Single production of vector-like $T$ quark at future high-energy linear $e^{+}{e}^{-}$ collider

Based on a model-independent framework including the vector-like top partner (VLQ-$T$), we investigate the prospect of discovering the singlet or doublet VLQ-$T$ via the single production process $e^{-}{e}^{+}\to T\bar{t}+t\bar{T}$ with the $T\to Zt$ decay channel at future high energy linear $e^{+}{e}^{-}$ collider with $\sqrt{s}=3$ TeV. We focus on the hadronic decay of the top quark and two types of decay channel for the $Z$ boson: $Z\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$ and $Z\to \nu \bar{\nu }$. By carrying out a full simulation for the signals and the relevant SM backgrounds, the $2\sigma$ exclusion limit and $5\sigma$ discovery prospects are, respectively, obtained on the VLQ-$T$ mass and the coupling strength $g^{\ast }$ with the integrated luminosity of 5 ab${}^{-1}$. In addition, we considered the initial state radiation and beamstrahlung effects as well as the systematic uncertainty effects of backgrounds, which are found to reduce the excluding or discovery capability.