Large spin corrections in ${\cal N}=4$ SYM sl(2): still a linear integral equation

TL;DR

This paper shows that for large spin in ${ m ext{N}=4}$ SYM, the complex non-linear integral equations simplify to linear ones, providing accurate approximations for anomalous dimensions and conserved charges at high spins.

Contribution

The authors derive a linear integral equation approximation for large spin in ${ m ext{N}=4}$ SYM, extending the validity of linear formulas beyond previous limits.

Findings

Non-linear terms vanish as spin increases, simplifying calculations.

Linear equations accurately describe anomalous dimensions at high spins.

Non-linear corrections decay faster than any inverse logarithm power.

Abstract

Anomalous dimension and higher conserved charges in the $sl(2)$ sector of ${\cal N}=4$ SYM for generic spin $s$ and twist $L$ are described by using a novel kind of non-linear integral equation (NLIE). The latter can be derived under typical situations of the SYM sectors, i.e. when the scattering need not depend on the difference of the rapidities and these, in their turn, may also lie on a bounded range. Here the non-linear (finite range) integral terms, appearing in the NLIE and in the dimension formula, go to zero as $s\to \infty$. Therefore they can be neglected at least up to the $O(s^0)$ order, thus implying a linear integral equation (LIE) and a linear dimension/charge formula respectively, likewise the 'thermodynamic' (i.e. infinite spin) case. Importantly, these non-linear terms go faster than any inverse logarithm power $(\ln s)^{-n}$, $n>0$, thus extending the linearity validity.