Searches for the Anomalous FCNC Top-Higgs Couplings with Polarized Electron Beam at the LHeC

In this paper, we study the single top and Higgs associated production $\rm e^- p\rightarrow \nu_e \bar{t} \rightarrow \nu_e h \bar{q}(h\rightarrow b\bar{b})$ in the top-Higgs FCNC couplings at the LHeC with the electron beam energy of $E_e$ = 60 GeV and $E_e$ = 120 GeV, combination of a 7 TeV and 50 TeV proton beam. With the possibility of e-beam polarization ($p_e$ = 0, $\pm 0.6$), we distinct the Cut-based method and the Multivariate Analysis (MVA) based method, and compare with the current experimental and theoretical limits. It is shown that the branching ratio Br $(\mathrm{t}\to \mathrm{u}\mathrm{h})$ can be probed to 0.113 (0.093) $\mathrm{\%}$, 0.071 (0.057) $\mathrm{\%}$, 0.030 (0.022) $\mathrm{\%}$ and 0.024 (0.019) $\mathrm{\%}$ with the Cut-based (MVA-based) analysis at ($E_p$, $E_e$) = (7 TeV, 60 GeV), ($E_p$, $E_e$) = (7 TeV, 120 GeV), ($E_p$, $E_e$) = (50 TeV, 60 GeV) and ($E_p$, $E_e$) = (50 TeV, 120 GeV) beam energy and 1$\sigma$ level. With the possibility of e-beam polarization, the expected limits can be probed down to 0.090 (0.073) $\mathrm{\%}$, 0.056 (0.045) $\mathrm{\%}$, 0.024 (0.018) $\mathrm{\%}$ and 0.019 (0.015) $\mathrm{\%}$, respectively.