Fully heavy pentaquarks

Very recently, the LHCb collaboration reported a fully charmed tetraquark state $X(6900)$ in the invariant mass spectrum of $J/\psi$ pairs. If one $J/\psi$ meson is replaced with a fully charmed baryon, we obtain a fully charmed pentaquark candidate. In this work, we perform a systematical study on the mass spectra of the S-wave fully heavy pentaquark $QQQQ\bar{Q}$ in the framework of the chromomagnetic interaction model. Based on our results in two different schemes, we further investigate the decay behaviors for them. We hope that our study will be helpful to search for such types of the exotic pentaquark states in experiment in the future.


I. INTRODUCTION
At the birth of quark model [1][2][3], Gell-Mann and Zweig indicated that hadronic states with the qqqq and qqqqq quark configurations should exist in nature. Such exotic states were further investigated with some phenomenological models soon afterwards. For example, Jaffe adopted the quark-bag model to study q 2q2 hadrons, where the mass spectrum and dominant decay behavior were predicted [4]. In 1979, Strottman calculated the masses of q 4q and q 5q2 in the framework of MIT bag model [5]. The name pentaquark was firstly proposed by Lipkin in 1987 [6], where he studied anticharmed strange pentaquarks. Since 2003, with the accumulation of experimental data, more and more charmonium-like XY Z states were reported in experiment. Especially, the observation by the LHCb Collaboration confirms the existence of pentaquark P c states [7][8][9]. In the past about twenty years, great progresses have been made on studying exotic multiquarks [10][11][12][13].
Recently, the LHCb collaboration studied the invariant mass spectrum of J/ψ pairs, and they reported a narrow structure around 6.9 GeV and a broad structure in the mass range 6.2-6.8 GeV. The global significance for these two structures are larger than 5σ. Such distinct structures are expected to be with the cccc configuration [14].
The discoveries of fully heavy tetraquark states and P c states make one speculate that the pentaquark state with fully heavy quarks QQQQQ may also exist. If the mass is above the baryon-meson thresholds, the heavy pentaquark state may allow the strong decays into the corresponding two body. The study of the masses and decay properties would help to search for the heavy pentaquark states in experiment.
The strong interaction in a fully heavy multiquark state is not clear at present. The Chromomagnetic Interaction (CMI) model provides us a simply picture to quantitatively understand the spectrum of multiquark states. In the framework of CMI model [56], the strong interaction between quarks via gluon-exchange force is parameterized into effective quark masses and quark coupling parameters. Despite its simple Hamiltonian, this model can catch the basic features of hadron spectra, since the mass splittings between hadrons reflect the basic symmetries of their inner structures [27]. This model has been widely adopted to study the mass spectra of multiquark systems [47,[57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74]. In this work, we systematically study the S-wave QQQQQ pentaquark system within the framework of CMI model to calculate the mass spectra, the relative partial decay widths and find possible stable pentaquark state.
This paper is organized as follows. In Sec. II, we introduce the CMI model and determine the relevant parameters used in the CMI model. The f lavor ⊗ color ⊗ spin wave functions are constructed and the CMI Hamiltonian elements are calculated for the QQQQQ pentaquark system in Sec. III. In Sec. IV, we present the mass spectra, the mass splittings, possible strong decay channels, and the relative partial decay widths, and also discuss the stability for the pentaquark states. A short summary is followed by Sec. V. Finally, some useful expressions are presented in Appendix. A.

II. THE CHROMOMAGNETIC INTERACTION MODEL
The masses of the ground hadrons can be obtained by the effective Hamiltonian at quark level where the H CMI denotes the Hamiltonian of the chromomagnetic interaction [56]. σ i and λ i are the Pauli matrices and the Gell-Mann matrices, respectively. For antiquark, the λ i should be replaced with the −λ * i . m i is the effective mass of the i-th constituent quark. In the above Hamiltonian, the chromoelectric interaction and color confinement effect are also incorporated in the effective quark mass m i . C ij is the effective coupling constant between the i-th quark and j-th quark. The effective quark mass m i and the coupling constant C ij can be determined from the experimental hadron masses.
As indicated in Ref. [57][58][59][60][61][62][63][64][65][66], the predicted hadron masses obtained from Eq. (1) are generally overestimated. The main reason is that the dynamical effects inside hadrons can not simply be absorbed into the effective quark masses. Thus, in order to take such effective interaction into account, we replace the sum of m i term in Eq. (1) Here, we choose the baryon-meson thresholds as the mass scales, where the reference baryon-meson system should have the same constituent quarks with the studied pentaquark state. In this way, the dynamical effects that are not incorporated in the original approach are somehow phenomenologically compensated in this procedure [60]. We label this method as the reference mass scheme.
In addition, we introduce another scheme to estimate the masses of pentaquark states. We separate the two-body chromoelectric effects out of the effective quark masses and generalize the chromomagnetic interaction model by writing the chromoelectric term explicitly [47,[72][73][74] i.e., where the omitted operator nullifies the color-singlet physical states, and This treatment has been successfully adopted in ref. [28,47,[72][73][74][75][76]. The parameters m ij and v ij are also determined from the experimental hadron masses. In this work, we label this method as the modified CMI model scheme.
To estimate the masses of the QQQQQ pentaquark states, we need some hadron masses as input to fit the effective coupling parameters C ij , m ij , and v ij [77]. These conventional hadrons are listed in Table I. Because some of the heavy flavor baryons are not yet observed, we introduce the theoretical results in Refs. [74,78] as our input, and enclose the theoretical values of masses for these baryons with parentheses in Table I. Now we can fit the effective coupling parameters C ij , m ij , and v ij in the reference mass and modified CMI model schemes by applying Eq. (2) and Eq. 3, respectively. We present the obtained effective coupling parameters of QQQQQ pentaquark states in Table II. One can refer to ref. [13,73,74] for more details.

III. THE QQQQQ PENTAQUARK WAVE FUNCTIONS AND THE CMI HAMILTONIAN
In order to study systematically the mass spectra of the QQQQQ pentaquark system, we need construct the wave function of QQQQQ pentaquark first. We exhaust all the possible color ⊗ spin wave functions of pentaquark states, and combine them with the corresponding flavor wave functions. The constructed pentaquark wave functions should be constrained appropriately by Pauli principle. After that, we can use these pentaquark wave functions to calculate the mass spectra of the corresponding pentaquark states.
The total wave function of S-wave QQQQQ pentaquark can be described by the direct product of flavor, color, and spin wave functions  [77]. The masses of not-yet-observed baryons and B * c in parentheses are taken from Refs. [74] and [78], and others are from experiments.
In the flavor space, we divide the QQQQQ pentaquark system into three groups of subsystems according to their symmetries, i.e., (1) the first four quarks are identical: the ccccc, ccccb, bbbbc, and bbbbb pentaquark subsystems, (2) the first three quarks are identical: the cccbc, cccbb, bbbcc, and bbbcb pentaquark subsystems, (3) there are two pairs of identical quarks: the ccbbc and ccbbb pentaquark subsystems.
The color wave functions should be singlets due to the color confinement. The color wave functions for QQQQQ pentaquark system can be deduced from the following direct product , Here, the subscript labels the irreducible representation of SU(3). Then, by combining the antitriplet from antiquark with the deduced three color triplets in Eq. (7), we obtain three color singlets for QQQQQ pentquark system and In the spin space, the direct product of five fermions rep-resented in terms of Young tableau can be written as For the pentaquark states with total spin J = 5/2, the spin state can be represented in terms of one-dimensional Young tableau [5] as For the pentaquark states with total spin J = 3/2, the spin state can be represented in terms of four-dimensional Young tableau [4,1] as Similarly, for the pentaquark states with total spin J = 1/2, the spin state can be represented in terms of fivedimensional Young tableau [3,2] as Since the particle 5 is an antiquark, we can isolate this antiquark and discuss the symmetry property of the first four quarks 1, 2, 3, and 4 in color ⊗ spin space.
When the antiquark 5 is separated from the spin wave functions, the spin states represented in Young tableaus without the antiquark 5 can be directly obtained from Eqs. (12)(13)(14) as We can identify the spin states in Eq. (15) with the Young-Yamanouchi bases for Young tableau [4], [3,1], and [2,2].
With the above preparation, we can start to construct the flavor ⊗ color ⊗ spin wave functions of QQQQQ pentaquark states.
Then we combine the flavor wave functions with the color ⊗ spin wave functions. We present the Young-Yamanouchi bases allowed by exchange symmetry in Table VI of Appendix. A, and the pentaquark wave functions satisfied Pauli Principle can then be obtained with the table.
Based on the possible ψ f lavor ⊗ ψ color ⊗ ψ spin bases of the QQQQQ pentaquark system, we calculate the CMI matrices for the corresponding pentaquark states. In Table VII of Appendix. A, we only present the expressions of CMI Hamiltonians for the ccccc, cccbc, and ccbbc pentaquark subsystems. The expressions of CMI matrices for the ccccb, bbbbc, bbbbb, cccbb, bbbcc, bbbcb, and ccbbb pentaquark subsystems can be obtained from those of the the ccccc, cccbc, and ccbbc pentaquark subsystems according to their similar symmetry properties.

IV. MASS SPECTRA AND DECAY BEHAVIORS
The interacting Hamiltonians can be diagonalized and one can thus obtain the eigenvalues as well as eigenvectors for the corresponding pentaquark systems. According to our results, we discuss the masses gaps, decay behaviors, and stabilities of all the QQQQQ penaquark states.
Based on the two schemes proposed in Sec. II, we present the mass spectra for all the QQQQQ pentaquark subsystems in Table III. Take the cccbc pentaquark subsystem as an example. In the reference mass scheme, we use two types of baryon-meson reference systems (Ω ccc + B c and Ω ccb + η c ) to estimate the masses of cccbc states. Some results calculated from the two reference systems differ by more than a hundred MeV for the cccbc pentaquark states. However, the gaps with different reference systems are still same. Thus, if one pentaquark state were observed, its partner states may be searched for with the relative positions presented in Table III. Such a study can be used to test our calculation. Here, we need to emphasis that as a rough estimation, the dynamics and contributions from other terms in the interacting potential are not elaborately considered in Eq. (3) [60].
The modified CMI model scheme takes the chromoelectric interaction explicitly compared to the reference mass scheme, and therefore we use the results in this scheme for the following analysis. According to the modified CMI model scheme, we present the masses of the QQQQQ pentaquark states and the relevant baryon-meson thresholds in Fig. 1. In Fig. 1, we label the possible total angular momenta of the S-wave baryon-meson states. When the spin of an initial pentaquark state is equal to the total angular momentum of the channel below, it may decay into that baryon-meson channel through S wave.
Here we define the relatively 'stable' pentaquarks as those which cannot decay into the S wave baryon-meson states. We label these stable pentaquark states with " " in the figure and tables.
In addition to the mass spectra, the eigenvectors of pentaquark states will also provide important information about the two-body strong decay of multiquark states [4,5,47,71,73,85]. Thus we calculate the overlaps of wave functions between a fully-heavy pentaquark state and a particular baryon ⊗ meson state, and show them in Table IV.
We can further study the decay of the fully-heavy pentaquark states into the baryon ⊗ meson channels. Here, we take the ccccc pentaquark states as an example to describe our calculation. We transform the eigenvectors of the pentaquark states into the ccc ⊗ cc configuration. Normally, the ccc and cc components inside the pentaquark can be either of color-singlet or of color-octet. The former one can easily dissociate into an S-wave baryon and meson (the so-called Okubo-Zweig-Iizuka (OZI)-superallowed decays [4]), while the latter one cannot fall apart without the gluon exchange force. For simplicity, in this work, we only focus on the OZI-superallowed pentaquark decay process. The colorsinglet can be described by the direct product of a meson wave function and a baryon wave function. For each decay mode, the branching fraction is proportional to the square of the coefficient of the corresponding component in the eigenvectors, and the strong decay phase space.
For the two body decay via L-wave process, the expression describing partial decay width can be parameterized as [47,73] where α is an effective coupling constant, m is the mass of the initial state, k is the momentum of the final states in the rest frame. c i is the coefficient related to the corre-sponding baryon-meson component, which is the overlaps of wave functions shown in Table IV. For the decay processes that we are interested in, (k/m) 2 is of O(10 −2 ) or even smaller. Thus we only consider the S-wave decays. γ i represents other factors that contribute to the decay widths Γ i . For each process, Γ i depends on also the spatial wave functions of the initial pentaquark state and final meson and baryon. In the quark model in the heavy quark limit, the spatial wave functions of the ground S-wave pseudoscalar and vector meson are the same [73]. As a rough estimation, we introduce the following approximations to calculate the relative partial decay widths of the pentaquark states.
We present k · |c i | 2 for each decay process in Table V. From Table V, one can roughly estimate the relative decay widths between different decay processes of different initial pentaquark states if neglecting the γ i differences.
We divide the QQQQQ pentaquark system into the following three groups: A. The ccccQ and bbbbQ pentaquark subsystems; B. The cccbQ and bbbcQ pentaquark subsystems; C. The ccbbQ pentaquark subsystem.
We discuss the mass spectra and strong decay properties of QQQQQ pentaquark system group by group. For simplicity, we use P content (Mass, I, J P ) to label a specific pentaquark state.

A. The ccccQ and bbbbQ pentaquark states
We first discuss the fully heavy pentaquark states with ccccc, ccccb, bbbbc, and bbbbb flavor configurations. The ccccb and bbbbc states are the absolute exotic states which have the different flavor quantum numbers from the conventional baryons. Because of the strong symmetrical constraint from Pauli principle, i.e., fully antisymmetric among the first four charm quarks, we only find two ccccc states: an I(J P ) = 0(3/2 − ) state: P c 4c (7864, 0, 3/2 − )  and an I(J P ) = 0(1/2 − ) state: P c 4c (7949, 0, 1/2 − ). Similarly, there are also only two pentaquark states in the ccccb, bbbbc, and bbbbb subsystems.
From Fig. 1 (a)-(d), the J P = 3/2 − states generally have smaller masses than the J P = 1/2 − states in the ccccQ and bbbbQ pentaquark subsystems. Meanwhile, from Fig. 1 (a)-(d), the masses of all the ccccQ and bbbbQ pentaquark states are larger than the thresholds of  for the ccccc, ccccb, bbbbc, bbbbb, cccbc, cccbb, bbbcc, bbbcb, ccbbc, and ccbbb pentaquark states labeled with solid lines. The dotted lines denote various S-wave baryon-meson thresholds, and the superscripts of the labels, e.g. (ΩcccJ/ψ) 5/2,3/2,1/2 , represent the possible total angular momenta of the channels. We mark the relatively stable pentaquarks, unable to decay into the S-wave baryon-meson states, with " " after their masses. We mark the pentaquark whose wave function overlaps with that of one special baryon-meson state more than 90% with " " after their masses.
the lowest possible baryon-meson systems. Thus, no stable pentaquark state exists in the ccccQ and bbbbQ pentaquark subsystems. The lowest baryon-meson channels are their dominant decay modes. In the future, searching for exotic signals in these baryon-meson strong decay channels would be an interesting topic.
The ccccc subsystem has one decay mode: ccc ⊗ cc, which could be Ω ccc J/ψ or Ω ccc η c . However, each ccccc state has only one decay channel from Table V. The J P = 1/2 − ccccc pentaquark state cannot decay into S-wave Ω ccc η c because of the constraint of angular conservation law, while the J P = 3/2 − one cannot decay into Ω ccc J/ψ since the mass is below the threshold.
According to Table IV, the I(J P ) = 0(5/2 − ) cccbb pentaquark state P c 3 bb (14246, 0, 5/2 − ) couples completely to the Ω ccc Υ system, which can be written as a direct product of a baryon Ω ccc and a meson Υ. Meanwhile, the P c 3 bb (14246, 0, 3/2 − ), P c 3 bb (14182, 0, 3/2 − ) , and P c 3 bb (14238, 0, 1/2 − ) states couple almost completely to the Ω ccc Υ, Ω ccc η b , and Ω ccc Υ baryon-meson systems, respectively. This kind of pentaquark behaves similar to the ordinary scattering state made of a baryon and meson if the inner interaction is not strong, but could also be a resonance or bound state dynamically generated by the baryon and meson with strong interaction. These kinds of pentaquarks deserve a more careful study with some hadron-hadron interaction models in future. We label these states with " " in the figure and tables. The P b 3 cb (20578, 0, 3/2 − ) has 94.2% of Ω bbb B c component, and the pentaquarks like P b 3 cb (20578, 0, 3/2 − ) are also marked with " ".
According to Fig. 1 (j), the P c 2 b 2b(17477, 0, 5/2 − ) does not have S-wave strong decay channels, and thus this state is expected to be narrow. It can still decay into the D-wave final states of Ω ccb η b and Ω cbb B c .
Meanwhile, the lowest J P = 3/2 − ccbbb pentaquark state P c 2 b 2b(17416, 0, 3/2 − ) is below all allowed strong decay channels. It should decay through the electromagnetic and weak interactions rather than the strong interaction. Thus, this state is considered as a good stable pentaquark.
The lowest J P = 1/2 − state P c 2 b 2b(17405, 0, 1/2 − ) can only decay into Ω ccb η b , and its mass is slightly larger than the Ω ccb η b threshold. Thus, its width should be narrow due to the small decay phase space.

V. SUMMARY
More and more exotic multiquark candidates are lastingly discovered in experiment these years [10][11][12][13]. The P c (4312), P c (4440), and P c (4457) states and the fullycharmed tetraquark candidate X(6900) reported from the LHCb collaboration motivate us to discuss the possible pentaquark states with QQQQQ configuration in the framework of CMI model.
In this work, we firstly construct the wave functions ψ f lavor ⊗ ψ color ⊗ ψ spin based on the SU(2) and SU(3) symmetry and Pauli Principle. Then, we extract the effective coupling constants from the conventional hadrons. After that, we systematically calculate the CMI Hamiltonian for the QQQQQ pentaquark states and obtain the corresponding mass spectra in the reference system scheme. In the modified CMI scheme, the effect of chromoelectric interaction is explicitly added.
The mass spectra is studied for the QQQQQ pentaquark system. In addition, we also provide the eigenvectors to extract useful information about the decay properties for the QQQQQ pentaquark systems. The overlaps for the pentaquark state with a particular baryon ⊗ meson state are obtained. Finally, we analyze the stability, possible quark rearrangement decay channels, and relative partial decay widths for all the QQQQQ pentaquark states.
According to our calculations and analysis, we only find two ccccc states due to the constraint from Pauli principle: a J P = 3/2 − state P c 4c (7864, 0, 3/2 − ) and a J P = 1/2 − state P c 4c (7949, 0, 1/2 − ), and there exists no ground J P = 5/2 − ccccc pentaquark state. The same situation also happen in the ccccb, bbbbc, and bbbbb subsystems. From the obtained tables and figures for the QQQQQ pentaquark system, we find one good stable candidates: the P c 2 b 2b(17416, 0, 3/2 − ). It lies only below the allowable decay channel Ω * ccb η b 4 MeV, and thus can only decay through electromagnetic or weak interactions. Meanwhile, the P c 2 b 2b(17477, 0, 5/2 − ) is also a relatively stable pentaquark since it is lower than all possible S-wave strong decay channels. It can still decay into Ω ccb η b and Ω cbb B c final states via D-wave.
Our systematical study can provide some understanding toward these pentaquark systems. We find some fully heavy pentaquark states can be very narrow and stable. If they do exist, identifying them may not be difficult from their exotic quantum numbers and masses. The X(6900) is found in the invariant mass spectrum of J/ψ pairs, where two pairs of cc are produced. In our calculation, the lowest fully-heavy pentaquark state is the J P = 3/2 − ccccc state. To produce the lightest ccccc pentaquark state, one needs to simultaneously produce at least four pairs of cc, this seems to be a difficult task in experiment. More detailed dynamical investigations on QQQQQ pentaquark systems are still needed. We hope that our study may inspire experimentalists to pay attention to this kind of pentaquark system.