On the high spin expansion in the sl(2) N=4 SYM theory

We study the form of the high spin expansion of the minimal anomalous dimension for long operators belonging to the sl(2) sector of N=4 SYM. Keeping fixed the ratio j between the twist and the logarithm of the spin, the minimal anomalous dimension expands as γ(g,j,s)=f(g,j)lns+f(0)(g,j)+O(1/lns). This particular double scaling limit is efficiently described, including the desired accuracy O((lns)0), in terms of a linear integral equation. By its use, we are able to evaluate both at weak and strong coupling the subleading scaling function f(0)(g,j) as a series in j, up to the order j5. 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.