Singularities of the Magnon Boundstate S-Matrix

TL;DR

This work analyzes the singularity structure of the conjectured exact S-matrix for magnon boundstate scattering in planar ${\cal N}=4$ SYM, showing that all poles near the real axis are accounted for by physical, on-shell intermediate BPS states. By constructing the boundstate S-matrix via fusion and examining simple and double poles from both the BDS and BES dressing factors, the authors connect each physical pole to concrete Landau diagrams and Bethe-string configurations. They identify which simple poles and which double-pole branches are physical in the giant-magnon, plane-wave, and Heisenberg limits, and provide a detailed Landau-diagram decoding (box, bow-tie) that matches the pole spectrum. The results reinforce the consistency of the boundstate S-matrix within the all-loop integrable framework and offer a platform for future tests, including residue computations and checks against gauge theory or worldsheet string theory.

Abstract

We study the conjectured exact S-matrix for the scattering of BPS magnon boundstates in the spin-chain description of planar N=4 SUSY Yang-Mills. The conjectured S-matrix exhibits both simple and double poles at complex momenta. Some of these poles lie parametrically close to the real axis in momentum space on the branch where particle energies are positive. We show that all such poles are precisely accounted for by physical processes involving one or more on-shell intermediate particles belonging to the known BPS spectrum.

Figures (9)

• Figure 1: Constructing boundstate S-matrix by fusion. Each boundstate is represented by an equally spaced sequence of Bethe roots (Bethe string).
• Figure 2: Building blocks of physical processes. The "double line" notation of Dorey:2007xn is employed, and time flows from bottom to top. The dotted line indicates the corresponding particles carry negative charges.
• Figure 3: (Examples of) diagrams describing four simple poles $Y_{1}^{+}=Y_{2}^{-}$ , $Y_{1}^{+}=1/Y_{2}^{-}$ , $Y_{1}^{+}=Y_{2}^{+}$ and $Y_{1}^{+}=1/Y_{2}^{+}$ . They correspond to the diagrams (a) - (d) respectively. The diagrams (a) - (c) describe physical processes, whereas (d) is not an allowed process.
• Figure 4: "Box" processes that give rise to double poles. In (A), the absolute value of $X_{1}^{\pm}$ is greater than one, while in (B) it is smaller than one (see Figure \ref{['fig:rap']}). In Section \ref{['sec:Bethe String']}, the two sets of time-slices, (A-1,2) and (B-1,2), will be interpreted as two different ways of viewing the same Bethe root configurations.
• Figure 5: "Bow-tie" process that also rises to double poles Dorey:2007xn. There is a blob in the centre of the diagram, therefore special care has to be paid for the case both the intermediate plane wave magnons carry charge $Q_{1}+Q_{2}$ .