Cohomological analysis of the Epstein-Glaser renormalization

A cohomological analysis of the renormalization freedom is performed in the Epstein-Glaser scheme on a flat Euclidean space. We study the deviation from commutativity between the renormalization and the action of all linear partial differential operators. It defines a Hochschild 1-cocycle and the renormalization ambiguity corresponds to a nonlinear subset in the cohomology class of this renormalization cocycle. We have shown that the related cohomology spaces can be reduced to de Rham cohomologies of the so called "(ordered) configuration spaces". We have also found cohomological differential equations that exactly determine the renormalization cocycles up to the renormalization freedom. This analysis is a first step towards a new approach for computing renormalization group actions. It can be also naturally extended to manifolds as well as to the case of causal perturbation theory.