Magnon Bound-state Scattering in Gauge and String Theory

TL;DR

This work develops a bootstrap/fusion framework for constructing bound-state S-matrices in the gauge/string AdS/CFT context and applies it to the BDS and AFS-type sectors, revealing a rich analytic structure including simple and double poles. By mapping two-magnon bound states to world-sheet solitons in the complex sine-Gordon theory, the paper shows that the large-$\lambda$ (strong coupling) limit of bound-state scattering agrees with semiclassical soliton scattering, and that the dressing phase has a well-defined world-sheet interpretation as soliton–antisoliton scattering. The analysis covers both fixed-charge and large-charge scaling regimes of bound states, derives explicit fused S-matrices for bound states in several sectors, and demonstrates consistency with the Landau–Lifshitz description in the small-momentum limit. Together, these results reinforce the integrability-based description of the spectrum in ${\cal N}=4$ SYM and its string dual, and illuminate how quantum corrections and higher-charge sectors fit into the binding and scattering of magnons.

Abstract

It has been shown that, in the infinite length limit, the magnons of the gauge theory spin chain can form bound states carrying one finite and one strictly infinite R-charge. These bound states have been argued to be associated to simple poles of the multi-particle scattering matrix and to world sheet solitons carrying the same charges. Classically, they can be mapped to the solitons of the complex sine-Gordon theory. Under relatively general assumptions we derive the condition that simple poles of the two-particle scattering matrix correspond to physical bound states and construct higher bound states ``one magnon at a time''. We construct the scattering matrix of the bound states of the BDS and the AFS S-matrices. The bound state S-matrix exhibits simple and double poles and thus its analytic structure is much richer than that of the elementary magnon S-matrix. We also discuss the bound states appearing in larger sectors and their S-matrices. The large 't Hooft coupling limit of the scattering phase of the bound states in the SU(2) sector is found to agree with the semiclassical scattering of world sheet solitons. Intriguingly, the contribution of the dressing phase has an independent world sheet interpretation as the soliton-antisoliton scattering phase shift. The small momentum limit provides independent tests of these identifications.