Doubly heavy pentaquarks

# Doubly heavy pentaquarks

## Abstract

Motivated by the observation of two pentaquark-like resonances and the doubly charmed baryon at LHCb, in this work, we systematically investigate the mass spectra of (; ) pentaquark systems in a color-magnetic interaction model. One finds that stable or narrow exotic states are possible and their rearrangement decay patterns are shown. Hopefully, the study is helpful to the experimental search for such pentaquark states.

###### pacs:
14.20.Pt, 12.39.Jh

## I Introduction

Nowadays it is still a hot topic to identify multiquark states from both theoretical side and experimental side since the proposal of the quark model (1); (2). More and more exotic states observed by experiments in recent years (4); (8); (5); (10); (11); (9); (3); (13); (6); (7); (15); (14); (12) are considered as possible tetraquark candidates (16); (17); (18); (19); (20); (21); (22); (23); (24). With one more quark component, the intriguing pentaquark states were also studied in various colliders. Although the subsequent experiments (25) did not confirm the light pentaquark with component claimed by the LEPS Collaboration (26), the LHCb experiment brought us new findings in the heavy quark realm in 2015 (27). Two hidden-charm pentaquark-like resonances and are extracted in the invariant mass distribution of the decay into . This observation stimulated further studies on pentaquark states (20); (28); (29). In this paper, we pay attention to the systems, where and , and estimate the masses of such pentaquark states roughly.

In the quark model, the doubly charmed baryon ( or ) is in a 20-plet representation of the flavor classification (30). Although its study started 40 years ago (31), its existence is confirmed very recently (32); (33); (34). The confirmation from LHCb motivates further theoretical studies on the possible stable () states, which had been predicted in various models. Both the baryon and the meson contain a heavy diquark. Now we would like to add one more light quark component and discuss the spectra of the doubly heavy pentaquarks within a simple model. The so-called heavy diquark-antiquark symmetry was used to relate the mass splittings of and in Ref. (35). Hopefully, the present investigation can also be helpful to further study on such a symmetry in multiquark systems.

Compared to the baryon, the pentaquark state should be heavier. However, the complicated interactions within multiquark systems may lower the mass, which probably makes it difficult to distinguish experimentally a conventional baryon from a pentaquark baryon just from the mass consideration. One example for this feature is the five newly observed states (36); (37). They can be accommodated in both configuration (38); (39); (40); (41); (42); (43); (44); (45); (46) and configuration (47); (48); (49); (50); (51); (52); (53) and much more measurements are needed to resolve their nature. As a theoretical prediction, the basic features for the pentaquark spectra may be useful for us to understand possible structures of heavy quark hadrons.

For the doubly heavy five-quark systems, we have a compact configuration and two baryon-meson moleucle configurations, and . As for the latter molecule configuration, there are theoretical studies in the meson exchange methods (54); (55); (56). Here, we discuss the mass splittings of the compact pentaquark states by considering the color-magnetic interactions between quarks and estimate their rough positions. It is still an open question how to distinguish the two configurations. For example, if we compare the prediction for the -type hidden charm state in the molecule picture (57) and the estimation for the mass of the lowest compact pentaquark (58), one gets consistent results. However, the numbers of possible states in these two pictures are different. The present study should be useful in looking for genuine pentaquark states rather than molecules.

This paper is organised as follows. In Sec. II, we construct the wave functions for the pentaquark states. In Sec. III, the relevant Hamiltonians for various systems are presented. In Sec. IV, we give numerical results and discuss the mass spectra of the pentaquark states and their strong decay channels. Finally, we present a summary in Sec. V.

## Ii Color-magnetic interaction and wave functions

For the ground state hadrons with the same quark content, e.g. and , their mass splitting is mainly determined by the color-magnetic interaction (CMI). We here study the mass splittings of the pentaquark states by adopting a color-magnetic model in which the Hamiltonian reads

 H = ∑imi+HCM, HCM = −∑i

where () are the Gell-Mann matrices for the -th quark and () are the Pauli matrices for the -th quark. For antiquarks, the is replaced with . The effective mass for the th quark includes the constituent quark mass and contributions from color-electric interactions and color confinements. The effective coupling constants depend on the quark masses and the ground state spatial wave functions. We can estimate the values of and from the known hadron masses. For the studied systems, we need to determine seventeen coupling parameters , , , , , , , , , , , , , , , , and , where represents or .

Obviously, we can calculate the color-magnetic matrix elements and investigate the mass spectra for the systems if the wave functions were constructed. Now we move on to the construction of the flavor-color-spin wave function of a system, which is a direct product of flavor wave function, color wave function, and spin wave function. We construct these wave functions separately and then combine them together by noticing the possible constraint from the Pauli principle. We will use the diquark-diquark-antiquark bases to construct the wave function. In principle, the selection of wave function bases is irrelevant with the final results since we will diagonalize the Hamiltonian in this CMI model. Here, the notation “diquark” only means two quarks and it does not mean a compact substructure.

In flavor space, the heavy quarks are treated as singlet states and the light diquark may be in the flavor antisymmetric or symmetric representation. For the case of the antisymmetric (symmetric) light diquark, the representations of the pentaquarks are and ( and ). We plot the weight diagrams for the systems in Fig. 1. The explicit wave functions are similar to the tetraquark states presented in Ref. (59). Because of the unequal quark masses, we consider symmetry breaking and the flavor mixing among different representations occurs. The resulting systems we consider are: , , , , , and .

In color space, the Young diagrams tell us that the pentaquark systems have three color singlets. Then we have three color wave functions. The direct product for the representations can be written as

 (3c⊗3c)⊗(3c⊗3c)⊗¯3c=(¯3c⊕6c)⊗(¯3c⊕6c)⊗¯3c (2) = (¯3c⊗¯3c⊗¯3c)⊕(¯3c⊗6c⊗¯3c)⊕(6c⊗¯3c⊗¯3c).

In the last line, the representations in the parentheses are for the heavy diquark, light diquark, and antiquark, respectively. Then the color-singlet wave functions can be constructed as

 ϕAA = [(Q1Q2)¯3c(q3q4)¯3c¯q], ϕAS = [(Q1Q2)¯3c(q3q4)6c¯q], ϕSA = [(Q1Q2)6c(q3q4)¯3c¯q], (3)

where () means antisymmetric (symmetric) for the diquarks. Explicitly, we have

 ϕAA = 12√6[(rbbg−rbgb+brgb−brbg+gbrb−gbbr+bgbr−bgrb)¯b (4) +(rbrg−rbgr+brgr−brrg+grrb−grbr+rgbr−rgrb)¯r +(gbrg−gbgr+bggr−bgrg+grgb−grbg+rgbg−rggb)¯g], ϕAS = 14√3[(2rgbb−2grbb−rbgb−rbbg+brgb+brbg+gbrb+gbbr−bgrb−bgbr)¯b (5) +(2gbrr−2bgrr−rbrg−rbgr+brrg+brgr−grrb−grbr+rgrb+rgbr)¯r +(2brgg−2rbgg+gbrg+gbgr−bgrg−bggrg−grgb−grbg+rggb+rgbg)¯g], ϕSA = 14√3[(2bbgr−2bbrg+gbrb−gbbr+bgrb−bgbr−rbgb+rbbg−brgb+brbg)¯b (6) +(2rrbg−2rrgb+rgrb−rgbr+grrb−grbr+rbgr−rbrg+brgr−brrg)¯r Missing or unrecognized delimiter for \Big

The spin wave functions for the pentaquark states are

 χSS : Unknown environment '% (7) χSA : Unknown environment '% (8) χAS : Unknown environment '% (9) χAA : χ10=[(Q1Q2)0(q3q4)0¯q]120. (10)

Here in the symbol , is the total spin of the first four quarks. The superscript of means that the first two quarks are symmetric and the second two quarks are antisymmetric. Other superscripts are understood similarly.

Considering the Pauli principle, we obtain twelve types of total wave functions , , , , , , , , , , , and . Here, when the first two quarks are identical, or else . When the two light quarks are antisymmetric (symmetric) in the flavor space, (), or else (). Then the considered pentaquark states are categorized into six classes:

1.The , and states with ;

2.The and states with ;

3.The and states with and ;

4.The states with and ;

5.The and states with and ;

6.The states with .
In the following discussions, we also use the notation to denote the total wave function.

## Iii The Hamiltonian expressions

With the constructed wave functions, we calculate color-magnetic matrix elements on various bases. In this section, we present the obtained Hamiltonians in the matrix form. To simplify the expressions, we use the following definitions: , , , , , , , , , , and .

### iii.1 (ccnn)I=1¯q, (bbnn)I=1¯q, (ccss)¯q, and (bbss)¯q states in the first class

Three types of basis vectors are involved in calculating the relevant matrix elements: , , and .

For the states, there is only one basis vector . The obtained Hamiltonian is

 ⟨HCM⟩J=52=23(4α+β+2λ+2ν). (11)

For the states, we have four basis vectors, , , , and . The resulting Hamiltonian is

 ⟨HCM⟩J=32=23⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝4α+β−3(λ+ν)√5(ν−λ)3√5ν3√5λ√5(ν−λ)4α−β+λ+ν3(β−ν)3(λ−β)3√5ν3(β−ν)12(9α−θ)−λ−32β3√5λ3(λ−β)−32β12(9α+τ)−ν⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (12)

For the states, the basis vectors are , , , and and the Hamiltonian reads

 Unknown environment '% (13)

### iii.2 (ccnn)I=0¯q and (bbnn)I=0¯q states in the second class

In this case, we also have three types of basis vectors to consider: , , and .

For the states, the involved basis vector is and the obtained Hamiltonian is

 ⟨HCM⟩J=52=13(3τ−α+5β−2λ+10ν). (14)

For the states, there are three basis vectors , , and . We can get the following Hamiltonian,

 ⟨HCM⟩J=32=13⎛⎜ ⎜⎝3τ−α+5β+3λ−15ν√5(λ+5ν)6√5ν√5(λ+5ν)3τ−α−5β−λ+5ν6(β−ν)6√5ν6(β−ν)4(2θ+λ)⎞⎟ ⎟⎠. (15)

For the states, we have four basis vectors , , , and . Then the Hamiltonian

 ⟨HCM⟩J=12=13⎛⎜ ⎜ ⎜ ⎜ ⎜⎝3τ−α−5β+2λ−10ν6(β+2ν)02√2(λ+5ν)6(β+2ν)8(θ−λ)6√6λ−6√2ν06√6λ3(3θ+α)3√3β2√2(λ+5ν)−6√2ν3√3β3τ−α−10β⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ (16)

can be obtained.

### iii.3 (cbnn)I=1¯q and (cbss)¯q states in the third class

Now, one does not need to consider the constraint for the heavy diquark from the Pauli principle and we then have six types of basis vectors, , , , , , and .

For the states, two basis vectors, and , are involved and the obtained Hamiltonian is

 ⟨HCM⟩J=52=13(2(4α+β+2λ+2ν)3√2(γ−2μ)3√2(γ−2μ)5β+10λ−2ν−α−3θ). (17)

For the states, there are seven basis vectors, , , , , , , and . One obtains the Hamiltonian as follows,

 Unknown environment 'pmatrix% (18)

For the states, eight basis vectors are involved, , , , , , , , and . The resulting Hamiltonian is

 ⟨HCM⟩J=12=23⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝(4α−β−2λ−2ν)2√2(ν−λ)√2(2μ−γ)3(β+2ν)03√2(2μ−γ)6μ−3(β+2λ)2√2(ν−λ)2(2α−β)2μ−3√2ν−3√62γ6μ−3√2γ−3√2λ√2(2μ−γ)−3√2ν−4(τ+ν)3√2γ3√6ν−3(β+2λ)−3√2λ03(β+2ν)−3√62γ3√2γ12(9α−θ)+2λ√3μ−3√2γ0−32β0−3√62γ3√6ν√3μ32(α−3τ)03√32β03√2(2μ−γ)−3√2γ−3(β+2λ)−3√2γ0(−12(α+3θ)−52β−5λ+ν)−√2(5λ+ν)5√2(2μ−γ)6μ−3√2γ−3√2λ03√32β−√2(5λ+ν)−12(α+3θ)−5β5μ−3(β+2λ)−3√2λ0−32β05√2(2μ−γ)5μ12(9α+τ)+2ν⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (19)

### iii.4 (cbnn)I=0¯q states in the fourth class

In this case, we also have six types of basis vectors, , , , , , and .

For the states, there is only one basis vector . The obtained Hamiltonian is

 ⟨HCM⟩J=52=13(3τ−α+5β−2λ+10ν). (20)

For the states, the involved basis vectors are , , , , and . The Hamiltonian can be written as

 ⟨HCM⟩J=32=23⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝4θ+2λ3√5ν3(β−ν)3√2γ−3√2μ3√5ν12(3τ−α+5β+3λ−15ν)√52(λ+5ν)√102μ03(β−ν)√52(λ+5ν)12(3τ−α−5β−λ+5ν)1√2(μ−5γ)−3√2γ3√2γ√102μ1√2(μ−5γ)5ν−12(9α+5τ)−32β−3√2μ0−3