Anomalous quartic $W^+W^-\gamma \gamma $ couplings in $e^-p$ collisions at the LHeC and the FCC-he

We conduct a study on measuring $W^+W^-$ production and on the sensitivity limits at 95% confidence level on thirteen anomalous couplings obtained by dimension-8 operators related to the anomalous quartic $W^+W^-\gamma \gamma $ couplings. We consider the main $e^-p \rightarrow e^-\gamma ^*\gamma ^*p \rightarrow e^-W^+W^-p$ reaction with the sub-process $\gamma ^*\gamma ^* \rightarrow W^+W^-$ at the Large Hadron electron Collider (LHeC) and the Future Circular Collider-hadron electron (FCC-he). For the LHeC, energies of the $e^-$ beams are taken to be $E_e =60$ and 140 GeV and the energy of the p beams is taken to be $E_p = 7$ TeV. For the FCC-he, energies of the $e^-$ beams are taken to be $E_e =60$ and 140 GeV and the energy of the p beams is taken to be $E_p = 50$ TeV, respectively. It is interesting to note that the LHeC and the FCC-he will lead to model-independent limits on the anomalous quartic $W^+W^-\gamma \gamma $ couplings which are one order of magnitude more stringent than the CMS Collaboration limits and are competitive with other limits reported in the literature.