Triply heavy tetraquarks $\bar{b}c\bar{q}c$ and $\bar{c}b\bar{q}b$ in a constituent quark model

Abstract: A systematic investigation of $S$-wave triply heavy tetraquark systems with quark content $\bar{b}c\bar{q}c$ and $\bar{c}b\bar{q}b$ $(q = u,\,d,\,s)$, spin-parity quantum numbers $J^P=0^+$, $1^+$, $2^+$ and isospin $I=0,\,\frac{1}{2}$, is carried out within the constituent quark model framework. The four-body bound and resonance states are determined by solving the Schr\"odinger equation employing the high-precision and efficient Gaussian Expansion Method (GEM) in conjunction with the powerful Complex Scaling Method (CSM). Besides, a comprehensive coupled-channel analysis of the $S$-wave tetraquark systems is performed, taking into account meson-meson, diquark-antidiquark and K-type configurations, as well as all allowed color structures. Several narrow resonant states are identified in each $I(J^P)$ channel. In particular, resonances for the $\bar{b}c\bar{q}c$ system are found in the mass range of $8.87-9.36$ GeV, while those for the $\bar{c}b\bar{q}b$ system lie between $12.13-12.45$ GeV. Most of the predicted resonances are compact tetraquark states with sizes smaller than $1.0$ fm, although four states exhibit more extended structures with sizes around $1.3$ fm, suggesting a more loosely bound nature. Magnetic moments and dominant wave function components of these exotic states are also analyzed. The results indicate that K-type configurations play a major role in the structure of the observed resonances. Finally, possible strong decay channels (golden modes) for these states are theoretically proposed.