On the Differential equations of the characters for the Renormalization group

TL;DR

This paper explores the mathematical structure of renormalization in quantum field theory by deriving differential equations for characters and counterterms, linking renormalization groups to integrable systems.

Contribution

It introduces differential equations for renormalized characters and counterterms, establishing a connection between renormalization groups and integrable systems.

Findings

Derived differential equations for renormalized characters and counterterms.

Provided a new proof for the scattering formula of the counterterm.

Defined the hierarchy of renormalization groups as integrable systems.

Abstract

Owing to the analogy between the Connes-Kreimer theory of the renormalization and the integrable systems, we derive the differential equations of the unit mass for the renormalized character $\phi_+$ and the counter term $\phi_-$. We give another proof for the scattering type formula of $\phi_-$. The differential equation of the counter term of coordinate on ${\mathbb P}^1$ is also given. The hierarchy of the renormalization groups is defined as the integrable systems.