Exploring the Anomalous Top-Higgs FCNC Couplings at the electron proton colliders

We perform an updated analysis on the searches for the anomalous FCNC Yukawa interactions between the top quark, the Higgs boson, and either an up or charm quark ($\rm tqh,\ q=u,\ c$). We probe the observability of the FCNC top-Higgs couplings through the processes $\rm e^- p\rightarrow \nu_e \bar{t} \rightarrow \nu_e h \bar{q}$ (signal.I) and $\rm \ e^- p \to \nu_e h b$ (singal.II) at the proposed electron proton (ep) colliders, where the Higgs boson decays to a $\rm b\bar{b}$ pair. We find that at the high luminosity (1 $\rm ab^{-1}$) ep colliders where the electrons have a polarisation of $\rm 80\%$ and electron energy is typical 60 GeV, the 2$\sigma$ upper limit on $\rm Br(t\to uh)$ are $0.15\times 10^{-2}$($2.9\times 10^{-4}$) at the 7TeV@LHeC(50TeV@FCC-eh) for signal.I and $0.15\times 10^{-2}$($2.2\times 10^{-4}$) for signal.II. We also give an estimate on how the sensitivity (take signal.I as examples) would change when we reduce the electron beam energy from 60 GeV to 50 GeV or even 40 GeV due to the cost reason. The conclusion is that the discovery potential reduce $8.7\%$($29.4\%$) if the electron beam change from 60GeV to 50(40) GeV at the 7TeV LHeC, and $16.8\%$($19.8\%$) at the 50 TeV FCC-eh.