Kinematic Singularities of Partial-Wave Scattering Amplitudes

Abstract

Those singularities (and zeros) introduced by partial-wave projection of unequal-mass scattering amplitudes are investigated. The particular points discussed are: (1) Threshold zeros, especially at the crossed threshold. We find that unitarity corrections vanish like ${k}^{L+{L}^{\ensuremath{'}}}$ at the crossed as well as the direct threshold, and thus it is appropriate to associate ${k}^{L+{L}^{\ensuremath{'}}}$ behavior of the amplitude with both thresholds. (2) The limit as $s\ensuremath{\rightarrow}0$ is related to the backward asymptotic behavior in the crossed $u$ and $t$ channels. The partial-wave amplitude goes like ${s}^{\ensuremath{-}\ensuremath{\alpha}}$ for all $L$ as $s\ensuremath{\rightarrow}0$, where $\ensuremath{\alpha}$ is the asymptotic power of the crossed-channel backward amplitudes. In Regge-pole theory, this would be leading direct-channel Regge trajectory. (3) Spin effects are shown to lead to $\ensuremath{\surd}s$ singularities for bosons as well as fermions. Generalized MacDowell relations are derived, relating negative-energy amplitudes to positive-energy amplitudes for any spin and determining the conditions under which there will be singularities.