Hopf algebra of graphs and the RG equations

TL;DR

This paper explores the Hopf algebra structure of graphs to derive and solve renormalization group equations, linking algebraic properties to Feynman integrals and their logarithmic coefficients.

Contribution

It introduces a Hopf algebra framework for RG equations, providing explicit solutions and connections to the parquet approximation.

Findings

RG equations are equivalent to those on Feynman integrals

Exponentiation of the beta-function yields coefficients of leading logs

Results apply to one-loop beta functions and parquet approximation

Abstract

We study the renormalization group equations following from the Hopf algebra of graphs. Vertex functions are treated as vectors in dual to the Hopf algebra space. The RG equations on such vertex functions are equivalent to RG equations on individual Feynman integrals. The solution to the RG equations may be represented as an exponent of the beta-function. We explicitly show that the exponent of the one-loop beta function enables one to find the coefficients in front of the leading logarithms for individual Feynman integrals. The same results are obtained in parquet approximation.