Study on the anomalous quartic $W^+W^-\gamma \gamma $ couplings of electroweak bosons in $e^-p$ collisions at the LHeC and the FCC-he

In this paper, a study is carried out on the $e^-p \rightarrow e^-\gamma ^* p \rightarrow p W^-\gamma \nu _e$ production to probe quartic $W^+W^-\gamma \gamma $ couplings at the Large Hadron electron Collider (LHeC) with $\sqrt{s}= 1.30, 1.98\,\hbox {TeV}$ and at the Future Circular Collider-hadron electron (FCC-he) with $\sqrt{s}= 3.46, 5.29\,\hbox {TeV}$. Production cross-sections are determined for both at leptonic and hadronic decay channels of the W-boson. With the data from future $e^-p$ colliders, it is possible to obtain sensitivity measures at $95\%$ C.L. on the anomalous $f_ {M,i}/\Lambda ^4$ and $ f_ {T,j}/\Lambda ^4$ couplings which are competitive with the limits obtained by the LHC, as well as with others limits reported in the literature. The production mode $e^-p \rightarrow e^-\gamma ^* p \rightarrow p W^-\gamma \nu _e $ in $e^-p$ collisions offers a window for study the quartic $W^+W^-\gamma \gamma $ electroweak bosons couplings at the LHeC and the FCC-he, which provides a much cleaner collision environment than the LHC.