Hidden-charm pentaquarks and their hidden-bottom and -like partner states
In the framework of the color-magnetic interaction, we have systematically studied the mass splittings of the possible hidden-charm pentaquarks () where the three light quarks are in a color-octet state. We find that i) the LHCb states fall in the mass region of the studied system; ii) most pentaquarks should be broad states since their -wave open-charm decays are allowed while the lowest state is the -like pentaquark with probably the suppressed decay mode only; and iii) the states do not decay through -wave and their widths are not so broad. The masses and widths of the two LHCb baryons are compatible with such pentaquark states. We also explore the hidden-bottom and -like partners of the hidden-charm states and find the possible existence of the pentaquarks which are lower than the relevant hadronic molecules.
In 2015, the LHCb Collaboration (1) reported two pentaquark-like resonances and in the process with the same decay mode . The decay channel indicates that their minimal quark content is (). The resonance parameters are MeV, MeV and MeV, MeV. The preferred angular momenta are and , respectively and their parities are opposite. Later, the and were confirmed by the reanalysis with a model-independent method (2). Recently, these two states were also observed in the decay (3).
In fact, the theoretical exploration of the hidden-charm pentaquarks was performed before the observation of two states by LHCb. In Refs. (4); (5), the authors predicted two states and four states, where their masses, decay behaviors and production properties were given in a coupled-channel unitary approach. Possible molecular states composed of a charmed baryon and an anticharmed meson were systematically studied with the one-boson-exchange (OBE) model in Ref. (6) and the chiral quark model in Ref. (7). More investigations can be found in Refs. (8); (9); (10); (11); (12); (13); (14); (15). In addition, Li and Liu indicated the existence of hidden-charm pentaquarks by the analysis of a global group structure (16).
After the announcement of the and states by LHCb, these two states were interpreted as , , or molecules (17); (18); (19); (20); (21); (22); (23); (28); (24); (25); (26); (27), bound states or resonances of charmonium and nucleon (29); (30); (31); (32), diquark-diquark-antiquark states (33); (34); (35); (36); (37); (38), diquark-triquark states (39); (40), compact pentaquark states (41); (42); (43), kinematical effects due to rescattering (44), due to triangle singularity (45); (46); (47), or due to a -soliton (48), or a bound state of the colored baryon and meson (49). Their decay and production properties were studied in Refs. (50); (51); (52); (53); (54); (55); (56); (57); (58); (59); (60); (61); (62); (63); (64); (65); (66).
The observation of these resonances also stimulated the arguments for more possible pentaquarks (67); (68); (69); (70); (71). Productions of another and were discussed in Ref. (72) and Refs. (73); (74); (75), respectively. For the detailed overview on the hidden-charm pentaquarks, the readers may refer to Refs. (76); (77).
The dynamical calculations of the bound states are relatively easier if one treats the system as two clusters. The investigation at the quark level is also simplified when one assumes the existence of substructures in a five-body system. If a hidden-charm pentaquark really exists, its spin partners with the same flavor content should also exist. The existence of substructures certainly results in less pentaquarks. Needless to say, configurations with various substructures (baryon-meson, diquark-diquark-antiquark, or diquark-triquark) lead to different results. From symmetry consideration, a physical pentaquark state should be a mixture of all these configurations with various color structures. We here would like to explore a pentaquark structure without the assumption of its substructure.
The masses of the ’s are both above the threshold of . Because the interaction between the and nucleon is very weak, the scattering resonances in this channel are not appropriate interpretations for the observed states. We focus on the possible pentaquark configurations where either the three light quark or the pair is a color octet state. We investigate whether the lower state can be assigned as a tightly bound five-quark state and explore its possible partner states. Recently, there appeared a preliminary quark model study on the hidden color-octet baryons (43).
In principle, a dynamical calculation for a five-body problem is needed in order to calculate their masses. In this work, we calculate their mass splittings with a simple color-magnetic interaction from the one-gluon-exchange (OGE) potential. For example, the baryon and the proton have the same quark content and color structure and their mass difference mainly arises from the color-magnetic interaction. With the calculated mass splittings and a reference threshold, one can estimate the pentaquark masses roughly.
This paper is organized as follows. In Sec. II, we construct the wave functions of the hidden-charm pentaquark states and calculate the matrix elements for the color magnetic interaction in the symmetric limit. Then we consider the flavor breaking case in Sec. III and give numerical results in Sec. IV. In Sec. V, we explore the heavier pentaquarks. We discuss our results and summarize in the final section.
Ii Wave functions and color-magnetic interaction
The color-magnetic Hamiltonian reads
where the -th Gell-Mann matrix should be replaced with for an antiquark. In the Hamiltonian, is the effective mass of the -th quark and is the effective coupling constant between the -th quark and the -th quark. The values of the parameters for light quarks and those for heavy quarks are different and they will be extracted from the known hadron masses. For the hidden-charm systems, we have four types of coupling parameters , , , and with . More parameters need to be determined for the other pentaquarks (). In Ref. (78), we have estimated the mass of another not-yet-observed but plausible exotic meson with this simple model. In Refs. (79); (80); (81), we discussed the mass splittings for the , , and systems, respectively.
To calculate the required matrix elements of the color-magnetic interaction (CMI), we here construct the flavor-color-spin wave functions of the ground state pentaquark systems. These wave functions will also be useful in the study of other properties of the pentaquak states in quark models. In Ref. (8), a study with the color-magnetic interaction is also involved but the wave functions are constructed with flavor symmetry. Now we consider flavor symmetry and treat the heavy (anti)quark as a flavor singlet state.
Because the three light quarks must obey Pauli principle, it is convenient to discuss the constraint with flavor-spin symmetry. The three-quark colorless ground baryons belong to the symmetric representation. Its decomposition gives and therefore flavor singlet baryon in is forbidden. Now the color-octet must belong to the mixed representation. The decomposition gives . So the flavor singlet pentaquark is allowed and we have two flavor octets with different spins. There is no symmetry constraint for the heavy quark pair and one finally gets three pentaquark decuplets with , 1/2, and 3/2, three octets with , 1/2, and 3/2, four octets with , 1/2, 3/2, and 5/2, and three singlets with , 1/2, and 3/2.
|Multiplet||Space||Wave function||Wave function|
To get a totally antisymmetric wave function, we need the components presented in Tab. 1 and need to make appropriate combinations. The notation () means that the first two quarks are symmetric (antisymmetric) when they are exchanged in corresponding space and the superscript () means that the wave function is totally symmetric (antisymmetric). When one combines the color and spin (or flavor and color, or flavor and spin), a wave function with a required symmetry is determined by the relative sign between different components. For example, is symmetric for the exchange of color and spin indices simultaneously for any two quarks. If one uses a minus sign, the wave function is symmetric only for the first two quarks, i.e. a mixed type color-spin wave function. One has to adopt a self-consistent convention for the SU(3) Clebsch-Gordan (C.G.) coefficients when constructing the wave functions. We here take the convention convenient for use (82); (83). For clarity, we present the color wave functions of in Fig. 1 and those of in Fig. 2. The flavor octet wave functions are easy to obtain with the replacements , , and . The spin wave functions for the spin-half case are , , , and . There is no confusion for the totally symmetric flavor and spin wave functions and we do not show them explicitly. With explicit calculation, we find the totally antisymmetric wave functions of the colored states and show them in Tab. 2.
|Multiplet||Flavor-color-spin wave function|
where the baryon symbols with the subscript ’C’ are borrowed from flavor octet and represent the color wave functions in Fig. 1. The structure of the full wave functions is the same as that in Tab. 2 by adding a subscript “penta” to each wave function.
Since the color-spin interaction is the same for baryons in the same flavor multiplet in the SU(3) limit, it is enough to consider only pentaquarks with flavor content in decuplet, in octet, and in singlet. After some calculations we get the results for the as follows,
where or , and is the quadratic Casimir operator specified by the Young diagram
The Young diagram for color symmetry is and those for color-spin symmetry can be found in Tab. 1. The part for can be verified with the formula
The heavy quark symmetry is strict in the limit , which leads to the irrelevance of the heavy quark spin (and flavor) for the interaction between a heavy quark and a light quark. In this limit, the interaction within the heavy quark pair is also irrelevant with their spin (but not the flavor) and the spin-flip between the case and the case is suppressed. The color-magnetic interaction obviously violates HQSS. This means that only terms in Eqs. (3)–(6) are important in the heavy quark limit and there are four degenerate multiplets with the mass ordering: , , , and from high to low.
After the heavy quark mass correction is included, all the terms involving in Eqs. (3)–(6) contribute. Since now the spin-flip between the case and the case is considered, the mixing between states with the same occurs, which results from the term proportional to . Then one should determine the final ’s for mixed states by diagonalizing the specified matrix.
Usually, the flavor mixing between different multiplet representations occurs once the symmetry breaking is considered. In the present case, even in the limit, the mixing between the two octets is nonvanishing, which complicates the color magnetic interactions. To be convenient, we now collect the averages of the CMI in a matrix form for the () and cases. For the other cases containing the quark, we show results in the next section.
For states (3 baryons in ),
One gets similar expressions for the states (3 baryons in ) by replacing with .
For states (7 baryons in ), the results read
where the bases for and are , , and , , , respectively.
Up to now, we have not considered the breaking. Once the mass difference between the strange quark and the quarks is included, the general mixing between flavor multiplets appears. Such an effect is included in the color-magnetic interaction and we now discuss this case. The systems we need to consider additionally are and . They are classified into two categories according to the symmetry for the first two quarks in flavor space: symmetric () and () and antisymmetric (). In the following, we use the symbol like to denote the base states. In this example, the subscript means that the color representation for the is and the color wave function is . The subscript 8 is the color representation for the . The superscript means that the spin wave function for the is and the spin is 1/2. The superscript () indicates the spin of the (pentaquark).
There is only one base state . For the case , we use the base vector , , , and get
where , , , , , , and . For the case , we have
with the base vector , , , , .
Now we present the results for the antisymmetric category. For the case , the base state is and the matrix element is
For the case, we use the base vector , , , and the obtained matrix is
where , , , , , , and . For the case, we have
with the base vector , , , , .
One may use the matrices in this section to numerically reproduce the ’s for the systems after diagonalization. For the case, both Eq. (12) and Eq. (15) give the same formula and thus the same result when . For the case , Eq. (13) and Eq. (16) result in different eigenvalues by assuming . However, one finds that the common numbers of the two sets of eigenvalues are just the results for the systems. The remaining value given by Eq. (13) is the result for the system while that given by Eq. (16) can be thought as a forbidden number because of the Pauli principle. The case has similar features with the case. Probably these features can be used to simplify the calculation for multiquark systems.
Iv Numerical results for the hidden-charm systems
By calculating the CMI matrix elements for ground state baryons and mesons (88), we extract the effective coupling parameters presented in Tab. 3. In determining , one may also use the mass difference between and . The resulting pentaquark masses would be around 10 MeV lower, which is a not a large number in the present method of estimation. Since we also discuss hidden-bottom and -like pentaquark states, Tab. 3 displays relevant parameters, too. Because there is no experimental data for the meson, we determine the value of to be 3.3 MeV from a quark model calculation MeV (89).