Precise measurement of CP violating $τ$ EDM through $e^+ e^- \to γ^*, ψ(2s) \to τ^+ τ^-$

A nonzero electric dipole moment of a tauon, $d_τ$, signals CP violation and provides an important probe for new physics. We study methods to measure $d_τ$ at low energy $e^+ e^-$ colliders through the processes $e^+e^- \to γ^*, ψ(2S) \to τ^+τ^-$ with $τ^\pm$ decays into a charged hadron and a tau neutrino. We point out that, with measuring energies of the charged hadron, Im$(d_τ)$ can be measured. On the other hand, selecting events of $τ$ decays after traveling more than the detector resolution distance, Re$(d_τ)$ can also be determined. We find that the precision at Super Tau-Charm Facility (STCF) running at the center energy of $m_{ψ(2S)}$ for 10 year data accumulation, the precision of Im$(d_τ)$ and Re$(d_τ)$ are found to be 1.8 and 11 in unit of $ 10^{-18}~e\,\text{cm}$, respectively. The sensitivity for $d_τ$ measurement precision at the STCF can be reached its optimum at a central energy of $6.3~\text{GeV}$, achieving a precision of $0.7$ for Im$(d_τ)$ and $2.8$ for Re$(d_τ)$ in unit of $ 10^{-18}~e\,\text{cm}$.