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

We study the the high spin expansion of the anomalous dimension for long operators belonging to the $sl(2)$ sector of $\mathcal{N}=4$ SYM. Keeping the ratio $j$ between the twist and the logarithm of the spin fixed, the anomalous dimensions expand as $\gamma =f(g,j)\ln s+f^{(0)}(g,j)+O(1\mathrm{/}\ln s)$. 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.