Integrable Renormalization I: the Ladder Case

Abstract

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underlying the target space of the regularized Hopf algebra characters. Working in the rooted tree Hopf algebra, the simple case of the Hopf subalgebra of ladder trees is treated in detail. The extension to the general case, i.e. the full Hopf algebra of rooted trees or Feynman graphs is briefly outlined.

16 pages

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Cited by in corpus (10)

• Mixable Shuffles, Quasi-shuffles and Hopf Algebras
• Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
• Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators
• Shuffle relations for regularised integrals of symbols
• On Products and Duality of Binary, Quadratic, Regular Operads
• From Rota-Baxter Algebras to Pre-Lie Algebras
• An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
• From Hurwitz numbers to Feynman diagrams: counting rooted trees in log gravity
• An operator approach to the rational solutions of the classical Yang-Baxter equation
• The Epstein-Glaser approach to pQFT: graphs and Hopf algebras