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 tqh, q = u, c). We probe the observability of the FCNC top-Higgs couplings through the processes e(-)p -> nu(e)(t) over bar -> nu(e)h (q) over bar signal I) and e(-)p -> nu(e)hb signal II) at the proposed electron proton ep colliders, where the Higgs boson decays to a b (b) over bar pair. We find that at the high-luminosity (1 ab(-1)) ep colliders where the electrons have a polarization of 80% and electron energy is typical 60 GeV, the 2 sigma upper limits on Brt -> uh) are 0.15 x 10(-2) 2.9 x 10(-4)) at the 7TeV@LHeC 50TeV@FCC-eh) for signal I and 0.15 x 10(-2) 2.2 x 10(-4)) for signal II. We also give an estimate on how the sensitivity (taking signal I as example) would change when we reduce the electron beam energy from 60 to 50 GeV or even 40 GeV due to the cost. The conclusion is that the discovery potential is reduced 8.7% (29.4%) if the electron beam changes from 60 to 50 (40) GeV at the 7 TeV LHeC, and 16.8% (19.8%) at the 50 TeV FCC-eh.