The virtual scaling function of AdS/CFT

TL;DR

This work addresses finite-order corrections to the large-spin scaling of twist operators in planar $\mathcal{N}=4$ SYM by deriving an all-loop, finite-order NLIE for the density $\hat{\sigma}(t)$ and expressing the finite part $B_L(g)$ entirely in terms of the BES equation. The authors show these corrections are not affected by wrapping and reproduce strong-coupling string predictions, connecting $B_L(g)$ to the cusp anomalous dimension via $f(g)=2\Gamma_{\mathrm{cusp}}(g)$. They provide an explicit strong-coupling expansion, relate $\gamma_1^{(1)}(g)$ to the generalized scaling function $\epsilon_1(g)$ with $\epsilon_1(g)=-1+O(e^{-\pi g})$, and determine subleading corrections for twist $L=2$ using all-loop quantization, including constants $c_p^{\pm}$. For $L=2$, the NLIE reproduces the expected string-energy form up to one loop and yields subleading spin corrections that agree with string theory and reciprocity, while wrapping effects appear only at higher orders beyond the NLIE's reach. Overall, the paper clarifies how finite-order, all-loop corrections to large-spin scaling can be extracted from the integrable structure and aligns these results with string-theoretic predictions, strengthening the link between Wilson-line cusps, collinear anomalous dimensions, and AdS/CFT dynamics.

Abstract

We write an integral equation that incorporates finite corrections to the large spin asymptotics of N=4 SYM twist operators from the non-linear integral equation. We show that these corrections are an all-loop result, not affected by wrapping effects, and agree, after determining the strong coupling expansion, with string theory predictions.