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

TL;DR

This paper analyzes the highest anomalous dimension operators in the SU(2) sector of planar ${ m extbf{N}=4}$ SYM, deriving integral equations and scaling laws for commuting charges at all loops, especially at strong coupling.

Contribution

It introduces a linear integral equation for Bethe root density and determines the scaling of charges at strong coupling, extending results to finite lengths and excited states.

Findings

Derived a linear integral equation for Bethe root density.

Established the scaling law for charges as \\lambda^{1/4 - r/2}.

Extended analysis to finite lengths and excited operators.

Abstract

We consider the highest anomalous dimension operator in the SU(2) sector of planar ${\cal N}=4$ 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 $\lambda ^{1/4-r/2}$, 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.