Two-Variable Expansions and theK→3πDecays

Previously suggested two-variable expansions of three-body decay amplitudes in terms of harmonic functions of an O(4) group are discussed and applied to analyze the Dalitz-plot distribution of over 3.2 million $K^{\pm }\to {\pi }^{\pm }{\pi }^{\pm }{\pi }^{\mp }$ decay events. Among the general features of the O(4) expansions we wish to stress that they are written in the c.m. system of two of the final particles, the angular momentum of which is displayed explicitly, and that each term in the expansion has a good behavior at the threshold, pseudothreshold, and at the boundary of the physical region. We analyze the recent data of Ford et al. on charged $K\to 3\pi$ decays, using both O(4) expansion and the standard power-series expansion in terms of the Dalitz-Fabri variables. In both cases it is perfectly adequate to keep four terms in the corresponding expansion. The ${\chi }^2$ fit is marginally better for the O(4) expansion. We conclude that the $K\to 3\pi$ Dalitz plot has too little structure in it to provide a real test of the advantages or disadvantages of different treatments. It is thus most desirable to apply the O(4) expansions to Dalitz plots of other processes, like $\eta \to 3\pi$ or $\overline{p}n\to \pi \pi \pi$. No conclusive evidence is found for $\mathrm{CP}$ violation. However, the "linear" term in the O(4) expansion of the difference between the squared matrix elements for $K^{+}$ and $K^{-}$ decays does differ from zero by more than two standard deviations. The effect is stable with regard to the number of terms kept in the expansions. An important distinctive feature of the O(4) expansions is their intimate relation to two-variable O(3, 1) expansions of physical scattering amplitudes.