Searching for the vector-like X-quark at the future electron-proton colliders

In a simplified model including an exotic vector-like X-quark (VLQ-X) with electric charge 5/3, we investigate the process e+p ->nu<overline>eX+nu<overline>eXt<overline>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e<^>{+}p \rightarrow \bar{\nu }_{e}X+\bar{\nu }_{e}X\bar{t}$$\end{document} induced by the couplings between the VLQ-X and the first- and third-generation quarks at the future ep colliders. We further explore the prospects of discovering the VLQ-X via the X -> tW\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X\rightarrow tW$$\end{document} decay model at the electron-hadron Future Circular Collider (FCC-eh) with its proposed 60 GeV electron beam and 50 TeV proton beam. We focus our analysis on the final states, including two same-sign lepton (electrons or muons) or mu mu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \mu $$\end{document} pairs, at least one b-tagged jet, and large missing transverse momentum. By performing a detailed signal-to-background analysis and detector simulations, the 95% confidence level exclusion limits and 5 sigma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5\sigma $$\end{document} discovery reaches are presented in terms of parameter space regions.