Anomalous quartic $WW\gamma \gamma$ couplings in $ep$ collisions at the LHeC and the FCC-he

We conducted a study on measuring $W^{+}{W}^{-}$ production and on the sensitivity limits at $95\mathrm{\%}$ Confidence Level on thirteen anomalous couplings obtained by dimension-8 operators which are related to the anomalous quartic $WW\gamma \gamma$ couplings. We consider the main $e^-p \to e^-\gamma^\gamma^p \to e^-W^+W^-p$reactionwiththesub-process$\gamma^\gamma^ \to W^+W^-$ 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 notice that the LHeC and the FCC-he will lead to model-independent limits on the anomalous quartic $WW\gamma \gamma$ couplings which are one order of magnitude stringent than the CMS Collaboration limits, in addition to being competitive with other limits reported in the literature.