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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Main Authors: Mrigankamauli Chakraborty, Franz Herzog
Format: Article
Language:English
Published: SpringerOpen 2025-01-01
Series:Journal of High Energy Physics
Subjects:
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.
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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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AT mrigankamaulichakraborty asymptotichopfalgebraoffeynmanintegrals
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