Analyzing $t\bar{t}Z$-couplings at the future $e^-p$ collider

The proposed Large Hadron Electron Collider (LHeC), with center-of-mass energy of $\sqrt{s}\approx 1.3$ TeV, provides a clean and sensitive environment to probe the top quark's neutral current interactions with the $Z$ boson via the process $e^- p \to e^- t \bar{t}$. We investigate the precision with which the Standard Model (SM) $t\bar{t}Z$ couplings-the vector and axial-vector components ($\Delta C_{1V}$, $\Delta C_{1A}$)-can be measured, along with possible new physics effects parameterized by higher-dimensional operators inducing weak electric and magnetic dipole-like interactions ($C_{2V}$, $C_{2A}$). Focusing on the semileptonic decay channel, where either the top quark or anti-top decays leptonically to a positively charged lepton ($\ell^+ = e^+, \mu^+$), we utilize the azimuthal angle difference $\Delta \phi$ between the scattered electron and the charged lepton as the key observable. Using a one-parameter multi-bin $\chi^2$-analysis of this differential distribution, we find that constraints on $\Delta C_{1V}$ and $\Delta C_{1A}$ improve from order $10^{-1}$ at 50 fb$^{-1}$ to order $10^{-2}$ at 1000 fb$^{-1}$, corresponding to approximately 50% and 6% precision relative to their SM values. The anomalous tensor couplings $C_{2V}$ and $C_{2A}$ are constrained at the $10^{-1}$ level even at low luminosity and improve moderately with high luminosity. While the two-parameter analysis broadens the allowed regions due to parameter correlations, it retains competitive sensitivity, particularly for SM-like couplings. A systematic uncertainty of 5% is assumed throughout. These results highlight the LHeC's potential to provide complementary and competitive sensitivity to top-$Z$ couplings compared to current and future hadron and lepton collider capabilities.