Crossed-channel pole in diagonal factorized scattering ($t(i\alpha )=m_a^2+m_b^2-2m_a{m}_b\cos \alpha$)

A physical-strip pole of a diagonal S-matrix can come from exchange of an existing particle in the crossed channel. For external masses $m_a,m_b$, use the momentum difference, rather than the sum, to identify the exchanged mass through the displayed invariant. Crossing symmetry turns the pole at $i\alpha$ into a direct pole at $i(\pi -\alpha )$ of the particle-antiparticle channel. For $m_B=2m\cos (u/2)$, the pole of $S_{BA}$ at $iu/2$ has $t=m^2$ and exchanges the existing charge-one particle; it is not a new charge-three bound state.