Searching for the charged-current non-standard neutrino interactions at the $e^{-}p$ colliders

Considering the theoretical constraints on the charged-current (CC) non-standard neutrino interaction (NSI) parameters in a simplified $W'$ model, we study the sensitivities of the Large Hadron electron Collider (LHeC) and Future Circular Collider-hadron electron (FCC-he) to the CC NSI parameters through the subprocess $e^{-}q \to W' \to v_{e}q'$. Our results show that the LHeC with $\mathcal{L} = 100\;{\rm fb}^{-1}$ is a little more sensitive to the CC NSI parameters than that of the high-luminosity (HL) LHC, and the absolute values of the lower limits on the CC NSI parameters at the FCC-he are smaller than those of the HL-LHC about two orders of magnitude.