Spectrum of $S$- and $P$-wave $cc\overline{q}{\overline{q}}^{\prime }$ $(\overline{q},{\overline{q}}^{\prime }=\overline{u},\overline{d},\overline{s})$ 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 $cc\overline{q}{\overline{q}}^{\prime }$ $(\overline{q},{\overline{q}}^{\prime }=\overline{u},\overline{d},\overline{s})$ 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 Schrödinger 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 $I{J}^P=01^{+}$, and quark constituent $cc\overline{u}\overline{d}$, in agreement with the experimentally observed $T_{cc}^{+}(3875)$. This state is approximately at the calculated $D{D}^{*}$ threshold and has a root-mean-square radius of about 0.48 fm. These demonstrate 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 $cc\overline{q}{\overline{q}}^{\prime }$ $(\overline{q},{\overline{q}}^{\prime }=\overline{u},\overline{d},\overline{s})$ states lie about 100 to a 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.

Figure 1

The mass spectra of $S$- and $P$-wave $cc\overline{q}{\overline{q}}^{\prime }$ $(\overline{q},{\overline{q}}^{\prime }=\overline{u},\overline{d},\overline{s})$ tetraquark states. Subfigures (a)–(c) correspond to the mass spectra of $cc\overline{n}{\overline{n}}^{\prime }$, $cc\overline{n}\overline{s}$, and $cc\overline{s}\overline{s}$ $(\overline{n},{\overline{n}}^{\prime }=\overline{u},\overline{d})$ states, respectively. For $P$-wave states, only those with lower masses are shown for the sake of simplicity.

Figure 2

Dependence of the mass for the lowest $cc\overline{u}\overline{d}$ state with $I{J}^P=01^{+}$ on parameter values. The vertical and horizontal axes indicate the multiples of the mass and parameter values, respectively, compared to their original values.