 Sine-Gordon breather fusion amplitude

The second breather is a bound pair of first breathers with constituent shifts $\pm ia$. The bound-state fusion of factorized S-matrices gives $S_{2,1}=S_{1,1}(\vartheta +ia)S_{1,1}(\vartheta -ia)$, which factors as displayed. The nearest physical-strip pole at $\vartheta =ia$ is a crossed-channel exchange of the first breather. At $n=3$ the more distant center pole is double, without changing that nearest-pole interpretation.