On the Differential equations of the characters for the Renormalization group

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 ${\varphi }_{+}$ and the counter term ${\varphi }_{-}$. We give another proof for the scattering type formula of ${\varphi }_{-}$. 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.

5 pages, no figures, AMS-LaTeX, minor corrections