Searching for beauty-fully bound tetraquarks using lattice nonrelativistic QCD

Motivated by multiple phenomenological considerations, we perform the first search for the existence of a $\overline{b}\overline{b}bb$ tetraquark bound state with a mass below the lowest noninteracting bottomonium-pair threshold using the first-principles lattice nonrelativistic QCD methodology. We use a full $S$-wave color/spin basis for the $\overline{b}\overline{b}bb$ operators in the three $0^{++}$, $1^{+-}$ and $2^{++}$ channels. We employ four gluon field ensembles at multiple lattice spacing values ranging from $a=0.06–0.12\mathrm{fm}$, all of which include $u$, $d$, $s$ and $c$ quarks in the sea, and one ensemble which has physical light-quark masses. Additionally, we perform novel exploratory work with the objective of highlighting any signal of a near threshold tetraquark, if it existed, by adding an auxiliary potential into the QCD interactions. With our results we find no evidence of a QCD bound tetraquark below the lowest noninteracting thresholds in the channels studied.

Figure 1

There are four connected Wick contractions for the two-meson-type correlator when the quarks have the same flavor. The grey region represents a color neutral meson, the blue line a quark and the red line an antiquark. We call these the (a) Direct1 contraction where each meson propagates to itself, (b) Xchange2 where an antiquark is exchanged between the meson pair, (c) Direct3 where each meson propagates to the other, and (d) Xchange4 where a quark is exchanged between the meson pair.

Figure 2

There are four connected Wick contractions for the diquark-antidiquark-type correlator when the quarks have the same flavor. The blue shaded region represents a diquark, the red shaded region the antidiquark, a blue line a quark and the red line an antiquark. The uncrossing of the lines in Fig. 2 to produce Fig. 2 gives a $\pm$, as discussed in the text, which enforces the Pauli-exclusion principle.

Figure 3

The effective mass plot for the ${\eta }_b$ and $\mathrm{{\rm Y}}$ on the superfine ensemble (set 4 listed in Table 3). The effective mass plots on the other ensembles are qualitatively identical.

Figure 4

Assuming a tetraquark exists 100 MeV below the $2{\eta }_b$ threshold, different normally distributed couplings in $\overline{b}\overline{b}bb$ correlator mock data produce different effective mass curves on the superfine ensemble (set 4 listed in Table 3) as described in the text.

Figure 5

The $\overline{b}\overline{b}bb$ effective masses for the $0^{++}$ $(M_1,M_1)\to (M_2,M_2)$ correlators where $M_1$, $M_2$ are the ${\eta }_b$ or $\mathrm{{\rm Y}}$. $E^{\mathrm{eff}}$ and $E^{\mathrm{eff},t}$ are the single- and two-particle effective masses defined in Eqs. (18) and (19) respectively. The mesons are separated by a distance $r_x$ in the $x$-direction when constructing the two-meson interpolating operator as given in Eq. (4). Gray points are not used when fitting the data (cf., Appendix pp1).

Figure 6

The $\overline{b}\overline{b}bb$ effective masses for the $0^{++}$ diquark-antidiquark $\overline{3}\times \overline{3}$ and $6\times \overline{6}$ correlators. $E^{\mathrm{eff}}$ and $E^{\mathrm{eff},t}$ are the single- and two-particle effective masses defined in Eqs. (18) and (19) respectively. The diquarks are separated by a distance $r_x$ in the $x$-direction when constructing the diquark-antidiquark interpolating operator as given in Eq. (4).

Figure 7

The $\overline{b}\overline{b}bb$ effective masses for the $1^{+-}$ $\mathrm{{\rm Y}}{\eta }_b$ and diquark-antidiquark $\overline{3}\times \overline{3}$ correlators. $E^{\mathrm{eff}}$ and $E^{\mathrm{eff},t}$ are the single- and two-particle effective masses defined in Eqs. (18) and (19) respectively.

Figure 8

The $\overline{b}\overline{b}bb$ effective masses for the $2^{++}$ $\mathrm{{\rm Y}}\mathrm{{\rm Y}}$ and diquark-antidiquark $\overline{3}\times \overline{3}$ correlators subduced into the $T_2$ irrep. $E^{\mathrm{eff}}$ and $E^{\mathrm{eff},t}$ are the single- and two-particle effective masses defined in Eqs. (18) and (19) respectively.

Figure 9

A summary of the $\overline{b}\overline{b}bb$ ground-state energies with the lowest noninteracting bottomonium-pair threshold subtracted, across the different lattice ensembles listed in Table 3, statistical error only. Note, as shown in Table 3, fewer configurations were used on the $a=0.06\mathrm{fm}$ ensemble than on the others.

Figure 10

The individual Wick contraction effective masses of the $2{\eta }_b\to 2{\eta }_b$ correlators. The upper figure is the Direct1 and the lower is the Xchange2 contraction [each shown diagrammatically in Figs. 1 and 1]. $E^{\mathrm{eff}}$ and $E^{\mathrm{eff},t}$ are the single- and two-particle effective masses defined in Eqs. (18) and (19) respectively.

Figure 11

The excluded region for the ratio of tetraquark/$2{\eta }_b$ overlaps, $Z_{4b}/Z_{2{\eta }_b}$, onto the ${\mathcal{O}}_{({\eta }_b,{\eta }_b)}$ operator, assuming a tetraquark with mass, $E_{4b}$, lying below the $2{\eta }_b$ threshold, $E_{2{\eta }_b}$. The red hashed region is excluded at $5\sigma$ by the data as described in the text. The $1\sigma -3\sigma$ and $3\sigma -5\sigma$ exclusion bands are also shown for reference.

Figure 12

The root mean square distance $r_{rms}$ of the ${\eta }_b$ and the $2{\eta }_b$ as a function of the harmonic oscillator strength calculated from a potential model with the different ensemble parameters listed in Table 3 as discussed in the text.

Figure 13

The effective mass plot for the ${\eta }_b$ when including the harmonic oscillator potential on set 3. $E^{\mathrm{eff}}$ is given by Eq. (16) while $E^{\mathrm{eff},\xi }$ removes the leading $1+e^{-2\omega t}$ dependence from the correlator (24) to enable a better comparison with the data when no harmonic oscillator potential is included.

Figure 14

The lattice ${\eta }_b$ energy $M(\omega {)}_{{\eta }_b}+3\omega /2$ when including the harmonic oscillator potential with $M(\omega =0{)}_{{\eta }_b}$ subtracted compared to the model predictions as discussed in the text.

Figure 15

The $\overline{b}\overline{b}bb$ effective masses for the $0^{++}$ correlators when including the harmonic oscillator potential on set 3. $E^{\mathrm{eff}}$ is given by Eq. (16), while $E^{\mathrm{eff},‡}$ removes the leading $1-e^{-4\omega t}$ dependence from the correlator (25) to enable a better comparison with the data when no harmonic oscillator potential is included.

Figure 16

The effective masses for the individual $2{\eta }_b\to 2{\eta }_b$ Wick contraction correlator data when including a harmonic oscillator potential on set 3. The upper figure is the Direct1 and the lower is the Xchange2 contraction [each shown diagrammatically in Figs. 1 and 1].

Figure 17

A comparison of our result for the $\overline{b}\overline{b}bb$ ground-state energy in the $0^{++}$ channel (statistical error only) and predictions from phenomenological models. The hatched region indicates the exclusion of a bound tetraquark with an energy $E_{4b}$ subject to the value of its nonperturbative overlap as given in Fig. 11. In this comparison, we take our ground-state energy obtained on the superfine ensemble (set 4 in Table 3) as a representative because it has the smallest discretization effects and the statistical error encompasses the results on the other ensembles as shown in Fig. 9. The $y$-axis labels results from different phenomenological models [7, 8, 9, 10, 11, 12, 13] by last initial of authors and year of publication. An error is plotted if given in the reference. The two results from $\mathrm{WLCZ}16^{†}$ differ by how the mass scale was set. The result $\mathrm{VVR}13^{\xi }$ finds no bound tetraquark and we indicate this by placing their result on threshold.

Figure 18

The integrands of the moments given in Eq. (a8) at multiple times. The crosses represent the discrete finite-volume momentum contributions on the coarse (set 1) ensemble as discussed in the text. Due to the Gaussian time dependence, the integrand peak moves towards the origin for larger times.

Figure 19

The difference between the discrete finite-volume and infinite-volume continuum moments (upper) and which moments can be neglected compared to our statistical precision (lower) as discussed in the text.