  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 $\gamma(z), | z |=1$ of
elements of a complex Lie group G the general procedure is given by evaluation
of $ \gamma_{+}(z)$ at z=0 after performing the Birkhoff decomposition $
\gamma(z)=\gamma_{-}(z)^{-1} \gamma_{+}(z)$ where $ \gamma_{\pm}(z) \in G$ are
loops holomorphic in the inner and outer domains of the Riemann sphere (with
$\gamma_{-}(\infty)=1$). 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.
