Feasibility study of $CP$ Violation in $\tau^{-}\to K_{S}\pi^{-}\nu_{\tau}$ decays at Super Tau Charm Facility.

We report a feasibility study of $CP$ violation of $\tau^{-}\rightarrow K_{S}\pi^{-} \nu_{\tau}$ decay at a Super Tau Charm Facility~(STCF).With an expected luminosity of 1~ab$^{-1}$ collected by STCF per year at a center-of-mass energy of 4.26 GeV, the statistical sensitivity for the $CP$ violation is determined to be of order $9.7\times10^{-4}$ by measuring the decay-rate difference between $\tau^{+}\rightarrow K_{S}\pi^{+}\bar{\nu}_{\tau}$ and $\tau^{-}\rightarrow K_{S}\pi^{-} \nu_{\tau}$. The analysis is performed using a reliable fast simulation software package, which can describe the detector responses properly and vary the responses flexibly for further optimization. Moreover, the energy-dependent efficiencies for reconstructing $\tau^{-}\rightarrow K_{S}\pi^{-} \nu_{\tau}$ are presented and the expected $CP$ sensitivity is proportional to $1/\sqrt{\mathcal{L}}$ in the energy region from 4.0 to 5.0 GeV. The sensitivity of $CP$ violation is of order $3.1\times10^{-4}$ with 10~ab$^{-1}$ integrated luminosity, which is equivalent to ten years data taking in this energy region at STCF.