Study of Hidden-Charm and Hidden-Bottom Pentaquark Resonances in the Quark Model

: Inspired by the LHCb observation of hidden-charm pentaquarks, i.e

Four years later, the LHCb collaboration reported on three new pentaquarks, namely P c (4312), P c (4440), and P c (4457) [25]. P c (4312) was discovered with a significance of 7.3σ by analyzing the J/ψp invariant mass spectrum. The previously reported P c (4450) structure, with a significance of 5.4σ, was resolved into two narrow states: the P c (4440) and P c (4457). The masses and widths of these states are as follows:

The Quark Delocalization Color Screening Model
The details of QDCSM used in the present work can be found in the references [37,39]. In the following, only the Hamiltonian and parameters are given.
where T c is the kinetic energy of the center of mass; S ij is the quark tensor operator; Y(x) and H(x) are standard Yukawa functions [43]; 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 broad energy range from light to heavy quarks, an effective scale-dependent quark-gluon coupling is introduced, α s (µ) [44]: where µ is the reduced mass of two interacting quarks. All other symbols have their usual meanings. Moreover, a phenomenological color screening confinement potential is used, and µ ij is the color screening parameter. The light-flavor quark system is determined by fitting the deuteron properties, NN scattering phase shifts, and NΛ and NΣ scattering phase shifts, respectively, with µ uu = 0.45, µ us = 0.19, and µ ss = 0.08, satisfying the relation µ 2 us = µ uu µ ss . When extended to the heavy quark case, we took it as an adjustable parameter, µ cc = 0.01 ∼ 0.0001 fm −2 , and we found that the results were not sensitive to the value of µ cc . Thus, we take parameter µ cc = 0.01 in this work, and µ uc is obtained by utilizing the relation µ 2 uc = µ uu µ cc . All other parameters are taken from our previous work [6].
In QDCSM, the color screening is associated with the color structures of the system under consideration. For the three-quark baryon and quark-antiquark meson, the unscreened confinement is enough, especially for low-lying states, and the two-body interaction works well. However, it is different in the multiquark system. Lattice QCD calculations show string-like structures [45,46]. The confinement is a genuine multibody interaction, and, in general, one does not expect it to be described by a sum of two-body interactions. To simplify the calculations, the two-body interaction forms are still employed to evaluate the matrix elements of the Hamiltonian. The main physics introduced is the recognition that the confining interaction between two quarks in different nucleons might be different from those within one nucleon. Thus, we model the confinement as follows: the interaction is in its normal, unscreened form (quadratic in r ij ) when the interacting quark pair consistently remains in the same cluster orbit before and after interaction; otherwise, the interaction takes on the screening form. Although this has not been demonstrated to be correct, it is more sophisticated than the usual, simple, two-body confining interaction; it is expected that it includes some nonlocal, non-perturbative effects of QCD, which is missing in the three-quark baryons and quark-antiquark mesons. In addition, the screened confinement permits the development of quark delocalization in the QDCSM.
The quark delocalization in QDCSM is realized by specifying the single particle orbital wave function of QDCSM as a linear combination of left and right Gaussians. More details can be seen in Equation (A15) in Appendix A, in which the mixing parameter is not an adjusted one but is determined variationally by the dynamics of the multi-quark system itself. In this way, the multi-quark system chooses a favorable configuration in the interacting process. This mechanism has been used to explain the crossover transition between the hadron phase and quark-gluon plasma phase [47].
In this work, the resonating group method (RGM) [48], a well-established method for studying bound-state or scattering problems, is used to investigate the hidden-charm and hidden-bottom pentaquark systems. The details of RGM are shown in Appendix A.

Wave Functions
The orbital wave functions are shown in the appendix. The flavor, spin, and color wave functions are constructed in two steps. First, the wave functions for the meson and baryon clusters are constructed individually. Then, the wave functions of the two clusters are coupled together to form the wave function for the pentaquark system. Here, we only list the wave functions we used in this work. The flavor wave functions for a meson cluster are: and for a baryon cluster: where the superscript of the χ is the index of the flavor wave function for a meson or a baryon, and the subscript indicates the isospin I and the third component I z . The spin wave functions for a meson cluster are: and for a baryon cluster are: The color wave function for a meson cluster is: and for a baryon cluster is: Then, the wave functions for the pentaquark system can be obtained by coupling the wave functions of the meson and the baryon clusters. Finally, multiplying the wave functions ψ L , ψ σ , ψ f , and ψ c , according to the definite quantum number of the system, we can acquire the total wave functions of the system. To save space, we take the I J P = 1 2 1 2 P Σ c D as an example. The total wave function of this state is as follows:

The Results and Discussions
In this work, we investigated the hidden-charm pentaquark resonances in the crosssection of the J/ψp scattering. The peak position and the half-width of the bell shape correspond to the mass and the decay width of the resonance.
We calculate the cross-section of the J/ψp scattering channel with the effect of the channel-coupling. The channels involved are listed in Table 1. The cross-section of the J/ψp channel with J P = 1 2 − and J P = 3 2 − is shown in Figure 1. Obviously, there are four sharp peaks in the cross-section of the J/ψp scattering, two with J P = 1 2 − and another two with J P = 3 2 − , which indicates that there are four narrow resonance states. The first one is located at a mass of 4307.9 MeV with a very narrow partial width of about 1.2 MeV. The mass is under the threshold of the Σ c D, so it is the Σ c D molecular pentaquark. Moreover, both the mass and decay width are close to the experimental values of the P c (4312), and the peak corresponds to the peak appearing in the experimental J/ψp invariant mass distribution, which indicates that the reported P c (4312) state could be identified as the Σ c D molecular pentaquark with J P = 1  Generally, the bound state with the smaller spin should have a lower mass. It seems that this order is reversed here. This is due to the channel-coupling effect. For the J P = 1 2 − system, bound states exist in the single channels of Σ c D, Σ c D * , and Σ * c D * . It may cause an increase or decrease in the mass of the Σ c D * bound state by coupling with the channels shown in Table 1; this is because there is a bound state Σ c D below the Σ c D * and a Σ * c D * bound state above the Σ c D * state. The case is similar for the J P = 3  Table 1 because of the stronger channel-coupling effect with the lower Σ c D state. Whereas the mass of the Σ c D * bound state of J P = 3 2 − is pushed down to 4445.7 MeV from 4451 MeV in the single-channel calculation; this is a result of the coupling with the channels listed in Table 1 due to the weaker channel-coupling effect with the lower Σ * c D state. Hence, it is understandable that the mass of the Σ c D * bound state of J P = 1 2 − is higher than the one of J P = 3 2 − . However, the mass difference is only 14 MeV here, and the model uncertainty cannot assure the mass order. Therefore, the two-peak structure can be reproduced at about 4450 MeV in our theoretical cross-section of J/ψp scattering; these peaks arise from the resonances of Σ c D * molecular states of J P = 1 2 − and J P = 3 2 − . However, it is not easy to confirm the more exact spin parity because of the small mass difference between these two states.
Moreover, regarding the three resonances discussed above, one may see the fourth peak, as shown in Figure 1b. It is the molecular pentaquark Σ * c D with J P = 3 2 − . The mass of this resonance is 4376.4 MeV, which is very close to the reported P c (4380). However, the two-body decay width (with heavy hadron in the final state) is only 2.4 MeV, which is much smaller than the experimental value. We propose conducting an experiment to search for any narrow resonance near the P c (4380). In addition, in Figure 1a, we also find that there is a cusp near the mass of 4527 MeV, which is near the threshold of the Σ * c D * . Our previous calculation showed that the single channel Σ * c D * state of J P = 1 The reason is that the coupling with the channel pushes the higher state above the threshold. All of these hidden-bottom pentaquarks have similar properties with hidden-charm pentaquarks, so we can call them P c −like molecular pentaquarks P b , which are also worth investigating in experiments. In particular, the nature of these hidden-charm and hidden-bottom pentaquarks is similar to the corresponding N * cc and N * bb states predicated in References [49,50], which definitely cannot be accommodated by the conventional 3q quark models, and should form part of the heavy island for the quite stable N * baryons.

Summary
To summarize, our study is inspired by the observation of three narrow pentaquarks, namely P c (4312), P c (4440), and P c (4457), observed in the process of Λ 0 b → J/ψK − p, as reported by LHCb. We investigated the cross-section of J/ψp scattering in the quark delocalization color screening model. Four peaks were found in the cross-section. The mass and width of the first one was close to the P c (4312), which indicates that the P c (4312) can be interpreted as the hidden-charm molecular pentaquark Σ c D with J P = 1 2 − . The two-peak structure can be reproduced at about 4450 MeV in our theoretical cross-section of J/ψp scattering, corresponding to P c (4440) and P c (4457). They correspond to the resonances of the molecular pentaquarks Σ c D * of J P = 1 2 − and J P = 3 2 − . The more exact spin parity should be confirmed through both theoretical and experimental studies in the future. Another molecular pentaquark Σ * c D with J P = 3 2 − is also suggested in our calculation, the mass of which is close to the P c (4380), but with a width much smaller than it. Moreover, the Σ * c D * of both J P = 1 2 − and J P = 3 2 − are possible molecular pentaquarks. All of these narrow pentaquarks are worth searching for or will be confirmed in future experiments. Note that all of the P c state measurements have been made by LHCb. The only independent evidence for the P c states to date comes from the D0 collaboration, which observes events that are consistent with the unresolved P c (4440) → J/ψp structure at 3.2σ [51]. The gluonic excitation (GlueX) experiment measured the total cross-sections for the reaction γp → J/ψp and reported no evidence for the LHCb pentaquark candidates P c [52]. New higher-statistic GlueX measurements of the threshold's total cross-section, for the reaction γp → J/ψp [53], motivated an alternative search for the LHCb exotics. There is still much that remains unknown about the pentaquark states. Higher statistics data and more theoretical works in the future might provide more information for the hidden-heavy flavor pentaquark states. CLAS12 at Jefferson Lab (JLab) may be a possible place to observe the P c states in J/ψ photoproduction [54,55]. Its spin and photocoupler may be measured with future data, too. The search for the LHCb pentaquark in the photo-production process at Hall C, JLab, has received approval [56]. Moreover, pentaquarks with charm quarks can also be observed by the PANDA/FAIR [57].
For hidden-bottom pentaquarks, we predict several P c −like molecular pentaquarks P b above 11 GeV with narrow widths in the Υp scattering process. We hope that the proposed electron-ion collider (EIC) [58] and the upgraded facilities at JLab [59] can play important roles in discovering these interesting super-heavy pentaquarks.

Data Availability Statement:
This manuscript has no associated data or the data will not be deposited.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. Resonating Group Method for Bound-State and Scattering Problems
We use the resonating group method (RGM) to carry out a dynamic calculation. For a bound-state problem; we write the wave function of the baryon-meson system as (A1) where [σ] = [222] gives the total color symmetry and all other symbols have their usual meanings. The symbol A is the anti-symmetrization operator. With the SU(4) extension, both the light and heavy quarks are considered identical particles. Thus, A is written as where 1, 2, and 3 stand for the quarks in the baryon cluster, and 4 stands for the quark in the meson cluster.φ A andφ B are the anti-symmetrized internal cluster wave functions of the baryon A and meson B: where η I A S A and χ c (A) are the internal flavor-spin and color wave functions of the baryon cluster A. The Jacobi coordinates are defined as follows: From the variational principle, after variation with respect to the relative motion wave function χ(R) = ∑ L χ L (R), one obtains the following RGM equation where the Hamiltonian kernel H(R , R ) and normalization kernel N(R , R ) can, respectively, be calculated by For a bound-state problem, the energies and the wave functions χ(R) are obtained by solving the RGM equation. In practice, it is not convenient to work with the RGM expressions. We introduce generator coordinates S i to expand the Lth relative motion wave function χ L (R): with where C i represents the expansion coefficients, n represents the number of Gaussian bases, which is determined by the stability of the results, and j L represents the Lth spherical Bessel function. Then the relative motion wave function χ(R) is After including the motion of the center of mass, the total wave function Equation (A1) can be rewritten as where φ α (S i ) and φ β (−S i ) are the single-particle orbital wave functions with different reference centers: With the reformulated Equation (A12), the RGM Equation (A6) becomes an algebraic eigenvalue equation: where N L i,j and H L,L i,j are the wave function (A12) overlaps and Hamiltonian matrix elements (without the summation over L ), respectively. By solving the generalized eigenproblem, we obtain the energies of the 5-quark systems and their corresponding wave functions. In our calculation, the Gaussian distribution is chosen to ensure the stability of the results. If the numerical results stabilize as the size of the space and the number of Gaussians increase, it indicates that they have converged to the true energies of the Hamiltonian. In the present work, the results are stable when the largest distance between the baryon and meson clusters is around 6 fm. To keep the matrix dimensions manageable, the baryon-meson separation is taken to be less than 6 fm.
In the QDCSM, the single-particle orbital wave functions are delocalized. To implement this here, we modify Equation (A13) as follows: For a scattering problem, the relative wave function is expanded as where u L is from Equation (A9), h ± L represents the Lth spherical Hankel functions, k is the momentum of relative motion, calculated as k = 2µE cm , µ is the reduced mass of two hadrons (A and B) in the open channel, E cm is the incident energy, and R C is the cutoff radius beyond which all strong interactions can be disregarded. Moreover, α i and s i are complex parameters that are determined by the smoothness condition at R = R C , and C i satisfies ∑ n i=1 C i = 1. After performing the variational procedure, an Lth partial-wave equation for the scattering problem can be deduced as n ∑ j=1 L L ij C j = M L i (i = 0, 1, · · ·, n − 1), and By solving Equation (A18), we can obtain the expansion coefficients C i . Then the S−matrix element S L and the phase shifts δ L are given by Finally, the cross-section can be obtained from the scattering phase shifts by the following formula: