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).

35 pages, formulas unchanged, but some comments on the UV asymptotics corrected, physical realizations of the resonant case and new references added

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• On the geometry of operator mixing in massless QCD-like theories
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Cited by in corpus (4)

• UV asymptotics of $n$-point correlators of twist-$2$ operators in SU($N$) Yang-Mills theory
• Nonresonant renormalization scheme for twist-$2$ operators in SU($N$) Yang-Mills theory
• Operator mixing, UV asymptotics of nonplanar/planar $2$-point correlators, and nonperturbative large-$N$ expansion of QCD-like theories
• Superfield twist-$2$ operators in $\mathcal{N}=1$ SCFTs and their renormalization-group improved generating functional in $\mathcal{N}=1$ SYM theory