Relativistic Two-Variable Expansions for Three-Body Decay Amplitudes

Three-body decays 1→2+3+4 are considered using a frame of reference analogous to the c.m. system for scattering. The physical decay region is mapped onto an O(4) sphere, so that the decay amplitude $f(\alpha ,\theta )$ is a function on this sphere (depending only on two of the three angles $\alpha$, $\theta$, and $\phi$). The amplitude is then expanded in terms of the basis functions of O(4) and we obtain two-variable expansions, in which all the dependence on the kinematic parameters is explicitly displayed in special functions. These expansions make it possible to treat decays and scattering on the same footing, in that they are intimately related to O(3,1) expansions of scattering amplitudes, considered previously. Some analyticity properties are built in, so that each partial wave has the correct behavior at the threshold ${(m_3+m_4)}^2$ and pseudothreshold ${(m_1-m_2)}^2$. The expansions make it possible to perform a kinematically (or group-theoretically) motivated harmonic analysis of Dalitz-plot distributions for $K\to 3\pi$ and $\eta \to 3\pi$ decays (the results will be presented separately).