The Hopf algebra structure of the R-operation


We give a Hopf-algebraic formulation of the R-operation, which is a canonical way to render UV and IR divergent Euclidean Feynman diagrams finite. Our analysis uncovers a close connection to Brown’s Hopf algebra of motic graphs. Using this connection we are able to provide a verbose proof of the long observed ‘commutativity’ of UV and IR subtractions. We also give a new duality between UV and IR counterterms, which, entirely algebraic in nature, is formulated as an inverse relation on the group of characters of the Hopf algebra of log-divergent scaleless Feynman graphs. Many explicit examples of calculations with applications to infrared rearrangement are given.

A preprint version of the article is available at ArXiv.


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Beekveldt, R., Borinsky, M. & Herzog, F. The Hopf algebra structure of the R-operation. J. High Energ. Phys. 2020, 61 (2020).

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  • Renormalization Regularization and Renormalons
  • Scattering Amplitudes
  • Quantum Groups
  • Differential and Algebraic Geometry