The asymptotic Hopf algebra of Feynman integrals
Abstract The method of regions is an approach for developing asymptotic expansions of Feynman Integrals. We focus on expansions in Euclidean signature, where the method of regions can also be formulated as an expansion by subgraph. We show that for such expansions valid around small/large masses and...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2025-01-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP01(2025)006 |
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| author | Mrigankamauli Chakraborty Franz Herzog |
| author_facet | Mrigankamauli Chakraborty Franz Herzog |
| author_sort | Mrigankamauli Chakraborty |
| collection | DOAJ |
| description | Abstract The method of regions is an approach for developing asymptotic expansions of Feynman Integrals. We focus on expansions in Euclidean signature, where the method of regions can also be formulated as an expansion by subgraph. We show that for such expansions valid around small/large masses and momenta the graph combinatorial operations can be formulated in terms of what we call the asymptotic Hopf algebra. This Hopf algebra is closely related to the motic Hopf algebra underlying the R ∗ operation, an extension of Bogoliubov’s R operation, to subtract both IR and UV divergences of Feynman integrals in the Euclidean. We focus mostly on the leading power, for which the Hopf algebra formulation is simpler. We uncover a close connection between Bogoliubov’s R operation in the Connes-Kreimer formulation and the remainder R $$ \mathcal{R} $$ of the series expansion, whose Hopf algebraic structure is identically formalised in the corresponding group of characters. While in the Connes-Kreimer formulation the UV counterterm is formalised in terms of a twisted antipode, we show that in the expansion by subgraph a similar role is played by the integrand Taylor operator. To discuss the structure of higher power expansions we introduce a novel Hopf monoid formulation. |
| format | Article |
| id | doaj-art-aa0a02e6bb04433ea1807a547ceff038 |
| institution | Kabale University |
| issn | 1029-8479 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj-art-aa0a02e6bb04433ea1807a547ceff0382025-01-19T12:06:50ZengSpringerOpenJournal of High Energy Physics1029-84792025-01-012025113910.1007/JHEP01(2025)006The asymptotic Hopf algebra of Feynman integralsMrigankamauli Chakraborty0Franz Herzog1II. Institute for Theoretical Physics, Hamburg UniversityHiggs Centre for Theoretical Physics, School of Physics and Astronomy, The University of EdinburghAbstract The method of regions is an approach for developing asymptotic expansions of Feynman Integrals. We focus on expansions in Euclidean signature, where the method of regions can also be formulated as an expansion by subgraph. We show that for such expansions valid around small/large masses and momenta the graph combinatorial operations can be formulated in terms of what we call the asymptotic Hopf algebra. This Hopf algebra is closely related to the motic Hopf algebra underlying the R ∗ operation, an extension of Bogoliubov’s R operation, to subtract both IR and UV divergences of Feynman integrals in the Euclidean. We focus mostly on the leading power, for which the Hopf algebra formulation is simpler. We uncover a close connection between Bogoliubov’s R operation in the Connes-Kreimer formulation and the remainder R $$ \mathcal{R} $$ of the series expansion, whose Hopf algebraic structure is identically formalised in the corresponding group of characters. While in the Connes-Kreimer formulation the UV counterterm is formalised in terms of a twisted antipode, we show that in the expansion by subgraph a similar role is played by the integrand Taylor operator. To discuss the structure of higher power expansions we introduce a novel Hopf monoid formulation.https://doi.org/10.1007/JHEP01(2025)006Renormalization and RegularizationHigher-Order Perturbative CalculationsScattering AmplitudesQuantum Groups |
| spellingShingle | Mrigankamauli Chakraborty Franz Herzog The asymptotic Hopf algebra of Feynman integrals Journal of High Energy Physics Renormalization and Regularization Higher-Order Perturbative Calculations Scattering Amplitudes Quantum Groups |
| title | The asymptotic Hopf algebra of Feynman integrals |
| title_full | The asymptotic Hopf algebra of Feynman integrals |
| title_fullStr | The asymptotic Hopf algebra of Feynman integrals |
| title_full_unstemmed | The asymptotic Hopf algebra of Feynman integrals |
| title_short | The asymptotic Hopf algebra of Feynman integrals |
| title_sort | asymptotic hopf algebra of feynman integrals |
| topic | Renormalization and Regularization Higher-Order Perturbative Calculations Scattering Amplitudes Quantum Groups |
| url | https://doi.org/10.1007/JHEP01(2025)006 |
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