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Search for anomalous $WWγ$ and $WWZ$ couplings with polarized $e$-beam at the LHeC

We examine the potential of the $ep\toν_{e}qγX$ and $ep\toν_{e}qZX$ processes to search for the anomalous couplings of $WWγ$ 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, ($δΔκ_γ$, $δλ_γ$) and ($δΔκ_{Z}$, $δλ_{Z}$) 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 ($δΔκ_γ, δλ_γ$) and ($δΔκ_{Z}, δλ_{Z}$), 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 ($Δκ_{Z}$, $λ_{Z}$) through the process $ep\toν_{e}qZX$ at the LHeC can further improve the current experimental limits.