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We present the first determination of the energy dependence of the$B$-$\bar{D}$ and $B^*$-$\bar{D}$ isospin-0, $S$-wave scattering amplitudesboth below and above the thresholds using lattice QCD, which allows us toinvestigate rigorously whether mixed bottom-charm $\bar{b}\bar{c}ud$tetraquarks exist as bound states or resonances. The scattering phase shiftsare obtained using L\"uscher's method from the energy spectra in two differentvolumes. To ensure that no relevant energy level is missed, we use large,symmetric $7 \times 7$ and $8 \times 8$ correlation matrices that include, atboth source and sink, $B^{(*)}$-$\bar{D}$ scattering operators with the lowestthree or four possible back-to-back momenta in addition to local$\bar{b}\bar{c}ud$ operators. We fit the energy dependence of the extractedscattering phase shifts using effective-range expansions. We observe sharppeaks in the $B^{(*)}$-$\bar{D}$ scattering rates close to the thresholds,which are associated with shallow bound states, either genuine or virtual, afew MeV or less below the $B^{(*)}$-$\bar{D}$ thresholds. In addition, we findhints for resonances with masses of order $100$ MeV above the thresholds anddecay widths of order $200$ MeV.

Evidence for shallow bound states and hints for broad resonances with  quark content $\bar{b}\bar{c}ud$ in $B$-$\bar{D}$ and $B^*$-$\bar{D}$  scattering from lattice QCD