Single production of vector-like top partner decaying to $Wb$ in the leptonic channel at $ep$ colliders in the LHT model

In the littlest Higgs model with T-parity(LHT), we study the single production of vector-like top partner with the subsequent decay $T_{+}\to Wb$ in the leptonic channel at the $ep$ colliders. Focus on the LHeC ($ \sqrt{s} $ = 1.98 TeV) and FCC-eh ($ \sqrt{s} $ = 5.29 TeV), we investigate the observability of the single top partner production with the unpolarized and polarized electron beams, respectively. As a result, the statistical significance can be enhanced by the polarized electron beams. Under the current constraints, the search for $T_{+}$ in the $Wb$ channel at the LHeC cannot provide a stronger limit on the top partner mass. By contrast, the search for the $T_{+}$ in this channel at the FCC-eh with polarized $ e^- $ beams can exclude the top partner mass up to 1350 GeV, 1500 GeV and 1565 GeV with integrated luminosities of 100 fb$^{-1}$, 1000 fb$^{-1}$ and 3000 fb$^{-1}$ at the 2$\sigma$ level, which is an improvement with respect to the current indirect searches and the LHC direct searches. Furthermore, we also give an extrapolation to the high-luminosity LHC with $\sqrt{s}=14$ TeV and $ L=3000~\rm{fb}^{-1} $. Our results show that the LHT model is still a natural solution to the shortcomings of the electroweak and scalar sector although it has been constrained severely.