Exploring dark $Z_d$-boson in future Large Hadron-electron collider

The interaction between the dark $U(1)_d$ sector with the visible Standard Model (SM) sector takes place through the kinetic mixing between the dark photon $U(1)_d$ field $Z_d^\mu$ and the SM $U(1)_Y$ gauge field $B_\mu$. After the electroweak and $U(1)_d$ symmetry breaking, the dark photon $Z_d^\mu$ acquires a mass and mixes with the SM neutral vector boson $Z_\mu$. This mixing leads to parity-violating coupling between the $Z_d^\mu$ and SM. The coupling between the dark photon and SM can be explored in low energy phenomenology as well as in collider experiments. The Lorentz structure of dark photon interaction with SM fermions is explored in the proposed high energy future Large Hadron-electron collider, which would provide efficient energy and a clean environment using cross-section and asymmetries associated with polarisation observable of the dark photon in leptons decay. A $\chi^2$-analysis is performed to compare the strength of various variables for both the charge- and neutral-current processes. Based on this analysis, $90\%$ confidence level (C.L.) contours in the $\epsilon$-$m_{Z_d}$ and $\epsilon$-$g_V$ plane are obtained to put limits on the $Z_d^\mu$ mass up to $100$ GeV, coupling strength $\epsilon$ and on the Lorentz structure of dark photon coupling with the SM fermions ($g_V$) at $\sqrt{s} \approx 1.3$ TeV.