= Crossed-channel pole in diagonal factorized scattering
{title2=$t(i\alpha)=m_a^2+m_b^2-2m_am_b\cos\alpha$}
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>.
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