Probing the vector-like $X$ quark via the $tW$ channel at future muon-proton colliders

We investigate the discovery potential for the vector-like $X$-quark (VLX) at future muon--proton ($\mu p$) colliders through the process $\mu^+ p \to \bar{\nu}_\mu X \to \bar{\nu}_\mu t W^+$. A simplified effective model is adopted in which the production and decay of the VLX are governed by the coupling strength $g^{*}$, the generation-mixing parameter $R_{L}$, and the VLX mass $m_X$. A comprehensive Monte Carlo analysis is performed at $\sqrt{s}=5.29$, $6.48$, and $9.16\ \mathrm{TeV}$, considering four complementary decay channels: the Fully Leptonic (FL), Fully Hadronic (FH), and two Semi-Leptonic (SL1 and SL2) modes. An $80\%$ polarized muon beam together with boosted-object reconstruction based on fat-jet techniques is employed to improve the signal sensitivity. The expected exclusion and discovery reaches are evaluated using the Asimov significance. We find that the sensitivity can be improved substantially with increasing center-of-mass energy and larger values of $R_L$. Among the four channels, the FH mode provides the strongest sensitivity, reaching a $2\sigma$ exclusion limit of $m_X\simeq8.3\ \mathrm{TeV}$ with $g^* = 0.009$ for $R_L=0.1$ at $\sqrt{s}=9.16\ \mathrm{TeV}$, whereas the FL mode gives the weakest reach because of its smallest branch ratio. These results demonstrate that future $\mu p$ colliders can offer significant sensitivity to heavy VLX over a broad region of parameter space.