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 ${e}^{-}p\to {e}^{-}{\gamma }^{\ast }p\to p{W}^{-}\gamma {\nu }_{e}$ production to probe quartic ${W}^{+}{W}^{-}\gamma \gamma $ couplings at the Large Hadron electron Collider (LHeC) with $\sqrt{s}=1.30,1.98\phantom{\rule{0.166667em}{0ex}}\text{TeV}$ and at the Future Circular Collider-hadron electron (FCC-he) with $\sqrt{s}=3.46,5.29\phantom{\rule{0.166667em}{0ex}}\text{TeV}$ . 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 ${e}^{-}p\to {e}^{-}{\gamma }^{\ast }p\to p{W}^{-}\gamma {\nu }_{e}$ 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.