Heavy flavor pentaquarks with four heavy quarks

In this work, we carry out the study of heavy flavor pentatuarks with four heavy quarks, which have typical $QQQQ\bar q$ configuration. Within the Chromomagnetic Interaction model, the mass spectrum of these discussed $QQQQ\bar q$ pentaquarks is given. In addition to the mass spectrum analysis, we also illustrate their two-body strong decay behavior by estimating some ratios of decay channels. By these effort, we suggest that future experiment should pay attention to this kind of pentaquark.

Obviously, it is not the end of the exploration of exotic hadronic matter. With the accumulation of experimental data, more and more charmoniumlike XY Z states have been discovered since 2013, which again inspired theorist's extensive interest in investigating exotic hadronic states [32][33][34][35][36][37][38][39]. Especially, in 2015, the LHCb Collaboration measured the Λ 0 b → J/ψK − p decay and observed two hidden-charm pentaquark-like resonances P c (4380) and P c (4450) in the J/ψp invariant mass spectrum, which indicates that they have a minimal quark content of uudcc [5,6]. In 2019, LHCb found three narrow P c structures in the J/ψp invariant mass spectrum of Λ b →J/ψK − p [7], where this new measurement shows that P c (4450) is actually composed of two * anht14@lzu.edu.cn † chenk 10@pku.edu.cn ‡ liuzhanwei@lzu.edu.cn § xiangliu@lzu.edu.cn substructures, P c (4440) and P c (4457) with 5.4σ significance. The characteristic mass spectrum of P c provides the strong evidence of existence of hidden-charm molecular pentaquarks [11][12][13][14][15]. At present, searching for exotic hadronic states is still full of challenge and opportunity. As theorists, we should provide valuable prediction, which requires us to continue to find some crucial hint for exotic hadronic states.
We may borrow some idea of proposing stable tetraquark state with QQqq configuration. Stimulated by the observation of double charm baryon Ξ ++ cc (3620) [40], Refs. [41][42][43] studied the possible stable tetraquark state with the QQqq configuration. Here, the QQqq configuration can be obtained by replacing light quark q of double heavy baryon QQq withqq pair since the color structure of q andqq can be the same. Along this line, we may continue to replaceq of QQqq with a QQ pair and get the QQQQq configuration, which is a typical pentaquark configuration (see Fig. 1).
This paper is organized as follows. After introduction, the adopted CMI model will be introduced in In Sec. II. In Sec. III, we present the mass spectra of S-wave QQQQq pentaquarks and estimate the ratios of possible strong decay widths. Finally, a short summary is followed in Sec. IV.

II. THE CHROMOMAGNETIC INTERACTION MODEL
The effective Hamiltonian involved in the estimated mass spectrum of these discussed pentaquarks where m i denotes the effective mass of the i-th constituent quark when considering these effects from kinetic energy, color confinement, and so on. H CMI , as the chromomagnetic interaction Hamiltonian, is composed of the Pauli matrices σ i and the Gell-Mann matrices λ i . Here, λ i should be replaced by −λ * i for antiquark, and C ij denotes the effective coupling constant between the i-th quark and j-th quark, which depends on the quark masses and the ground particle's spatial wave function. As input, C ij is fixed by the involved hadron masses.

With deduction
where the color operator i λ i nullifies any colorless physical state, we have the Hamiltonian of the modified CMI model In the above expression, V C ij = λ i · λ j and V CMI ij = λ i · λ j σ i · σ j denote the color and color-magnetic interactions between quarks, respectively. A new mass parameter of quark pair should be defined, i.e., By these conventional hadron masses (see Table I), we may fix the parameters m ij and v ij (see Table II). Interested readers can refer to Refs. [27,[59][60][61] for more details. With this modified CMI model scheme, we can give mass spectrum of the QQQQq pentaquarks. When calculating the mass of the pentaquark states, we need the information of the total wave function, which is composed of space, flavor, color, and spin wave functions, i.e., Since we only focus on the low-lying S-wave pentaquark states, the symmetrical constraint from spatial pentaquark wave function is trivial. For the discussed QQQQq pentaquarks, their ψ f lavor ⊗ ψ color ⊗ ψ spin wave functions of pentaquark states should be fully antisymmetric when exchanging identical quarks. All the possible flavor combinations for the QQQQq pentaquark system are shown in Table III, by which we further determine ψ f lavor ⊗ψ color ⊗ψ spin wave functions, which satisfy {1234}, {123}, and {12}{34} symmetry. Here, we use the notation {1234} to label that the 1st, 2nd, 3rd, and 4th quarks have antisymmetry property, which will be applied in the following discussions. For identifying the pentaquark configuration with certain exchanging symmetry, we use the approach of Young diagram and Young tableau, which represents the irreducible bases of the permutation group. The color wave functions for QQQQq pentaquark system are expressed a direct product Due to the requirement of color confinement, the color wave function must be a singlet. Therefore, the four heavy quarks should be in the color triplet states, and Here, the subscript labels the irreducible representation of SU(3). Then, by combining the antitriplet from light anti-quark with the deduced three color triplets in Eq. (9), we obtain three color singlets for all the studied pentquark systems and In Eq. (16), the spin states can be identified with the Young-Yamanouchi basis vectors for partitions [4], [31], and [22].
For constructing the ψ f lavor ⊗ψ color ⊗ψ spin wave functions of QQQQq pentaquark system, we should combine the partition [211] of the color singlets in Eq. (9) with partitions [4], [31], [22] of the spin states in Eq. (16) by the inner product of the permutation group. Thus, the ψ color ⊗ ψ spin wave functions with a certain symmetry can be constructed. We get the corresponding Young diagram representations of ψ color ⊗ ψ spin bases [64][65][66] By using the Clebsch-Gordan (CG) coefficient of the permutation group S n , one obtains the coupling scheme designed to construct the ψ color ⊗ ψ spin states. Here, any CG coefficient of S n can be factorized into an isoscalar factor, called K matrix, and the Clebsch-Gordan (CG) coefficient of S n−1 [65]. The expression of Clebsch-Gordan (CG) coefficient of S n reads as (18) Here, S in the left-hand (right-hand) side denotes a CG coefficient of S n (S n−1 ) and the [f ] ([f p ]) is a Young tableau of S n (S n−1 ) with [f ]pqy as a specific Young-Yamanouchi basis vector. And, p (q) is the row of the n (n−1)-th particle in the Young-Yamanouchi basis vector, while y is the distribution of the n − 2 remaining particles. In fact, a similar application is also used in Refs. [66][67][68][69][70]. According to the isoscalar factors for S 3 and S 4 in Tables 6.2 and 6.3 of Ref. [65], the corresponding Clebsch-Gordan (CG) coefficient of S 4 can be obtained. The corresponding Young-Yamanouchi basis vector obtained from the ψ color ⊗ ψ spin coupling (see Eq.(17)) are collected in Eq. (A1) of Appendix. A.
We can combine the flavor wave functions with the ψ color ⊗ ψ spin wave functions (see Eq. A1) of QQQQq pentaquark states to deduce the symmetry allowed ψ f lavor ⊗ ψ color ⊗ ψ spin pentaquark wave functions. In Table. VII of Appendix. A, all the symmetrically allowed Young-Yamanouchi basis vector for the different flavor wave functions are listed.
By constructing all the possible ψ f lavor ⊗ψ color ⊗ψ spin bases satisfied for {1234}, {123}, and {12}{34} symmetry, we can calculate the CMI matrices for the studied pentaquark states. Here, we only present the expressions of CMI Hamiltonians for the ccccn, cccbn, and ccbbn pentaquark subsystems in Table VIII of Appendix. A. According to their similar symmetry properties, the expressions of CMI matrices for other pentaquark subsystems can be deduced for those of the the ccccc, cccbc, and ccbbc pentaquark subsystems.

III. MASS SPECTRA AND DECAY BEHAVIORS
As shown in Table III, according to symmetry properties, we can divide the QQQQq pentaquark system into three groups: A. The ccccq and bbbbq pentaquark subsystems; B. The cccbq and bbbcq pentaquark subsystems; C. The ccbbq pentaquark subsystem.
For the ccccq and bbbbq subsystems, the wave functions should be to antisymmetric for the exchange between the 12, 13, 14, 23, 24, or 34 particles. The cccbq and bbbcq wave functions are antisymmetric when exchanging the 12, 13, or 23 particles. However, for the ccbbq subsystem, the antisymmetry is considered only for exchanging the 12 or 34 particles. The fewer restrictions lead to more allowed wave functions. Therefore, the number of basis states in Table VII increases from ccccn to cccbn and next to ccbbn due to the Pauli principle. In the following, we will discuss the mass spectra and strong decay properties of QQQQq pentaquark system group by group. All of them are explicitly exotic states. If such pentaquark states could be observed in experiment, its pentaquark state nature could be easily identified. For simplicity, we use P content (Mass, I, J P ) to label a particular pentaquark state.

The ccccq and bbbbq pentaquark states
Here, we first discuss the ccccq and bbbbq pentaquark subsystems. Because of the symmetrical constraint from Pauli principle, i.e., fully antisymmetric among the first four charm quarks, the ground J P = 5/2 − pentaquark state with ccccq and bbbbq can not exist. We only find two ground states: a J P = 3/2 − state and a J P = 1/2 − state for these subsystems.
Diagonalizing the Hamiltonians in Table VIII with the corresponding parameters in Table II, we can obtain the corresponding mass spectra for ccccq and bbbbq pentaquark subsystems and present them in Table IV. For the reference mass scheme, the only combination of meson-baryon reference systems are (Ω ccc )+(D), (Ω ccc )+(D s ), (Ω bbb )+(B), and (Ω bbb )+(B s ) for the ccccn, ccccs, bbbbn, and bbbbs subsystems, respectively. 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. Based on the results calculated from the modified CMI model scheme, we plot the mass spectra of the ccccn, ccccs, bbbbn, and bbbbs subsystems in Fig. 2 (a)-(d), respectively. Moreover, we also plot all the baryon-meson thresholds which they can decay to through quark rearrangement in Fig. 2 (a)-(d). Meanwhile, we label the spin of the baryon-meson states with superscript. When the spin of an initial pentaquark state is equal to the number in the superscript of a baryon-meson state, it may decay into that baryonmeson channel through S wave. Here, we define the relatively "stable" pentaquark states as those which cannot decay into the S wave baryon-meson states. Meanwhile, we label these stable pentaquark states with " " in the relevant figure and tables.
Based on the obtained ccccq and bbbbq pentaquark spectra, we can discuss the possible decay patterns by considering different rearrangement of quarks in the corresponding pentaquark states. The discussion of possible decay patterns for these pentaquark states would be helpful for the observation in experiments. From Fig. 2 (a)-(d), we can easily find that the J P = 3/2 − state generally have smaller masses than that of the J P = 1/2 − state in the ccccq and bbbbq pentaquark subsystems. We also find that all the ccccq and bbbbq pentaquark states have strong decay channels, which indicates that in the modified CMI model, no stable ccccq and bbbbq pentaquark exists.
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 [60][61][62][71][72][73]. The overlap for the pentaquark with a specific baryon ⊗ meson state can be calculated by transforming the eigenvectors of the pentaquark states into the baryon ⊗ meson configuration. In the QQQ ⊗ Qq configuration, the color wave function of the pentaquark falls into two categories: the color-singlet |(QQQ) 1c (Qq) 1c and the color-octet |(QQQ) 8c (Qq) 8c . The former one can easily dissociate into an S-wave baryon and meson (the so-called Okubo-Zweig-Iizuka (OZI)-superallowed decays), while the latter one cannot fall apart without the gluon exchange force. In this work, we only focus on the OZI-superallowed pentaquark decay process. It means that only the C 3 color part in Eq. (12) is considered in the color space.
For the two body decay via L-wave process, the expression describing partial decay width can be parameterized as [60,61] 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 overlap between the pentaquark wave function and the meson + baryon wave function. Take P c 4n(6761, 1/2, 3/2 − ) → Ω ccc D * as an example, = (ccc) 1c S=1/2 ⊗ (cn) 1c S=1 |P c 4n(6761,  We show all possible overlaps between a pentaquark state and its possible |(QQQ) 1c (Qq) 1c and |(QQQ) 1c (Qq) 1c components in Table V. 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.
As for the γ i , it depends on the spatial wave functions of initial and final states, and may not be the same for different decay processes. In the quark model, the spatial wave functions of the ground state scalar and vector meson are the same [61]. As a rough estimation, we introduce the following approximations to calculate the relative partial decay widths of the ccccq and bbbbq pen- We present k · |c i | 2 for each decay process in Table VI  for the ccccn, ccccs, bbbbn, bbbbs, cccbn, cccbs, bbbcn, bbbcs, ccbbn, and ccbbs pentaquark states labeled with solid lines. The dotted lines denote various S-wave baryon-meson thresholds, and the superscripts of the labels, e.g. (ΩcccD * ) 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.
Such approximation have already been applied in Refs. [55,56]. We emphasised that we are only interested in its relative partial decay widths between different decay modes for a particular pentaquark state. Based on Eqs. (19)- (22), we can calculate relative partial decay widths for the ccccq and bbbbq pentaquark subsystems. For the J P = 1/2 − ccccn pentaquark state, it cannot decay into S-wave Ω ccc D because of the constraint of angular conservation law. Same situation also happen in the ccccs, bbbbn, and bbbbs subsystems.

The cccbq and bbbcq pentaquark states
Next we consider the cccbq and bbbcq pentaquark subsystems. The cccbq and bbbcq pentaquark subsystems include three identical heavy quarks.
The masses of cccbq and bbbcq pentaquark states can be determined in two schemes and shown in Table IV. In the reference mass scheme, we can exhaust two types of baryon-meson reference systems. Specifically, we can use the Ω ccc +B (B s ) and Ω ccb +D (D s ) as reference systems to estimate the masses of the cccbn (cccbs) pentaquark subsystem. Similarly, the meson-baryon reference systems Ω bbb +D (D s ) and Ω bbc +B (B s ) are used to calculate the masses of the bbbcn (bbbcs) pentaquark subsystem. The obtained eigenvalues and masses of cccbn (s) and bbbcn (s) pentaquark states calculated from two types of reference systems are presented in the third and fourth columns of Table IV, respectively. We easily find that the mass spectra came from two different reference systems differ by more than 100 MeV for some studied subsystems. The reason is that the dynamics and contributions from other terms in conventional meson and baryon potential are not elaborately considered in this model [50]. However, the mass gaps under 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 IV.
Based on the results listed in the last column of Table  IV, we plot the mass spectra and relevant quark rearrangement decay patterns for the cccbn, cccbs, bbbcn, and bbbcs subsystems in Fig. 2 (e)-(h), respectively. Moreover, according to the modified CMI model, we can obtain the overlaps for cccbn (cccbs) and bbbcn (bbbcs) pentaquark states with different baryon ⊗ meson bases, and the results are shown in Table V.
From Table V, the P c 3 bn (10110, 1/2, 5/2 − ) state couples completely to the Ω cccB * system , which can be written as a direct product of a baryon Ω ccc and a mesonB * . Moreover, for the P c 3 bn (10118, 1/2, 3/2 − ), P c 3 bn (10078, 1/2, 3/2 − ), and P c 3 bn (10134, 1/2, 1/2 − ) states, they strongly couple to the Ω cccB * , Ω cccB , and Ω cccB * bases, 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. Thus, we label them with " " in Tables V, VI, and Fig. 2. Moreover, we find that the J P = 5/2 − QQQQ q pentaquark states all have only one component Ω QQQ B * (s) (D * (s) ). Therefore, the J P = 5/2 − ground states are regarded as the states made of two hadrons.
For cccbq pentaquark states, they have two types of decay modes: ccc − bq and ccb − cq. Similarly, the bbbcq pentaquark states also have two types of decay modes: bbb − cq and bbc − bq. In the heavy quark limit, Ω * ccb (Ω * bbc ) and Ω ccb (Ω bbc ) have the same spatial wave function. Thus, for a cccbq or bbbcq pentaquark state, we use the following approximations Based on Table V, we obtain k · |c i | 2 for each cccbq and bbbcq pentaquark state and present them in Table VI.
As an example, we only concentrate on the cccbn pentaquark subsystem.
For the P c 3 bn (10062, 1/2, 1/2 − ) state, we have its relative partial decay width ratios as Γ Ω * ccb D * : Γ Ω ccb D * : Γ Ω ccb D = 24 : 24 : 1, which suggests that the partial decay width of the Ω * ccb D * channel is nearly equal to that of the Ω ccb D * channel. Note that if a state would be observed in the decay pattern Ω * ccb D * , Ω * ccb D, Ω * ccb D * , and Ω ccb D , it is a good candidate of a cccbn pentaquark state.

The ccbbq pentaquark states
The last group of the QQQQq system is the pentaquark states with the ccbbq configuration. The ccbbq pentaquark states have two pairs of identical heavy quarks, the cc pair and bb pair. When we construct the wave functions of ccbbq pentaquark states, the Pauli principle should be satisfied simultaneously for these two pairs of heavy quarks.
In the reference mass scheme, there are also two types of meson-baryon reference systems for the ccbbn (ccbbs) pentaquark subsystem, i.e., the Ω ccb +B and Ω cbb +D (Ω ccb +B s and Ω cbb +D s ).
Based on the results obtained from the modified CMI model in Table IV, we plot the mass spectra and possible decay patterns via rearrangement of constituent quarks in ccbbn and ccbbs pentaquark states in Fig. 2 (i)-(j). According to Fig. 2 (i)-(j), we find that all ccbbn and ccbbs pentaquark states have strong decay channels. i.e., from the modified CMI model analysis, there is no stable pentaquark state in ccbbn and ccbbs pentaquark subsystems.
To calculate the strong decay widths of the ccbbn and ccbbs pentaquark subsystems, we can use the following approximations By introducing the above relations, k · |c i | 2 for ccbbn (ccbbs) pentaquark states can be obtained and we present them in Table VI.
To discuss the strong decay behaviors of the ccbbn (s) pentaquark states, we mainly focus on the relative partial decay widths of the ccbbn subsystem, and the ccbbs subsystem can be analyzed in a similar way.
Obviously, its dominant decay modes in cbb−cn and ccb− bn sectors are Ω * cbb D * and Ω * ccb B * channels, respectively. Other three J P = 3/2 − and three J P = 1/2 − states only have cbb − cn decay mode. The ccb − bn decay mode are strongly suppressed by the corresponding phase space.

Comparison of other pentaquark systems
In 2020, the LHCb collaboration studied the invariant mass spectrum of J/ψ pairs, and they reported a narrow structure around 6.9 GeV [74]. Take this as an opportunity, the heavy flavored pentaquarks with four heavy quarks (QQQQq) and fully heavy pentaquarks (QQQQQ) are systematically discussed in this work and Ref. [57] within the modify CMI model. Here we can compare the differences between these two pentaquark systems.
Firstly, we discuss the mass differences among the ground states of QQQQn, QQQQs, QQQQc, and QQQQb with the same J P . From Table V and Table  IV of Ref. [57], the masses of ccccn, ccccs, ccccc, and ccccb ground states with J P = 3/2 − are 6761, 6864, 7864, and 11130 MeV, respectively. Moreover, relative to the the J P = 1/2 − states, their corresponding mass gaps are 106 MeV, 108 MeV, 85 MeV, and 47 MeV, respectively. Other subsystems also have similar situations. Thus, compared with the QQQQQ system, the QQQQq system has lighter masses and bigger mass gaps when a heavy antiquark is replaced by a light antiquark because v ij ∝ 1/m i m j .
Secondly, we study the relations between the pentaquark and their corresponding baryon-meson channels. The fully heavy pentaquarks QQQQQ are more likely below all possible strong-decays channels and thus more stable compared to the QQQQq systems. We have found two relatively stable states P c 2 b 2b (17416, 0, 3/2 − ) and P c 2 b 2b (17477, 0, 5/2 − ), which are below all allowed rearrangement decay channels in QQQQQ system. However, we do not find any stable state for the QQQQq multiquark systems. When both QQQQQ and QQQQq ground states are above their corresponding baryonmeson channel, the QQQQQ mass would be closer to the threshold. For example, the ccccn, ccccs, ccccc, and ccccb states are above the corresponding minimum threshold (Ω ccc + pseudoscalar meson) 106 MeV, 110 MeV, 94.5 MeV, and 69.5 MeV, respectively.
Unlike fully pentaquark QQQQQ, all QQQQq pentaquark states can never mix with a triquark baryon and thus are explicit exotic states. Accurate measurement in future experiment and the comparison may help us understand the QQ annihilation effects in the hadron spectrum.

IV. SUMMARY
The observation of the P c (4312), P c (4440), and P c (4457) states achieved by the LHCb Collaboration and the study of the possible stable QQqq tetraquark states give us strong confidence to study the mass spectra of the QQQQq pentaquark system within the framework of CMI model. Similar to the fully-heavy QQQQ tetraquark system [74], the QQQQq system consist of four heavy quarks are dominantly bounded by the gluon exchange interaction, and can hardly be considered as molecular states.
In this work, by including the flavor SU(3) breaking effect, we firstly construct the ψ f lavor ⊗ ψ color ⊗ ψ spin wave functions 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 chromomagnetic interaction Hamiltonian for the discussed pentaquark states and obtain the corresponding mass spectra. As a modification to the CMI model, the effect of chromoelectric interaction is added in the modified CMI model. So, we mainly discussed the results of mass spectra for the QQQQq pentaquark system obtained from the modified CMI model. The results from the reference mass scheme are presented for comparison. In addition to the eigenvalues, we also provide the eigenvectors to extract useful information about the decay properties for the studied pentaquark systems. Finally, we analyze the stability, possible quark rearrangement decay channels and relative partial decay widths for all the QQQQq pentaquark states.
Due to the constraint from Pauli principle, there is no ground J P = 5/2 − ccccq and bbbbq pentaquark states. From the obtained Tables and Figs for the QQQQq pentaquark system, we find no stable candidate in the modified CMI model. However, due to the uncertainty of the modified CMI model, some of them may not truly be unstable states, and further dynamical calculations may help us to clarify their natures. Especially, for some unstable states which are a little higher than the mesonbaryon thresholds of lowest strong decay channels, they can be considered as narrow pentaquark states, and have opportunities to be found in future experiment. Meanwhile, the whole mass spectra has a slight shift or down due to the mass deviation of constituent quarks. While the mass gaps between different pentaquark states are relatively stable, if one pentaquark states are observed in experiment, we can use these mass gaps to predict their corresponding multiplets.
Among the studied QQQQq pentaquark states, all of them are explicit exotic states. If such pentaquark states are observed, their exotic nature can be easily identified. However, up to now, none of QQQQq pentaquark states is found. More detailed dynamical investigations on these pentaquark systems are still needed. Producing a QQQQq pentaquark state seems to be a difficult task in experiment. Our systematical study may provide theorists and experimentalists some preliminary understanding toward this pentaquark system. We hope that the present study may inspire experimentalists and theorists to pay attention to this kind of pentaquark system.  Table VII and some CMI Hamiltonians in Table. VIII.