# On interpolating anomalous dimension of twist-two operators with general spins

## TL;DR

This paper introduces multivariate interpolating functions to analyze anomalous dimensions of twist-two operators in ${
m extbf{N}=4}$ SYM across different spins and coupling regimes, providing a unified non-perturbative framework.

## Contribution

It develops a novel approach using multivariate interpolating functions to study twist-two anomalous dimensions at finite and infinite spins, incorporating non-perturbative effects and S-duality.

## Key findings

- Constructed modular invariant interpolating functions for finite N.
- Analyzed level crossing between twist-two and twist-four operators.
- Provided unified non-perturbative estimates for anomalous dimensions.

## Abstract

We study non-perturbative interpolating functions to probe the physics of anomalous dimensions associated with twist-two operators in ${\cal N}=4$ SYM of finite and infinite spin. Compared to previous studies, the novel result of this paper is to introduce single multivariate functions of both coupling $g$ and spin $j$ to approximate such anomalous dimensions. We provide a unified framework to study such operators in interim ranges of the parameters which so far has eluded previous results. Explicitly, we consider twist-two anomalous dimensions in two distinct scenarios using interpolating functions. For the large $N$ case, we stick to simple Pad\'{e} approximants and its generalizations . For the finite $N$ case, ${\cal N}=4$ SYM is expected to be S-dual invariant, hence the observables are expected be modular invariant. To probe the finite $N$ physics, we take into account the non-planar and instanton contributions by constructing modular invariant interpolating functions to approximate the cusp and twist-two anomalous dimensions. We also consider interpolating functions for the twist-four operators and study level crossing phenomenon between the twist-two and twist-four operators.