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Probing model-independent limits on $W^+W^-γ$ triple gauge boson vertex at the LHeC and the FCC-he

We study the cross-section of production of a single $W^-$ boson in association with a neutrino through the process $e^-p \to e^-γ^*p \to ν_eW^- p$. Additionally, we obtain the anomalous couplings $Δκ_γ$ and $λ_γ$ of the $W^+W^-γ$ vertex at the Large Hadron Electron Collider (LHeC) and the Future Circular Collider Hadron-Electron (FCC-he). The impact of the polarized beam due to the electron is also analyzed. Our best limits for $Δκ_γ$ and $λ_γ$ at the $95\%$ C.L. are: $Δκ_γ= \pm 0.0017$, $λ_γ= \pm 0.0053$ (unpolarized electron beam) and $Δκ_γ= \pm 0.0013$, $λ_γ= \pm 0.0046$ (polarized electron beam) identifying the $W^-$ boson through the hadronic decay channel. In addition, the $e^-γ^* \to ν_eW^-$ collision is one of the clean, pure and simple process to investigate the $W^+W^-γ$ coupling without the complications of QCD backgrounds.