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Rare two-body decays of the top quark into a neutral bottom-quark meson plusan up- or charm-quark: $t\to {\overline B}^0+ u, c$; $t\to {\overline B}^0_{s}+c,u$; and $t \to \Upsilon(nS)+ c,u$, are studied for the first time. Thecorresponding partials widths are computed at leading order in thenon-relativistic QCD framework. The sums of all two-body branching ratiosamount to $\mathcal{B}(t \to {\overline B}^0+ {\rm jet}) \approx \mathcal{B}(t\to {\overline B}^0_{s}+ {\rm jet}) \approx 4.2\cdot 10^{-5}$ and$\mathcal{B}(t \to \Upsilon(nS)+ {\rm jet}) \approx 2\cdot 10^{-9}$,respectively. The feasibility to observe the $t\to {\overline B}^0_{(s)}+{\rmjet}$ decay is estimated in top-pair events produced in proton-protoncollisions at $\sqrt{s} = 14, 100$ TeV at the LHC and FCC, respectively.Combining many exclusive hadronic ${\overline B}^0_{(s)}$ decays, with $J/\psi$or $D^{0,\pm}$ final states, about 50 (16000) events are expected in 3 (20)ab$^{-1}$ of integrated luminosity at the LHC (FCC), after typical selectioncriteria, acceptance, and efficiency losses. An observation of the two-bodytop-quark decay can also be achieved in the interesting $t\tob(\rm{jet})+c(\rm{jet})$ dijet final state, where the ${\overline B}^0_{(s)}$decay products are reconstructed as a jet, with 5300 and 1.4 million signalevents above backgrounds expected after selection criteria at the LHC and FCC,respectively. Such unique final states provide a new direct method to preciselymeasure the top-quark mass via simple 2-body invariant mass analyses.

Rare two-body decays of the top quark into a bottom meson plus an up or charm quark