Self-dualities and Galois symmetries in Feynman integrals
Abstract It is well-known that all Feynman integrals within a given family can be expressed as a finite linear combination of master integrals. The master integrals naturally group into sectors. Starting from two loops, there can exist sectors made up of more than one master integral. In this paper...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2024-09-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP09(2024)084 |
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| author | Sebastian Pögel Xing Wang Stefan Weinzierl Konglong Wu Xiaofeng Xu |
| author_facet | Sebastian Pögel Xing Wang Stefan Weinzierl Konglong Wu Xiaofeng Xu |
| author_sort | Sebastian Pögel |
| collection | DOAJ |
| description | Abstract It is well-known that all Feynman integrals within a given family can be expressed as a finite linear combination of master integrals. The master integrals naturally group into sectors. Starting from two loops, there can exist sectors made up of more than one master integral. In this paper we show that such sectors may have additional symmetries. First of all, self-duality, which was first observed in Feynman integrals related to Calabi-Yau geometries, often carries over to non-Calabi-Yau Feynman integrals. Secondly, we show that in addition there can exist Galois symmetries relating integrals. In the simplest case of two master integrals within a sector, whose definition involves a square root r, we may choose a basis (I 1, I 2) such that I 2 is obtained from I 1 by the substitution r → −r. This pattern also persists in sectors, which a priori are not related to any square root with dependence on the kinematic variables. We show in several examples that in such cases a suitable redefinition of the integrals introduces constant square roots like 3 $$ \sqrt{3} $$ . The new master integrals are then again related by a Galois symmetry, for example the substitution 3 $$ \sqrt{3} $$ → − 3 $$ -\sqrt{3} $$ . To handle the case where the argument of a square root would be a perfect square we introduce a limit Galois symmetry. Both self-duality and Galois symmetries constrain the differential equation. |
| format | Article |
| id | doaj-art-6c9c4288b8624d639da867b9fb808211 |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-6c9c4288b8624d639da867b9fb8082112024-12-08T12:15:35ZengSpringerOpenJournal of High Energy Physics1029-84792024-09-012024913910.1007/JHEP09(2024)084Self-dualities and Galois symmetries in Feynman integralsSebastian Pögel0Xing Wang1Stefan Weinzierl2Konglong Wu3Xiaofeng Xu4PRISMA Cluster of Excellence, Institut für Physik, Staudinger Weg 7, Johannes Gutenberg-Universität MainzPhysik Department, TUM School of Natural Sciences, Technische Universität MünchenPRISMA Cluster of Excellence, Institut für Physik, Staudinger Weg 7, Johannes Gutenberg-Universität MainzDeutsches Elektronen-Synchrotron DESYPRISMA Cluster of Excellence, Institut für Physik, Staudinger Weg 7, Johannes Gutenberg-Universität MainzAbstract It is well-known that all Feynman integrals within a given family can be expressed as a finite linear combination of master integrals. The master integrals naturally group into sectors. Starting from two loops, there can exist sectors made up of more than one master integral. In this paper we show that such sectors may have additional symmetries. First of all, self-duality, which was first observed in Feynman integrals related to Calabi-Yau geometries, often carries over to non-Calabi-Yau Feynman integrals. Secondly, we show that in addition there can exist Galois symmetries relating integrals. In the simplest case of two master integrals within a sector, whose definition involves a square root r, we may choose a basis (I 1, I 2) such that I 2 is obtained from I 1 by the substitution r → −r. This pattern also persists in sectors, which a priori are not related to any square root with dependence on the kinematic variables. We show in several examples that in such cases a suitable redefinition of the integrals introduces constant square roots like 3 $$ \sqrt{3} $$ . The new master integrals are then again related by a Galois symmetry, for example the substitution 3 $$ \sqrt{3} $$ → − 3 $$ -\sqrt{3} $$ . To handle the case where the argument of a square root would be a perfect square we introduce a limit Galois symmetry. Both self-duality and Galois symmetries constrain the differential equation.https://doi.org/10.1007/JHEP09(2024)084Differential and Algebraic GeometryHigher Order Electroweak CalculationsHigher-Order Perturbative CalculationsScattering Amplitudes |
| spellingShingle | Sebastian Pögel Xing Wang Stefan Weinzierl Konglong Wu Xiaofeng Xu Self-dualities and Galois symmetries in Feynman integrals Journal of High Energy Physics Differential and Algebraic Geometry Higher Order Electroweak Calculations Higher-Order Perturbative Calculations Scattering Amplitudes |
| title | Self-dualities and Galois symmetries in Feynman integrals |
| title_full | Self-dualities and Galois symmetries in Feynman integrals |
| title_fullStr | Self-dualities and Galois symmetries in Feynman integrals |
| title_full_unstemmed | Self-dualities and Galois symmetries in Feynman integrals |
| title_short | Self-dualities and Galois symmetries in Feynman integrals |
| title_sort | self dualities and galois symmetries in feynman integrals |
| topic | Differential and Algebraic Geometry Higher Order Electroweak Calculations Higher-Order Perturbative Calculations Scattering Amplitudes |
| url | https://doi.org/10.1007/JHEP09(2024)084 |
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