A differential-geometry approach to operator mixing in massless QCD-like theories and Poincaré-Dulac theorem

Abstract

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 Poincaré-Dulac theorem. We show that the matrix A(g)=-\frac{\gamma(g)}{\beta(g)}=\frac{\gamma_0}{\beta_0}\frac{1}{g}+\cdotsA(g)=−γ(g)β(g)=γ0β01g+⋯ determines which different cases of operator mixing can occur, and we review their classification. We derive a sufficient condition for A(g)A(g) to be set in the one-loop exact form A(g) = \frac{\gamma_0}{\beta_0}\frac{1}{g}A(g)=γ0β01g. Finally, we discuss the consequences of the unitarity requirement in massless QCD-like theories, and we demonstrate that \gamma_0γ0 is always diagonalizable if the theory is conformal invariant and unitary in its free limit at g =0g=0.