  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.
