Renormalization in Quantum Field Theory and the Riemann–Hilbert problem

We show that renormalization in quantum field theory is a special instance of a general mathematical procedure of multiplicative extraction of finite values based on the Riemann–Hilbert problem. Given a loop γ(z), |z| = 1 of elements of a complex Lie group G the general procedure is given by evaluation of γ+(z) at z = 0 after performing the Birkhoff decomposition γ(z) = γ−(z) γ+(z) where γ±(z) ∈ G are loops holomorphic in the inner and outer domains of the Riemann sphere (with γ−(∞) = 1). We show that, using dimensional regularization, the bare data in quantum field theory delivers a loop (where z is now the deviation from 4 of the complex dimension) of elements of the decorated Butcher group (obtained using the Milnor-Moore theorem from the Kreimer Hopf algebra of renormalization) and that the above general procedure delivers the renormalized physical theory in the minimal substraction scheme.