Nonresonant renormalization scheme for twist-$2$ operators in $\mathcal{N}=1$ SUSY SU($N$) Yang-Mills theory
Francesco Scardino
TL;DR
The paper proves a number-theoretic nonresonant condition for the eigenvalues of $\frac{\gamma_0}{\beta_0}$ for twist-2 operators in $\mathcal{N}=1$ SUSY SU($N$) Yang–Mills theory, establishing the existence of a diagonal, nonresonant renormalization scheme where the operator-mixing matrix is one-loop exact. It builds on a geometric interpretation of operator mixing and uses Bertrand's postulate and harmonic-number properties to show that all eigenvalue gaps are never integers multiples of 2, i.e., $\frac{\gamma_{0\,n}-\gamma_{0\,m}}{\beta_0} \neq 2k$, for $n>m$ in all operator sectors (unbalanced, balanced even/odd). The analysis covers explicit expressions for the anomalous dimensions, including balanced/unbalanced superfields, and provides a comprehensive set of bounds (standard and generalized) to handle all spin cases. Consequently, the UV asymptotics derived in prior work applies to the full twist-2 sector, and a diagonal nonresonant scheme exists for renormalization, with $Z(\lambda)$ given by $Z_{\mathcal{O}_i}(\lambda) = (g(\mu)/g(\mu/\lambda))^{\gamma_{0\mathcal{O}_i}/\beta_0}$.
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 $-γ(g)/ β(g)$ reduces to $γ_0/ (β_0\,g)$ at all orders in perturbation theory, where $γ_0$ is diagonal, $γ(g) = γ_0 g^2+\ldots$ is the anomalous-dimension matrix, and $β(g) = -β_0 g^3+\ldots$ is the beta function. The method is founded on a new geometric interpretation of operator mixing [3], assuming that the eigenvalues of the matrix $γ_0/ β_0$ meet the nonresonant condition $λ_i-λ_j\neq 2k$, with the eigenvalues $λ_i$ 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.