Tribaryons with lattice QCD and one-boson exchange potentials

Motivated by the existence of two-body hadronic molecules composed of $\Omega\Omega$, $\Omega_{ccc}\Omega_{ccc}$ and $\Omega_{bbb}\Omega_{bbb}$ predicted by lattice QCD simulations, we use the Gaussian expansion method to investigate whether three-body systems composed of $\Omega\Omega\Omega$, $\Omega_{ccc}\Omega_{ccc}\Omega_{ccc}$ and $\Omega_{bbb}\Omega_{bbb}\Omega_{bbb}$ can bind with the two-body $^1S_0$ interactions provided by lattice QCD. Our results show that none of the three-body systems bind. On the other hand, we find that with the one-boson exchange potentials the $\Omega\Omega\Omega$ system develops a bound state, for which the $^5S_2$ interaction plays an important role. Our studies support the existence of the $\frac{3}{2}^+$ $\Omega\Omega\Omega$ bound state and the nonexistence of the $\frac{3}{2}^+$ $\Omega_{ccc}\Omega_{ccc}\Omega_{ccc}$ and $\Omega_{bbb}\Omega_{bbb}\Omega_{bbb}$ bound states, due to the suppressed $^5S_2$ interactions in heavier systems.

Introduction.-Thequark model, as a classification scheme for light-flavor hadrons, was proposed by Gell-Mann [1] and Zweig [2] in 1964, which was established when the predicted Ω baryon with the highest strangeness number was observed experimentally [3].It is often viewed as the first stage in hadron physics.Since 2003, we have witnessed a new stage in hadron physics with the observation of many new hadronic states, such as the charmoniumlike XY Z states and the pentaquark states [4][5][6][7][8][9][10][11][12], which have stimulated extensive studies, both theoretically and experimentally.Although remarkable progress has been made, a unified understanding of exotic hadronic states is still missing.At present, it is widely acknowledged that one should pay more attention to new configurations, exotic quantum numbers, and special systems in order to better understand the nature of exotic hadronic matter and the nonperturbative strong interaction.
In recent years, fully strange and fully heavy dibaryon systems have attracted considerable attention.With increasing computational power, lattice QCD has become the primary force to derive hadron-hadron interactions in a quantitative way from first principles.In Ref. [13], the authors investigated the ΩΩ interaction in the 1 S 0 channel, and concluded that there exists a weakly bound state regardless of the Coulomb interaction, which is even shallower than the deuteron.In Ref. [14], the existence of a 1 S 0 Ω ccc Ω ccc shallow bound state is predicted while it disappears once the Coulomb interaction is taken into account.Very recently, the existence of a deeply bound 1 S 0 Ω bbb Ω bbb state was also predicted [15].For the ΩΩ, Ω ccc Ω ccc , and Ω bbb Ω bbb systems, some of us developed an extended one-boson-exchange (OBE) model to derive their interactions in Ref. [16], and obtained results consistent with those of lattice QCD [13][14][15].In Ref. [17], the authors found the existence of fully heavy dibaryon bound states, Ω ccc Ω ccc and Ω bbb Ω bbb , in the constituent quark model, while the corresponding fully heavy hexaquark states are found to be above the Ω ccc Ω ccc and Ω bbb Ω bbb mass thresholds in both the constituent quark models [18][19][20] and the QCD sum rules [21].
On the experimental side, studies of fully heavy multiquarks have made important breakthroughs.In 2020, the LHCb Collaboration reported the observation of the first fully heavy tetraquark state, X(6900) [22].It was later confirmed by the CMS Collaboration with a statistical significance of 9.4σ, and in addition, two new states X(6600) and X(7200) were observed [23].The ATLAS Collaboration further confirmed the discovery of the LHCb Collaboration [24].Clearly, the existence of fully heavy multiquark states can be considered as firmly established.
In this paper, motivated by the remarkable progress achieved on studies of the fully heavy multiquark states from both lattice QCD [13][14][15] and experiments [22][23][24], we study the ΩΩΩ system, as well as the Ω ccc Ω ccc Ω ccc and Ω bbb Ω bbb Ω bbb systems.Notice that this study differ from the previous works.Regarding the works we mentioned above, they have either different species of baryons (ΩN N , ΩΩN , ...) or different flavors or charges (pnn, ∆∆∆, ΞΞΞ, ...).To the best of our knowledge, it is the first time that three-body systems that are composed of fully identical flavored baryons have been studied.The systems we study contain only one species of baryons that are composed of only one species of quarks and, therefore, have the highest symmetries.This means that we need only the interactions of a pair of identical baryons and the number of allowed configurations is much reduced as well, thus allowing for more robust predictions.Most strange tribaryon.-Weadopt the Gaussian expansion method (GEM) [50][51][52] to study the ΩΩΩ system.To solve the Schrödinger equation with GEM, one needs to derive the two-body interactions and construct the three-body wave functions.We note that three-body interactions may play an important role in many-body systems, such as the nucleus.Unfortunately, no empirical information on the three-body interactions is available for the three identical baryons we have studied.Thus, in this exploratory work, we only consider twobody interactions.
The ΩΩ interaction has been derived in lattice QCD [13], where it was shown that the S-wave ΩΩ system can bind with a binding energy of 1.6(6) +0.7 −0.6 MeV (without taking into account the Coulomb interaction).In addition to lattice QCD, other methods such as the extended OBE model can also provide the ΩΩ interaction [16].The light meson exchange, including the pseudoscalar(π), scalar(σ) and vector(ρ, ω) mesons, can well describe many hadron-hadron interactions, which is naively extended to the Ω ccc Ω ccc system by invoking the exchange of the charmonium states η c , χ c0 and J/ψ.The couplings between Ω ccc and the charmonium states are assumed to be proportional to the couplings between the nucleon and the light mesons utilizying the quark model.We emphasize that although the OBE model constructed in this way suffers from relatively large uncertainties, these can be minimized by fitting to the lattice QCD binding energies of the ΩΩ, Ω ccc Ω ccc , and Ω bbb Ω bbb bound states.In this work, we utilize both interactions to study the ΩΩΩ three-body system, see Fig. 1 for the potentials.
The ΩΩ lattice QCD potentials for the 1 S 0 channel are expressed with three Gaussian functions ΩΩΩ systems.The Coulomb potential between a pair of ΩΩ is V C (r) = −α/r, where α = 1/137 is the electromagnetic fine structure constant.The lattice QCD simulations provided only the 1 S 0 potential between the ΩΩ pair.As we see later, the 5 S 2 potential plays an important role in the three-body system as well.
In GEM, a three-body system is studied by solving the three-body Schrödinger equation with the three-body wave functions and the Hamiltonian in Jacobi coordinates.For the ΩΩΩ three-body system, the Schrödinger equation is as follows where c = 1 − 3 denote the three Jacobi channels, r c (R c ) are the Jacobi coordinates.T is the kinetic-energy operator and V ΩΩ is the two-body ΩΩ interaction.For the details on how to construct the Jacobi coordinates and the three-body kineticenergy operator, please refer to Ref. [51].
The ΩΩΩ three-body wave function can be written as a sum of three Jacobi channels where A c α is the expansion coefficients and α is the set of quantum numbers characterizing the wave function in each Jacobi channel.The wave function of each Jacobi channel reads as where χ i is the spin wave function of the ith particle, H c s,S = [[χ i χ j ] s χ k ] S is the spin wave function of Jacobi channel c, ψ(r i )ϕ(R i ) is the spatial wave function, s is the spin of the sub-ΩΩ two-body system, S = 3/2 is the total spin of ΩΩΩ, l i (L i ) is the orbit angular momentum corresponding to r i (R i ), Λ is the total orbit angular momentum built from l and L, and J is the total angular momentum built from Λ and S.
Fermi-Dirac statistics dictates that only the 1 S 0 and 5 S 2 interactions contribute to the formation of an ΩΩΩ 3 2 + state.
The spin coupling coefficients of different spin configurations between Jacobi channels i and j for i ̸ = j are shown in Table I.Note that for i = j, the matrix is orthogonal.TABLE I. Coupling coefficients of different spin configurations between Jacobi channels i and j (i ̸ = j).Here, H c s,S is the spin function, s = {0, 2} are alternative spin values of ΩΩ, and S = 3/2 is the total spin of ΩΩΩ.
It is important to point out that for the ΩΩΩ system, the 5 S 2 potential can play a very important role, even more important than the 1 S 0 potential.This is because the 5 S 2 partial wave is more strongly coupled to the three-body spin-3/2 state than the 1 S 0 partial wave.As shown in Table I, the spin coupling coefficient of different Jacobi channels i and j in the 5 S 2 partial wave is , which means that in the spin space, the coupling between channels i and j in the 5 S 2 partial wave is 9 times larger than that in the 1 S 0 partial wave.
Once the wave functions are obtained , with either the lattice QCD or OBE ΩΩ interactions, one can adopt the GEM [51] to obtain the binding energies and root-meansquare (rms) radius of the ΩΩΩ system.
The results for the two-body ΩΩ system are summarized in Table II, which show that the binding energies and rms radii obtained with the OBE potentials are consistent with those of lattice QCD.With both lattice QCD and OBE potentials, the ΩΩ system can bind with a binding energy of 1.4 +0.9 −0.4 MeV.The uncertainties are determined by multiplying a scaling factor to the lattice QCD potential so that the binding energy varies from 1.0 to 2.3 MeV, consistent with the lattice QCD result 1.6 +0.7 −0.6 MeV [13].From the analysis given above, we know that both 1 S 0 and 5 S 2 interactions contribute to the 3/2 ΩΩΩ system.Given that the lattice QCD provided only the 1 S 0 interaction, we first consider only the 1 S 0 two-body interaction and find that the ΩΩΩ system does not bind.But this result should not be taken too seriously since the 5 S 2 partial wave plays an important role in the spin configuration (⟨H c=i 2,3/2 |H c=j 2,3/2 ⟩ i̸ =j = 3 4 ) and has a significant correlation with the 1 S 0 partial wave 4 ) in the three-body case.Actually, with the OBE 1 S 0 and 5 S 2 potentials, we find that the three-body ΩΩΩ system binds with a binding energy of 5.8 +2.5 −1.2 MeV and rms radius 1.9 +0.1 −0.2 fm.Note that the binding energy per baryon of the ΩΩΩ system is larger than that of the ΩΩ system, and consequently its rms radius is smaller than that of the ΩΩ bound state.This is understandable because for the ΩΩΩ system the 5 S 2 potential plays an important role, while only the 1 S 0 potential is relevant for the ΩΩ system.
The weights of partial waves and Hamiltonian expectation values of the predicted ΩΩΩ bound state are given in Table III, which clearly show that the 5 S 2 interaction plays a significantly important role in the ΩΩΩ system.More specifically, the weights of the 1 S 0 and 5 S 2 partial waves are about 22% and 78%, respectively.
As we mentioned above, since the Ω baryon is charged, the impact of the Coulomb interaction is worth discussing.We find that the Coulomb interaction in this three-body system affects the binding energy by about 2-3 MeV but does not change the conclusion.Considering the Coulomb interaction, the binding energy and rms radius of the ΩΩΩ bound state predicted by the OBE model are 2.0 MeV and 2.3 fm, respectively.
It is important to discuss where to search for the predicted ΩΩ and ΩΩΩ bound states.In Ref. [48], the production yield of the ΩΩ bound state was estimated using a dynamical coalescence mechanism for the relativistic heavy-ion collisions at √ s N N = 200 GeV and 2.76 TeV, which turn out to be of the order of 10 −6 .In Ref. [41], the production yields of N N Ω and N ΩΩ were estimated to be 10 −7 and 10 −9 , respectively.Comparing these results, one can estimate the ΩΩΩ production rate for the relativistic heavy-ion collisions at √ s N N = 200 GeV and 2.76 TeV, which is of the order of 10 −11 .the 1 S 0 potentials [14,15] and their OBE counterparts also exist [16].Note that in Ref. [15] no analytic form of the Ω bbb Ω bbb potential was provided.We fitted the lattice QCD potential with a sum of three Gaussian functions as done in Ref. [14].All the lattice QCD potentials and the corresponding OBE potentials are shown in Fig. 2. We note that although the interaction strengths of the lattice QCD potential and those of the OBE potentials are different, the positions where they become the most attractive are almost the same.The same can be said about the ΩΩ potentials shown in Fig. 1.Such a coincidence indicates that the OBE model must have captured some essential features of the baryon-baryon potentials.
With the above lattice QCD and the OBE potentials, we can study the two-body and three-body systems composed of Ω ccc and Ω bbb .As shown in Table IV, with only the strong interaction the Ω ccc Ω ccc bound system can be formed, but it dissolves once the Coulomb interaction is taken into account.On the other hand, the Ω bbb Ω bbb system is always bound regardless of the Coulomb interaction.Furthermore, we note that the results obtained with the lattice QCD potentials and those with the OBE potentials are similar.Nonetheless, none of the Ω ccc Ω ccc Ω ccc and Ω bbb Ω bbb Ω bbb three-body systems can bind, mainly because of the much weaker 5 S 2 interactions, which are nontrivial predictions of the present work.We found that the ΩΩ, Ω ccc Ω ccc , and Ω bbb Ω bbb systems can also bind with the OBE potentials, with binding energies and rms radii consistent with those of lattice QCD simulations.The repulsive Coulomb interactions plays an important role in these systems especially in the Ω ccc Ω ccc system, which is strong enough to break the Ω ccc Ω ccc pair bound by the strong force.
For the three-body systems, we find that the 5 S 2 partial wave plays a very important role in forming the 3 2 + threebody state.With only the 1 S 0 lattice QCD potentials, the ΩΩΩ, Ω ccc Ω ccc Ω ccc , and Ω bbb Ω bbb Ω bbb three-body systems do not bind.With the OBE potentials both in the 1 S 0 and 5 S 2 partial waves, the ΩΩΩ system becomes bound, while the Ω ccc Ω ccc Ω ccc and Ω bbb Ω bbb Ω bbb systems remain unbound mainly due to the much suppressed attractive 5 S 2 interaction in the two-body Ω ccc Ω ccc and Ω bbb Ω bbb systems.To verify the existence of the ΩΩΩ bound state, lattice QCD studies of the 5 S 2 interactions of the ΩΩ system will be the key.We hope that the predicted ΩΩΩ bound state can be searched for in present and future hadron-hadron colliders.
A particularly interesting discovery of the present work is that even the two-body interactions are attractive and strong enough to form two-body bound states, the three-body systems do not necessarily bind.This is because in three-body systems, spin-spin interactions can play an important role.The three highly symmetric systems studied in the present work provide an ideal platform to understand the relevance of spin-spin interactions in forming few-body bound states.

TABLE II .
Binding energies (BE) and root-mean-square radii (⟨r⟩) of the ΩΩ and ΩΩΩ bound states obtained with lattice QCD (with only 1 S0) and OBE potentials (with both 1 S0 and 5 S2).BE in MeV and radius ⟨r⟩ in fm.

TABLE III
Most charming and beautiful tribaryons.-It is straightforward to extend the above study to the Ω ccc Ω ccc and Ω bbb Ω bbb systems, for which the lattice QCD simulations already provided FIG.2.OBE and lattice QCD potentials of the ΩcccΩccc (top) and Ω bbb Ω bbb (bottom) systems.The blue dashed, orange dashed and green solid lines denote the 1 S0 OBE, the 5 S2 OBE and the 1 S0 lattice QCD potentials, respectively.

TABLE IV .
Binding energies (BE) and root-mean-square radii (⟨r⟩) of the ΩcccΩccc and Ω bbb Ω bbb bound states obtained with OBE and LQCD potentials (BE in MeV and radius ⟨r⟩ in fm.).NC means that the Coulomb interaction is not taken into account, while C means that the Coulomb interaction is considered.Motivated by the existence of ΩΩ, Ω ccc Ω ccc , and Ω bbb Ω bbb bound states predicted by lattice QCD simulations, we studied the 3 2 + ΩΩΩ, Ω ccc Ω ccc Ω ccc , and Ω bbb Ω bbb Ω bbb three-body systems with the lattice QCD and OBE potentials.