Hidden-flavor four-quark states in the charm and bottom region

We discuss the spectrum and the internal composition of ground and excited four-quark states in the charm and bottom energy region. To this end we extend previous calculations within the framework of the relativistic four-body Faddeev-Yakubovsky equation to include quantum numbers with $J^{PC}=0^{++},0^{-+},1^{--},1^{+-}$ and $1^{++}$ and study their internal composition in terms of heavy-light meson pairs, hadroquarkonia and diquark-antidiquark clusters. We observe similar patterns in the charm and bottom energy region with different compositions of the four-quark states depending on $J^{PC}$ quantum numbers. Most notably, we find that all states with $C·P=+1$ are dominated by heavy-light meson contributions, whereas for axial-vector states with $J^{PC}=1^{+-}$ including the $Z_c(3900)$ we find a much more complicated picture depending on the flavor content. We systematically compare our results for the spectrum with existing experimental results and provide predictions for future analyses.

Figure 1

Four-quark BSE for a generic hidden-flavor $Qq\overline{q}\overline{Q}$ system in the (12)(34) topology; the permutations (13)(24) and (14)(23) are not shown here. The green half-circles denote the Bethe-Salpeter amplitudes, blue boxes represent the two-body interaction kernels and the blobs denote fully-dressed quark propagators.

Figure 2

Graphical representation of the Bethe-Salpeter amplitude in the physical basis. The diagrams on the r.h.s. represent the direct product of the internal clusters spanning the physical basis. The first diagram shows the meson-meson configuration (${\mathcal{M}}_1$), with the heavy-light meson clusters depicted as blue half-circles. The second diagram is the hadroquarkonium contribution (${\mathcal{M}}_2$), with individual components shown as violet half-circles, and the last diagram is the diquark-antidiquark configuration ($\mathcal{D}$), where each diquark is depicted by an orange half-circle.

Figure 3

Current-mass evolution of the $cq\overline{q}\overline{c}$ (crosses) and $bq\overline{q}\overline{b}$ ground states (dots) in the $0(1^{++})$ and $1(1^{+-})$ channels. The gray vertical-dashed lines mark the position of the $q=n$, $s$, $c$, $b$ current-quark masses, where $n=u/d$. The dotted, dashed and dash-dotted curves represent the meson-meson and diquark-antidiquark thresholds for the charm and bottom four-quark system, respectively. The colors of the thresholds comply with Fig. 2, i.e., blue for the ${\mathcal{M}}_1$, purple for the ${\mathcal{M}}_2$ and orange for the $\mathcal{D}$ cluster. The gray bands are the fits to the data.

Figure 4

Current-quark mass evolution of the $cq\overline{q}\overline{c}$ and $bq\overline{q}\overline{b}$ ground states in the $0(0^{++})$ channel; see Fig. 3 for details.

Figure 5

Current-quark mass evolution of the $cq\overline{q}\overline{c}$ and $bq\overline{q}\overline{b}$ ground states in the $0(1^{--})$ and $0(0^{-+})$ channels; see Fig. 3 for details.

Figure 6

Hidden-charm (left) and hidden-bottom spectrum (right) for the ground and first excited four-quark states compared to experiment [34]. The colored boxes are our results, where the height of the boxes stands for the error of the extracted masses. The gray and black boxes are the PDG masses (real parts of the pole positions) for conventional and exotic hadrons, respectively. The pale gray colored states are not yet well-established.

Figure 7

Current-quark mass evolution of the norm contributions for the $cq\overline{q}\overline{c}$ ground states in the $0(1^{++})$ channel. The color coding is the same as in Fig. 8.

Figure 8

Graphical illustration of the norm contribution matrix. Each entry in the matrix is an overlap integral that contributes to the normalization of the four-quark state shown in the denominator. The diagonal terms correspond to the norm contribution coming from the ${\mathcal{M}}_1$, ${\mathcal{M}}_2$ and $\mathcal{D}$ topologies and the off-diagonal terms arise from the mixing of the topologies. Note that the matrix is symmetric, so that the contributions with the same background color are summed up.

Figure 9

Norm contributions for the $cn\overline{n}\overline{c}$, $bn\overline{n}\overline{b}$ and $bc\overline{c}\overline{b}$ states. In the scalar channel we show the results for the excited states and in all other channels those for the ground states. The ${\tilde{f}}_0$ here corresponds to the $f_0(1370)$. The bars sum up to 100%.

Figure 10

Example distribution of extrapolated eigenvalues for the $I(J^{PC})=0(1^{++})$ state with quark configuration $cq\overline{q}\overline{c}$ and $m_q=2750\mathrm{MeV}$. The extrapolation results are centred around a value of $M_{cq\overline{q}\overline{c}}\approx 9.4\mathrm{GeV}$ with a spread that resembles a Gaussian distribution.

Figure 11

Current-mass evolution of the $cq\overline{q}\overline{c}$ (crosses) and $bq\overline{q}\overline{b}$ first radial excited states (dots) for the investigated quantum numbers in this work; see Fig. 3 for details.