Evidence for charm-bottom tetraquarks and the mass dependence of heavy-light tetraquark states from lattice QCD

We continue our study of heavy-light four-quark states and find evidence from lattice QCD for the existence of a strong-interaction-stable $I(J^P)=0(1^{+})$ $ud\overline{c}\overline{b}$ tetraquark with mass in the range of 15 to 61 MeV below $\overline{D}{B}^{*}$ threshold. Since this range includes the electromagnetic $\overline{D}B\gamma$ decay threshold, current uncertainties do not allow us to determine whether such a state would decay electromagnetically or only weakly. We also perform a study at fixed pion mass, with non-relativistic QCD (NRQCD) for the heavy quarks, simulating $q{q}^{\prime }{\overline{b}}^{\prime }\overline{b}$ and $q{q}^{\prime }{\overline{b}}^{\prime }{\overline{b}}^{\prime }$ tetraquarks with $q$, $q^{\prime }=ud$ or $\ell s$ and variable, unphysical $m_{b^{\prime }}$ in order to investigate the heavy mass dependence of such tetraquark states. We find that the dependence of the binding energy follows a phenomenologically expected form and that, though NRQCD breaks down before $m_{b^{\prime }}=m_c$ is reached, the results at higher $m_{b^{\prime }}$ clearly identify the $ud{\overline{b}}^{\prime }\overline{b}$ channel as the most likely to support a strong-interaction-stable tetraquark state at $m_{b^{\prime }}=m_c$. This observation serves to motivate the direct $ud\overline{c}\overline{b}$ simulation. Throughout we use dynamical $n_f=2+1$ ensembles with pion masses $m_{\pi }=415$, 299, and 164 MeV reaching down almost to the physical point, a relativistic heavy quark prescription for the charm quark and NRQCD for the bottom quark(s).

Figure 1

The dependence on the heavy-quark mass ratio, $r=m_{\mathrm{bare}}^b/m_{\mathrm{bare}}^{b^{\prime }}$, of the binding energies for the $ud{\overline{b}}^{\prime }\overline{b}$, $ud{\overline{b}}^{\prime }{\overline{b}}^{\prime }$, $\ell s{b}^{\prime }b$, and $\ell s{b}^{\prime }{b}^{\prime }$ channels. The results for each channel are separately fit to the phenomenologically motivated $\mathrm{\Delta }E(m_Q)$ ansatze detailed in Eqs. (1)–(6) of the text.

Figure 2

$ud\overline{c}\overline{b}$ tetraquark results for binding correlator eigenvalues (left panels) and binding energies (right panels) on $E_H$ (top), $E_M$ (center), and $E_L$ (bottom). Red squares and blue circles denote $3\times 3$ GEVP ground and first excited state results, respectively, green vertical dashes and green diagonal crosses ground and first excited state $2\times 2$ GEVP results. The grey bands depict the final binding energies, derived from single-exponential fits to the first eigenvalues. Further details may be found in the text, and supplementary material in Appendix pp3. The $2\times 2$ GEVP results are offset slightly in $t$ for visual clarity.

Figure 3

Examples of potentially observable $ud\overline{c}\overline{b}$ tetraquark decay channels. Channels accessible if only weak decays are possible: (left) into two mesons (e.g., to ${\overline{D}}^0{\overline{D}}^0$), and (center) into three mesons (e.g., to ${\pi }^{+}{\overline{D}}^0{D}^{-}$). Right: decay to $\overline{D}B\gamma$, expected to dominate if electromagnetic decay is possible.

Figure 4

Binding correlators (left) and effective energies (right) for the four channels $\ell s{\overline{b}}^{\prime }\overline{b}$, $ud{\overline{b}}^{\prime }\overline{b}$, $\ell s{\overline{b}}^{\prime }{\overline{b}}^{\prime }$, and $ud{\overline{b}}^{\prime }{\overline{b}}^{\prime }$ (top to bottom). Fit results are given as shaded bands (left) or boxes (right).

Figure 5

$ud\overline{c}\overline{b}$ tetraquark results on $E_H$ (top), $E_M$ (center), and $E_L$ (bottom). The left panels show the GEVP eigenvalues, and the right panels the corresponding energies. The red and blue symbols show, respectively, the ground and first excited state results obtained from the $3\times 3$ GEVP analyses. The green vertical dashes and green diagonal crosses similarly denote the ground and first excited state results obtained from the $2\times 2$ GEVP analyses. The two-meson $\overline{D}{B}^{*}$ thresholds, shown for comparison, are given by the brown bands in the right-hand panels. The grey bands in those same panels depict the final energies obtained from the single exponential fits to the eigenvalues described above. Further details are given in the text. The $2\times 2$ GEVP results have been offset in $t$ for visual clarity.

Figure 6

Fit window dependence for fitting the binding correlator GEVP solutions and those obtained from the particle correlators for the ensembles $E_H$ (top), $E_M$ (center), and $E_L$ (bottom). We obtain our final results from this selection of accepted fits in the stable region that is representative of both procedures.