Sensitivity analysis for the anomalous $W^+W^-\gamma$ couplings at the LHeC and the FCC-he

We study the anomalous $W^+W^-\gamma$ couplings in performing the production of the W-pair through the process $e^- p\,\rightarrow\,e^-\gamma^*\gamma^* p\,\rightarrow\,e^-W^+W^-p$ at the Large Hadron electron Collider (LHeC) and the Future Circular Collider-hadron electron (FCC-he). We examine the pure hadronic, the semi-leptonic and the pure leptonic decay channels of the produced W-pair. We assume the center-of-mass energies of $\sqrt{s}=1.30$ and $1.98$ TeV and the integrated luminosities of 10-100 fb$^{-1}$ for the LHeC and the center-of-mass energies of $\sqrt{s}=3.46$ and $5.29$ TeV and the integrated luminosities of 100-2000 fb$^{-1}$ for the FCC-he. The best limits at 95$\%$ confidence level obtained for the anomalous couplings $\Delta\kappa_{\gamma}$ and $\lambda_{\gamma}$ are $\Delta\kappa_{\gamma}=[-0.00208; 0.00206]$ and $\lambda_{\gamma}=[-0.00174; 0.00834]$. By comparing these limits with the experimental limits, we present the potentials of the process and future $ep$ colliders.