Prediction of NΩ -like dibaryons with heavy quarks

Possible $N\Omega$-like dibaryons $N\Omega_{ccc}$ and $N\Omega_{bbb}$ with quantum numbers $IJ^P=\frac{1}{2}2^+$ are investigated within the framework of quark delocalization color screening model. We find both of these two states are bound, and the binding energy increases as the quarks of the system become heavier. The attraction between $N$ and $\Omega_{ccc}$ (or $\Omega_{bbb}$) mainly comes from the kinetic energy term due to quark delocalization and color screening. The effect of the channel-coupling provides more effective attraction to $N\Omega_{ccc}$ and $N\Omega_{bbb}$ systems. Besides, the scattering length, the effective range, and the binding energy, obtained from the calculation of the low-energy scattering phase shifts, also supports the existence of the $N\Omega_{ccc}$ and $N\Omega_{bbb}$ states. All these properties can provide necessary information for experimental search for the $N\Omega$-like dibaryons with heavy quarks. And the experimental progress can also check the mechanism of the intermediate-range attraction of the baryon-baryon interaction in quark models.


I. INTRODUCTION
The existence of dibaryons is one of the long standing problems in hadron physics. Although the dibaryon searches experienced several ups and downs in their long history, they received renewed interest in recent years. For the nonstrange dibaryon, the best-known candidate is the ∆∆ resonance state, which was predicted by Dyson and Xuong in 1964 [1] and later also by Goldman et al., who called it the "inevitable dibaryon" d * due to its unique symmetry features [2]. Recently, the WASA-at-COSY collaboration reported the discovery of this ∆∆ resonance state with M = 2.37 GeV, Γ ≈ 70 MeV, and IJ P = 03 + [3][4][5]. The detail of this dibaryon observation can be found in Ref. [6]. The quark model calculations [7][8][9][10], as well as the relativistic three-body calculations [11,12] all described properly the characteristics of this resonance.
For the strange dibaryon, the H-dibaryon with J = 0, S = −2, the N Ω with J = 2, S = −3, and the ΩΩ with J = 0, S = −6 are particularly interesting, since the Pauli blocking among valence quarks do not operate in these systems. Above all, the progress of the N Ω searches in experiment attracted more and more attention for its accessible. Very recently, the measurement of the pΩ correlation function was conducted in Au + Au collisions by the STAR experiment at the Relativistic Heavy-Ion Collider (RHIC) [13], and the result indicated that the scattering length is positive for the pΩ interaction and favored the pΩ bound state hypothesis. On the theoretical side, the N Ω state has been investigated by several groups. Goldman et al. predicted that the S = −3, I = 1/2, J = 2 dibaryon state N Ω might be a narrow res-onance in a relativistic quark model [14]. Oka proposed that there should be a quasi-bound state with IJ P = 1 2 2 + by using a constituent quark model [15]. Recent study of (2 + 1)-flavor lattice quantum chromodynamics (QCD) simulations by HAL QCD Collaboration reported that the N Ω was indeed a bound state at pion mass of 875 MeV [16] and later with nearly physical quark masses (m π ≃ 146 MeV and m K ≃ 525 MeV) [17]. K. Morita et al. studied the two-pair momentum correlation functions of the dibaryon candidate N Ω in relativistic heavy-ion collisions by employing the interactions obtained from the (2 + 1)-flavor lattice QCD simulations [18,19]. Besides, this state has also been observed to be bound in several relativistic quark models [20][21][22][23][24].
For the dibaryons with heavy quarks, the N Λ c system and the H-like dibaryon state Λ c Λ c were both studied on hadron level [25,26] and on quark level [27,28]. The possibility of existing deuteron-like dibaryons with heavy quarks, such as N Σ c , N Ξ ′ c , N Ξ cc , ΞΞ cc and so on, were investigated by several realistic phenomenological nucleon-nucleon interaction models [29,30]. Recently many near-threshold charmonium-like states called "XY Z" particles were observed, triggering lots of studies on the molecule-like bound states containing heavy quarks. Such studies will give further information on the hadron-hadron interactions. In the heavyquark sector, the large masses of the heavy quarks reduce the kinetic energy of the system, which makes them easier to form bound states. Very recently, the Lattice QCD also studied the deuteron-like dibaryons with valence quark contents: Σ c Ξ cc (uucucc), Ω c Ω cc (sscscc), Σ b Ξ bb (uububb), Ω b Ω bb (ssbsbb), and Ω ccb Ω cbb (ccbcbb), and with spin parity J P = 1 + [31]. They also found that the binding of these dibaryons became stronger as they became heavier in mass. Therefore, the dibaryons with heavy quarks are also possible multiquark states, and the study of such system will help us to understand the hadron-hadron interactions and search for exotic quark states in temporary hadron physics.
It is known to all that QCD is the fundamental theory of the strong interaction. However, in the low-energy region of strong interaction, it is difficult to directly use QCD to study the complicated systems such as hadronhadron interactions and multiquark states because of the nonperturbative complication. Various QCD-inspired models have been developed to get physical insights into the multiquark systems. The quark delocalization color screening model (QDCSM), developed in the 1990s with the aim of explaining the similarities between nuclear and molecular forces [32], is the representative of the quark models. In this model, quarks confined in one nucleon are allowed to delocalize to a nearby baryon and the confinement interaction between quarks in different baryon orbits is modified to include a color screening factor. The latter is a model description of the hidden color channel coupling effect [33]. The delocalization parameter is determined by the dynamics of the interacting quark system, this allows the quark system to choose the most favorable configuration through its own dynamics in a larger Hilbert space. The model gives a good description of N N and Y N interactions and the properties of deuteron [34,35]. It is also employed to study the dibaryon candidates: d * [7,8] and N Ω [20][21][22][23], and dibaryons with heavy quarks: the N Λ c system and the Λ c Λ c system [27,28].
In this work, we continue to investigate dibaryons with heavy quarks by using QDCSM. In our previous work, we have shown N Ω is a narrow resonance in ΛΞ Dwave scattering process [22]. However, the Λ-Ξ scattering data analysis is quite complicated experimentally. Then we calculated N Ω scattering length, effective range and binding energy based on the low-energy scattering phase shifts. These information can be observed by the N -Ω correlation analysis with RHIC and LHC data, or by the new developed automatic scanning system at J-PARC [23]. It is interesting to extend such study to the dibaryons with heavy quarks. So we investigate whether the N Ω-like dibaryons: N Ω ccc and N Ω bbb exist or not in QDCSM. The low-energy scattering phase shifts, scattering length, effective range and binding energy of N Ω ccc and N Ω bbb are calculated, which are useful for the experimental search of such heavy dibaryons.
The structure of this paper is as follows. A brief introduction of QDCSM is given in section II. Section III devotes to the numerical results and discussions. The summary is shown in the last section.

II. THE QUARK DELOCALIZATION COLOR SCREENING MODEL (QDCSM)
The detail of QDCSM used in the present work can be found in the references [32,34]. Here, we just present the salient features of the model. The model Hamiltonian is: Where S ij is quark tensor operator; Y (x) and H(x) are standard Yukawa functions; T c is the kinetic energy of the center of mass; α ch is the chiral coupling constant, determined as usual from the π-nucleon coupling constant; α s is the quark-gluon coupling constant. In order to cover the wide energy range from light, strange, to heavy quarks, an effective scale-dependent quark-gluon coupling α s (u) was introduced [36]: The other symbols in the above expressions have their usual meanings. All parameters, which are fixed by fitting the masses of baryons with light flavors and heavy flavors, are taken from our previous work [28]. The values of those parameters are listed in Table I. The masses of light flavor baryons are shown in our former work [23], here we list the masses of the charmed and bottom baryons in Table II.
bb Ω b Ω bbb Expt. 5811 5832 5619 5792 5955 · · · · · · 6046 · · · Model 5809 5817 5619 5888 5971 10455 10464 6131 15169 The quark delocalization in QDCSM is realized by specifying the single particle orbital wave function of QD-CSM as a linear combination of left and right Gaussians, the single particle orbital wave functions used in the ordinary quark cluster model, Here s i , i = 1, 2, ..., n are the generating coordinates, which are introduced to expand the relative motion wave function. The delocalization parameter ǫ(s i ) is determined by the dynamics of the quark system rather than adjusted parameters. It has been used to explain the cross-over transition between hadron phase and quarkgluon plasma phase [38].

III. THE RESULTS AND DISCUSSIONS
In this work, we investigate the N Ω-like dibaryons with heavy quarks: uudccc and uudbbb systems with quantum numbers IJ P = 1 2 2 + in QDCSM. The channel coupling effects are also considered. The labels of all coupled channels are listed in Table III.
In order to check whether or not there is any bound state, a dynamical calculation is needed. The resonating group method (RGM) [39] is employed here. By expanding the relative motion wave function between two clusters in the RGM equation by gaussians, the integrodifferential equation of RGM can be reduced to an algebraic equation, which is the generalized eigen-equation. Then by solving the eigen-equation, the energy of the system can be obtained. Besides, to keep the matrix dimension manageably small, the baryon-baryon separation is taken to be less than 6 fm in the calculation. The binding energies of every channel and the channel-coupling of the uudccc and uudbbb systems are listed in Table IV, where B stands for the binding energy, c.c. means the result of channel-coupling calculation, and ub means that the state is unbound.
For the uudccc system, the single channel calculation shows that the both Σ c Ξ * cc and N Ω ccc is bound with very small binding energies. By doing a channel-coupling calculation, the lowest energy of the system is 30.9 MeV lower than the threshold of N Ω ccc , which means that this heavy quark dibaryon N Ω ccc is a bound state in QDCSM. Obviously, the effect of the channel-coupling is important for providing more effective attraction to the N Ω ccc systems. Besides, the bound state Σ c Ξ * cc disappears, because the channel-coupling calculation pushes the energy of this state above its threshold. For the N Ω bbb system, the result is similar to the N Ω ccc system. The individual N Ω bbb channel is bound with binding energy −2.8 MeV. The channel-coupling calculation lower the energy of the system further, whose energy is 50.7 MeV below the threshold of N Ω bbb , is obtained. By comparing with the dibaryon N Ω [23], we find that the binding energy increases as the quark of the system becomes heavier. This conclusion is consistent with the recent work of lattice QCD [31], in which they studied the deuteron-like dibaryons with heavy quarks and found that the stability of dibaryons increases as they become heavier.
To investigate the interaction between N and Ω ccc (or Ω bbb ), we calculate the effective potentials, V eff = E(s) − E(s = ∞), E(s) is the energy of the system with N − Ω ccc separation s, as well as the contribution of each interaction term to the energy of the system. The results for N Ω ccc and N Ω bbb states are similar. To save space, we take the effective potentials between N and Ω ccc as an example. The total effective potentials and the contributions of all interaction terms to the effective potential, including the kinetic energy (V vk ), the confinement (V con ), the one-gluon-exchange (V oge ), the π-exchange (V π ), and the η-exchange (V η ), are shown in Fig. 1. We notice that due to the special quark content of N Ω ccc system, the effective interaction have very small contribution from the one-gluon-exchange interaction. The attraction between N and Ω ccc mainly comes from the kinetic energy term due to the quark delocalization, other terms provide repulsive potentials, which reduce the total attraction of the N Ω ccc potential. By comparing with the interaction between N and Ω, the behavior is similar, because we use the same model, and the mechanism of the intermediaterange attraction in this model is the quark delocalization and color screening, which work together to provide short-range repulsion and intermediate-range attraction.
In our previous work, we also calculated the low-energy scattering phase shifts, scattering length, and effective range of N Ω, which can provide necessary information for the N -Ω correlation analysis with RHIC and LHC data. Naturally, we do the same calculation for the N Ω ccc and N Ω bbb systems. The low-energy scattering phase shifts are calculated by using the well developed Kohn-Hulthen-Kato(KHK) variational method [39]. The wave function of the dibaryon system is of the form where ξ 1 and ξ 2 are the internal coordinates for the baryon cluster A, and ξ 3 and ξ 4 are the internal coordinates for another baryon cluster B. R AB = R A − R B is the relative coordinate between the two clusters. The symbol A is the anti-symmetrization operator. Theφ A andφ B are the internal cluster wave functions of two baryons A and B, and χ L (R AB ) is the relative motion wave function between two clusters. For a scattering problem, χ L (R AB ) is expanded as with S i is called the generating coordinate, C i is expansion coefficients, n is the number of the gaussians, which is determined by the stability of the results. h ± L is the L-th spherical Hankel functions, k AB is the momentum of relative motion with k AB = √ 2µ AB E cm , µ AB is the reduced mass of two hadrons (A and B) of the open channel, E cm is the incident energy, and R C is a cutoff radius beyond which all the strong interaction can be disregarded. α i and s i are complex parameters which are determined by the smoothness condition at R AB = R C and C i satisfy n i=1 C i = 1. j L is the L-th spherical Bessel function. After performing variational procedure, a L-th partialwave equation for the scattering problem can be deduced as with and By solving Eq.(8), we can obtain the expansion coefficients C i . Then the scattering matrix element S L and the phase shifts δ L are given by Then, we can extract the scattering length a 0 and the effective range r 0 from the low-energy phase shifts by using the formula: Finally, the binding energy B ′ is calculated according to the relation: where α is the wave number which can be obtained from the relation [40]: Please note that here we use another method to calculate the binding energy, so we label it as B ′ . The lowenergy phase shifts are shown in Fig. 2, and the scattering length, the effective range, as well as the binding energy B ′ are listed in Table V. All results are obtained with the five channels coupling calculation in QDCSM.
It is obvious that the scattering phase shifts of both N Ω ccc and N Ω bbb states go to 180 degrees at E c.m. ∼ 0 and rapidly decreases as E c.m. increases, which implies the existence of the bound states. The change rule is similar to the N Ω state [23]. Besides, the results are consistent with the bound state calculation shown above. From Table V, we can see that the scattering length of both N Ω ccc and N Ω bbb states are all positive, which implies again that these two states are bound states. The binding energies B ′ of these two states are close to the binding energy B shown above. It indicates that the binding energy from the two methods are coincident with each other.

IV. SUMMARY
In this work, we investigate the N Ω-like dibaryons with heavy quarks: uudccc and uudbbb systems with quantum numbers IJ P = 1 2 2 + in the framework of QD-CSM. Our results show that both of these two states are bound. In quark model, the hadron-hadron interaction usually depends critically upon the contribution of the color-magnetic interaction. However, due to the special quark content of N Ω ccc and N Ω bbb systems, the effective interaction have very small contribution from the color-magnetic interaction. The attraction between N and Ω ccc (or Ω bbb ) mainly comes from the kinetic energy term due to quark delocalization and color screening. Besides, the channel-coupling also plays an important role for providing more effective attraction to the N Ω ccc and N Ω bbb systems. This rule is similar to the N Ω system. Searching for the N Ω bound state has made considerable headway by the STAR experiment. If the existence of N Ω state can be confirmed by more experiments, it will be a signal showing that the mechanism of quark delocalization and color screening is really responsible for the intermediate range attraction of baryon-baryon interaction.
On the other hand, we find that the binding of these dibaryons becomes stronger as they become heavier in mass, which indicates that it is more possible for the N Ω ccc and N Ω bbb states to be bound. So it is worth looking for such N Ω-like dibaryons in the experiment, although it will be a challenging subject. Besides, the lowenergy scattering phase shifts, the scattering length, the effective range, and the binding energy (obtained from the scattering length) also support the existence of the N Ω ccc and N Ω bbb states. All these characteristic can provide necessary information for experimental search for the N Ω-like dibaryons with heavy quarks.