Model independent study for the anomalous quartic $WW\gamma\gamma$ couplings at Future Electron-Proton Colliders

The Large Hadron Electron Collider and the Future Circular Collider-hadron electron with high center-of-mass energy and luminosity allow to better understand the Standard Model and to examine new physics beyond the Standard Model in the electroweak sector. Multi-boson processes permit for a measurement of the gauge boson self-interactions of the Standard Model that can be used to determine the anomalous gauge boson couplings. For this purpose, we present a study of the process $ep \rightarrow \nu_{e} \gamma \gamma j$ at the Large Hadron Electron Collider with center-of-mass energies of 1.30, 1.98 TeV and at the Future Circular Collider-hadron electron with center-of-mass energies of 7.07, 10 TeV to interpret the anomalous quartic $WW\gamma\gamma$ gauge couplings using a model independent way in the framework of effective field theory. We obtain the sensitivity limits at $95\%$ Confidence Level on 13 different anomalous couplings arising from dimension-8 operators. The best limit in $f_{Mi}/\Lambda^{4}$ ($i=0,1,2,3,4,5,7$) parameters is obtained for $f_{M2}/\Lambda^{4}$ parameter while the best sensitivity derived on $f_{Ti}/\Lambda^{4}$ ($i=0,1,2,5,6,7$) parameters is obtained for $f_{T5}/\Lambda^{4}$. In addition, this study is the first report on the anomalous quartic couplings determined by effective Lagrangians at $ep$ colliders.