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We review recent progress on operator mixing in the light of the theory ofcanonical forms for linear systems of differential equations and, inparticular, of the Poincar\'e-Dulac theorem. We show that the matrix $A(g) =-\frac{\gamma(g)}{\beta(g)} =\frac{\gamma_0}{\beta_0}\frac{1}{g} + \cdots $determines which different cases of operator mixing can occur, and we reviewtheir classification. We derive a sufficient condition for $A(g)$ to be set inthe one-loop exact form $A(g) = \frac{\gamma_0}{\beta_0}\frac{1}{g}$. Finally,we discuss the consequences of the unitarity requirement in massless QCD-liketheories, and we demonstrate that $\gamma_0$ is always diagonalizable if thetheory is conformal invariant and unitary in its free limit at $g =0$.

A differential-geometry approach to operator mixing in massless QCD-like theories and Poincar\'e-Dulac theorem