Spectrum of $S$- and $P$-wave $cc\bar{q}\bar{q}'$ $(\bar{q},\bar{q}' = \bar{u}, \bar{d}, \bar{s})$ systems in a chiral SU(3) quark model

Inspired by the resonance $T_{cc}^+(3875)$ recently observed by the LHCb Collaboration, we systematically explore the $S$- and $P$-wave $cc\bar{q}\bar{q}'$ $(\bar{q},\bar{q}' = \bar{u}, \bar{d}, \bar{s})$ systems in a chiral SU(3) quark model. The Hamiltonian contains the kinetic energy, the one-gluon-exchange (OGE) potential, the confinement potential, and the one-boson-exchange (OBE) potential stemming from the coupling of quark and chiral fields. The Schr\"odinger equation is solved by use of the variational method with the spacial trial wave functions chosen as Gaussian functions. It is found that the lowest state has a mass $3879$ MeV, isospin and spin-parity $IJ^P=01^+$, and quark constituent $cc\bar{u}\bar{d}$, in agreement with the experimentally observed $T_{cc}^+(3875)$. This state is approximately at the calculated $DD^\ast$ threshold, and has a root-mean-square radius about $0.48$ fm. These demonstrates that the $T_{cc}^+(3875)$ can be accommodated as a stable and compact tetraquark sate in the chiral SU(3) quark model. All the other $S$- and $P$-wave $cc\bar{q}\bar{q}'$ $(\bar{q},\bar{q}' = \bar{u}, \bar{d}, \bar{s})$ states lie about one hundred to few hundreds MeV higher than the corresponding meson-meson thresholds, and thus are not suggested to be candidates of stable and compact tetraquark states due to their fall-apart decays to two mesons.