The $T_{bc}$ tetraquarks near the $B\bar{D}$ threshold

We study the doubly heavy open-flavor tetraquarks $T_{bc}^{(0)}$ ($J^{P}=0^{+}$) and $T_{bc}^{(1)}$ ($J^{P}=1^{+}$) in the dynamical diquark model, describing the system as a heavy antidiquark--light diquark pair interacting through the lattice-QCD $\Sigma_g^+(1S)$ Born--Oppenheimer potential. Solving the radial Schr\"odinger equation yields $M(T_{bc}^{(0)}) = 7.143$--$7.158$ GeV and $M(T_{bc}^{(1)}) = 7.217$--$7.222$ GeV, with hyperfine splittings of $\Delta_{HF}\simeq 59$--$79$ MeV. The splitting is driven mainly by the mass difference between symmetric and antisymmetric heavy-antidiquark configurations, while the chromomagnetic interaction contributes linearly with $\partial\Delta_{HF}/\partial\kappa_{\bar b\bar c}=2$, consistent with heavy-antidiquark spin algebra. The mean separation, $\langle r\rangle\simeq 0.45$--$0.46$ fm, and inverse radius, $\langle 1/r\rangle^{-1}\simeq 0.33$--$0.34$ fm, exhibit weak parameter dependence and support a compact diquark--antidiquark interpretation. Relative to open-flavor thresholds, the scalar state lies essentially at the $B\bar D$ threshold and may appear either as a weakly decaying bound tetraquark or as a narrow near-threshold resonance. In contrast, the axial-vector state is consistently predicted as an $S$-wave resonance located $23$--$28$ MeV above $B^{*}\bar D$ and about $70$ MeV below $B\bar D^{*}$, implying a line shape strongly influenced by the nearby $B^{*}\bar D$ threshold.