arXiv 2103.162202021-03hep-th

无质量类QCD理论中的算子混合和Poincare '-Dulac定理

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

Matteo Becchetti, Marco Bochicchio

备注：: 35 pages, formulas unchanged, but some comments on the UV asymptotics corrected, physical realizations of the resonant case and new references added
DOI：: 10.1140/epjc/s10052-022-10551-2

摘要：

最近，在arXiv：2103.15527中提出了一种几何方法来处理无质量QCD类理论中的算子混合--涉及基于Poincare '-Dulac定理的线性系统的规范形式，该线性系统定义了坐标表示$Z(x,μ)$中的重正化混合矩阵。其结果是，一个分类的运营商混合在四种情况下-根据规范形式的$- \frac{γ(g)}{β(g)}$，与$γ(g)=γ_0 g^2+\cdots$的矩阵的异常尺寸和$β(g)=-β_0 g^3 + \cdots$的β功能-已提出：（I）非共振$\frac{γ_0}{β_0}$ diagonalizable，（II）共振$\frac{γ_0}{β_0}$ diagonalizable，（III）非共振$\frac{γ_0}{β_0}$ nondiagonalizable，（IV）共振$\frac{γ_0}{β_0}$ nondiagonalizable。特别是，在arXiv：2103.15527中，提供了对情况（I）的详细分析-其中算子混合将微扰理论的所有阶数减少到乘法重正化的情况-。在本文中，我们按照上述方法，在其余三种情况下，$- \frac{γ(g)}{β(g)}$的所有阶微扰理论的标准形，相应的UV渐近$Z(x,μ)$，和物理解释。我们还详细地研究了情形（I）和（II）的物理实现。

英文摘要：

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 $Z(x,μ)$ -- has been advocated in arXiv:2103.15527 . As a consequence, a classification of operator mixing in four cases -- depending on the canonical forms of $- \frac{γ(g)}{β(g)}$, with $γ(g)=γ_0 g^2+\cdots$ the matrix of the anomalous dimensions and $β(g)=-β_0 g^3 + \cdots$ the beta function -- has been proposed: (I) nonresonant $\frac{γ_0}{β_0}$ diagonalizable, (II) resonant $\frac{γ_0}{β_0}$ diagonalizable, (III) nonresonant $\frac{γ_0}{β_0}$ nondiagonalizable, (IV) resonant $\frac{γ_0}{β_0}$ 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 $- \frac{γ(g)}{β(g)}$ to all orders of perturbation theory, the corresponding UV asymptotics of $Z(x,μ)$, and the physics interpretation. We also work out in detail physical realizations of the cases (I) and (II).

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