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

Abstract

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 $-γ(g)/ β(g)$ reduces to $γ_0/ (β_0\,g)$ to all orders of perturbation theory, with $γ_0$ diagonal, $γ(g) = γ_0 g^2+\ldots$ the anomalous-dimension matrix and $β(g) = -β_0 g^3+\ldots$ 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 $γ_0/ β_0$ satisfy the nonresonant condition $λ_i-λ_j\neq 2k$, with $λ_i$ 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.