Nonresonant renormalization scheme for twist-$2$ operators in SU($N$) Yang-Mills theory

Recently, the short-distance asymptotics of the generating functional of $n$-point correlators of twist-$2$ operators in SU($N$) Yang-Mills (YM) theory has been worked out in [1]. The above computation relies on a basis change of renormalized twist-$2$ operators, where $-\gamma(g)/ \beta(g)$ reduces to $\gamma_0/ (\beta_0\,g)$ to all orders of perturbation theory, with $\gamma_0$ diagonal, $\gamma(g) = \gamma_0 g^2+\ldots$ the anomalous-dimension matrix and $\beta(g) = -\beta_0 g^3+\ldots$ the beta function. The construction is based on a novel geometric interpretation of operator mixing [2], under the assumption that the eigenvalues of the matrix $\gamma_0/ \beta_0$ satisfy the nonresonant condition $\lambda_i-\lambda_j\neq 2k$, with $\lambda_i$ in nonincreasing order and $k\in \mathbb{N}^+$. The nonresonant condition has been numerically verified up to $i,j=10^4$ in [1]. In the present paper we provide a number theoretic proof of the nonresonant condition for twist-$2$ operators essentially based on the classic result that Harmonic numbers are not integers. Our proof in YM theory can be extended with minor modifications to twist-$2$ operators in $\mathcal{N}=1$ SUSY YM theory, large-$N$ QCD with massless quarks and massless QCD-like theories.