Sine-Gordon breather spectrum at reflectionless couplings ($m_k=2M\sin [k\pi /(2n)],k=1,\dots ,n-1$)

At renormalized coupling $\gamma =8\pi /n$, the kink-antikink transmission poles lie at $\vartheta =i\pi (1-k/n)$. The relativistic bound-state mass from a rapidity pole gives the displayed increasing breather masses. The kink mass is $M=mn/\pi$, and $k=n$ is an excluded threshold state. No breathers occur at $n=1$; scattering involving a physical second breather requires $n\ge 3$.

For the transmitting product with factors $\cosh [(\theta -i\pi j/N)/2]/\cosh [(\theta +i\pi j/N)/2]$ and prefactor $(-1{)}^N$, the continuous phase on $\theta \ge 0$ is $\pi N-2{\sum }_{j=1}^{N-1}\arctan [\tanh (\theta /2)\tan (\pi j/(2N))]$. Its derivative is $-{\sum }_j\sin (\pi j/N)/[\cosh \theta +\cos (\pi j/N)]$. A Riemann sum at fixed positive rapidity gives $(2N/\pi )\log \tanh (\theta /2)$ at leading order. The endpoints depend on the stated unwrapped branch; setting the high-energy value to zero silently changes the constant.

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.

At the closest physical-strip pole of $S_{2,1}$, the difference of the analytically continued external momenta has squared mass $m_1^2$. Using $m_2=2m_1\cos a$ makes the identity immediate. Thus the exchanged t-channel one-particle state is the lightest Sine-Gordon breather, not a new member of the spectrum.