Kinematic Singularities and Threshold Relations for Helicity Amplitudes

The kinematic singularities of two-body helicity amplitudes at thresholds and the concomitant relations among these amplitudes are discussed in a direct and elementary way, without recourse to the singularity structure of the crossing matrix. The tools are those of nonrelativistic quantum mechanics, as befits a situation where $p\to 0$, with spins combined into channel spins S and Russell-Saunders coupling of L+S=J. The kinematic singularities are shown to follow from a mismatch between $J$ and $L$ for each term in the partial-wave series. The method is applicable at pseudothresholds ${(m_1-m_2)}^2$ as well as normal thresholds ${(m_1+m_2)}^2$ with two formal changes involving an intrinsic parity and a helicity-dependent phase. The relations among the different helicity amplitudes at the thresholds are shown to result from the presence at threshold of fewer Russell-Saunders amplitudes than there are independent helicity amplitudes. The use of invariant amplitudes is shown to be an alternative which automatically yields the kinematic singularities and also the threshold relations among the helicity amplitudes. A discussion is given of dynamical exceptions to the threshold constraints, resulting from less singular than standard behavior at a threshold. The threshold relations are important constraints on the amplitudes, and must be satisfied by any realistic model. In the use of $t$-channel amplitudes for peripheral processes in the $s$ channel, the explicit imposition of all the relations at the $t$-channel thresholds is necessary in order to assure a differential cross section without spurious, polelike singularities in $t$ whose variation could in some circumstances completely control the $t$ dependence. The reactions $\pi N\to \mathrm{KY}$ and $\pi N\to {\pi }^{\prime }\Delta$ are used as illustrations. The latter process is especially illuminating because its $t$-channel amplitudes have a pole (rather than a simple inverse-square-root singularity) at the $\overline{N}\Delta$ pseudothreshold, $t=0.09\mathrm{} {\left(\frac{\mathrm{GeV}}{c}\right)}^2$. The proximity of this point to the physical region of the $s$ channel means that the threshold relations there are of crucial importance. The consequences of these constraints on the cross-section and decay density matrix of the $\Delta$ are discussed within the framework of the Regge-pole model. Comparison with experiment implies that the dynamics make the amplitudes for $\pi {\pi }^{\prime }\to \overline{N}\Delta$ have less than the standard kinematic singularity at $\overline{N}\Delta$ pseudothreshold and so avoid almost all the threshold constraints. Examples are cited from the literature where use of Regge-pole formulas possessing the spurious kinematic factors has led to incorrect inferences concerning the dynamic behavior of Regge residues.