Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem II: TheBeta-Function, Diffeomorphisms and the Renormalization Group

摘要

The authors showed in part I that the Hopf algebra H of Feynman graphs in a graphQFT is the algebra of coordinates on a complex infinite dimensional Lie group G. The authors show in this paper that the group G acts naturally on the complex space X of dimensionless coupling constants of the theory. 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 epsilon. It also allows to lift both the renormalization group and the beta-function as the asymptotic scaling in the group G.