Renormalization and Mellin transforms

TL;DR

This paper explores renormalization in quantum field theory using Hopf algebra and Mellin transforms, providing a combinatorial framework and analyzing the scaling behavior of Feynman graphs.

Contribution

It introduces a combinatorial description of renormalized amplitudes via Mellin transform coefficients within a Hopf algebraic setting.

Findings

Recovered known results in the Hopf algebraic renormalization framework

Described automorphisms of the Hopf algebra from Mellin transform actions

Revealed the scaling behavior of individual Feynman graphs in scalar QFT

Abstract

We study renormalization in a kinetic scheme using the Hopf algebraic framework, first summarizing and recovering known results in this setting. Then we give a direct combinatorial description of renormalized amplitudes in terms of Mellin transform coefficients, featuring the universal property of rooted trees H_R. In particular, a special class of automorphisms of H_R emerges from the action of changing Mellin transforms on the Hochschild cohomology of perturbation series. Furthermore, we show how the Hopf algebra of polynomials carries a refined renormalization group property, implying its coarser form on the level of correlation functions. Application to scalar quantum field theory reveals the scaling behaviour of individual Feynman graphs.