Searching for the vector-like X-quark at the future electron–proton colliders Yan-Ju Zhang, Yong-Tao Zhu, Lin Han, Yan-Ping Bi and Tai-Gang Liu Eur. Phys. J. C, 84 11 (2024) 1184 Published online: 19 November 2024 DOI: 10.1140/epjc/s10052-024-13475-1