Kinematic Constraints, Crossing, and the Reggeization of Scattering Amplitudes

We investigate the constraints among helicity amplitudes which hold at the boundary of the crossed-channel physical region, like the one discovered for nucleon-nucleon scattering by Goldberger, Grisaru, MacDowell, and Wong, and we develop a general scheme for computing these rules with arbitrary external spins. For equal-mass elastic scattering these constraints hold at $s=0\mathrm{}$; a catalog of them for several low-spin cases is presented. Complete knowledge of these conditions is needed to discuss, for a general scattering amplitude, a question first raised by Gell-Mann et al. in connection with electrodynamics, namely, under what conditions must a given expression for an $S$ matrix (e.g., the perturbation expansion of a field theory) be analytic in the angular momentum? Mandelstam has argued that this "Reggeization" of the vectorspinor amplitude is a consequence of kinematic constraints, and hence independent of the perturbation expansion. He discussed only the equal-mass, $j=\frac{1}{2}$ case. We generalize his argument to unequal masses and to arbitrary spins and angular momenta, and discuss, case by case, the necessity of Reggeization of a list of low-spin amplitudes. We find in particular that under rather general assumptions a spin-½ particle must lie on a Regge trajectory in a large class of amplitudes, and that the ${\pi }^{+}p$ amplitude is analytic in $j$ in perturbation theory.