Strong coupling limit of Bethe Ansatz equations

TL;DR

This work develops a systematic framework to analyze the strong-coupling limit of Bethe ansatz equations for the planar ${\cal N}=4$ SYM spectrum in the rank-one sectors ${\rm su}(2)$, ${\rm su}(1|1)$, and ${\rm sl}(2)$ using an elliptic parametrization that becomes hyperbolic at large $g$. By formulating the Bethe equations as integral equations with a set of kernels, including a BES/BHL dressing kernel, the authors derive leading magnon densities and energies for highest excited states, both with and without dressing. They show that the BES dressing kernel yields well-behaved strong-coupling results and reproduce known leading terms such as $f(g)\sim 4g$ for twist-two operators, with explicit energies $E^{d}_{\text{su(2)}}\sim\sqrt{\pi g}\,L$ and $E^{d}_{\text{su(1|1)}}\sim\sqrt{2\pi g}\,L$ in certain sectors; the analysis also highlights the importance of near-flat-space regions for subleading corrections. The framework provides a controlled route to extract systematic $1/g$ expansions and offers insights into the underlying degrees of freedom at strong coupling, informing tests of the AdS/CFT correspondence and dressing-phase conjectures.

Abstract

We develop a method to analyze the strong coupling limit of the Bethe ansatz equations supposed to give the spectrum of anomalous dimensions of the planar ${\cal N}=4$ gauge theory. This method is particularly adapted for the three rank-one sectors, $su(2)$, $su(1|1)$ and $sl(2)$. We use the elliptic parametrization of the Bethe ansatz variables, which degenerates to a hyperbolic one in the strong coupling limit. We analyze the equations for the highest excited states in the su(2) and $su(1|1)$ sectors and for the state corresponding to the twist-two operator in the $sl(2)$ sector, both without and with the dressing kernel. In some cases we were able to give analytic expressions for the leading order magnon densities. Our method reproduces all existing analytical and numerical results for these states at the leading order.