Renormalization in quantum field theory and the Riemann-Hilbert problem II: The beta-function, diffeomorphisms and the renormalization group

We showed in Part I that the Hopf algebra H of Feynman graphs in a given QF T 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 epsilon = 0 the holomorphic part gamma (+) (epsilon) of t he Riemann-Hilbert decomposition gamma-(epsilon)(-1)gamma (+)(epsilon) of t he loop gamma(epsilon) epsilon 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 form ula go = gZ(1)Z(3)(-3/2) for the effective coupling constant, when viewed a s a formal power series, does define a Hopf algebra homomorphism between th e Hopf algebra of coordinates on the group of formal diffeomorphisms to the Hopf algebra H. This allows first of all to read off directly, without usi ng the group G, the bare coupling constant and the renormalized one from th e Riemann-Hilbert decomposition of the unrenormalized effective coupling co nstant viewed as a loop of formal diffeomorphisms. This shows that renormal ization is intimately related with the theory of non-linear complex bundles on the Riemann sphere of the dimensional regularization parameter epsilon. It also allows to lift both the renormalization group and the beta -functi on as the asymptotic scaling in the group G. This exploits the full power o f the Riemann-Hilbert decomposition together with the invariance of gamma-( epsilon) under a change of unit of mass. This not only gives a conceptual p roof of the existence of the renormalization group but also delivers a scat tering formula in the group G for the full higher pole structure of minimal subtracted counterterms in terms of the residue.