The Strong Coupling Limit of the Scaling Function from the Quantum String Bethe Ansatz
P. Y. Casteill, C. Kristjansen
TL;DR
This work uses the quantum string Bethe Ansatz to derive the strong-coupling behavior of the scaling function for low-twist, high-spin operators by analyzing a folded string with spins $S$ and $J$ in $AdS_3\times S^1$. The one-loop energy splits into analytic (spin-chain finite-size) and non-analytic (Hernandez-Lopez phase) parts, both computed exactly as functions of $z=(\sqrt{\lambda}/\pi J)\log(S/J)$, and their sum reproduces the string theory result. In the large-$z$ regime, the scaling function expands as $f(\lambda) = \frac{\sqrt{\lambda}}{\pi} - \frac{3\log(2)}{\pi} + O(\lambda^{-1/2})$, with the famous constant arising from a delicate cancellation between the two contributions. This confirms the consistency of the Bethe Ansatz approach with string theory for this observable and points to extensions to higher-loop orders.
Abstract
Using the quantum string Bethe ansatz we derive the one-loop energy of a folded string rotating with angular momenta (S,J) in AdS_3 x S^1 inside AdS_5 x S^5 in the limit 1 << J << S, z=λ^(1/2) log(S/J) /(πJ) fixed. The one-loop energy is a sum of two contributions, one originating from the Hernandez-Lopez phase and another one being due to spin chain finite size effects. We find a result which at the functional level exactly matches the result of a string theory computation. Expanding the result for large z we obtain the strong coupling limit of the scaling function for low twist, high spin operators of the SL(2) sector of N=4 SYM. In particular we recover the famous -3 log(2)/π. Its appearance is a result of non-trivial cancellations between the finite size effects and the Hernandez-Lopez correction.