  We showed in part I (hep-th/9912092) that the Hopf algebra ${\cal H}$ of
Feynman graphs in a given QFT is the algebra of coordinates on a complex
infinite dimensional Lie group $G$ and that the renormalized theory is obtained
from the unrenormalized one by evaluating at $\ve=0$ the holomorphic part
$\gamma_+(\ve)$ of the Riemann-Hilbert decomposition
$\gamma_-(\ve)^{-1}\gamma_+(\ve)$ of the loop $\gamma(\ve)\in G$ provided by
dimensional regularization. We show in this paper that the group $G$ acts
naturally on the complex space $X$ of dimensionless coupling constants of the
theory. More precisely, the formula $g_0=gZ_1Z_3^{-3/2}$ for the effective
coupling constant, when viewed as a formal power series, does define a Hopf
algebra homomorphism between the Hopf algebra of coordinates on the group of
formal diffeomorphisms to the Hopf algebra ${\cal H}$. This allows first of all
to read off directly, without using the group $G$, the bare coupling constant
and the renormalized one from the Riemann-Hilbert decomposition of the
unrenormalized effective coupling constant viewed as a loop of formal
diffeomorphisms. This shows that renormalization is intimately related with the
theory of non-linear complex bundles on the Riemann sphere of the dimensional
regularization parameter $\ve$. It also allows to lift both the renormalization
group and the $\beta$-function as the asymptotic scaling in the group $G$. This
exploits the full power of the Riemann-Hilbert decomposition together with the
invariance of $\gamma_-(\ve)$ under a change of unit of mass. This not only
gives a conceptual proof of the existence of the renormalization group but also
delivers a scattering formula in the group $G$ for the full higher pole
structure of minimal subtracted counterterms in terms of the residue.
