Crossed-channel lightest-breather exchange ($t(ia)=m_2^2+m_1^2-2m_2{m}_1\cos a=m_1^2$)

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.