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 $e^-p \to e^-\gamma^*p \to \nu_eW^- p$\mathrm{.}Additionally,weobtaintheanomalouscouplings$\Delta\kappa_\gamma$and$\lambda_\gamma$ofthe$W^+W^-\gamma$ 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 $\mathrm{\Delta }{\kappa }_{\gamma }$ and ${\lambda }_{\gamma }$ at the $95\mathrm{\%}$ C.L. are: $\mathrm{\Delta }{\kappa }_{\gamma }=\pm 0.0017$, ${\lambda }_{\gamma }=\pm 0.0053$ (unpolarized electron beam) and $\mathrm{\Delta }{\kappa }_{\gamma }=\pm 0.0013$, ${\lambda }_{\gamma }=\pm 0.0046$ (polarized electron beam) identifying the $W^{-}$ boson through the hadronic decay channel. In addition, the $e^{-}{\gamma }^{\ast }\to {\nu }_e{W}^{-}$ collision is one of the clean, pure and simple process to investigate the $W^{+}{W}^{-}\gamma$ coupling without the complications of QCD backgrounds.