Title: Nonresonant renormalization scheme for twist-$2$ operators in $\mathcal{N}=1$ SUSY SU($N$) Yang-Mills theory Show Chinese title

Title: 非共振重正化方案在$2$扭角算符的$\mathcal{N}=1$超对称SU($N$)杨-米尔斯理论中的应用

Abstract: The short-distance asymptotics of the generating functional for $n$-point correlators of twist-$2$ operators in $\mathcal{N}=1$ supersymmetric (SUSY) SU($N$) Yang-Mills (SYM) theory were recently calculated in [1,2]. This calculation depends on a change of basis for renormalized twist-$2$ operators, in which $-\gamma(g)/ \beta(g)$ reduces to $\gamma_0/ (\beta_0\,g)$ at all orders in perturbation theory, where $\gamma_0$ is diagonal, $\gamma(g) = \gamma_0 g^2+\ldots$ is the anomalous-dimension matrix, and $\beta(g) = -\beta_0 g^3+\ldots$ is the beta function. The method is founded on a new geometric interpretation of operator mixing [3], assuming that the eigenvalues of the matrix $\gamma_0/ \beta_0$ meet the nonresonant condition $\lambda_i-\lambda_j\neq 2k$, with the eigenvalues $\lambda_i$ ordered nonincreasingly and $k\in \mathbb{N}^+$. This nonresonant condition was numerically verified for $i,j$ up to $10^4$ in [1,2]. In this work, we employ techniques initially developed in [4] to present a number-theoretic proof of the nonresonant condition for twist-$2$ operators, fundamentally based on the classic result that Harmonic numbers are not integers. Show Chinese abstract

Abstract: 生成泛函在 $n$ 点扭结 $2$ 算符的短距离渐进行为在 $\mathcal{N}=1$ 超对称 (SUSY) SU($N$) 杨-米尔斯 (SYM) 理论中最近在 [1,2] 中被计算出来。 此计算依赖于重整化扭角$2$算符的基变换，在此变换中，$-\gamma(g)/ \beta(g)$在微扰理论的所有阶数中都简化为$\gamma_0/ (\beta_0\,g)$，其中$\gamma_0$是对角的，$\gamma(g) = \gamma_0 g^2+\ldots$是奇异维度矩阵，而$\beta(g) = -\beta_0 g^3+\ldots$是β函数。 该方法基于对算子混合的新几何解释 [3]，假设矩阵$\gamma_0/ \beta_0$的特征值满足非共振条件$\lambda_i-\lambda_j\neq 2k$，其中特征值$\lambda_i$按非递增顺序排列且$k\in \mathbb{N}^+$。该非共振条件在 [1,2] 中针对$i,j$直到$10^4$进行了数值验证。在本工作中，我们采用最初在 [4] 中开发的技术，提出一种数论证明，用于证明扭转-$2$算子的非共振条件，其基础是经典结果：调和数不是整数。