Physical rapidity strip ($0<\mathrm{Im}\vartheta <\pi$)

For relativistic two-body scattering in one spatial dimension, this strip is the standard domain containing bound-state and crossed-channel poles. A stable-particle interpretation requires appropriate pole kinematics and residues; not every singularity automatically represents a new particle. Crossing relates rapidity $\vartheta$ to $i\pi -\vartheta$.

At relative rapidity $iu$, the analytically continued sum of the constituent four-momentum vectors is on the bound-state mass shell with the displayed mass. For equal constituents, rapidities $\chi \pm iu/2$ sum to $2m_a\cos (u/2)(\cosh \chi ,\sinh \chi )$. A pole at $u=0$ is at threshold rather than a strictly bound state.