Discrete two-variable expansions of scattering amplitudes for particles with spin

Abstract

Scattering amplitudes for two-body reactions involving particles with arbitrary spins are expanded in terms of functions of the energy and scattering angle. The expansions are in terms of the representation matrices of the O(4) group and are a modification of previously obtained two-variable expansions in terms of the representations of the internal Lorentz group. The new expansions involve only sums, rather than integrals. They can be interpreted as partial-wave expansions, supplemented by a further expansion of each partial wave in terms of the O(4) $d$ functions. The partial waves automatically have the correct threshold behavior and the O(4) amplitudes (the expansion coefficients) have simple properties with respect to the usual discrete symmetries (parity, etc.). The expansions can be applied to analyze two-body scattering data simultaneously for all angles $0\ensuremath{\le}\ensuremath{\theta}\ensuremath{\le}\ensuremath{\pi}$, $0\ensuremath{\le}\ensuremath{\varphi}l2\ensuremath{\pi}$ and all energies from threshold to some a priori chosen maximal energy.