Geometrically relating momentum cut-off and dimensional regularization

TL;DR

This paper explicitly relates the beta functions of scalar field theories under momentum cut-off and dimensional regularization schemes using Hopf algebra gauge transformations, clarifying their connection.

Contribution

It introduces a gauge transformation framework linking the beta functions of different regularization schemes via Hopf algebras.

Findings

Established a formal relation between regularization schemes

Used Hopf algebra gauge transformations to connect beta functions

Clarified scheme dependence in scalar field theories

Abstract

The $\beta$ function for a scalar field theory describes the dependence of the coupling constant on the renormalization mass scale. This dependence is affected by the choice of regularization scheme. I explicitly relate the $\beta$-functions of momentum cut-off regularization and dimensional regularization on scalar field theories by a gauge transformation using the Hopf algebras of the Feynman diagrams of the theories.