Complex powers of analytic functions and meromorphic renormalization in QFT

In this article, we study functional analytic properties of the meromorphic families of distributions (∏ᵢ₌₁ᵖ (fⱼ+i0)λⱼ)(λ₁,,λₚ) ∈ Cᵖ using Hironaka's resolution of singularities, then using recent works on the decomposition of meromorphic germs with linear poles, we renormalize products of powers of analytic functions ∏ᵢ₌₁ᵖ(fⱼ+i0)kⱼ, kⱼ ∈ Z in the space of distributions. We also study microlocal properties of (∏ᵢ₌₁ᵖ (fⱼ+i0)λⱼ)(λ₁,,λₚ)ᵖ and ∏ᵢ₌₁ᵖ (fⱼ+i0)kⱼ, kⱼ ∈ Z. In the second part, we argue that the above families of distributions with regular holonomic singularities provide a universal model describing singularities of Feynman amplitudes and give a new proof of renormalizability of quantum field theory on convex analytic Lorentzian spacetimes as applications of ideas from the first part.