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