Wilsonian renormalization, differential equations and Hopf algebras

TL;DR

This paper develops an algebraic framework combining Hopf algebras and differential equations to analyze renormalization and diagrammatic expansions in quantum field theory, with applications to the Tutte polynomial and Schwinger-Dyson equations.

Contribution

It introduces a novel algebraic formalism inspired by numerical analysis and renormalization, connecting Hopf algebras, differential equations, and diagrammatic methods.

Findings

Recovered the universality of the Tutte polynomial.

Applied techniques to solve Polchinski's exact renormalization group equation.

Established a bijection between planar φ³ diagrams and decorated rooted trees.

Abstract

In this paper, we present an algebraic formalism inspired by Butcher's B-series in numerical analysis and the Connes-Kreimer approach to perturbative renormalization. We first define power series of non linear operators and propose several applications, among which the perturbative solution of a fixed point equation using the non linear geometric series. Then, following Polchinski, we show how perturbative renormalization works for a non linear perturbation of a linear differential equation that governs the flow of effective actions. Then, we define a general Hopf algebra of Feynman diagrams adapted to iterations of background field effective action computations. As a simple combinatorial illustration, we show how these techniques can be used to recover the universality of the Tutte polynomial and its relation to the $q$-state Potts model. As a more sophisticated example, we use ordered diagrams with decorations and external structures to solve the Polchinski's exact renormalization group equation. Finally, we work out an analogous construction for the Schwinger-Dyson equations, which yields a bijection between planar $\phi^{3}$ diagrams and a certain class of decorated rooted trees.