Operator mixing in massless QCD-like theories and Poincare'-Dulac theorem

Abstract

Recently, a geometric approach to operator mixing in massless QCD-like theories -- that involves canonical forms based on the Poincare'-Dulac theorem for the linear system that defines the renormalized mixing matrix in the coordinate representation $Z(x,\mu)$ -- has been advocated in arXiv:2103.15527 . As a consequence, a classification of operator mixing in four cases -- depending on the canonical forms of $- \frac{\gamma(g)}{\beta(g)}$, 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 -- has been proposed: (I) nonresonant $\frac{\gamma_0}{\beta_0}$ diagonalizable, (II) resonant $\frac{\gamma_0}{\beta_0}$ diagonalizable, (III) nonresonant $\frac{\gamma_0}{\beta_0}$ nondiagonalizable, (IV) resonant $\frac{\gamma_0}{\beta_0}$ nondiagonalizable. In particular, in arXiv:2103.15527 a detailed analysis of the case (I) -- where operator mixing reduces to all orders of perturbation theory to the multiplicatively renormalizable case -- has been provided. In the present paper, following the aforementioned approach, we work out in the remaining three cases the canonical forms for $- \frac{\gamma(g)}{\beta(g)}$ to all orders of perturbation theory, the corresponding UV asymptotics of $Z(x,\mu)$, and the physics interpretation. We also work out in detail physical realizations of the cases (I) and (II).