O(4)Expansion of Off-Shell Scattering Amplitudes

Using an off-mass-shell approach, we develop a method for the $O\left(4\right)\mathrm{}$ expansion of the scattering amplitude for two unequal-mass, arbitrary-spin particles. General-spin spherical harmonics for $O\left(4\right)\mathrm{}$ are constructed. With the aid of these spherical harmonics, the $M$ amplitude is decomposed into $O\left(4\right)\mathrm{}$ partial waves. The prescription for obtaining the on-mass-shell helicity states from the $O\left(4\right)\mathrm{}$ partial waves is given. This $O\left(4\right)\mathrm{}$ expansion, which is valid even away from $t=0\mathrm{}$, is useful in simplifying the Bethe-Salpeter equation.