On the high spin expansion in the $sl(2)$ ${\cal N}=4$ SYM theory

TL;DR

This paper analyzes the high spin limit of anomalous dimensions in the $sl(2)$ sector of ${ m N}=4$ SYM, deriving integral equations to evaluate sub-leading functions at weak and strong coupling, with implications for the O(6) sigma model.

Contribution

It introduces a method using linear integral equations to compute sub-leading scaling functions in the high spin limit of ${ m N}=4$ SYM, extending analysis to weak and strong coupling regimes.

Findings

Derived integral equations for the high spin expansion

Evaluated sub-leading scaling function $f^{(0)}(g,j)$ up to $j^5$

Connected results to the O(6) non-linear sigma model

Abstract

We study the the high spin expansion of the anomalous dimension for long operators belonging to the $sl(2)$ sector of ${\cal N}=4$ SYM. Keeping the ratio $j$ between the twist and the logarithm of the spin fixed, the anomalous dimensions expand as $\gamma= f(g,j)\ln s + f^{(0)}(g,j)+O(1/\ln s)$. This particular double scaling limit is efficiently described, up to the desired accuracy $O(\ln s ^0)$, in terms of linear integral equations. By using them, we are able to evaluate both at weak and strong coupling the sub-leading scaling function $f^{(0)}(g,j)$ as series in $j$, up to the order $j^5$. Thanks to these results, the possible extension of the liaison with the O(6) non-linear sigma model may be tackled on a solid ground.