Search for anomalous $WW\gamma$ and $WWZ$ couplings with polarized $e$-beam at the LHeC

We examine the potential of the $ep\to\nu_{e}q\gamma X$ and $ep\to\nu_{e}qZX$ processes to search for the anomalous couplings of $WW\gamma$ and $WWZ$ vertices at the LHeC with the electron beam energy of $E_{e}=60$ GeV and $E_{e}=140$ GeV. The difference of maximum and minimum bounds on the anomalous couplings, ($\delta\Delta\kappa_{\gamma}$, $\delta\lambda_{\gamma}$) and ($\delta\Delta\kappa_{Z}$, $\delta\lambda_{Z}$) are obtained as ($0.990$, $0.122$) and ($0.362$, $0.012$) without electron beam polarization at the beam energy of $E_{e}=140$ GeV for an integrated luminosity of $L_{int}=100$ fb$^{-1}$, respectively. With the possibility of $e-$beam polarization, we obtain more improved results as ($0.975$, $0.118$) and ($0.285$, $0.009$) for ($\delta\Delta\kappa_{\gamma}, \delta\lambda_{\gamma}$) and ($\delta\Delta\kappa_{Z}, \delta\lambda_{Z}$), respectively. The results are found to be comparable with the current experimental limits obtained from two-parameter analysis at the lepton and hadron colliders. It is found that the limits on the anomalous couplings ($\Delta\kappa_{Z}$, $\lambda_{Z}$) through the process $ep\to\nu_{e}qZX$ at the LHeC can further improve the current experimental limits.