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arXiv 2103.15527hep-thmath-phmath.MP
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无质量类QCD理论中算符混合的几何性质

On the geometry of operator mixing in massless QCD-like theories

Marco Bochicchio

备注：
10 pages, paper shortened, minor changes, typos corrected
发表信息：
Eur. Phys. J. C (2021) 81:749
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摘要：

我们重新审视了无质量QCD类理论中的算子混合。特别是，我们解决的问题，确定在何种条件下存在的重整化方案的重整化混合矩阵的坐标表示，$Z(x, μ)$，是对角化的所有微扰订单。作为关键的一步，我们提供了重整化的微分几何解释，使我们能够将庞加莱-杜拉克定理应用于上述问题：我们把重整化方案的变换解释为一个（形式）全纯规范变换，把$-\frac{γ(g)}{β(g)}$解释为在$g=0$处具有Fuchsian奇点的（形式）亚纯联络，把$Z(x,μ)$解释为Wilson线，其中$γ(g)=γ_0 g^2 + \cdots$为反常维数的矩阵，$β(g)=-β_0 g^3 +\cdots$为β函数。作为庞加莱-杜拉克定理的结果，如果矩阵$\frac{γ_0}{β_0}$的特征值$λ_1, λ_2, \cdots $，在非增阶$λ_1 \geq λ_2 \geq \cdots$中，满足非共振条件$λ_i -λ_j -2k \neq 0$，其中$i\leq j$和$k$为正整数，则存在重正化方案，其中$-\frac{γ(g)}{β(g)} = \frac{γ_0}{β_0} \frac{1}{g}$对所有微扰阶都是单圈精确的。如果另外$\frac{γ_0}{β_0}$是可对角化的，则$Z(x, μ)$也是可对角化的，并且混合本质上简化为乘法可重整化的情况。我们还分类的其余情况下的操作混合的庞加莱-Dulac定理。

英文摘要：

We revisit the operator mixing in massless QCD-like theories. In particular, we address the problem of determining under which conditions a renormalization scheme exists where the renormalized mixing matrix in the coordinate representation, $Z(x, μ)$, is diagonalizable to all perturbative orders. As a key step, we provide a differential-geometric interpretation of renormalization that allows us to apply the Poincaré-Dulac theorem to the problem above: We interpret a change of renormalization scheme as a (formal) holomorphic gauge transformation, $-\frac{γ(g)}{β(g)}$ as a (formal) meromorphic connection with a Fuchsian singularity at $g=0$, and $Z(x,μ)$ as a Wilson line, 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. As a consequence of the Poincaré-Dulac theorem, if the eigenvalues $λ_1, λ_2, \cdots $ of the matrix $\frac{γ_0}{β_0}$, in nonincreasing order $λ_1 \geq λ_2 \geq \cdots$, satisfy the nonresonant condition $λ_i -λ_j -2k \neq 0$ for $i\leq j$ and $k$ a positive integer, then a renormalization scheme exists where $-\frac{γ(g)}{β(g)} = \frac{γ_0}{β_0} \frac{1}{g}$ is one-loop exact to all perturbative orders. If in addition $\frac{γ_0}{β_0}$ is diagonalizable, $Z(x, μ)$ is diagonalizable as well, and the mixing reduces essentially to the multiplicatively renormalizable case. We also classify the remaining cases of operator mixing by the Poincaré-Dulac theorem.

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