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The spin density matrix of the $\omega$ has been determined for the reaction$\bar{p} p \, \rightarrow \, \omega \pi^0$ with unpolarized in-flight datameasured by the Crystal Barrel LEAR experiment at CERN. The two main decaymodes of the $\omega$ into $\pi^0 \gamma $ and $\pi^+ \pi^- \pi^0$ have beenseparately analyzed for various $\bar{p}$ momenta between 600 and 1940 MeV/c.The results obtained with the usual method by extracting the matrix elementsvia the $\omega$ decay angular distributions and with the more sophisticatedmethod via a full partial wave analysis are in good agreement. A strong spinalignment of the $\omega$ is clearly visible in this energy regime and allindividual spin density matrix elements exhibit an oscillatory dependence onthe production angle. In addition, the largest contributing orbital angularmomentum of the $\bar{p}p$ system has been identified for the different beammomenta. It increases from $L^{max}_{\bar{p}p}$ = 2 at 600 MeV/c to$L^{max}_{\bar{p}p}$ = 5 at 1940 MeV/c.

Spin Density Matrix of the $\omega$ in the Reaction $\bar{p} p \, \rightarrow \, \omega \pi^0$