Two-Variable Expansions for Three-Body Decays Involving Particles with Arbitrary Spins

Helicity amplitudes for the decay process $1\to 2+3+4$ involving particles with arbitrary spins are expanded in terms of matrix elements of the O(4) group transformations. The resulting expansions display the entire dependence on the kinematic parameters (energies and angles) explicitly and can be interpreted as standard O(3) partial-wave expansions of the amplitudes in a center-of-mass-like frame of reference, supplemented by O(4) expansions of the partial-wave helicity amplitudes. Restrictions on the O(4) amplitudes due to parity conservation are established, the physical meaning of all the quantum numbers figuring in the expansion is clarified, and the expansions are shown to have a reasonable threshold and pseudothreshold behavior. The expansion formula, as well as its inverse, involves amplitudes defined in the physical decay region only. The O(4) expansions of this paper are a modification of O(3, 1) expansions of helicity amplitudes for the scattering of particles with arbitrary spins, suggested in the preceding paper. For spinless particles the corresponding expansions have been derived and applied earlier by Hicks and Winternitz.