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 $\mathrm{SO}(1,3)\mathrm{}$ and $\mathrm{SO}(1,2)\mathrm{}$ crossed-channel partial-wave analyses. The $\mathrm{SO}(1,2)\mathrm{}$ expansion, used when the momentum transfer $Q$ is spacelike, is studied in detail in the limit $Q_{\mu }\to 0$. The connection between the $\mathrm{SO}(1,3)\mathrm{}$ and $\mathrm{SO}(1,2)\mathrm{}$ partial-wave amplitudes at $Q_{\mu }=0\mathrm{}$ is obtained explicitly, as is the familiar result that a Toller pole is equivalent to an infinite sequence of integrally spaced Regge poles at $Q_{\mu }=0\mathrm{}$.