Prospects for discovering strongly decaying doubly heavy $T_{bc}$ tetraquark states at LHCb

We investigate the discovery potential of the $T_{bc}$ state with $J^P=0^{+}$ in proton-proton ($pp$) collisions at LHCb at a center-of-mass energy of $\sqrt{s}=13\mathrm{T}\mathrm{e}\mathrm{V}$. The study focuses on the decay channel $T_{bc}\to B^{-}{D}^{+}$. A phenomenological approach is employed to construct the background model based on the associated production of $B$ and $D$ mesons, incorporating previously published LHCb results. Background processes are simulated using $\mathtt{\text{MadGraph5\_aMC@NLO}}$ and $\mathtt{\text{Pythia8.3}}$. We explore the parameter space of the $T_{bc}$ mass, width, production cross section, and the effective double-parton scattering cross section (${\sigma }_{\mathrm{e}\mathrm{f}\mathrm{f}}$) relevant for the $BD$ meson background. The integrated luminosity required for a $5\sigma$ discovery at LHCb is evaluated under various assumptions. In particular, we consider three representative $T_{bc}$ production cross section scenarios: an optimistic estimate of $103\mathrm{n}\mathrm{b}$, an intermediate value of $18\mathrm{n}\mathrm{b}$ obtained by scaling from the $T_{cc}^{+}$ production cross section, and a conservative lower bound of $0.3\mathrm{n}\mathrm{b}$. We find that a $5\sigma$ observation is achievable for a production cross section of $103\mathrm{n}\mathrm{b}$, which is expected to be within reach during Run~4. In contrast, the more realistic cross section estimate of $18\mathrm{n}\mathrm{b}$ requires the full Run~5 dataset ($300{\mathrm{f}\mathrm{b}}^{-1}$) under the most favorable parameter choices. For the conservative scenario, no significant signal would be observable even with $300{\mathrm{f}\mathrm{b}}^{-1}$. In addition, we estimate the minimum observable $\sigma (T_{bc})\times BR(T_{bc}\to B^{-}{D}^{+})$ for a $5\sigma$ discovery under different luminosity scenarios, providing guidance for future experimental searches at LHCb.