SciPost Phys. Proc. 7, 032 (2022) ·
published 21 June 2022

· pdf
We review recent progress on operator mixing in the light of the theory of canonical forms for linear systems of differential equations and, in particular, of the Poincar\'eDulac 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 review their classification. We derive a sufficient condition for $A(g)$ to be set in the oneloop exact form $A(g) = \frac{\gamma_0}{\beta_0}\frac{1}{g}$. Finally, we discuss the consequences of the unitarity requirement in massless QCDlike theories, and we demonstrate that $\gamma_0$ is always diagonalizable if the theory is conformal invariant and unitary in its free limit at $g =0$.