Two-Variable Lorentz-Group Expansions of Physical Scattering Amplitudes for Particles with Arbitrary Spins

Two-variable expansions of relativistic scattering amplitudes that have previously been suggested for the scattering and decays of spinless particles are generalized to the case of two-body scattering of particles with arbitrary spins. The usual helicity amplitudes are expanded in terms of the transformation matrices of the homogenous Lorentz group in a basis, corresponding to the group reduction $\mathrm{O}\mathrm{}(3,1)\mathrm{}\supset \mathrm{O}\mathrm{}\left(3\right)\mathrm{}\supset \mathrm{O}\mathrm{}\left(2\right)\mathrm{}$. The expansion can be interpreted as the usual Jacob and Wick partial-wave expansion, in which the energy dependence of the partial-wave helicity amplitudes is further expanded in terms of the $\mathrm{O}\mathrm{}(3,1)\mathrm{} d$ functions. Restrictions due to parity and time-reversal invariance are discussed. The O(3,1) expansions are shown to have the correct threshold behavior "term by term". Further generalizations of the formalism to include O(2,1) expansions (and thus Regge-pole theory) are discussed as well as applications to particle decays (these will be presented separately).