Study of $I=0$ bottomonium bound states and resonances in $S$, $P$, $D$, and $F$ waves with lattice QCD static-static-light-light potentials

In this paper we study $I=0$ bottomonium in $S$, $P$, $D$ and $F$ waves considering five coupled channels, one confined quarkonium and four open $B^{(*)}{\overline{B}}^{(*)}$ and $B_s^{(*)}{\overline{B}}_s^{(*)}$ meson-meson channels. To this end we use and extend a recently developed novel approach utilizing lattice QCD string breaking potentials for the study of quarkonium bound states and resonances. This approach is based on the Born-Oppenheimer approximation and the unitary emergent wave method and allows to compute the poles of the $\mathrm{T}$ matrix. We compare our results to existing experimental results for $I=0$ bottomonium and discuss masses, decay widths and the assignment of angular momentum quantum numbers. Moreover, we determine the quarkonium and meson-meson composition of these states to clarify, which of them are ordinary quarkonium, and which of them should rather be interpreted as tetraquarks.

Figure 1

Data points for $V_{\overline{Q}Q}(r)$, $V_{\overline{M}M,\parallel }(r)$ and $V_{\mathrm{mix}}(r)$. The curves represent the parametrizations (4)–(6).

Figure 2

Eigenphase sum ${\delta }_{\tilde{J}}$ as function of the energy $E$. Narrow resonances are indicated by pronounced steps. We also show the real parts of the positions of the poles of ${\mathrm{T}}_{\tilde{J}}$ using green dotted lines (the line at $\approx 10.798\mathrm{GeV}$ in the lower left plot does not correspond to the real part of a pole, but to a fit to the eigenphase sum; see the technical discussion in Sec. 3c1). A pale red dotted line indicates the spin averaged $B_s^{(*)}{B}_s^{(*)}$ threshold, the light blue solid line $E_{\text{threshold}}$. Results for energies inside the light-blue shaded region, i.e. above $\approx 11.025\mathrm{GeV}$, should not be trusted, since we neglect decay channels containing a negative and a positive parity heavy-light meson.

Figure 3

Positions of the poles of the $\mathrm{T}$ matrix (19) in the complex energy plane for $\tilde{J}=0$ (top left), $\tilde{J}=1$ (top right), $\tilde{J}=2$ (bottom left) and $\tilde{J}=3$ (bottom right) representing bound states and resonances up to 11.2 GeV. Colored point clouds represent 1000 independent computations with resampled lattice data, while black dots correspond to the mean values and error bars. The pale red dotted lines indicate the spin-averaged $B^{(*)}{B}^{(*)}$ and $B_s^{(*)}{B}_s^{(*)}$ thresholds, the light blue solid line $E_{\text{threshold}}$. Results for energies inside the light-blue shaded region, i.e. above $\approx 11.025\mathrm{GeV}$, should not be trusted, since we neglect decay channels containing a negative and a positive parity heavy-light meson.

Figure 4

Graphical summary of theoretical predictions and experimental results for masses of $I=0$ bottomonium with ${\tilde{J}}^{PC}=0^{++},1^{--},2^{++},3^{--}$. We also show the quarkonium and meson-meson composition as discussed in Sec. 3d: $\%\overline{Q}Q$ in orange, $(\%\overline{M}M{)}_{\tilde{J}-1}+(\%\overline{M}M{)}_{\tilde{J}+1}$ in blue and $(\%{\overline{M}}_s{M}_s{)}_{\tilde{J}-1}+(\%{\overline{M}}_s{M}_s{)}_{\tilde{J}+1}$ in green.

Figure 5

Fits of $\alpha +\beta \arctan ((2/\mathrm{\Gamma })(E-m))$ to the data points for the eigenphase sum ${\delta }_2$ in the region of the steps at $\approx 10.798\mathrm{GeV}$ and $\approx 11.038\mathrm{GeV}$.

Figure 6

Percentages of quarkonium and of meson-meson pairs for $I=0$ bottomonium with ${\tilde{J}}^{PC}=0^{++}$ as functions of $R_{\max }$. Note that $R_{\max }$ denotes the upper bound of the integral in Eqs. (26)–(28) and not the quark-antiquark separation.

Figure 7

Percentages of quarkonium and of meson-meson pairs for $I=0$ bottomonium with ${\tilde{J}}^{PC}=1^{--}$ as functions of $R_{\max }$. Note that $R_{\max }$ denotes the upper bound of the integral in Eqs. (26)–(28) and not the quark-antiquark separation.

Figure 8

Percentages of quarkonium and of meson-meson pairs for $I=0$ bottomonium with ${\tilde{J}}^{PC}=2^{++}$ as functions of $R_{\max }$. Note that $R_{\max }$ denotes the upper bound of the integral in Eqs. (26)–(28) and not the quark-antiquark separation.

Figure 9

Percentages of quarkonium and of meson-meson pairs for $I=0$ bottomonium with ${\tilde{J}}^{PC}=3^{--}$ as functions of $R_{\max }$. Note that, $R_{\max }$ refers to the upper bound of the integral in Eqs. (26)–(28) and is not the quark-antiquark separation.