Tests of $CP$ symmetry in entangled ${\Xi }^{0}-{\overline{\Xi }}^{0}$ pairs

The $J/\psi \to {\Xi }^{0}{\overline{\Xi }}^{0}$ process and subsequent decays are investigated using $\left(10087±44\right)×{10}^{6}\text{}\text{}J/\psi $ events collected at the BESIII experiment. The decay parameters of ${\Xi }^{0}$ and ${\overline{\Xi }}^{0}$ are simultaneously measured to be ${\alpha }_{\Xi }=-0.3750±0.0034±0.0016$, ${\overline{\alpha }}_{\Xi }=0.3790±0.0034±0.0021$, ${\varphi }_{\Xi }=0.0051±0.0096±0.0018\text{}\text{}\mathrm{rad}$, ${\overline{\varphi }}_{\Xi }=-0.0053±0.0097±0.0019\text{}\text{}\mathrm{rad}$ with unprecedented accuracies, where the first and the second uncertainties are statistical and systematic, respectively. The most precise values for $CP$ asymmetry observables of ${\Xi }^{0}$ decay are obtained to be ${A}_{CP}^{\Xi }=\left(-5.4±6.5±3.1\right)×{10}^{-3}$ and $\Delta {\varphi }_{CP}^{\Xi }=\left(-0.1±6.9±\phantom{\rule{0ex}{0ex}}0.9\right)×{10}^{-3}\text{}\text{}\mathrm{rad}$. For the first time, the weak and strong phase differences are determined to be ${\xi }_{P}-{\xi }_{S}=\left(0.0±1.7±0.2\right)×{10}^{-2}\text{}\text{}\mathrm{rad}$ and ${\delta }_{P}-{\delta }_{S}=\left(-1.3±1.7±0.4\right)×{10}^{-2}\text{}\text{}\mathrm{rad}$, which are the most precise results for any weakly decaying baryon. These results will play important roles in the studies of the $CP$ violations and polarizations for the strange, charmed, and beauty baryons.