to its corresponding physical loop integrand after being summed over all ordered subspaces that admit it. |
| first_indexed | 2024-12-22T01:50:16Z |
| format | Article |
| id | doaj.art-a7d56c85467e45bb93898bca96ef6e8e |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-22T01:50:16Z |
| publishDate | 2018-06-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-a7d56c85467e45bb93898bca96ef6e8e2022-12-21T18:42:57ZengSpringerOpenJournal of High Energy Physics1029-84792018-06-012018613910.1007/JHEP06(2018)023All-loop Mondrian diagrammatics and 4-particle amplituhedronYang An0Yi Li1Zhinan Li2Junjie Rao3Zhejiang Institute of Modern Physics, Zhejiang UniversityZhejiang Institute of Modern Physics, Zhejiang UniversityZhejiang Institute of Modern Physics, Zhejiang UniversityZhejiang Institute of Modern Physics, Zhejiang UniversityAbstract Based on 1712.09990 which handles the 4-particle amplituhedron at 3-loop, we have found an extremely simple pattern, yet far more non-trivial than one might naturally expect: the all-loop Mondrian diagrammatics. By further simplifying and rephrasing the key relation of positivity in the amplituhedron setting, remarkably, we find a completeness relation unifying all diagrams of the Mondrian types for the 4-particle integrand of planar N $$ \mathcal{N} $$ = 4 SYM to all loop orders, each of which can be mapped to a simple product following a few plain rules designed for this relation. The explicit examples we investigate span from 3-loop to 7-loop order, and based on them, we classify the basic patterns of Mondrian diagrams into four types: the ladder, cross, brick-wall and spiral patterns. Interestingly, for some special combinations of ordered subspaces (a concept defined in the previous work), we find failed exceptions of the completeness relation which are called “anomalies”, nevertheless, they substantially give hints on the all-loop recursive proof of this relation. These investigations are closely related to the combinatoric knowledge of separable permutations and Schröder numbers, and go even further from a diagrammatic perspective. For physical relevance, we need to further consider dual conformal invariance for two basic diagrammatic patterns to correct the numerator for a local integrand involving one or both of such patterns, while the denominator encoding its pole structure and also the sign factor, are already fixed by rules of the completeness relation. With this extra treatment to ensure the integrals are dual conformally invariant, each Mondrian diagram can be exactly translated to its corresponding physical loop integrand after being summed over all ordered subspaces that admit it.http://link.springer.com/article/10.1007/JHEP06(2018)023Differential and Algebraic GeometryScattering AmplitudesSupersymmetric Gauge Theory |
| spellingShingle | Yang An Yi Li Zhinan Li Junjie Rao All-loop Mondrian diagrammatics and 4-particle amplituhedron Journal of High Energy Physics Differential and Algebraic Geometry Scattering Amplitudes Supersymmetric Gauge Theory |
| title | All-loop Mondrian diagrammatics and 4-particle amplituhedron |
| title_full | All-loop Mondrian diagrammatics and 4-particle amplituhedron |
| title_fullStr | All-loop Mondrian diagrammatics and 4-particle amplituhedron |
| title_full_unstemmed | All-loop Mondrian diagrammatics and 4-particle amplituhedron |
| title_short | All-loop Mondrian diagrammatics and 4-particle amplituhedron |
| title_sort | all loop mondrian diagrammatics and 4 particle amplituhedron |
| topic | Differential and Algebraic Geometry Scattering Amplitudes Supersymmetric Gauge Theory |
| url | http://link.springer.com/article/10.1007/JHEP06(2018)023 |
| work_keys_str_mv | AT yangan allloopmondriandiagrammaticsand4particleamplituhedron AT yili allloopmondriandiagrammaticsand4particleamplituhedron AT zhinanli allloopmondriandiagrammaticsand4particleamplituhedron AT junjierao allloopmondriandiagrammaticsand4particleamplituhedron |