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Anomalous quartic $WWγγ$ 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 $95\%$ Confidence Level on thirteen anomalous couplings obtained by dimension-8 operators which are related to the anomalous quartic $WWγγ$ couplings. We consider the main $e^-p \to e^-γ^*γ^*p \to e^-W^+W^-p$ reaction with the sub-process $γ^*γ^* \to W^+W^-$ 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 $WWγγ$ 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.