Entanglement measures and Bell-type spin-correlation observables in tau-lepton pairs at the Super Tau-Charm Facility

Within the framework of quantum field theory and the Standard Model (SM), we investigate the prospects of studying entanglement measures and Bell-type spin-correlation observables in the electroweak process $e^{-}{e}^{+}\to {\tau }^{-}{\tau }^{+}$ at the center-of-mass (COM) energies of $\sqrt{s}=3.670,4.630$, and $7.000$ GeV at the proposed Super Tau-Charm Facility (STCF) in China. Focusing on the hadronic decay channel ${\tau }^{\pm }\to {\pi }^{\pm }\nu$, we determine the spin-correlation coefficients of the ${\tau }^{-}{\tau }^{+}$ system within the SM framework using measurable production kinematics, namely the $\tau$ velocities and scattering angles in the COM frame. From these correlation coefficients, we construct concurrence and a Bell-type correlation combination as defined in the literature. Assuming an integrated luminosity of 1 ab${}^{-1}$ for each considered COM energy, and incorporating realistic detector efficiencies as well as statistical and systematic uncertainties, we estimate the expected sensitivity to these observables at STCF. Our results indicate that, under the SM hypothesis, the corresponding Bell-type correlation combinations could be resolved with high statistical significance at STCF.