On the commuting charges for the highest dimension SU(2) operators in planar $\mathcal{N}=4$ SYM

We consider the highest anomalous dimension operator in the SU(2) sector of planar $\mathcal{N}=4$ SYM at all-loop, though neglecting wrapping contributions. In any case, the latter enter the loop expansion only after a precise length-depending order. In the thermodynamic limit we write both a linear integral equation for the Bethe root density and a linear system obeyed by the commuting charges. Consequently, we determine the leading strong coupling contribution to the density and from this an approximation to the leading and sub-leading terms of any charge $Q_r$: it scales as ${\lambda }^{1\mathrm{/}4-r\mathrm{/}2}$, which generalises the Gubser-Klebanov-Polyakov energy law. In the end, we briefly extend these considerations to finite lengths and ‘excited’ operators by using the idea of a non-linear integral equation.