Nonresonant renormalization scheme for twist-$2$ operators in N=1 SUSY SU(N) Yang-Mills theory

The short-distance asymptotics of the generating functional for n-point correlators of twist-$2$ operators in 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 $-γ(g)/ β(g)$ reduces to γ₀/ (β₀\,g) at all orders in perturbation theory, where γ₀ is diagonal, γ(g) = γ₀ g²+… is the anomalous-dimension matrix, and β(g) = -β₀ g³+… is the beta function. The method is founded on a new geometric interpretation of operator mixing [3], assuming that the eigenvalues of the matrix γ₀/ β₀ meet the nonresonant condition λᵢ-λⱼ≠ 2k, with the eigenvalues λᵢ ordered nonincreasingly and k∈ N⁺. This nonresonant condition was numerically verified for $i,j$ up to 10⁴ 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.