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

Abstract

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 Schrödinger 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, $⟨r⟩\simeq 0.45$--$0.46$ fm, and inverse radius, $⟨1/r{⟩}^{-1}\simeq 0.33$--$0.34$ 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 $B^{\ast }\bar{D}$ and about $70$ MeV below $B{\bar{D}}^{\ast }$, implying a line shape strongly influenced by the nearby $B^{\ast }\bar{D}$ threshold.

Comments: 13 pages, 1 figure