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

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 $-{\hat{I}}^3(g)/{\hat{I}}^2(g)$ reduces to ${\hat{I}}_0^3/({\hat{I}}_0^{2}g)$ at all orders in perturbation theory, where ${\hat{I}}_0^3$ is diagonal, ${\hat{I}}^3(g)={\hat{I}}_0^3{g}^2+\dots$ is the anomalous-dimension matrix, and ${\hat{I}}^2(g)=-{\hat{I}}_0^2{g}^3+\dots$ is the beta function. The method is founded on a new geometric interpretation of operator mixing [3], assuming that the eigenvalues of the matrix ${\hat{I}}_0^3/{\hat{I}}_0^2$ meet the nonresonant condition $\hat{I}{»}_i-\hat{I}{»}_j\ne 2k$, with the eigenvalues $\hat{I}{»}_i$ 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.