Measuring CP nature of top-Higgs couplings at the future Large Hadron electron collider

We investigate the sensitivity of top-Higgs coupling by considering the associated vertex as CP phase ($\zeta_t$) dependent through the process $p\, e^- \to \bar t \,h \,\nu_e$ in the future Large Hadron electron collider. In particular the decay modes are taken to be $h \to b\bar b$ and $\bar t \to$ leptonic mode. Several distinct $\zeta_t$ dependent features are demonstrated by considering observables like cross sections, top-quark polarisation, rapidity difference between $h$ and $\bar t$ and different angular asymmetries. Luminosity ($L$) dependent exclusion limits are obtained for $\zeta_t$ by considering significance based on fiducial cross sections at different $\sigma$-levels. For electron and proton beam-energies of 60 GeV and 7 TeV respectively, at $L = 100$~fb$^{-1}$, the regions above $\pi/5 < \zeta_t \leq \pi$ are excluded at 2$\sigma$ confidence level, which reflects better sensitivity expected at the Large Hadron Collider. With appropriate error fitting methodology we find that the accuracy of SM top-Higgs coupling could be measured to be $\kappa = 1.00 \pm 0.17 (0.08)$ at $\sqrt{s} = 1.3 (1.8)$ TeV for an ultimate $L = 1 \,\rm{ab}^{-1}$.