Exotic pentaquark states with the $qqQQ\bar{Q}$ configuration

In the framework of the color-magnetic interaction model, we have systematically calculated the mass splittings for the S-wave triply heavy pentaquark states with the configuration $qqQQ\bar{Q}$ $(Q=c,b;q=u,d,s)$. Their masses are estimated and their stabilities are discussed according to possible rearrangement decay patterns. Our results indicate that there may exist several stable or narrow such states. We hope the present study can help experimentalists to search for exotic pentaquarks.

The studies of more possible pentaquarks were also stimulated by the observation of P c states [69][70][71][72][73]. After the experimental confirmation of the doubly charmed baryon Ξ ++ cc [74,75], the multiquark states with two or more heavy quarks were studied in many works [69][70][71][72][73]76]. For example, two possible triple-charm molecular pentaquarks Ξ cc D 1 and Ξ cc D * 2 were considered in Ref. [77]. In this paper, we systematically study the mass splittings of compact pentaquark states with the qqQQQ configuration (q = u, d, s; Q = c, b). If a heavy quark-antiquark pair forms an unflavored state, such pentaquarks look like excited qqQ baryons. Otherwise, they are explicitly exotic states. At present, it is still not easy to dynamically solve the multi-body problem. Here, we use the color-magnetic interaction (CMI) model to calculate the mass splittings and investigate the mass spectrum of the qqQQQ pantaquark states preliminarily. One may consult relevant studies with other methods in Refs. [78][79][80].
The Hamiltonian of the quark potential model consists of the one-gluon-exchange interaction part and nonperturbative scalar confining part, which was proposed by de Rujula, Georgi, and Glashow in Ref. [81]. For the ground state hadrons with the same quark content, such as ∆ and N , their mass splitting is mainly determined by the color-magnetic interaction [82]. When the spacial contributions are encoded into effective quark masses and coupling parameters, the Hamiltonian can be written as the form containing just the quark mass term and the color-spin interaction term and one gets the CMI model. There are many studies about the mass spectrum for multiquark systems within this model [83][84][85][86][87][88][89][90][91]. The qualitative properties of the obtained spectra are helpful for us to search for relevant exotic states. In the early stage studies on the pentaquark properties, colormagnetic effects were intensively considered as the primary contribution in an attempt to explain the narrow hadronic resonances, too [102].
This paper is organized as follows. In Sec. II, we introduce the CMI model and construct the f lavor ⊗ arXiv:1905.07858v1 [hep-ph] 20 May 2019 color ⊗ spin wave functions for the qqQQQ pentaquark states. In Sec. III, we calculate the relevant Hamiltonian elements and present the corresponding results. In Sec. IV, we give numerical results for the masses of the pentaquark states, illustrate their possible rearrangement decay channels, and discuss the stability of the states. Finally, we present a summary in Sec. V and an appendix in Sec. VII.

II. THE COLOR-MAGNETIC INTERACTION AND THE WAVE FUNCTIONS
The Hamiltonian of the CMI model has a simple form Here, M i represents the effective quark mass for the ith quark or antiquark and it takes account of effects from kinetic energy, color confinement, and other terms in realistic potential models. The effective constant C ij ∼ δ 3 ( r ij ) /(m i m j ) reflects the coupling strength between the i-th quark and the j-th quark, which depends on the quark masses and the spatial wave functions of the ground states [88]. The Pauli matrix σ i and Gell-Mann matrix λ i (−λ * i ) act on the spin and color wave functions of the i-th quark (antiquark), respectively.
To calculate the required matrix elements, we construct the wave functions of the ground qqQQQ pentaquark states. They are the direct products of SU(3) f flavor wave function, SU(3) c color wave function, and SU(2) s spin wave function. Here, we treat the heavy quark/antiquark as a flavor singlet state instead of constructing the wave function with flavor SU(4) f symmetry [92]. It is convenient to adopt the diquark-diquarkantiquark base in organizing the wave functions. The notion "diquark" only means two quarks and the meaning is different from that in the diquark model in Ref. [93] where the diquark is a strongly correlated quark-quark substructure with color=3 c and spin=0. The constructed wave functions may also be used to study properties of the qqQQQ states in dynamical quark models.
In the SU(3) f flavor space, the qqQQQ states belong to the flavor symmetric 6 f and antisymmetric3 f representations ( Fig. 1), which is similar to the situation for part of the QQqqq states [88]. For the nnQQQ (n = u, d) case, the isovector states (I = 1) and the isoscalar states (I = 0) do not mix since we do not consider isospin breaking effects. For the nsQQQ case, the fact m n = m s leads to SU(3) f breaking and thus the state mixing between 6 f and3 f . As a result, we need to consider four cases of states: nnQQQ (I = 1), nnQQQ (I = 0), nsQQQ (I = 1/2), and ssQQQ (I = 0). Note that the isovector and isoscalar nnQQQ states are not degenerate since the Pauli principle has impacts. In the color space, the wave functions can be analyzed with the SU(3) c group theory. The Young diagrams tell us that there are three color-singlet wave functions for the qqQQQ states. With the diquark-diquark-antiquark base, they are In the notation [(q 1 q 2 ) color1 (Q 3 Q 4 ) color2Q ], the color1 and color2 stand for the color representations of the light diquark and heavy diquark, respectively. The S (A) means "symmetric" ("antisymmetric") with quark exchanges. The explicit wave functions are the same as those for the QQqqq states studied in Ref. [88].
One can also use the baryon-meson base (qqQ-QQ or qQQ-qQ) to construct the wave functions. The relevant decomposition is Ref. [86] adopted this base in studying the hidden-charm pentaquark states. Although the final Hamiltonians are different for these two bases, the eigenvalues and mass spectrum would be identical after diagonalization. However, the baryon-meson base is not suitable to the present systems since two pairs of identical quarks may exist in a state like nnccQ.
In the spin space, the possible wave functions for the considered states in the diquark-diquark-antiquark base are In the notation [(q 1 q 2 ) spin1 (Q 3 Q 4 ) spin2Q ] spin4 spin3 , spin 1 and spin 2 represent the spins of the light and heavy diquarks, respectively, spin 3 represents the total spin of the four quarks, and spin 4 represents the total spin of the pentaquark. The diquark is symmetric (antisymmetric) when spin 1,2 is 1 (0).

III. THE CMI HAMILTONIAN EXPRESSIONS
With the constructed wave functions, we can calculate CMI Hamiltonian matrix elements. To simplify the expressions, we define the combinations of the effective couplings shown in Table II. For the pentqaurk states without constraints from the Pauli principle, e.g. nscbQ, all the color-spin wave function bases in Table I are involved. In the Appendix, we show the obtained CMI matrices for the cases J P = 5/2 − , 3/2 − , and 1/2 − in Tables XII, XIII, and XIV, respectively. For the pentaquark states having constraints from the Pauli Principle, relevant matrices can be extracted from these tables. Here, we take the nnccQ case as an example. When one considers the I(J P ) = 1(5/2 − ) state, one has δ S 12 = 0, δ A 12 = 1, and δ 34 = 0 and only the base φ 1 χ 1 is allowed. It is easy to read out the CMI Hamiltonian from Table XII, Similarly, when one considers the I(J P ) = 0(5/2 − ) state, only the wave function base φ 3 χ 1 is allowed because δ A 12 = 0, δ S 12 = 1, and δ 34 = 0. The extracted CMI Hamiltonian from Table XII is IV. THE qqQQQ PENTAQUARK MASS SPECTRA A. The determination of parameters and estimation strategy Now, we determine the values of the seventeen coupling parameters (C nn , C ns , C ss , C cn , C bn , C cs , C bs , C bc , C cc , C bb , C nc , C nb , C sc , C sb , C cc , C bc , C bb , and C cb ) in order to estimate the pentaquark masses. Most of them can be extracted from the measured masses of the conventional hadrons (see Table III). The related CMI expressions are where the two bases for the last matrix corresponds to the case of J q1q2 = 1 and that of J q1q2 = 0. We show the determined coupling parameters in Table IV where C qQ = C Qq is implied. Further, we use the approximation C QQ = 2/3C QQ [3] (C bb = 2/3C bb ≈ 1.8 MeV, C cc = 2/3C cc ≈ 3.3 MeV, and C bc = 2/3C bc ≈ 2.0 MeV) because only one doubly heavy baryon Ξ cc is observed in experiments. For the B * c mass, it has not been observed yet and we take a theoretical result.  [96]. The adopted masses of the not-yet-observed doubly heavy baryons are taken from Ref. [100]. The values in parentheses are obtained with the parameters in Ref. [89].

Mesons
Mesons Baryons Baryons Using the mass formula M = i M i + H CMI and the obtained parameters, one sees that the estimated masses of conventional hadrons are in general higher than the measured values, which is illustrated in Table V. The reason is that the adopted model and parameters could not account for the necessary attractions for all the hadrons. Overestimated masses with this approach were also obtained in various tetraquark and pentaquark states [83][84][85][86][87][88][89][90]. To make a more reasonable estimation, we use the improved mass formula by replacing i M i in Eq.

ment. Then
In the present study for pentaquark states, we choose the baryon-meson thresholds as the mass scales, where the reference baryon-meson system should have the same constituent quarks with a considered system. The attraction not incorporated in the original approach is somehow phenomenologically compensated in this procedure [88].
Before the detailed discussions about the qqQQQ pentaquark states, we emphasize that our results are only rough estimations. They should be updated once a qqQQQ pentaquark state is observed in future experiments and its mass can be chosen as a reference scale. Although the pentaquark masses may be changed largely, the mass splittings should not be affected significantly.
In the following parts, we only present the numerical values obtained with Eq. (4). Here, the involved masses of reference baryons and mesons have been given in Table  III. To understand the decay properties in the following discussions, we also show some masses of the not-yetobserved doubly heavy baryons in the table, which were obtained from several theoretical calculations. Since the spin of the Ξ cc observed by LHCb may be 1/2 or 3/2, we show results in both cases in Table III. B. The nnccQ, ssccQ, nnbbQ, and ssbbQ pentaquark states Substituting the parameters into the CMI matrices and diagonalizing them, the pentaquark masses are obtained. Here, we present the masses with corresponding reference systems for the nnccQ, ssccQ, nnbbQ, and ssbbQ states in Table VI. In these systems, a state is explicitly exotic if the flavor of Q is different from the heavy quarks.
For the nnccc states, there are two types of reference systems we can adopt, (cc)(cnn) and (cn)(ccn). The mass M Ξcc = 3621.4 MeV measured by the LHCb Collaboration is used in the latter case. We assume that the spin of Ξ cc is 1/2 although it has not been determined yet. If the spin is 3/2, the pentaquark masses estimated with the threshold relating to M Ξcc would be shifted downward by 64 MeV according to Ref. [89], but the gaps are the same. As for the nnbbQ, ssccQ, and ssbbQ systems, we can similarly adopt two types of refer-   Table VI, it is obvious that the pentaquark masses will change when one adopts different reference systems, which indicates that the estimation method with Eq. (4) should be further improved. If the adopted model can reproduce all the hadron masses accurately, different reference thresholds should lead to the same result.
Table VI shows us that the obtained nnccc (nnccb) masses with the reference threshold Σ c J/ψ (Σ c B c ) are lower than those with Ξ ccD (Ξ cc B). This feature is consistent with our anticipation since ∆ Σc + ∆ J/ψ > ∆ Ξcc +∆ D and ∆ Σc +∆ Bc > ∆ Ξcc +∆ B from Table V. At present, it is not clear which type threshold gives more reasonable masses. For a pentaquark, the effective attraction is probably not strong and maybe a higher mass would be more reasonable [88]. However, the choice of reference scale does not affect the mass splittings.
In showing the spectra in the figure form, we use the higher pentaquark masses although relevant estimations rely on the masses of the not-yet-observed QQq states. The diagrams of Figs. 2 and 3 illustrate relative po-sitions of the nnccc, nnbbb, nnbbc, nnccb, ssccc, ssbbb, ssbbc, and ssccb states in order. The selected masses are obtained with the reference systems Ξ ccD , Ξ bb B, and Ω cc B 0 s , respectively. The thresholds for relevant rearrangement decay patterns are also displayed. For the nnccc system, the I = 0 states have generally lower masses than the I = 1 states. The quantum numbers for both the lowest and the highest states are J P = 1/2 − . From the diagrams (b), (c), and (d) of Fig.  2, one sees similar features for the nnbbb, nnbbc, and nnccb systems.
As for the stability of the pentaquark states, their dominant decay modes should be related with the rearrangement mechanism. Now we move on to such decays. One has to consider the constraints from the angular momentum conservation, isospin conservation, parity conservation, and so on when discussing allowed decay channels. For convenience, we have marked the spin and isospin of the baryon-meson channels in the superscripts and subscripts of their symbols in Fig. 2, respectively. For the ssccQ and ssbbQ states, only one isospin is possible and no label is given explicitly. Of course, whether the decay can happen or not is also kinematically constrained by the pentaquark mass which depends on models. In the following discussions, we assume that the obtained masses shown in the figures are all reasonable.
For the nnccc states, they look like excited nnc baryons. Because only orbital or radial excitation energy cannot explain their high masses, the states once observed are good candidates of compact nnccc pentaquark states or hadronic molecules. To distinguish these two configurations, decay properties would be helpful. We here just discuss relevant rearrangement decay patterns. In the case of I(J P ) = 1(5/2 − ), the possible S-wave decay channels are Σ * c J/ψ and Ξ * ccD * . In the case of I(J P ) = 0(5/2 − ), the possible S-wave decay channel is only Ξ * ccD * . The I(J P ) = 0(5/2 − ) isoscalar pentaquark is a candidate of stable state. We mark it in Fig. 2(a) with a dagger. In the case of I(J P ) = 1(3/2 − ), the possible S-wave channels are Ξ * ccD * , Ξ ccD * , Σ * c J/ψ, Ξ * ccD , Σ c J/ψ, and Σ * c η c . In the case of I(J P ) = 0(3/2 − ), the possible S-wave channels are Ξ * ccD * , Ξ ccD * and Λ c J/ψ. In the case of I(J P ) = 1(1/2 − ), the possible S-wave channels are Ξ * ccD * , Ξ ccD * , Σ * c J/ψ, Σ c J/ψ, Ξ ccD , and Σ c η c . In the case of I(J P ) = 0(1/2 − ), the possible S-wave channels are Ξ * ccD * , Ξ ccD * , Ξ ccD , Ξ * ccD , Λ c J/ψ, and Λ c η c . The observation of any one of the mentioned decay patterns could provide hints for the existence of a nnccc pentaquark state. Because the lowest I(J P ) = 0(1/2) − state is much lower than the Ξ ccD threshold, if an observed state in Λ c η c (or Λ c J/ψ) is around 5.4 GeV, this state would be more likely to be a compact pentaquark than a Ξ ccD molecule. If the spin of the observed Ξ cc by LHCb is 3/2, Ξ cc → Ξ * cc and the estimated pentaquark masses will be reduced by 64 MeV. The stability of the pentaquark states is not affected. 96.9 28.0 12200.9 12132.0 12175.9 12106.9        When the isospin (spin) of an initial pentaquark state is equal to a number in the subscript (superscript) of a baryon-meson state, its decay into that baryon-meson channel through S-wave is allowed by the isospin (angular momentum) conservation. We have adopted the masses estimated with the reference thresholds of (a) ΞccD, The relatively stable states judged with the observed hadrons have been marked with a dagger.   have not yet been observed in experiments, we use the theoretical masses of B * c , Ξ bb , and Ξ * bb in Table III to check the pentaquark stability. Now, it is easy to see that the lowest-lying states with I(J P ) = 0(1/2 − ) and I(J P ) = 0(3/2 − ) are both stable. The situation for the nnccb (nnbbb) states can be analyzed similar to the nnccc (nnbbc) case, but now all of them are explicitly (implicitly) exotic.
For the ssccc, ssbbb, ssccb, and ssbbc states, their properties are similar to those of nnccQ (I = 1) and nnbbQ (I = 1). Here, we also use the theoretical masses of B * c , Ω cc , Ω * cc , Ω bb , and Ω * bb to discuss the possible decay channels. In the ssccc case shown in Fig. 3(a), any possible pentaquark is above their allowed rearrangement decay channels and thus there is no stable state. One does not find stable states in the ssbbb and ssccb cases, either. In the ssbbc system, the lowest-lying (J P ) = (3/2 − ) pentaquark is slightly above its decay channel Ω * bb D − s . Probably it is not a broad state.
C. The nnbcQ and sscbQ pentaquark states All these nnbcQ and ssbcQ states are implicitly exotic. To estimate their masses, we can use three types of reference systems, (qqc)-(bQ), (qqb)-(cQ), and (qbc)-(qQ). We present the obtained masses in Table VII and  Table VIII for the nnbcQ and ssbcQ states, respectively, where the theoretical masses M Ξ bc = 6820.0 MeV and M Ω bc = 6920.0 MeV given in Table III are adopted. From the tables, the results with these three types of reference systems are slightly different.
In Fig. 4, we plot the relative positions for the nnbcQ and ssbcQ pentaquark states. The masses we use are obtained with the reference thresholds of Σ b J/ψ, Ξ bc B, Ω b J/ψ, and Ω bc B s channels for the nnbcc, nnbcb, ssbcc, and ssbcb states, respectively. From the figure, 16 rearrangement decay channels are involved for the nnbcc and nncbb states and 12 channels are involved for the ssbcc and sscbb states.
We first check possible stable pentaquarks in the nnbcc case. The lowest J P = 1/2 − and J P = 3/2 − states                                       both have rearrangement decay channels and should not be very narrow. On the contrary, the lowest I(J P ) = 0(5/2) − state is below the possible decay channel Ξ * cbD * s and it is considered a relative state. Similarly, Fig. 4 (b) tells us that the only possible stable nncbb pentaquark has the quantum numbers I(J P ) = 0(5/2 − ) if the mass of Ξ * bc is larger than 6870 MeV. Lastly, it seems that there is no stable ssbcc or sscbb pentaquark state according to the diagrams (c) and (d) of Fig. 4. Here, the possible stable pentaquarks in Fig. 4 have been marked with a dagger.  Table III. The numerical results for the nsccQ and nsbbQ systems are presented in Table IX. In the diagrams (a), (b), (c), and (d) of Fig. 5, we show the relative positions for the nsccc, nsbbb, nsbbc, and nsccb states, respectively. The adopted masses are obtained with the Ω ccD , Ω cc B, Ω bbD , and Ω bb B thresholds.
Of these states, the nsccb and nsbbc pentaquarks are explicitly exotic. The observation of such a state in future measurements will be an important finding, in particular when the state is narrow. From Table IX, the nsccc pentaquark masses estimated with the (nsQ)-(QQ) type reference systems are lower than those with other type thresholds. Such states are easy to be identified as fivequark states because of their high masses, although they are implicitly exotic. The situation is different from the nsccn or nsccs states studied in Ref. [88]. It is not easy to distinguish such a pentaquark state from a 3q baryon once it is observed.
From Fig. 5 and the rough values of the doubly heavy 3q baryons in Table III, it seems that no stable pentaquark states exist in the nsccc, nsbbb, and nsccb systems. However, according to the diagram (c) of Fig. 5, the lowest J P = 1/2 − , J P = 3/2 − , and J P = 5/2 − states are below any possible rearrangement decay channels and they are possibly stable. Of course, if its mass is underestimated, they may also decay into Ξ bb D − s (and probably Ω bbD ), Ω * bbD , and Ω * bbD * , respectively.

E. The nsbcQ pentaquark states
The nsbcQ states are implicitly exotic. Their wave functions are not constrained from the Pauli principle.
The number of wave function bases for a pentaquark with given quantum numbers is bigger than that for other states. After diagonalizing the Hamiltonian, one gets numbers of possible pentaquark states. To estimate the nsbcc (nscbb) masses, we use four different types of reference systems, and Ω bcD (Ω bc B). The numerical results are presented in Table X where we use two theoretical values of masses, M Ω bc = 6920.0 MeV and M Ξ bc = 6820.0 MeV given in Table III. The spectra for the nsbcQ pentaquark states with the Ξ b J/ψ or Ω bc B threshold are shown in Fig. 6. From the figure and the masses given in Table III, it is difficult to find stable pentaquarks in these systems. Only the lowest nsbcc pentaquark is slightly above the Ξ c B − c threshold and is possibly a state without broad width. Of course, the nsbcc states can be searched for in the Ξ c B − c or Ξ b η c channel in future experiments. If such a state could be observed, its exotic nature can be easily identified, a situation different from the nsbcq case [88].

V. DISCUSSIONS AND SUMMARY
Recently, the observation of the P c (4312), P c (4440) and P c (4457) at LHCb [28] gave us significant evidence for the existence of pentaquak states, which motivates us to study the ground compact qqQQQ (q = u, d, s and Q = b, c) pentaquark states within the CMI model. In the considered pentaquark systems, the qqbbc, qqccb states are explicitly exotic and are easy to be identified. Other states can also be easily identified as exotic baryons because their large masses could not be understood without an excited QQ pair.
In this work, we have firstly constructed the flavorcolor-spin wave functions for the qqQQQ pentaquark states from the SU(3) and SU(2) symmetries and Pauli Principle. We extract the effective coupling constants from the mass splittings between conventional hadrons. Based on these, we systematically calculate the colormagnetic interaction for these pentaquark states and obtain the corresponding mass gaps. Then, various reference thresholds are used to estimate the masses of these states. Some theoretical results for the masses of the doubly heavy 3q baryons are adopted in our estimation. At last, we analyze the stability and possible rearrangement decay channels of the qqQQQ pentaquark states.
We have shown the mass spectra and rearrangement decay patterns in the figure form. Following Figs. 2, 3, 4, 5, and 6, we can see ten stable states are possible which are also collected in Table. XI. However, not all of them are really stable states. The reason is that the predicted pentaquarks in the current model may have mass deviations from the case they should be.
As a general feature, the high spin J P = (5/2) − pentaquark states should be usually narrow since they have many D-wave decay modes but one or two S-wave decay modes. This feature is similar to theQQqqq and QQqqq cases [86,88]. Now, the narrowQQqqq pentaquark states have been observed. It is also worthwhile to search for the exotic narrow qqQQQ pentaquark states in future experiments.
In the study of the pentaquark states, the number of color-spin structures may be more than ten. The mixing or channel-coupling effects could be important. Such ef-fects should be carefully considered in detail in further studies.
In short summary, we have systematically studied the exotic states with the structure qqQQQ (q = u, d, s and Q = c, b). They can be easily identified as explicitly exotic or implicitly exotic pentaquark states once observed. We hope the present study may stimulate further investi-   When the spin of an initial pentaquark state is equal to a number in the superscript of a baryon-meson state, its decay into that baryon-meson channel through S-wave is allowed by the angular momentum conservation. We have adopted the masses estimated with the reference thresholds of (a) Ξ b J/ψ and (b) Ω bc B.