Probing model-independent limits on $W^+W^-\gamma$ 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 $e^-p \to e^-\gamma^*p \to \nu_eW^- p$. Additionally, we obtain the anomalous couplings $\Delta\kappa_\gamma$ and $\lambda_\gamma$ of the $W^+W^-\gamma$ 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 $\Delta\kappa_\gamma$ and $\lambda_\gamma$ at the $95\%$ C.L. are: $\Delta\kappa_\gamma = \pm 0.0017$, $\lambda_\gamma = \pm 0.0053$ (unpolarized electron beam) and $\Delta\kappa_\gamma = \pm 0.0013$, $\lambda_\gamma = \pm 0.0046$ (polarized electron beam) identifying the $W^-$ boson through the hadronic decay channel. In addition, the $e^-\gamma^* \to \nu_eW^-$ collision is one of the clean, pure and simple process to investigate the $W^+W^-\gamma$ coupling without the complications of QCD backgrounds.