Polarization analysis in $e^+e^-\rightarrow J/\psi\rightarrow$ baryon-antibaryon pairs

We present a comprehensive polarization analysis for the process $e^{+}e^{-} \rightarrow J/\psi \rightarrow$ baryon-antibaryon pairs~$B_{1}\bar{B}_{2}$. All possible helicity amplitudes for $e^{+}e^{-} \rightarrow J/\psi$ and $J/\psi \rightarrow B_{1} \bar{B}_{2}$ are provided, along with their transformation properties under parity and $CP$ operations. Taking full account of beam polarization, we analyze the polarization of the $J/\psi$ produced in $e^{+}e^{-}$ annihilation and show that all eight independent polarization components can be nonzero. By introducing a polarization transfer matrix, we propose a novel method to compute the polarization transfer in the decay $J/\psi \rightarrow B_{1} \bar{B}_{2}$. Considering both hadronic and radiative baryon decay models, we derive the joint angular distribution of the final-state particles. Focusing on the process $J/\psi \rightarrow \Lambda \bar{\Sigma}^{0}$, we provide statistical uncertainty estimates for probing $CP$ violation in associated production of different baryons. This work establishes a complete framework for studying the parity and $CP$ violations present in baryon productions and decays within the helicity formalism.