Bounds on the non-standard $W^+W^-\gamma$ couplings at the LHeC and the FCC-he

We examine the potential of the $e^-p \to e^-\gamma^*p \to e^-W^-q'X$ ($\gamma^*$ is Weizsacker-Willams photon) reaction to probe the non-standard $W^+W^-\gamma$ couplings at the Large Hadron electron Collider (LHeC) and the Future Circular Collider-hadron electron (FCC-he). We find $95\%$ confidence level bounds on the anomalous coupling $\Delta\kappa_\gamma$ and $\lambda_\gamma$ parameters in the view of effective Lagrangian approach with various values of the integrated luminosity. We assume center-of-mass energies of the electron-proton system $\sqrt{s}= 1.30, 1.98, 7.07, 10\hspace{0.8mm}{\rm TeV}$ and luminosities ${\cal L} = 10-1000 \hspace{0.8mm}{\rm fb^{-1}}$. The best limits obtained from the process $e^-p \to e^-\gamma^*p \to e^-W^-q'X$ on the anomalous $\Delta\kappa_\gamma$ and $\lambda_\gamma$ couplings are $\Delta\kappa_\gamma = |0.00069|$ and $\lambda_\gamma = [-0.0099, 0.0054]$. These bounds show that the process under consideration is a good prospect for the searching of the non-standard $\Delta\kappa_\gamma$ and $\lambda_\gamma$ couplings at the LHeC and the FCC-he. In addition, our results provide complementary information on other results for $\Delta\kappa_\gamma$ and $\lambda_\gamma$ couplings.