  We revisit the operator mixing in massless QCD-like theories. In particular,
we address the problem of determining under which conditions a renormalization
scheme exists where the renormalized mixing matrix in the coordinate
representation, $Z(x, \mu)$, is diagonalizable to all perturbative orders. As a
key step, we provide a differential-geometric interpretation of renormalization
that allows us to apply the Poincar\'e-Dulac theorem to the problem above: We
interpret a change of renormalization scheme as a (formal) holomorphic gauge
transformation, $-\frac{\gamma(g)}{\beta(g)}$ as a (formal) meromorphic
connection with a Fuchsian singularity at $g=0$, and $Z(x,\mu)$ as a Wilson
line, with $\gamma(g)=\gamma_0 g^2 + \cdots$ the matrix of the anomalous
dimensions and $\beta(g)=-\beta_0 g^3 +\cdots$ the beta function. As a
consequence of the Poincar\'e-Dulac theorem, if the eigenvalues $\lambda_1,
\lambda_2, \cdots $ of the matrix $\frac{\gamma_0}{\beta_0}$, in nonincreasing
order $\lambda_1 \geq \lambda_2 \geq \cdots$, satisfy the nonresonant condition
$\lambda_i -\lambda_j -2k \neq 0$ for $i\leq j$ and $k$ a positive integer,
then a renormalization scheme exists where $-\frac{\gamma(g)}{\beta(g)} =
\frac{\gamma_0}{\beta_0} \frac{1}{g}$ is one-loop exact to all perturbative
orders. If in addition $\frac{\gamma_0}{\beta_0}$ is diagonalizable, $Z(x,
\mu)$ is diagonalizable as well, and the mixing reduces essentially to the
multiplicatively renormalizable case. We also classify the remaining cases of
operator mixing by the Poincar\'e-Dulac theorem.
