Search for Anomalous $WW\gamma $ and $WWZ$ Couplings with Polarized $e$-Beam at the LHeC

We examine the possibility of constraining the anomalous $WW\gamma$ and $WWZ$ couplings by measuring total cross sections of the $ep\to\nu_{e}q\gamma X$ and $ep\to\nu_{e}qZX$ processes at the LHeC collider with the electron beam energy $E_{e}=60$ GeV and $E_{e}=140$ GeV. We consider the cases of unpolarized and polarized electron beams. The difference of the upper and lower 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 $L_{\mathrm{int}}=100$~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 fits to the data collected in 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.