Generalized Scaling Function at Strong Coupling

TL;DR

The paper analyzes the $sl(2)$ sector of the AdS/CFT correspondence and derives the generalized scaling function $f(\lambda,\ell)$ in the double-scaling limit $S,J\to\infty$ with $\ell=\frac{\pi J}{\sqrt{\lambda}\log S}$ fixed. By reformulating the all-loop Bethe equations in terms of a resolvent and reducing them to a quadratic equation, the authors perform a strong-coupling expansion and extract the two-loop correction $f_{\rm 2-loop}(\ell)$, including finite behavior as $\ell\to 0$ due to cancellations with the dressing phase. The solution involves dissecting ${\cal F}(x)$ into classical, Hernandez–Lopez, and anomaly parts, computing leading and subleading contributions, and demonstrating that divergent pieces cancel to yield a finite result; they also obtain all leading logarithmic terms $\ell^{2m}\log^n\ell/\lambda^{n/2}$ to arbitrary order and reproduce the $m=1$ case known from Alday–Maldacena. The methodology provides a systematic framework for higher-order strong-coupling expansions and potential extensions to weak coupling, offering a nontrivial test of integrability and the dressing phase in the AdS/CFT setup.

Abstract

We considered folded spinning string in AdS_5 x S^5 background dual to the Tr(D^S Phi^J) operators of N=4 SYM theory. In the limit S,J-> \infty and l=pi J/\sqrtλ\log S fixed we compute the string energy with the 2-loop accuracy in the worldsheet coupling \sqrtλfrom the asymptotical Bethe ansatz. In the limit l-> 0 the result is finite due to the massive cancelations with terms coming from the conjectured dressing phase. We also managed to compute all leading logarithm terms l^{2m}\log^n l/λ^n/2 to an arbitrary order in perturbation theory. In particular for m=1 we reproduced results of Alday and Maldacena computed from a sigma model. The method developed in this paper could be used for a systematic expansion in 1/\sqrtλand also at weak coupling.