Coupled Channel Analysis of $\bar{p} p \rightarrow \pi^0 \pi^0 \eta$, $\pi^0 \eta \eta$ and $K^+ K^- \pi^0$ at 900 $MeV/c$ and of $\pi\pi$-Scattering Data

A partial wave analysis of antiproton-proton annihilation data in flight at 900 $MeV/c$ into $\pi^0 \pi^0 \eta$ , $\pi^0 \eta \eta$ and $K^+ K^- \pi^0$ is presented. The data were taken at LEAR by the Crystal Barrel experiment in 1996. The three channels have been coupled together with $\pi\pi$-scattering isospin I=0 S- and D-wave as well as I=1 P-wave data utilizing the K-matrix approach. Analyticity is treated using Chew-Mandelstam functions. In the fit all ingredients of the K-matrix, including resonance masses and widths, were treated as free parameters. In spite of the large number of parameters, the fit results are in the ballpark of the values published by the Particle Data Group. In the channel $\pi^0 \pi^0 \eta$ a significant contribution of the spin exotic $I^G=1^-$ $J^{PC}=1^{-+}$ $\pi_1$-wave with a coupling to $\pi^0 \eta$ is observed. Furthermore the contributions of $\phi(1020) \pi^0$ and $K^*(892)^\pm K^\mp$ in the channel $K^+ K^- \pi^0$ have been studied in detail. The differential production cross section for the two reactions and the spin-density-matrix elements for the $\phi(1020)$ and $K^*(892)^\pm$ have been extracted. No spin-alignment is observed for both vector mesons. The spin density matrix elements have been also determined for the spin exotic wave.