A search for events with one top quark and missing transverse momentum in the final state is presented. The fully hadronic decay of the top quark is explored by selecting events with a reconstructed boosted top-quark topology produced in association with large missing transverse momentum. The analysis uses 139 fb$^{-1}$ of proton-proton collision data at a centre-of-mass energy of $\sqrt{s}$=13 TeV recorded during 2015-2018 by the ATLAS detector at the Large Hadron Collider. The results are interpreted in the context of simplified models for Dark Matter particle production and the single production of a vector-like $T$ quark. In the absence of a significant excess with respect to the Standard Model expectations, 95% confidence-level upper limits on the corresponding cross-sections are obtained. The production of Dark Matter particles in association with a single top quark is excluded for masses of a scalar (vector) mediator up to 4.3 (2.3) TeV, assuming $m_\chi$=1 GeV and the model couplings $\lambda_q$=0.6 and $\lambda_\chi$=0.4 ($a$=0.5 and $g_\chi$=1). The production of a single vector-like $T$ quark is excluded for masses below 1.8 TeV assuming a coupling to the top quark $\kappa_T$=0.5 and a branching ratio for $T\to Zt$ of 25%.
95% CL upper limits on the cross-section of the considered signal models as a function of the DM scalar mediator $\phi$ mass (for fixed model parameters of $\lambda_q =0.6$, $y_\chi=0.4$ and $m_\chi=1$ GeV).
95% CL upper limits on the cross-section of the considered signal models as a function of the DM vector mediator $V$ mass (for $a=0.5$, $g_\chi=1$ and $m_\chi=1$ GeV).
95% CL upper limits on the cross-section of the considered signal models as a function of the vector-like $T$ quark mass (for $\kappa_T=0.5$).
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