Triply charm and bottom tetraquarks in a constituent quark model

Singly, doubly and fully charmed tetraquark candidates, \emph{e.g.}, $T_{c\bar{s}}(2900)$, $T^+_{cc}(3875)$ and $X(6900)$ have been recently reported by the LHCb collaboration. Therefore, it is timely to implement a theoretical investigation on triply heavy tetraquark systems; herein, the S-wave triply charm and bottom tetraquarks, $\bar{Q}Q\bar{q}Q$ $(q=u,\,d,\,s;\,Q=c,\,b)$, with spin-parity $J^P=0^+$, $1^+$ and $2^+$, isospin $I=0$ and $\frac{1}{2}$, are systematically studied in a constituent quark model. Besides, all tetraquark configurations, \emph{i.e.} meson-meson, diquark-antidiquark and K-type arrangements, along with any allowed color structure, are comprehensively considered. The Gaussian expansion method (GEM), in combination with the complex-scaling method (CSM), which is quite ingenious in dealing with either bound or resonances, is the approach adopted in solving the complex scaled Schr\"odinger equation. This theoretical framework has already been applied in various tetra- and penta-quark systems. In a fully coupled-channel calculation within the GEM$+$CSM, narrow resonances are found in each $I(J^P)$ channel of the charm and bottom sector. In particular, triply charm and bottom tetraquark resonances are obtained in $5.6-5.9$ GeV and $15.3-15.7$ GeV, respectively. We provide also some insights of the compositeness of these exotic states, such as the inner quark distance, magnetic moment and dominant wave function component. All this may help to distinguish them in future high energy nuclear and particle experiments.