Spin Density Matrix for $\Omega^-$ and its Polarization Alignment in $\psi(3686) \to \Omega^- \bar\Omega^+$

We investigate the spin density matrix of $\Omega^-$ in the Cartesian coordinate system of baryon-antibaryon pairs produced in $e^{+}e^{-}$ annihilation. Using the helicity formalism of Jacob and Wick, we derive the expression for the spin-3/2 density matrices. Our analysis is based on the angular distribution of the process $e^+e^-\rightarrow\psi(3686)\rightarrow\Omega^-\bar{\Omega}^+$ in the BESIII experiment. By decomposing the polarization state of $\Omega^-$ particles along different coordinate axes, we examine the polarization dependence of the cross-section. Our results demonstrate that $\Omega^-$ particles exhibit varying degrees of tensor polarization along the x-, y-, and z-axes, as well as weak vector polarization and rank-3 tensor polarization along the y-axis. To our knowledge, this is the first study to calculate the polarization dependence of the cross-section distributions for the annihilation process $e^+e^-\rightarrow\Omega^-\bar{\Omega}^+$. Our theoretical predictions are in good agreement with the experimental measurements.