The Hopf algebra structure of the $R^*$-operation

Understand

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.