Measurement of the spin density matrix for the $\rho^0$, $K^{*0}(892)$ and $\phi$ produced in $Z^0$ Decays

The spin density matrix elements for the $\rho^0$, K$^{*0}(892)$ and $\phi$ produced in hadronic Z$^0$ decays are measured in the DELPHI detector. There is no evidence for spin alignment of the K$^{*0}(892)$ and $\phi$ in the region $x_p \leq 0.3$ ($x_p = p/p_{beam}$), where $\rho_{00} = 0.33 \pm 0.05$ and $\rho_{00} = 0.30 \pm 0.04$, respectively. In the fragmentation region, $x_p \geq 0.4$, there is some indication for spin alignment of the $\rho^0$ and K$^{*0}(892)$, since $\rho_{00} = 0.43 \pm 0.05$ and $\rho_{00} = 0.46 \pm 0.08$, respectively. These values are compared with those found in meson-induced hadronic reactions. For the $\phi$, $\rho_{00} = 0.30 \pm 0.04$ for $x_p \geq 0.4$ and $0.55 \pm 0.10$ for $x_p \geq 0.7$. The off-diagonal spin density matrix element $\rho_{1-1}$ is consistent with zero in all cases.