Reanalysis of the most strange dibaryon within constituent quark models

The most strange dibaryon $\Omega\Omega$ with quantum numbers $S=-6$, $I=0$, and $J^{P}=0^{+},~1^{-},~2^{+},~3^{-}$ is reanalyzed in the framework of quark delocalization color screening model (QDCSM) and chiral quark model (ChQM). The $\Omega\Omega$ dibaryon with $J^{P}=0^{+}$ is bound, and the one with other quantum numbers $J^{P}=1^{-},~2^{+},~3^{-}$ are all unbound in our calculation. The low-energy scattering phase shifts, the scattering length, and the effective range of the $\Omega\Omega$ dibaryon with $J^{P}=0^{+}$ also support the existence of such strange dibaryon. This dibaryon is showed to be a shallow bound state in QDCSM, while the binding energy becomes much larger in the ChQM by including the effect of the hidden-color channel coupling. And the scalar nonet meson-exchange in the ChQM also provides more attraction for the $\Omega\Omega$ system. Experimental search for such most strange dibaryon will provide much information for understanding the hadron-hadron interactions in different quark models.


I. INTRODUCTION
As is commonly believed that quantum chromodynamics (QCD) is the fundamental theory of the strong interaction. However, the low energy physics of QCD, such as hadron structure, hadron-hadron interactions, and the structure of multiquark systems, is much harder to calculate directly from QCD. Various QCD-inspired quark models have been developed to get physical insights into the multiquark systems. There are the MIT bag model [1], cloudy bag model [2], Friedberg-Lee nontopological soliton model [3], Skyrme topological soliton model [4], the constituent quark model [5,6], etc. Different models use quite different effective degrees of freedom, which might be indicative of the nature of lowenergy QCD.
The constituent quark model has been quite successful in understanding hadron spectroscopy and hadronhadron interactions even though we have not yet derived the constituent quark model directly from QCD. De Rujula, Georgi, and Glashow [5] first put forward a quarkgluon coupling model based on constituent quark and gluon effective degrees of freedom. Isgur and Karl obtained a good description of hadron spectroscopy based on this model [6]. However, extension of the model to baryon-baryon interactions does not reproduce the nucleon-nucleon (N N ) intermediate and long-range interaction.
One modification is the addition of scalar meson exchange and Goldstone bosons exchange on the quark level [7][8][9][10], which provide the nucleon-nucleon intermediate and long-range interaction, respectively. A typical approach is the chiral quark model (ChQM) [11,12], in which the constituent quarks interact with each other through colorless Goldstone bosons exchange in addition to the colorful one-gluon-exchange and confinement. To obtain the immediate-range attraction of N N interaction, the chiral partner σ meson-exchange has to be introduced. The σ meson had been observed by BES collaboration as a ππ S-wave resonance [13]. However, the results found by three groups independently, show that the correlated two-pion exchange between two nucleons generates strong short-range repulsion and very moderate long-range attraction, which is quite different from the behavior of the σ meson which used in the ChQM [14]. Therefore, one may wonder that the σ meson used in the ChQM is the correlated ππ resonance or an effective one.
An alternative approach to study baryon-baryon interaction is the quark delocalization color screening model (QDCSM), which was developed in 1990s with the aim of explaining the similarities between nuclear and molecular forces [15]. Two new ingredients were introduced: quark delocalization (to enlarge the model variational space to take into account the mutual distortion or the internal excitations of nucleons in the course of their interactions) and color screening (assuming the quark-quark interaction dependent on quark states aimed to take into account the QCD effect which has not yet been included in the two-body confinement and effective one gluon exchange). The model gives a good description of N N and Y N interactions and the properties of deuteron [16,17]. It is also employed to calculate the baryon-baryon scattering phase shifts and predict the dibaryon candidates d * and N Ω [18][19][20][21][22][23].
The difference between the ChQM and QDCSM is the intermediate-range attraction mechanism, which is the σ meson-exchange in ChQM and the quark delocalization and color screening in QDCSM. These two models have been applied to the study of nucleon-nucleon (N N ) and the N Ω systems [17,18,22,23]. The results show that the intermediate-range attraction mechanism in the QDCSM is equivalent to the σ meson-exchange in the ChQM in these two systems. And the color screening is an effective description of the hidden-color channels coupling [24]. It is interesting to check this consistency in other systems, such as the most strange dibaryon ΩΩ.
The dibaryon ΩΩ with quantum numbers S = −6, I = 0, and J = 0 was predicted by Kopeliovich in the framework of the flavor SU (3) Skyrmion model [25]. Zhang et al. also suggested to search for this ΩΩ state in heavy ion collision experiments [26]. And this dibaryon was also showed to be a bound state in the QDCSM [27]. Very recently, the ΩΩ with S = −6, I = 0, and J = 0 was investigated by the HAL QCD Collaboration [28]. They studied this dibaryon on the basis of the (2+1)-flavor lattice QCD simulations with a nearly physical pion mass m π ≃ 146 MeV. The results showed that this ΩΩ state had an overall attraction and was located near the unitary regime. They suggested that such a system can be best searched experimentally by the pair-momentum correlation in relativistic heavy-ion collisions. Then Morita et al. [29] calculated the correlation functions of this ΩΩ state based on an expanding source model by using the interaction potentials from the lattice QCD calculations.
In this work, we reanalyze the most strange dibaryon ΩΩ with quantum numbers S = −6, I = 0, and J P = 0 + , 1 − , 2 + , 3 − in both ChQM and QDCSM. The binding energy, as well as the low-energy scatter-ing phase shifts, the scattering length, and the effective range, which are useful for the experimental search of this strange dibaryon, are investigated. By comparing the results within these two quark models, we can check the model dependence of this dibaryon. On the other hand, we can also inspect the consistency of the intermediaterange attraction mechanism of these two models in such strange dibaryon ΩΩ system.
The structure of this paper is as follows. A brief introduction of two quark models is given in section II. Section III devotes to the numerical results and discussions. The summary is shown in the last section.

II. TWO QUARK MODELS
A. Chiral quark model In this work, the Salamanca model was chosen as the representative of the ChQM, because the work of the Salamanca group covers the hadron spectra and the nucleon-nucleon interaction, and has been extended to the study of multiquark states. The model details can be found in Ref. [12]. Here only the Hamiltonian is given: Where α s is the quark-gluon coupling constant. In order to cover the wide energy scale from light to strange quark, one introduces an effective scale-dependent quark-gluon coupling constant α s (µ) [30], where µ is the reduced mass of the interacting quark-pair. The coupling constant g ch for chiral field is determined from the N N π coupling constant through The other symbols in the above expressions have their usual meanings.
For the most strange dibaryons, two versions of ChQM [31,32] are used here. One is the SU (2) ChQM, in which σ meson is restricted to exchange between u and/or d quark pair only; another is the SU (3) ChQM, where full SU (3) scalar nonet meson-exchange was used. These scalar potentials have the same functional form as the one of SU (2) ChQM but a different SU (3) operator dependence [31], that is,

B. Quark delocalization color screening model
The Hamiltonian of QDCSM is almost the same as that of ChQM but with two modifications [15,16]: First, there is no σ-meson exchange in QDCSM, and second, the screened color confinement is used between quark pairs resident in different baryon orbits. That is (12) where the color screening constant µ ij is determined by fitting the deuteron properties, N N scattering phase shifts and N Λ, N Σ scattering cross sections, µ uu = 0.45, µ us = 0.19 and µ ss = 0.08, which satisfy the relation, µ 2 us = µ uu µ ss . The single particle orbital wave functions in the ordinary quark cluster model are the left and right centered single Gaussian functions: The quark delocalization in QDCSM is realized by writing the single particle orbital wave function as a linear combination of the left and right Gaussians: where ǫ(S i ) is the delocalization parameter determined by the dynamics of the quark system rather than adjusted parameters. In this way, the system can choose its most favorable configuration through its own dynamics in a larger Hilbert space. The parameters of these models are from our previous work of N Ω system [23]. We list all parameters in Table I. The calculated baryon masses in comparison with experimental values are shown in Table II.   In this work, we investigate the most strange dibaryon ΩΩ with quantum numbers S = −6, I = 0, and J P = 0 + , 1 − , 2 + , 3 − in QDCSM, SU (2) ChQM and SU (3) ChQM. The partial wave of J P = 0 + , 2 + is S-wave, and the one of J P = 1 − , 3 − is P -wave. We calculate the effective potentials of the ΩΩ system, because an attractive potential is necessary for forming bound state or resonance. The effective potential between two clusters is defined as, V (S) = E(S)−E(∞), where E(S) is the diagonal matrix element of the Hamiltonian of the system in the generating coordinate. The effective potentials of J P = 0 + , 1 − , 2 + , 3 − in three quark models are shown in Fig. 1(a), (b), (c) and (d), respectively. From Fig. 1(a), we can see that the potentials are attractive for the J P = 0 + ΩΩ state in all quark models. It is obvious that the attraction in SU (3) ChQM is the largest one, followed by the attractions in SU (2) ChQM and QDCSM. For the ΩΩ state with J P = 1 − , the potential is repulsive in SU (2) ChQM and QDCSM, but in SU (3) ChQM, it is a little bit attractive. The case is similar for the ΩΩ state with J P = 2 + . For the ΩΩ state with J P = 3 − , the potentials are all repulsive in three quark models. Therefore, from the behavior of the effective potentials of the ΩΩ state, it is possible for the J P = 0 + ΩΩ dibaryon to form bound state, while for the ΩΩ dibaryon with other quantum numbers, it is nearly impossible to form any bound state because of the repulsive interaction between two Ωs.
In order to investigate the source of the attractions for the J P = 0 + ΩΩ state, we calculate the contribution of each interaction term to the total potential of the system. The potentials of various terms: the kinetic energy (V vk ), the confinement (V con ), the one-gluon-exchange (V oge ), the one-boson-exchange (V π and V K do not contribute to the effective potential because they do not exchange between two s quarks, so only V η contributes), and the scalar nonet meson-exchange (V σ , V a0 , V κ , and V f0 ) are shown in Fig. 2.
For the QDCSM, quark delocalization and color screening work together to provide short-range repulsion and intermediate-range attraction. We illustrate this mechanism by showing contributions of all interaction terms to the effective potential in Fig. 2(a). From which we see that the attraction of the ΩΩ system mainly comes from the kinetic energy term. The confinement interaction provides a little attraction, while other terms provide repulsive potentials, which reduce the total attraction of the ΩΩ potential.
For the SU (2) ChQM, the quadratic confinement do not contribute to the potential between two Ω's, because of the properties of two color singlets. Since the σ meson is restricted to exchange between the u and d quarks only in SU (2) ChQM, there is no σ meson-exchange interaction between two Ωs. Therefore, there are only kinetic energy, one-gluon-exchange and one-η-exchange contribute to the effective potentials. It is shown in Fig. 2(b) that the kinetic energy term provides the major attraction, while other two terms provide repulsive potentials, which decrease the total attractions.
For the SU (3) ChQM, the scalar nonet mesonexchange is included. Although a 0 and κ mesons do not contribute because they do not exchange between s quarks, both the f 0 meson-exchange and the σ mesonexchange introduce large attractions, which lead to the strong attraction between two Ωs.

B. Binding energy calculation
In order to see whether or not there is any bound state, we carry out a dynamic calculation. The resonat- ing group method (RGM), described in more detail in Ref. [33], is used here. Expanding the relative motion wavefunction between two clusters in the RGM by a set of gaussians, the integro-differential equation of RGM can be reduced to algebraic equation, the generalized eigenequation. The energy of the system can be obtained by solving the eigen-equation. In the calculation, the baryon-baryon separation (|s n |) is taken to be less than 6 fm (to keep the matrix dimension manageably small). In our calculation, the ΩΩ system with J P = 1 − , 2 + ,3 − are unbound in all quark models, which agree with the repulsive nature of the interaction of these states. While, the J P = 0 + ΩΩ state is bound in all quark models, due to the strong attractions in this system. Here, we discuss the J P = 0 + state in detail. The binding energies of J P = 0 + ΩΩ state in various quark models are listed in Table III, where B sc stands for the binding energy of the single channel ΩΩ, and B cc refers to the binding energy with the hidden-color channel coupling. The single channel calculation shows that the binding energy in QDCSM and SU (2) ChQM is very small, which indicates that the J P = 0 + ΩΩ is a shallow bound state. In contrast, the binding energy in the SU (3) ChQM is much larger due to the stronger attraction between two Ωs, which suggests that the ΩΩ is a deep bound state. By coupling the hidden-color channel, a much deeper binding energy is obtained in both the SU (2) ChQM and SU (3) ChQM, which reaches to −46.8 MeV and −103.3 MeV respectively. It indicates that the effect of the hidden-color channel coupling is important for the ΩΩ system in the ChQM. In QDCSM, since it contains hidden-color channels coupling effect already through the color screening [18,24], including the color-singlet channels is enough. Therefore, we find that the J P = 0 + ΩΩ appears as a shallow bound state in QDCSM, and this conclusion is consistent with that of the HAL QCD Collaboration [28], in which they showed that the J P = 0 + ΩΩ state had an overall attraction and was located near the unitary regime. However, the J P = 0 + ΩΩ state becomes a deeper bound state in the SU (2) ChQM, and even a much deeper bound state in the SU (3) ChQM.

C. The low-energy scattering phase shifts
For the purpose of providing more information for the experimental search of such most strange dibaryon, we calculate the low-energy scattering phase shifts, the scattering length, and the effective range of the ΩΩ dibaryon with J P = 0 + . In experiment, each Ω can be identified through a successive weak decay Ω − → Λ + K − → p + π − + K − . A large scattering length (not the existence of a bound state) is the important element for the correlation C(Q) to have characteristic enhancement at small relative momentum Q [34]. Here, the well developed Kohn-Hulthen-Kato (KHK) variational method is used to calculate the low-energy scattering phase shifts. The details can be found in Ref. [33]. Fig. 3 illustrates the scattering phase shifts of the J P = 0 + ΩΩ state. It is obvious that in all quark models, the scattering phase shifts go to 180 degrees at E c.m. ∼ 0 and rapidly decreases as E c.m. increases, which implies the existence of a bound state. The results are consistent with the the bound state calculation shown above. Besides, the behavior of the low-energy scattering phase shifts is also in agreement with that of the lattice QCD calculation [28].
Then, the scattering length a 0 and the effective range r 0 of the ΩΩ state can be extracted from the low-energy scattering phase shifts by the following formula: where δ is the low-energy scattering phase shifts, k is the momentum of the relative motion with k = √ 2µE cm , µ is the reduced mass of two baryons, and E cm is the incident energy. The binding energy B ′ can be calculated according to the relation: where α is the wave number, which can be obtained from the relation [35]: The results are listed in Table IV. From Table IV, we can see that in all quark models, the scattering length are all positive, which implies that the J P = 0 + ΩΩ dibaryon is a bound state here. The binding energies obtained by Eq. (16) is broadly consistent with that in Table III, which is obtained by the dynamic calculation. Here again, the scattering length a 0 and effective range r 0 of the J P = 0 + ΩΩ dibaryon in the QDCSM are in agreement with the results of the lattice QCD calculation [28], in which a 0 = 4.6(6)( +1.2 −0.5 ) fm and r 0 = 1.27(3)( +0.06 −0.03 ) fm.
(2) For the J P = 0 + ΩΩ, the attraction between two Ωs is strong enough to form a bound state in all quark models.
(3) The low-energy scattering phase shifts, the scattering length, and the effective range of the ΩΩ system with J P = 0 + also support that this most strange dibaryon is a bound state. (4) All the results in the QDCSM are consistent with that obtained by the HAL QCD method, which suggest that the J P = 0 + ΩΩ dibaryon is a shallow bound state. In contract, the binding energy is much deeper in both the SU (2) ChQM SU (3) ChQM, where the hidden-color channel coupling is employed.
The quark model study of the hadron interaction has experienced a long history. The mechanism of the intermediate-range attraction of the baryon-baryon interaction is one of the important issues in the study. In N N case we have shown that the phenomenological σ meson exchange in ChQM is equivalent to the quark delocalization and color screening in QDCSM [17] and the color screening effect in QDCSM is an effective description of hidden color channel coupling [24]. For the strange N Ω system, the QDCSM predicts a bound N Ω dibaryon with quantum numbers S = −3, I = 1 2 , J P = 2 + , while the ChQM cannot obtain the bound state if the σ-meson is not universally exchanged between any quark pair. However, the bound state was finally obtained by considering the the hidden-color channels coupling. Although the similar results are obtained in both models, the mechanism is different. In QDCSM, quark delocalization and color screening work together to provide short-range repulsion and intermediate-range attraction, the coupling of the color singlet channels is enough to form a bound state N Ω, while in the SU (3) ChQM, although the universal σ-meson exchange introduces large attraction, but it is canceled by the repulsive potentials of κ and f 0 exchange, and the bound N Ω state is obtained by coupling both color singlet and hidden-color channels. Extending to the most strange ΩΩ dibaryon, a shallow bound state is obtained in QDCSM. But this J P = 0 + ΩΩ become a much deeper bound state in both the SU (2) ChQM and SU (3) ChQM by coupling the hidden-color channel. Besides, in the SU (3) ChQM, the f 0 meson-exchange and the σ meson-exchange introduce large attractions, which also increase the total attraction of the ΩΩ potential.
To validate the intermediate attraction mechanism, more experimental data are needed. Experimental search for dibaryons may provide more information for this issue. Searching for the N Ω bound state has made considerable headway by the STAR experiment [36]. If experiment confirms the existence of N Ω dibaryon state, it will be a signal showing that the quark delocalization and color screening (an effective description of hidden color channels coupling) is effective way to describe the intermediate range attraction of baryon-baryon interaction. This mechanism is also preferred by the similarity between nuclear force and molecular force. Besides, from the phenomenological point of view, the ΩΩ system can be best searched by the measurement of pair-momentum correlation C(Q) with Q being the relative momentum between two baryons produced in relativistic heavy-ion collisions [37]. Experimental confirmation of the N Ω and ΩΩ dibaryons will provide other samples of six-quark system than the non-strange dibaryon d * [38][39][40]. We wish there will be more experimental collaborations to be involved in the search of such strange dibaryons.