Spectroscopy, Regge trajectories and decay properties of the $bc\bar{q}\bar{q}$ and $bq\bar{c}\bar{q}$ tetraquark systems

Bottom-charm tetraquarks probe the interplay of heavy-quark dynamics, light-quark correlations, color structure and nearby hadronic thresholds. We investigate the spectroscopy and strong-rearrangement properties of the $bc\bar{q}\bar{q}$ and $bq\bar{c}\bar{q}$ ($q=u,d$) systems in a Cornell-type diquark–antidiquark framework with nonrelativistic (NR) and semirelativistic (SR) kinetics. Scalar and axial-vector cluster configurations are treated explicitly, and model-parameter uncertainties are propagated into state-dependent mass errors. For the ${\bar{3}}_c\otimes 3_c$ axial-axial ground states, the $0^{+}$ masses are $7530.1\pm 75.4$ MeV (NR) and $7720.7\pm 78.8$ MeV (SR) for $bc\bar{q}\bar{q}$, and $7307.6\pm 90.5$ MeV (NR) and $7523.5\pm 93.6$ MeV (SR) for $bq\bar{c}\bar{q}$. The multiplets follow $0^{+}<1^{+}<2^{+}$, and radial spacings decrease with excitation. In the $bq\bar{c}\bar{q}$ sector several low-lying configurations approach the $B^{\ast }{D}^{\ast }$ threshold: the axial-axial $0^{+}$ state lies $25.7$ MeV below it in NR but $190.2$ MeV above it in SR, so its threshold assignment is model sensitive and not evidence of robust strong-interaction stability. Spin and color wavefunctions are recoupled independently, giving configuration-dependent $PP$, $PV$, $VP$ and $VV$ rearrangement strengths and distinct color-singlet projections for the ${\bar{3}}_c\otimes 3_c$ and $6_c\otimes {\bar{6}}_c$ sectors. The strong short-distance attraction of the sextet coefficient $k_s=-10\mathrm{/}3$ makes the finite SR expansion unreliable in that branch, so quantitative masses are restricted to the antitriplet sector. The results provide experimentally useful signatures and a baseline for coupled-channel and color-mixing studies.