Crossed-Channel Partial-Wave Expansions and the Bethe-Salpeter Equation

The Bethe-Salpeter equation for the direct-channel absorptive part of the scattering amplitude is analyzed in the ladder approximation to investigate the relationship between $\mathrm{SO}(1,3)$ and $\mathrm{SO}(1,2)$ crossed-channel partial-wave analyses. The $\mathrm{SO}(1,2)$ expansion, used when the momentum transfer $Q$ is spacelike, is studied in detail in the limit ${Q}_{\ensuremath{\mu}}\ensuremath{\rightarrow}0$. The connection between the $\mathrm{SO}(1,3)$ and $\mathrm{SO}(1,2)$ partial-wave amplitudes at ${Q}_{\ensuremath{\mu}}=0$ is obtained explicitly, as is the familiar result that a Toller pole is equivalent to an infinite sequence of integrally spaced Regge poles at ${Q}_{\ensuremath{\mu}}=0$.