⇦Chinese Physics Letters, 2021, Vol. 38, No. 10, Article code 101201 Prediction of an $\varOmega_{bbb}\varOmega_{bbb}$ Dibaryon in the Extended One-Boson Exchange Model Ming-Zhu Liu (刘明珠)1,2 and Li-Sheng Geng (耿立升)2,3,4* Affiliations 1School of Space and Environment, Beihang University, Beijing 102206, China 2School of Physics, Beihang University, Beijing 102206, China 3Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing 102206, China 4School of Physics and Microelectronics, Zhengzhou University, Zhengzhou 450001, China Received 7 August 2021; accepted 27 August 2021; published online 17 September 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 1210050997, 11975041, 11735003, and 11961141004).
*Corresponding author. Email: lisheng.geng@buaa.edu.cn Citation Text: Liu M Z and Geng L S 2021 Chin. Phys. Lett. 38 101201 Abstract Since Yukawa proposed that the pion is responsible for mediating the nucleon-nucleon interaction, meson exchanges have been widely used in understanding hadron-hadron interactions. The most studied mesons are the $\sigma$, $\pi$, $\rho$, and $\omega$, while other heavier mesons are often argued to be less relevant because they lead to short range interactions. However, whether the range of interactions is short or long should be judged with respect to the size of the system studied. We propose that one charmonium exchange is responsible for the formation of the $\varOmega_{ccc}\varOmega_{ccc}$ dibaryon, recently predicted by lattice QCD simulations. The same approach can be extended to the strangeness and bottom sectors, leading to the prediction on the existence of $\varOmega\varOmega$ and $\varOmega_{bbb}\varOmega_{bbb}$ dibaryons, while the former is consistent with the existing lattice QCD results, the latter remains to checked. In addition, we show that the Coulomb interaction may break up the $\varOmega_{ccc}\varOmega_{ccc}$ pair but not the $\varOmega_{bbb}\varOmega_{bbb}$ and $\varOmega\varOmega$ dibaryons. DOI:10.1088/0256-307X/38/10/101201 © 2021 Chinese Physics Society Article Text In 1935, Yukawa proposed the one-pion exchange theory to explain the strong nuclear force.[1] Then, Taketani et al. employed one-pion and two-pion exchange potentials to explain the then available nucleon-nucleon ($NN$) scattering data.[2,3] Later, it was realized that the exchange of heavier mesons, such as $\sigma$, $\rho$, and $\omega$, was needed to explain the rich experimental data, leading to the so-called one-boson exchange (OBE) model,[4] based on which the Bonn group[5] and Nijmegen group[6] constructed high-precision nuclear forces. Motivated by the success of the OBE model in describing the $NN$ interaction, Voloshin and Okun applied the OBE model to study the interaction between charmed hadrons forty years ago.[7] After $X(3872)$ was discovered by the Belle collaboration in 2003,[8] Swanson utilized the one-pion exchange potential to interpret it as a $\bar{D}D^{\ast}$ bound state.[9] Later, Liu et al. studied $X(3872)$ with the OBE model by considering $\sigma$, $\rho$, and $\omega$ exchanges.[10,11] In Ref. [12] the likely existence of bound states of $\bar{D}^{(\ast)}$ and $\Sigma_{c}$ was predicted in the OBE model (also in the unitary approach[13] and quark model[14]), which may correspond to the pentaquark states discovered by the LHCb collaboration.[15,16] Clearly, the OBE model has been an invaluable tool in describing hadron-hadron interactions containing not only light quarks but also heavy quarks. In the conventional OBE model, the interaction between two hadrons is mainly generated by exchanging mesons consisting of light up and down quarks. It is natural to expect that the exchange of mesons containing strange quarks may also play a role in certain systems, such as $K$, $K^{\ast}$, $\eta$, and $\eta'$. In Ref. [17], it was shown that the one-kaon exchange force is able to bind the $DD_{s0}(2317)$ system. Compared with the one-pion exchange, $\eta$ and $\eta'$ exchanges contribute little to systems containing only light quarks due to their large masses and small isospin factor,[18] but they could contribute more to hadrons containing strange quarks.[19–21] In recent years, to understand the nature of $Z_{c}(3900)$,[22,23] the one charmonium exchange has been explored to investigate the $\bar{D}D^{\ast}$ interaction.[24,25] For such systems, however, since the exchange of light mesons is also allowed, the one charmonium exchange plays a minor role because of the large masses of charmonia. To confirm the relevance of one charmonium exchange, one needs to study other systems where light meson exchanges are not allowed. Recently, the HAL QCD collaboration found one $\varOmega_{ccc}\varOmega_{ccc}$ dibaryon with a binding energy of $B=5.7$ MeV and an rms radius of $R=1.1$ fm,[26] which indicates that there exist strong attractive interactions between the $\varOmega_{ccc}\varOmega_{ccc}$ pair (we note that the quark model study[27] did not support the existence of an $\varOmega_{ccc}\varOmega_{ccc}$ dibaryon). In this work, we extend the conventional OBE model by allowing for exchanges of charmonia. We show indeed that the one charmonium exchange can generate attractive interactions strong enough to bind the $\varOmega_{ccc}\varOmega_{ccc}$ pair. Replacing charm quarks with strange quarks, we find that the one strangeonium exchange leads to the existence of an $\varOmega\varOmega$ bound state, consistent with the HAL QCD result.[28] Furthermore, we predict the existence of an $\varOmega_{bbb}\varOmega_{bbb}$ dibaryon in the extended OBE model with the exchange of bottomonia.
cpl-38-10-101201-fig1.png
Fig. 1. One charmonium exchange for the $\varOmega_{ccc}\varOmega_{ccc}$ system.
Theoretical Framework. In the OBE model, the interaction between the $\varOmega_{ccc}\varOmega_{ccc}$ pair can only be generated by exchanging charmonia, for which we choose the ground-state $\eta_{c}$, $J/\psi$, and $\chi_{c0}(1P)$ mesons. As shown in Fig. 1, they play the same role as the $\pi$, ($\rho$,$\omega$), and $\sigma$ mesons in the conventional OBE model for the $NN$ interaction. To derive the one charmonium exchange potential of Fig. 1, we need the following Lagrangians:[29] $$\begin{alignat}{1} \mathcal{L}_{\varOmega_{ccc}\varOmega_{ccc}\eta_{c}} ={}& g_{\eta_{c}} {\boldsymbol\varOmega}_{ccc}^† \cdot(\eta_{c} {\nabla} \times {\boldsymbol\varOmega}_{ccc}) ,~~ \tag {1} \end{alignat} $$ $$\begin{alignat}{1} \mathcal{L}_{\varOmega_{ccc}\varOmega_{ccc}\chi_{c0}} ={}& g_{\chi_{c0}} {\boldsymbol\varOmega}_{ccc}^†\cdot \chi_{c0} {\boldsymbol\varOmega}_{ccc} ,~~ \tag {2} \end{alignat} $$ $$\begin{alignat}{1} \mathcal{L}_{\varOmega_{ccc}\varOmega_{ccc}\psi} ={}&g_{\psi} {{\boldsymbol\varOmega}_{ccc}}^† \cdot \psi {\boldsymbol\varOmega}_{ccc} \\ &-\frac{f_{\psi}}{4 M}{\boldsymbol\varOmega}_{ccc}^{i†}(\partial_{i}{\boldsymbol\psi}_{j}-\partial_{j}{\boldsymbol\psi}_{i}){\boldsymbol\varOmega}_{ccc}^{j},~~ \tag {3} \end{alignat} $$ where ${\boldsymbol\varOmega}_{ccc}$ denotes the non-relativistic field of $\varOmega_{ccc}$; $\eta_{c}$, $\chi_{c0}$, and $\psi$ are the fields of exchanged mesons; $g_{\eta_{c}}$, $g_{\chi_{c0}}$, $g_{\psi}$, and $f_{\psi}$ are the corresponding couplings, and the mass scale $M$ renders the coupling $f_{\psi}$ dimensionless. The $\varOmega_{ccc}$ couplings to the charmonia are unknown experimentally, but they can be estimated with reasonable confidence as explained in the following. In the OBE $NN$ interaction,[5,30,31] one can reproduce the binding energy and rms radius of the deuteron with the following nucleon couplings to $\pi$, $\sigma$, and $\omega$: $g_{\pi}=6.84$, $g_{\sigma }=8.46$, $g_{v}=3.25$, $f_{v}=19.82$ together with $M=0.94$ GeV, and a cutoff $\varLambda=0.86$ GeV. Assuming that the OBE $\varOmega_{ccc}\varOmega_{ccc}$ interaction is similar to the $NN$ interaction by replacing light quarks with charm quarks, it is reasonable to expect that the couplings $g_{\eta_{c}}$, $g_{\chi_{c0}}$, and $g_{\psi}$ are proportional to $g_{\pi}$, $g_{\sigma}$ and $g_{v}$, respectively. For the sake of simplicity and without loss of generality, we assume that the ratios are the same, i.e., $g_{\eta_c}/g_\pi=g_{\chi_{c0}}/g_\sigma=g_\psi/g_v=f_\psi/f_v=r$, where $r$ is a free parameter in our model, less than or equal to 1. As in the conventional OBE model, we must take into account the finite size of exchanged mesons by supplementing the vertices of Fig. 1 with a monopolar form factor $$\begin{align} F(\varLambda,q)=\frac{\varLambda^2-m^2}{\varLambda^2-q^2}.~~ \tag {4} \end{align} $$ In the OBE model with only $\pi,\sigma,\rho/\omega$ exchanges, the cutoff $\varLambda$ is the only unknown parameter, which can be determined by reproducing the binding energies of some molecular candidates. From the physical perspective, the cutoff should be larger than the masses of exchanged bosons (if we replace the monopolar form factor by $\varLambda^2/(\varLambda^2 + {\boldsymbol q}^2)$, to reproduce the binding energies of $\varOmega\varOmega$ and $\varOmega_{ccc}\varOmega_{ccc}$ dibaryons obtained in lattice QCD simulations, we need a cutoff smaller than the mass of the lightest meson exchanged, in agreement with Ref. [32]). Thus, for the one charmonium exchange model, the cutoff should be larger than the mass of $\chi_{c0}(1P)$. In addition, the cutoff should not be too large to justify the neglect of exchange of heavier mesons. Using the above Lagrangians, the one-charmonium exchange potentials of the $\varOmega_{ccc}\varOmega_{ccc}$ system in coordinate space read $$\begin{alignat}{1} V_{\eta_{c}}({\boldsymbol r}) ={}&g_{\eta_{c}}^{2}\Big[- {\boldsymbol a}_{1} \cdot {\boldsymbol a}_{2}\delta({\boldsymbol r}) +{\boldsymbol a}_{1} \cdot {\boldsymbol a}_{2}m_{\eta_{c}}^3W_Y(m_{\eta_{c}} r) \\ &+S_{12}({\boldsymbol a}_{1},{\boldsymbol a}_{2},{\boldsymbol r})m_{\eta_{c}}^3 W_{T}(m_{\eta_{c}} r) \Big] ,~~ \tag {5} \end{alignat} $$ $$\begin{alignat}{1} V_{\chi_{c0}}({\boldsymbol r}) ={}& -{g_{\chi_{c0} }^2}m_{\chi_{c0}}W_Y(m_{\chi_{c0}} r) ,~~ \tag {6} \end{alignat} $$ $$\begin{alignat}{1} V_{\psi}({\boldsymbol r}) ={}&{g_{\psi}^2}m_{\psi}W_Y(m_{\psi} r)+\frac{f_{\psi}^{2}}{4M^2}\Big[-\frac{2}{3} {\boldsymbol a}_{1} \cdot {\boldsymbol a}_{2}\delta({\boldsymbol r}) \\ & +\frac{2}{3} {\boldsymbol a}_{1} \cdot {\boldsymbol a}_{2}m_{\psi}^3W_Y(m_{\psi} r)\\ & +\frac{1}{3}S_{12}({\boldsymbol a}_{1},{\boldsymbol a}_{2},{\boldsymbol r})\cdot m_{\psi}^3 W_{T}(m_{\psi} r)\Big] ,~~ \tag {7} \end{alignat} $$ where ${\boldsymbol a}_{1} \cdot {\boldsymbol a}_{2}$ and $S_{12}({\boldsymbol a}_{1},{\boldsymbol a}_{2},{\boldsymbol r})$ denote the spin-spin and tensor operators, respectively; ${\boldsymbol a}_{1}$ and ${\boldsymbol a}_{2}$ denote the spin operator of a spin-3/2 particle, which is the product of spin-1/2 and spin-1 operators. The functions $W_{Y}(mr)$, $\delta(r)$, and $W_{T}(mr)$ are $$\begin{align} W_{Y}(mr)={}&\frac{e^{-mr}-e^{-\varLambda r}}{4\pi m r} - \frac{\varLambda^2-m^2}{8 \pi \varLambda m}e^{-\varLambda r},~~ \tag {8} \\ \delta(r)={}&\frac{(\varLambda^2-m^2)^2}{8 \pi \varLambda}e^{-\varLambda r}, \\ W_{T}(mr)={}&\frac{3}{4}\frac{e^{-mr}-e^{-\varLambda r}}{\pi m^3 r^3} -\frac{3}{4}\frac{\varLambda e^{-\varLambda r}- m e^{-mr}}{\pi m^3 r^2} \\ +&\frac{1}{8\pi m^3r}\Big[(m^2-3\varLambda^2)e^{-\varLambda r}+2 m^2 e^{-mr}\Big] \\ -&\frac{\varLambda^2-m^2}{8\pi m^3}\varLambda e^{-\varLambda r}. \end{align} $$ It is well known that the $S$–$D$ coupling plays an important role in the description of the deuteron, thus we also consider the $S$–$D$ coupling in the extended OBE model (as all the lattice QCD simulations only considered $S$-wave interactions,[26,28] in the supplementary material, we discuss the results obtained in the extended OBE model with the $S$–$D$ coupling turned off). For $J=0$, the allowed orbital angular momenta and spins for the $\varOmega_{ccc}\varOmega_{ccc}$ system are $(L=0,S=0)$ and $(L=2,S=2)$, and the relevant matrix elements of the spin-spin and tensor operators are given in Table 2. The $\varOmega_{ccc}$ mass is taken from lattice QCD simulations.[26] The masses of other relevant charmonia from PDG[33] are collected in Table 1.
Table 1. Hadron masses (in units of GeV) of relevance in this work.
Hadron $\varOmega_{ccc}$[26] $\eta_{c}$ $\chi_{c0}(1P)$ $J/\psi$
Mass 4.7956 2.9839 3.414 3.096
Hadron $\varOmega$ $\eta$ $f_{0}(980)$ $\phi$
Mass 1.672 0.958 0.99 1.019
Hadron $\varOmega_{bbb}$[34] $\eta_{b}$ $\chi_{b0}(1P)$ $\varUpsilon(1S)$
Mass 14.371 9.3987 9.859 9.460
Table 2. Relevant spin-spin and tensor matrix elements for the $\varOmega_{ccc}\varOmega_{ccc}$ system.
State $J^{P}$ Partial wave $\langle {\boldsymbol a}_{1}\cdot {\boldsymbol a}_{2}\rangle$ $ S_{12}({\boldsymbol a}_{1}, {\boldsymbol a}_{2}, {\boldsymbol r})$
$\varOmega_{ccc}\varOmega_{ccc}$ $J=0$ $^1 S_{0}$–$^5 D_{0}$ $\begin{pmatrix} -15/4 & 0 \\ 0 & -3/4 \\ \end{pmatrix}$ $\begin{pmatrix} 0 & -3 \\ -3 & -3 \\ \end{pmatrix}$
In the same way, the $\varOmega\varOmega$ and $\varOmega_{bbb}\varOmega_{bbb}$ potentials can also be constructed in the extended OBE model. The relevant particles and their masses[33] are given in Table 1. The mass of $\varOmega_{bbb}$ is also taken from the lattice QCD simulations.[34] As explained above, although we do not know the exact values of the couplings and cutoff in the OBE potential due to lack of experimental data, it is reasonable to assume that the couplings are not larger than the couplings of the nucleon, simultaneously the cutoff is not smaller than the mass of the heaviest meson exchanged and is not too much larger either. With these two constraints, we check whether one can reproduce the binding energies of the $\varOmega_{ccc}\varOmega_{ccc}$ and $\varOmega\varOmega$ dibaryons obtained in lattice QCD simulations. With the OBE potentials, we solve the Schrödinger equation to search for bound states. Assuming that the ratio $r$ is 1 and with a cutoff $\varLambda=3.78$ GeV, the binding energy and rms radius for the $J^{P}=0^{+} \varOmega_{ccc}\varOmega_{ccc}$ dibaryon are found to be $B=5.1$ MeV and $R=1.1$ fm, respectively. Decreasing the ratio to $1/2$ and with a larger cutoff of $\varLambda=4.60$ GeV, the corresponding binding energy and rms radius become $B= 5.8$ MeV and $R=1.0$ fm, respectively. Referring to the cutoff in the traditional OBE model containing only light quarks, we expect that the cutoff in the $\varOmega_{ccc}\varOmega_{ccc}$ system should be larger than $m_{_{\scriptstyle \chi_{c0}(1P)}}=3.414$ GeV, but not too much larger. Therefore, the coupling ratio of $r=1$ is preferred. The results listed in Table 3 imply that an $\varOmega_{ccc}\varOmega_{ccc}$ bound state with charm number $C=6$ is likely to exist, corresponding to the dibaryon state discovered by the HAL QCD collaboration.[26] Using the same couplings and cutoff we did not find an $\varOmega_{ccc}\varOmega_{ccc}$ bound state with $J^{P}=2^{+}$ (for more discussions about $J^P=2^+$ configurations, see the Supplementary Material), consistent with the quark model study.[35] Because $\varOmega_{ccc}$ has an electric charge of $+2e$, the repulsive Coulomb interaction between the $\varOmega_{ccc}\varOmega_{ccc}$ pair should be taken into account. On top of the one charmonium exchange potential, we add the $\varOmega_{ccc}\varOmega_{ccc}$ Coulomb interaction $\frac{4\alpha}{r}$. The updated results show that the $\varOmega_{ccc}\varOmega_{ccc}$ system is no longer bound in the case of $r=1$, which is in agreement with the HAL QCD study.[26] However, varying the cutoff by only 0.1 GeV, we find that the $\varOmega_{ccc}\varOmega_{ccc}$ system binds with a binding energy of $B=13.8$ MeV. As the reasonable range of the cutoff in the OBE model is at least a few hundreds of MeV, we can not determine whether the $\varOmega_{ccc}\varOmega_{ccc}$ system is bound or not once the Coulomb interaction is taken into account. The bottom line is that the Coulomb interaction is important for the $\varOmega_{ccc}\varOmega_{ccc}$ system.
Table 3. Binding energy $B$ and rms radius $R$ of the $J^{P}=0^{+} {\varOmega}_{ccc}{\varOmega}_{ccc}$ bound states obtained with different ratios $r$ and cutoffs $\varLambda$.
Molecule $J^{P}$ Ratio $r$ $\varLambda$ (GeV) $B$ (MeV) $R$ (fm)
${\varOmega}_{ccc}{\varOmega}_{ccc}$ $0^{+}$ 1 3.78 $5.1$ $1.1$
1/2 4.60 5.8 1.0
We can test the above approach in the $\varOmega\varOmega$ system, which can exchange $\eta'$, $f_{0}(980)$, and $\phi$. The couplings between the triply strange baryon $\varOmega$ and the exchanged mesons are assumed to be proportional to those nucleon couplings as in the one-charmonium exchange potential. The cutoff for the $\varOmega\varOmega$ system should be greater than the mass $m_{\phi}=1.019$ GeV. With the ratio $r=1$ and cutoff $\varLambda=1.62$ GeV, we obtain one $J^{P}=0^{+} \varOmega\varOmega$ bound state with a binding energy of $B= 1.6$ MeV and an rms radius $r=3.1$ fm. The HAL QCD collaboration have investigated the $S$-wave interaction of the $\varOmega\varOmega$ system, and found a bound state with a binding energy of $B=1.6$ MeV and an rms radius $r=3$–4 fm.[28] If we decrease the ratio $r$ to 1/2, we need a cutoff $\varLambda=2.83$ GeV to reproduce the binding energy and rms radius of the lattice QCD study.[28] In this case, the cutoff value is quite close to $2m_{\phi}$, which does not seem very natural. This then implies that the ratio in the strangeness sector should be larger than 1/2 as in the charm sector. We note that allowing for mixing of light quarks with strange quarks, a larger coupling ratio than that of the charm sector is expected. Compared with the $\varOmega_{ccc}\varOmega_{ccc}$ dibaryon, the $\varOmega\varOmega$ dibaryon has a larger size, which can be easily understood because on the one hand the size of $\varOmega$ is larger than that of $\varOmega_{ccc}$ and on the other hand the exchanged mesons are lighter and therefore the interaction range is longer. From the above study, it is clear that the extended OBE model can be trusted to study baryon-baryon interactions containing only heavy quarks such as charm and strangeness. As a result, it is natural to ask how about the $\varOmega_{bbb}\varOmega_{bbb}$ system, which can exchange $\eta_{b}$, $\chi_{b0}(1P)$, and $\varUpsilon(1S)$. The couplings of the triply bottom baryon $\varOmega_{bbb}$ and bottomonia can also be estimated from the couplings of the nucleon with light mesons. A reasonable cutoff for the $\varOmega_{bbb}\varOmega_{bbb}$ system is expected to be larger than the mass of $m{\chi_{b0}(1P)}=9.859$ GeV. With a cutoff $\varLambda=10.073$ GeV and the corresponding ratio $r=1$ we obtain one $J^P=0^+$ bound state with a binding energy $B= 5.7$ MeV and an rms radius $R=0.55$ fm. If we decrease the ratio to 1/2 and increase the cutoff to $\varLambda=10.718$ GeV, the corresponding binding energy and rms radius become $B=5.7$ MeV and $R=0.55$ fm, respectively. Compared with the sizes of the $\varOmega_{ccc}\varOmega_{ccc}$ and $\varOmega\varOmega$ dibaryons, the size of the $\varOmega_{bbb}\varOmega_{bbb}$ dibaryon is small. Nonetheless, one can see that the ratio of the size of the $\varOmega_{bbb}\varOmega_{bbb}$ dibaryon to the size of $\varOmega_{bbb}$ is similar to the ratio of their charm counterparts, $R_{\varOmega_{ccc}\varOmega_{ccc}}/R_{\varOmega_{ccc}}$.
Table 4. Binding energies $B$ and rms radii $R$ of $J^{P}=0^{+} {\varOmega}{\varOmega}$ and $\varOmega_{bbb}\varOmega_{bbb}$ bound states obtained with different ratio $r$ and cutoff $\varLambda$. The numbers in the brackets represent the results obtained with the Coulomb interaction taken into account.
Molecule $J^{P}$ Ratio $r$ $\varLambda$ (GeV) $B$ (MeV) $R$ (fm)
$\varOmega\varOmega$ $0^{+}$ 1 1.62 $1.6(0.7)$ 3.1(4.0)
1/2 2.83 $1.6(0.6)$ $3.0(4.2)$
${\varOmega_{bbb}}{\varOmega_{bbb}}$ $0^{+}$ 1 10.073 $5.7$(0.5) 0.55(1.12)
1/2 10.718 $5.7$(0.4) 0.55(1.17)
cpl-38-10-101201-fig2.png
Fig. 2. One boson exchange potential for the $NN$, $\varOmega\varOmega$, $\varOmega_{ccc}\varOmega_{ccc}$, and $\varOmega_{bbb}\varOmega_{bbb}$ systems as functions of the distance in the baryon-baryon pair.
Both $\varOmega$ and $\varOmega_{bbb}$ contain one negative charge $-e$, which leads to a repulsive potential $\frac{\alpha}{r}$ between the $\varOmega\varOmega$ and $\varOmega_{bbb}\varOmega_{bbb}$ pairs. With the Coulomb interaction taken into account we update the results in Table 4. The numbers in the parentheses show that the $\varOmega\varOmega$ and $\varOmega_{bbb}\varOmega_{bbb}$ systems still bind, though less bound than in the case where only the OBE potentials are considered. In addition, we find that the impact of the Coulomb interaction on the $\varOmega\varOmega$ system is smaller than that on the $\varOmega_{bbb}\varOmega_{bbb}$ system, mainly because the $\varOmega\varOmega$ dibaryon is more spatially extended than the $\varOmega_{bbb}\varOmega_{bbb}$ dibaryon. With the coupling ratio $r=1$ and the relevant cutoffs given in Tables 3 and 4, we show the $S$-wave potential of $NN$, $\varOmega\varOmega$, $\varOmega_{ccc}\varOmega_{ccc}$, and $\varOmega_{bbb}\varOmega_{bbb}$ in Fig. 2. The line shapes of these potentials are similar, i.e., they all contain a short-range repulsive core and a medium- and long-range attractive tail. As the baryon mass becomes heavier, the baryon-baryon potential becomes shorter ranged and more attractive, which would generate a delta-like potential in the limit of infinite heavy baryon masses. As $r$ approaches 0, all the potentials become strongly repulsive, which is qualitatively similar to the lattice QCD results.[26,28] However, the magnitude of the repulsion is larger than that of the lattice QCD potential. This is understandable because for extremely short ranges, the OBE potential is not expected to work. We note that the $^3\!S_{1} NN$ potential is qualitatively consistent with those of more refined phenomenological models[6,30] as well as lattice QCD simulations.[36]
cpl-38-10-101201-fig3.png
Fig. 3. The rms radii of the dibaryons and their components. Those of the nucleon, $\varOmega$, $\varOmega_{ccc}$, and $\varOmega_{bbb}$ are 0.84 fm,[33] 0.57 fm,[37] 0.41 fm,[37] 0.25 fm,[34] while those of the $NN$, $\varOmega\varOmega$, $\varOmega_{ccc}\varOmega_{ccc}$, and $\varOmega_{bbb}\varOmega_{bbb}$ dibaryons calculated in the extended OBE model are 3.8 fm, 3.1 fm, 1.1 fm, and 0.55 fm, respectively, with the ratio $r=1$ and the cutoff $\varLambda=0.86,\, 1.62,\, 3.78,\, 10.073$ GeV, respectively.
In Fig. 3, we show the rms radii of deuteron, $\varOmega\varOmega$, $\varOmega_{ccc}\varOmega_{ccc}$, and $\varOmega_{bbb}\varOmega_{bbb}$ dibaryons calculated in the extended OBE model as well as those of their components (we do not distinguish charge radius from matter radius, which are assumed to be equal or similar for baryons/dibaryons studied here). It is clear that for all the dibaryons studied, the radius of the dibaryon is larger than the sum of those of their components. As a result, it is reasonable to refer to these dibaryons as molecular states. Moreover, the rms radius of the dibaryon state becomes smaller as the baryon mass increases, because the exchanged mesons become heavier. The ratio of the size of the dibaryon to its component $R_{\rm BB}/R_B$ is 4.5, 5.4, 2.7, 2.2, for the $NN$, $\varOmega\varOmega$, $\varOmega_{ccc}\varOmega_{ccc}$, and $\varOmega_{bbb}\varOmega_{bbb}$ dibaryons, respectively. These results exhibit approximate $SU(3)$ symmetry and heavy quark flavor symmetry, as is expected. It should be noted that the results obtained in the extended OBE model are cutoff dependent, as in the conventional OBE model. In fixing the cutoffs for the $\varOmega_{ccc}\varOmega_{ccc}$ and $\varOmega\varOmega$ systems, we have used the lattice QCD simulations as guidance. However, for the $\varOmega_{bbb}\varOmega_{bbb}$ system, we have used the binding energy of the $\varOmega_{ccc}\varOmega_{ccc}$ dibaryon as a reference. The resulting cutoffs all seem very reasonable, which strongly support the existence of $\varOmega\varOmega$, $\varOmega_{ccc}\varOmega_{ccc}$, and $\varOmega_{bbb}\varOmega_{bbb}$ dibaryons. In particular, for the $\varOmega_{bbb}\varOmega_{bbb}$ system, a naive application of heavy quark flavor symmetry to connect $\varOmega_{ccc}\varOmega_{ccc}$ to $\varOmega_{bbb}\varOmega_{bbb}$ would result in a binding energy of a few tens of MeV for the latter, which implies that the Coulomb interaction will not likely break up the $\varOmega_{bbb}\varOmega_{bbb}$ pair, further supporting our conclusion. Lastly, we briefly discuss about the $B\bar{B}$ system using the $\varOmega_{ccc}\bar{\varOmega}_{ccc}$ system as an example. If only the Coulomb interaction is considered, we obtain two bound states with binding energies $B=1.0$ MeV and $B=0.2$ MeV. The one charmonium exchange potential for the $\varOmega_{ccc}\bar{\varOmega}_{ccc}$ system is related to that for the $\varOmega_{ccc}\varOmega_{ccc}$ system via $C$-parity, analogous with the $G$-parity in the $NN$ system.[38] Therefore, the only difference between the $\varOmega_{ccc}\bar{\varOmega}_{ccc}$ potential and the $\varOmega_{ccc}\varOmega_{ccc}$ potential comes from the one $J/\psi$ exchange potential. With the same parameters as those used for the $\varOmega_{ccc}\varOmega_{ccc}$ system, we obtain a deeply bound $\varOmega_{ccc}\bar{\varOmega}_{ccc}$ state, because the $\varOmega_{ccc}\bar{\varOmega}_{ccc}$ potential is more attractive, similar to the nucleon-antinucleon case.[38] It is natural to expect that the $\varOmega_{ccc}\bar{\varOmega}_{ccc}$ system may generate a bound state after considering both the Coulomb potential and the one charmonium exchange potential. Summary and Outlook. Replacing light $u/d$ quarks with heavy (strangeness, charm, and bottom) quarks, we extended the conventional OBE model by allowing for exchanges of heavy $s\bar{s}$, $c\bar{c}$, $b\bar{b}$ ground-state mesons. Fixing the nucleon couplings to light mesons by reproducing the binding energy and rms radius of the deuteron, and assuming that the $\varOmega$, $\varOmega_{ccc}$, and $\varOmega_{bbb}$ couplings to their relevant exchanged mesons are equivalent to or less than the nucleon ones, we have investigated the possible existence of $\varOmega\varOmega$, $\varOmega_{ccc}\varOmega_{ccc}$, and $\varOmega_{bbb}\varOmega_{bbb}$ dibaryons via the extended OBE model. Our results show that the existence of $J^{P}=0^{+} \varOmega\varOmega$ and $\varOmega_{ccc}\varOmega_{ccc}$ bound states is supported by the OBE model. Furthermore, we predict the existence of one $J^{P}=0^{+} \varOmega_{bbb}\varOmega_{bbb}$ bound state. In addition, taking into account the Coulomb interaction, we find that the $\varOmega_{ccc}\varOmega_{ccc}$ system is at the verge of binding, while the $\varOmega\varOmega$ and $\varOmega_{bbb}\varOmega_{bbb}$ systems still bind. Our studies show that the spin-parities of such dibaryons favor $J^{P}=0^{+}$ rather than $J^{P}=2^{+}$. Given the lattice QCD simulations and the results presented in this work, future experimental searches for the $\varOmega\varOmega$, $\varOmega_{ccc}\varOmega_{ccc}$, and $\varOmega_{bbb}\varOmega_{bbb}$ dibaryons are strongly encouraged. We note that recently Zhang et al. estimated the production of the $\varOmega\varOmega$ dibaryon and pointed out that it could be searched for by the STAR and ALICE experiments.[39] In Ref. [40], Zhuang et al. showed that the $\varOmega_{ccc}$ production in Pb–Pb collisions is much larger (about two orders of magnitude) than in $pp$ collisions. Open charm hadron productions in high-energy nuclear collisions are also discussed in the multiphase transport model by Chen et al.[41] In Ref. [42] Qiu et al. studied the $J/\psi$ production mechanism at the future Electron-Ion Collider. Similar studies on the production of the $\varOmega_{ccc}\varOmega_{ccc}$ and $\varOmega_{bbb}\varOmega_{bbb}$ dibaryons should be pursued in the future. Experimental discovery of these dibaryons, together with recent observations of other exotic hadrons, such as $X_{0,1}(2900)$,[43] which may be a $D^{\ast}\bar{K}^{\ast}$ molecule,[44–46] could shed more light on the nature of the nonperturbative strong interaction, in addition to, e.g., the progress made in understanding the more conventional hadrons in the Dyson–Schwinger equations.[47] References On the Interaction of Elementary Particles. I *On the Method of the Theory of Nuclear ForcesThe Meson Theory of Nuclear Forces, I: The Deuteron Ground State and Low Energy Neutron-Proton ScatteringOne Boson Exchange Model in Proton-Proton ScatteringThe bonn meson-exchange model for the nucleon—nucleon interactionConstruction of high-quality NN potential modelsObservation of a Narrow Charmoniumlike State in Exclusive B ± → K ± π + π − J / ψ DecaysShort range structure in the X(3872)Is X(3872) really a molecular state?X(3872) and other possible heavy molecular statesPossible hidden-charm molecular baryons composed of an anti-charmed meson and a charmed baryonPrediction of Narrow N * and Λ * Resonances with Hidden Charm above 4 GeV Σ c D ¯ and Λ c D ¯ states in a chiral quark modelObservation of J / ψ p Resonances Consistent with Pentaquark States in Λ b 0 → J / ψ K − p DecaysObservation of a Narrow Pentaquark State, P c ( 4312 ) + , and of the Two-Peak Structure of the P c ( 4450 ) + Exotic doubly charmed D s 0 * ( 2317 ) D and D s 1 * ( 2460 ) D * moleculesSpin-parities of the P c ( 4440 ) and P c ( 4457 ) in the one-boson-exchange modelExotic resonances due to η exchange N Ω interaction: Meson exchanges, inelastic channels, and quasibound stateHidden-charm pentaquarks with triple strangeness due to the Ω c ( * ) D ¯ s ( * ) interactionsObservation of a Charged Charmoniumlike Structure in e + e − → π + π − J / ψ at s = 4.26     GeV Study of e + e − → π + π − J / ψ and Observation of a Charged Charmoniumlike State at BellePrediction of an I = 1 D D ¯ * state and relationship to the claimed Z c ( 3900 ) , Z c ( 3885 ) Z c ( 3900 ) as a resonance from the D D ¯ * interactionDibaryon with Highest Charm Number near Unitarity from Lattice QCDVery Heavy Flavored DibaryonsMost Strange Dibaryon from Lattice QCDHeavy baryon-antibaryon molecules in effective field theoryHigh-precision, charge-dependent Bonn nucleon-nucleon potentialExotic baryons from a heavy meson and a nucleon: Negative parity statesReview of Particle PhysicsPrediction of the Ω b b b mass from lattice QCDFull heavy dibaryonsNuclear Force from Lattice QCDLook inside charmed-strange baryons from lattice QCDAntinucleon–nucleon interaction at low energy: scattering and protoniumΩ-dibaryon production with hadron interaction potential from the lattice QCD in relativistic heavy-ion collisionsΩccc production in high energy nuclear collisionsStudy on open charm hadron production and angular correlation in high-energy nuclear collisionsExploring J/ψ Production Mechanism at the Future Electron-Ion ColliderModel-Independent Study of Structure in B + → D + D − K + DecaysX 0 (2900) and X 1 (2900): Hadronic Molecules or Compact TetraquarksStrong decays of ${\bar{D}}^{*}K^{*}$ molecules and the newly observed $X_{0,1}$ states X 0 ( 2866 ) as a D * K ¯ * molecular stateImpressions of the Continuum Bound State Problem in QCD
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