The scaling function at strong coupling from the quantum string Bethe equations
Matteo Beccaria, Gian Fabrizio De Angelis, Valentina Forini
TL;DR
The paper addresses the problem of determining the strong-coupling scaling function $f(g)$ governing the large-spin anomalous dimension of twist-two operators in ${\cal N}=4$ SYM by analyzing the quantum string Bethe equations in the $\mathfrak{sl}(2)$ sector and the strong-coupling Eden–Staudacher equation with the leading Arutyunov–Frolov–Staudacher (AFS) dressing. It combines numerical solutions of the string Bethe equations to test short-string (GKP, BMN) limits with analytical and perturbative studies of the strong-coupling ES equation, proving a unique asymptotic solution that agrees with semiclassical string predictions and Alday et al. The work shows that the strong-coupling dressing can be consistently incorporated and that the resulting scaling function matches expectations from AdS/CFT, providing a nontrivial cross-check between gauge theory and string theory descriptions. Overall, it strengthens the understanding of the dressing phase’s role at strong coupling and demonstrates the feasibility of extracting universal, high-precision results from asymptotic Bethe ansatz equations in this regime.
Abstract
We study at strong coupling the scaling function describing the large spin anomalous dimension of twist two operators in ${\cal N}=4$ super Yang-Mills theory. In the spirit of AdS/CFT duality, it is possible to extract it from the string Bethe Ansatz equations in the $\mathfrak{sl}(2)$ sector of the $\ads$ superstring. To this aim, we present a detailed analysis of the Bethe equations by numerical and analytical methods. We recover several short string semiclassical results as a check. In the more difficult case of the long string limit providing the scaling function, we analyze the strong coupling version of the Eden-Staudacher equation, including the Arutyunov-Frolov-Staudacher phase. We prove that it admits a unique solution, at least in perturbation theory, leading to the correct prediction consistent with semiclassical string calculations.