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

Abstract

Anomalous dimension and higher conserved charges in the $sl(2)$ sector of $\mathcal{N}=4$ 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.

Latex file, typos corrected, 2 figures, appendix, references and clarifications added

References in corpus (5)

• Dressing and Wrapping
• On the highest transcendentality in N=4 SUSY
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• The 2-loop generalized scaling function from the BES/FRS equation
• The integrable O(6) model and the correspondence: checks and predictions
• On the high spin expansion in the $sl(2)$ $\mathcal{N}=4$ SYM theory
• The virtual scaling function of twist operators in the N=6 Chern-Simons theory