PHYSICAL REVIEW D 101, 076025 (2020)

Pentaquark components in low-lying resonances

† K. Xu ,1,* A. Kaewsnod ,1 Z. Zhao,1 X. Y. Liu ,1,2 S. Srisuphaphon,3 A. Limphirat,1 and Y. Yan1, 1School of and Center of Excellence in High Energy Physics and Astrophysics, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand 2School of Mathematics and Physics, Bohai University, Jinzhou 121013, China 3Department of Physics, Faculty of Science, Burapha University, Chonburi 20131, Thailand

(Received 17 February 2020; accepted 17 April 2020; published 28 April 2020)

We study pentaquark states of both light q4q¯ and hidden heavy q3QQ¯ (q ¼ u; d; s in SU(3) flavor symmetry; Q ¼ c, b quark) systems with a general group theory approach in the constituent , and the spectrum of light baryon resonances in the ansatz that the l ¼ 1 baryon states may consist of the q3 as well as q4q¯ pentaquark component. The model is fitted to ground state and light baryon resonances which are believed to be normal three-quark states. The work reveals that the Nð1535Þ1=2− and Nð1520Þ3=2− may consist of a large q4q¯ component while the Nð1895Þ1=2− and Nð1875Þ3=2− are respectively their partners, and the Nþð1685Þ might be a q4q¯ state. By the way, a new set of color-- flavor-spatial wave function for q3QQ¯ systems in the compact pentaquark picture are constructed systematically for studying hidden pentaquark states.

DOI: 10.1103/PhysRevD.101.076025

Γ ≡ I. INTRODUCTION the review paper [9]. That the decay width of Nð1535Þ→Nη 65 25 Γ ≡ 67 5 Baryon resonance spectrum has been studied over ð MeVÞ is as large as Nð1535Þ→Nπ ð . 19 η decades, but theoretical results are still largely inconsistent MeVÞ [12] indicates that N(1535) may couple to the much more strongly than predicted by flavor sym- with experimental data. Except for the ground state − baryons, even the low-lying resonances, for example, the metry [10]. The component in Nð1535Þ1=2 is Roper resonance Nð1440Þ, Nð1520Þ, and Nð1535Þ have shown to account for the ordering of N(1440) and been of an ordering problem. Theoretical works in the N(1535) [13], and it is claimed that ss¯ pair contribution is three-quark picture always predict a larger mass for important to the properties of the in Ref. [14]. As for − the lowest positive- state Nð1440Þ than for the the other lowest orbital excited state Nð1520Þ3=2 , the Γ =Γ lowest negative-parity states Nð1520Þ and Nð1535Þ [1]. branching ratio of ηN tot is less than 1% [12] which Since the discoveries of Nð1895Þ1=2−, Nð1875Þ3=2−, reveals that there is little strange component contribution. It Δð1900Þ1=2−, and Δð1940Þ3=2− [2], these states and other is also stated that γN → Nð1520Þ form factors are domi- baryon resonance states near 1900 MeV have not been well nated by the meson cloud contributions which means explained in conventional constituent quark models [3–6]. N(1520) may not be pure q3 state but include the extrinsic By applying the new approaches of photoproduction and qq¯ pair contribution in the form of q4q¯ components [11]. electroproduction experiments, more baryon resonances And the baryon states including pentaquark components have been discovered and confirmed [2–7] and the internal have also been studied in the light quark sectors for Roper structures of some resonance states have been revealed by resonance [15], Nð1535Þ [16–18] to give a better explan- the properties including Breit-Wigner amplitudes, transi- ation of the experimental results like transitions amplitudes tions amplitudes, and form factors [2,8–11]. The Roper and form factors. electroproduction amplitudes [8] has proven us that it is In this work we study the role of pentaquark components mainly the nucleon first radial excitation as interpreted in in low-lying baryon resonance states. The constructions of light pentaquark wave functions in the Yamanouchi tech- – *[email protected] nique have been formulated in the previous works [19 22]. † [email protected] As a consequence, the light baryon resonance spectrum is newly reproduced by mixing three quark and pentaquark Published by the American Physical Society under the terms of components. And we extend the group theory approach to the Creative Commons Attribution 4.0 International license. hidden heavy pentaquarks in the SU(3) flavor symmetry, Further distribution of this work must maintain attribution to 3 ¯ the author(s) and the published article’s title, journal citation, where the pentaquark wave functions for the q QQ systems and DOI. Funded by SCOAP3. are systematically constructed in the harmonic oscillator

2470-0010=2020=101(7)=076025(10) 076025-1 Published by the American Physical Society K. XU et al. PHYS. REV. D 101, 076025 (2020)

C interaction and applied as complete bases to evaluate hidden λi in the above equations are the generators of color charm and bottom pentaquark mass spectra for all possible SU(3) group. quark configurations and interactions of other types. The model parameters are determined by fitting the The paper is organized as follows: We briefly review in theoretical results to the experimental data of the mass of all Sec. II the constituent quark model extensively described in the ground state baryons, namely, eight light baryon our previous work [22], and predetermined all the model states, seven charm baryon states, and six bottom baryon parameters by comparing the theoretical and experimental states as well as light baryon resonances of energy level of all the ground state baryons and low-lying q3 N ≤ 2, including the first radial excitation state Nð1440Þ baryon resonance states. The baryon masses in the q3 with mass at 1.5 GeV and a number of orbital excited l ¼ 1 picture are also presented in Sec. II. In Sec. III we derive the and l ¼ 2 baryons. All these baryons are believed to be mass spectra of light q4q¯ pentaquark states, and to mainly 3q states whose masses were taken from reproduce the negative-parity nucleon and Δ resonances Data Group [12]. The least squares method is applied to below 2 GeV by introducing light pentaquark components minimize the weighted squared distance δ2, in three-quark baryon states. The wave functions of q3QQ¯ systems are constructed in the harmonic oscillator inter- XN exp cal 2 2 ðM − M Þ action for all possible quark configurations and applied δ ¼ ωi 2 ð3Þ Mexp as complete bases to evaluate hidden heavy pentaquark i¼1 mass spectra in Sec. IV. A summary is given in Sec. V. 3 The details of q wave functions as well as the construction where ωi are weights being 1 for all the states except for of q3QQ¯ pentaquark wave functions are shown in the N(939) and Δð1232Þ which are set to be 100, Mexp and Mcal Appendices. are respectively the experimental and theoretical masses. Listed in Tables I–IV are the theoretical masses which are q3 II. THEORETICAL MODEL calculated in the Hamiltonian in Eq. (1) in the picture and fitted to the experimental data. Possible assignments of A group theory approach to construct the wave functions the theoretical results of excited nucleon and Δ resonances for baryon and pentaquark states has been described in below 2.2 GeV to all the known baryon states are presented Refs. [19–22], and we refer the readers to those works for details. Here, we just present the general Hamiltonian for multiquark systems, TABLE I. Ground state baryons applied to fit the model parameters. The last column shows the deviation between the ex- H H HOGE; D 100 Mexp −Mcal = ¼ 0 þ hyp perimental and theoretical mean values, ¼ ·ð Þ Mexp. Mexp are taken from PDG [12]. XN p2 H m k 0 ¼ k þ Baryon Mexp MeV Mcal D(%) 2mk ð Þ ðMeVÞ k¼1 N(939) 939 939 0 XN 3 B − λC λC A r − ij ; Δð1232Þ 1232 1232 0 þ 8 i · j ij ij r Λ 1116 −1 16 i

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TABLE II. Nucleon resonances of positive parity applied to fit mu ¼ md ¼ 327 MeV;ms ¼ 498 MeV; the model parameters. mc ¼ 1642 MeV;mb ¼ 4960 MeV; 2s 1 P P exp ðΓ; þ D; N; L Þ Status J M ðMeVÞ McalðMeVÞ 2 Cm ¼ 18.3 MeV;a¼ 49500 MeV ;b¼ 0.75 ð4Þ N 56; 28; 0; 0þ 1þ ð Þ **** 2 939 939 N 56; 28; 2; 0þ 1þ ð Þ **** 2 N(1440) 1499 Similar model parameters were obtained in the previous N 56; 28; 2; 2þ 5þ ð Þ **** 2 N(1720) 1655 work [22]. The parameters fixed in the work are slightly N 56; 28; 2; 2þ 3þ ð Þ **** 2 N(1680) 1655 different from the preliminary ones since charm and bottom N 20; 21; 2; 1þ 1þ ð Þ *** 2 N(1880) 1749 baryons are included and more accurate method is used for N 20; 41; 2; 1þ 3þ ð Þ 2 Missing 1749 the model fixing. And the u and d constituent quark mass is N 70; 210; 2; 0þ 1þ ð Þ **** 2 N(1710) 1631 closer to the quark mass, 330 MeV which was determined N 70; 410; 2; 0þ 3þ by the baryon magnetic moments [23]. ð Þ **** 2 N(1900) 1924 N 70; 210; 2; 2þ 3þ In general, all the ground state baryons are well described, ð Þ 2 Missing 1702 N 70; 210; 2; 2þ 5þ with the maximum deviation less than 3%. For excited ð Þ ** 2 N(1860) 1702 N 70; 410; 2; 2þ 1þ baryon states, the Roper resonance as the first radial excited ð Þ *** 2 N(2100) 1994 N 70; 410; 2; 2þ 3þ state gets a mass around 1.5 GeV which does not agree well ð Þ * 2 N(2040) 1994 4 5 with the pole mass on PDG [12], but has a 0.56 GeV gap Nð70; 10; 2; 2þÞ ** þ N(2000) 1994 2 between the ground state nucleon, close to the gap 0.55 GeV Nð70; 410; 2; 2þÞ ** 7þ N(1990) 1994 2 between the two lowest-magnitude JP ¼ 1=2þ poles in Refs. [8,9]. The lowest negative-parity nucleon states turn out to be lower than the Roper resonance just as other TABLE III. Resonances of negative-parity applied to fit the predictions of the conventional constituent quark models. model parameters. We assume that the lowest negative-parity baryon resonan- q3 q4q¯ ðΓ; 2sþ1D; N; LPÞ Status JP MexpðMeVÞ McalðMeVÞ ces may consist of the component as well as the pentaquark component. The spin-orbit interactions are not N 70; 210; 1; 1− **** 3− N(1520) 1380 ð Þ 2 included in this work, so the states in the same spatial-spin- N 70; 210; 1; 1− 1− ð Þ **** 2 N(1535) 1380 flavor configuration as shown in Appendix A have the same N 70; 410; 1; 1− **** 1− N(1650) 1672 2 ð Þ 2 mass value. Except for the two missing Δð70; 10; 2; 2þÞ N 70; 410; 1; 1− **** 5− N(1675) 1672 2 ð Þ 2 states and the two missing nucleon states Nð20; 1; 2; 1þÞ N 70; 410; 1; 1− *** 3− N(1700) 1672 2 ð Þ 2 and Nð70; 10; 2; 2þÞ, most positive-parity states are rea- Δ 70; 210; 1; 1− 1− Δ 1620 ð Þ **** 2 ð Þ 1380 sonably reproduced. Δ 70; 210; 1; 1− 3− Δ 1700 ð Þ **** 2 ð Þ 1380 III. LIGHT QUARK SPECTRUM in Tables II–IV following the SUð6ÞSF representations. The A. Mass of q4q¯ pentaquark states 3 orbital-spin-flavor wave functions of q baryon states are 4 3 The mass spectra of the ground state q q¯ and q ss¯ listed in Appendix A. pentaquarks are evaluated in the Hamiltonian in Eq. (1), The 3 model coupling constants and 4 constituent quark by applying the complete bases of the pentaquark wave masses are fitted, functions derived in our previous work [22]. Listed in Tables V and VI are the theoretical results, with the model parameters fixed in the previous section. Comparing to other Δ TABLE IV. resonance of positive parity applied to fit the works [24,25] for q4q¯ and q3ss¯ hidden strange pentaquark model parameters. states, the model here employs much less model parameters ðΓ; 2sþ1D; N; LPÞ Status JP MexpðMeVÞ McalðMeVÞ Δ 56; 48; 0; 0þ 3þ Δ 1232 q4q¯ ð Þ **** 2 ð Þ 1232 TABLE V. ground state pentaquark masses. Δ 56; 48; 2; 0þ *** 3þ Δð1600Þ 1791 ð Þ 2 q4q¯ JP M q4q¯ Δ 56; 48; 2; 2þ 5þ Δ 1905 configurations ð Þ (MeV) ð Þ **** 2 ð Þ 1947 4 1 csf 4 1− 3− Δ 56; 8; 2; 2þ **** þ Δð1910Þ 1947 Ψ ðq q¯Þ 2 , 2 2562, 2269 ð Þ 2 ½211C½31FS½4F½31S Δ 56; 48; 2; 2þ 3þ Δ 1920 csf 4 3− 5− ð Þ *** 2 ð Þ 1947 Ψ ðq q¯Þ 2 , 2 2025, 2269 ½211C½31FS½31F ½4S 4 þ 7þ Δ 1950 Δð56; 8; 2; 2 Þ **** 2 ð Þ 1947 csf 4 1− 3− Ψ 211 31 31 31 ðq q¯Þ 2 , 2 2123, 2049 Δð70; 210; 2; 0þÞ * 1þ Δð1750Þ 1631 ½ C½ FS½ F ½ S 2 Ψcsf q4q¯ 1− 2025 Δ 70; 210; 2; 2þ 3þ 211 31 31 22 ð Þ 2 ð Þ 2 Missing 1702 ½ C½ FS½ F ½ S 2 5 csf 4 1− 3− Δ 70; 10; 2; 2þ þ Ψ ðq q¯Þ 2 , 2 1683, 2049 ð Þ 2 Missing 1702 ½211C½31FS½22F ½31S

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3 TABLE VI. q ss¯ ground state pentaquark masses. where ψ 1 and ψ 2 are respectively the lower and higher negative-parity physical states, and the mixing angle θ q4q¯ JP M q3ss¯ configurations ð Þ (MeV) between the q3 and q4q¯ states is generally complex. The csf 3 1− 3− Ψ ðq ss¯Þ 2 , 2 2762, 2586 masses of the physical states, Mψ and Mψ are derived as ½211C½31FS½4F½31S 1 2 csf 3 3− 5− Ψ ðq ss¯Þ 2 , 2 2420, 2546 follows: ½211C½31FS½31F ½4S Ψcsf q3ss¯ 1− 3− 2448, 2414 2 2 211 31 31 31 ð Þ 2 , 2 M M 3 θ M 4 θ − m ; ½ C½ FS½ F ½ S ψ 1 ¼ q cos þ q q¯ sin δ csf 3 1− Ψ 211 31 31 22 ðq ss¯Þ 2 2393 2 2 ½ C½ FS½ F ½ S Mψ ¼ Mq3 sin θ þ Mq4q¯ cos θ þ mδ; csf 3 1− 3− 2 Ψ 211 31 211 31 ðq ss¯Þ 2 , 2 2032, 2243 ½ C½ FS½ F ½ S ðMq4q¯ − Mq3 Þ csf 3 1− m 2θ 2θ Ψ ðq ss¯Þ 2 2165 δ ¼ tan sin ð7Þ ½211C½31FS½211F ½22S 2 csf 3 1− 3− Ψ 211 31 22 31 ðq ss¯Þ 2 , 2 2135, 2354 ½ C½ FS½ F ½ S The mixing angle θ in Eq. (7) is determined by adjusting the lower negative-parity states ψ 1 to Nð1535Þ, Nð1520Þ, Δð1620Þ, and Δð1700Þ. With both the real and imaginary and predict relatively higher mass spectra. It is predicted part of the mixing angle in the domain of ð0; π=2Þ, the in the calculation that the pentaquark state with the M mixing angle and the ψ 2 can be determined without ½31FS½22F½31S configuration and the quantum numbers duplication from Eq. (7). Thus, one gets four pairs of mixed I JP 1 1− has the lowest mass, 1683 MeV which is ð Þ¼2 ð2 Þ states as shown in Table VII with all Re θ 0. 1=2 ð Þ¼ quite close to the mass of the isospin- narrow resonance Nð1520Þ3=2− and Nð1875Þ3=2− form a nonstrange pair, Nþ 1685 Nþ 1685 ð Þ. One may make a bold guess that this ð Þ and the Nð1535Þ1=2− and Nð1895Þ1=2− form a strange resonance could be the lowest pentaquark state. pair for the nucleon resonances while the Δð1620Þ1=2− and Δð1900Þ1=2− form a nonstrange pair, and the B. Possible mixtures of q3 and q4q¯ states Δð1700Þ3=2− and Δð1940Þ3=2− form a strange pair for Δ Ground state pentaquarks always have a negative parity, the resonances. For the pair of N(1520) and N(1875), l 1 Δ we have shown in Table VII the results with both the thus only ¼ nucleon and orbitally excited states could 4 4 q q¯ 31 4 q q¯ 22 31 mix with ground state pentaquarks. Considering the low pentaquark states ½ F½ S (2025 MeV) and ½ F½ S 3 theoretical masses for the Nð1535Þ and Nð1520Þ resonan- (2049 MeV) mixed with the q state. In the present model ces in the q3 picture and their quantum numbers, it is one can not rule out either of them. natural to assume that the two baryon resonances may The mass spectrum of the negative-parity nucleon and Δ 3 4 include both the q3 and q4q¯ pentaquark component con- resonances are listed in Table VIII in the q and q q¯ picture. tributions. The wave function of these baryon resonances Nð1650Þ, Nð1675Þ, and Nð1700Þ are assumed to be mainly 3 3 may be expressed as linear combinations of the q3 state pure q states since the q picture reproduces their masses and q4q¯ pentaquark states which have the same quantum well, as shown in Table III, and hence there is no mixing 3 N 1685 numbers as the q state, with pentaquark states. ð Þ could be the lowest pure X pentaquark state. The others are q3 and q4q¯ mixing states 3 4 α a0jq iþ aαjq q¯i : ð5Þ taken from Table VII. α The conventional constituent quark models have failed to describe the higher nucleon and Δ resonance states near In principle, one can determine the coefficients aα by 1900 MeV [3–6]. In this constituent quark model with a solving the coupled equations of all channels including not color dependent Cornell-like potential, however, we have only the coupling between the q3 and q4q¯ states and the given not only the possible theoretical interpretations coupling between the q4q¯ states, but also the contributions of hidden channels such as meson-baryon ones. The mass 3 4 matrix is usually not Hermitian but complex, thus the bare TABLE VII. The mixture of q and q q¯ components. All four 3 states and physical states cannot be linked by an unitary q states take the same mass, 1380 MeV. The chosen pentaquark 4 4 transformation. In this work we simplify the problem to the states and masses are listed as q q¯ configuration and q q¯ Mass (in simplest case that the q3 state mixes with only one q4q¯ MeV) from Tables V and VI. P 4 4 pentaquark state which has the lowest mass, eliminating ψ 1 State J θψ2 State q q¯ configuration q q¯ Mass other pentaquark states and meson-baryon channels. As a 1530 1− i35.2° 1882 q3ss¯ 2032 2 2 2 ½211F½31S result, the × mass matrix will be highly complex, which 3− 4 1515 i32.6° 1899 q q¯ 31 4 2025 may be eigendiagonalized by the transformation, 2 ½ F½ S i31 7 q4q¯ . ° 1914 ½22F½31S 2049 3 4 1− 4 ψ 1 θ q − θ q q¯ ; i46 4 q q¯ ¼ cos j i sin j i 1610 2 . ° 1893 ½31F½31S 2123 3− 3 3 4 1710 i51.5° 2024 q ss¯ 22 31 2354 ψ 2 ¼ sin θjq iþcos θjq q¯i; ð6Þ 2 ½ F½ S

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TABLE VIII. Masses of negative-parity resonances after in- TABLE IX. Ground hidden-charm pentaquark q3cc¯ mass cluding ground state pentaquark components. The theoretical spectrum, where the q3 and QQ¯ components are in the color masses of Nð1535Þ, Nð1520Þ, Δð1620Þ, and Δð1700Þ states take octet states. the mean values of their Breit-Wigner mass from [12]. q3QQ¯ configurations JP Mðq3cc¯ÞðMeVÞ JP Mexp Mcal Resonance Status ðMeVÞ ðMeVÞ csf 3 1− 3− Ψ 21 21 21 21 ðq cc¯Þ 2 , 2 4483, 4495 N 1520 3− ½ C½ FS½ F½ S ð Þ **** 2 1510-1520 1515 csf 3 1− 3− 1− Ψ 21 21 3 21 ðq cc¯Þ 2 , 2 4702, 4701 N 1535 **** 1515-1545 1530 ½ C½ FS½ F½ S ð Þ 2 csf 3− 5− 1 3 N 1650 − Ψ 21 21 21 3 ðq cc¯Þ 2 , 2 4556, 4598 ð Þ **** 2 1645-1670 1672 ½ C½ FS½ F½ S N 1675 5− ð Þ **** 2 1670-1680 1672 N 1685 1− ð Þ * 2 ? 1665-1675 1683 N 1700 3− q3bb¯ ð Þ *** 2 1650-1750 1672 TABLE X. Ground hidden-bottom pentaquark mass N 1875 3− q3 QQ¯ ð Þ *** 2 1850-1920 1899/1914 spectrum, where the and components are in the color N 1895 1− octet states. ð Þ **** 2 1870-1920 1882 1− Δð1620Þ **** 1590-1630 1610 3 P 3 2 q QQ¯ configurations J Mðq bb¯ÞðMeVÞ Δ 1700 3− ð Þ **** 2 1690-1730 1710 csf 3 ¯ 1− 3− Δ 1900 1− Ψ ðq bbÞ 2 , 2 10964, 10968 ð Þ *** 2 1840-1920 1893 ½21C½21FS½21F½21S Δ 1940 3− csf 3 ¯ 1− 3− ð Þ ** 1940-2060 2024 Ψ ðq bbÞ 2 , 2 11183, 11183 2 ½21C½21FS½3F½21S csf 3 ¯ 3− 5− Ψ ðq bbÞ 2 , 2 11037, 11051 ½21C½21FS½21F½3S for Nð1895Þ1=2−, Nð1875Þ3=2−, Δð1900Þ1=2−, and Δð1940Þ3=2− states as negative-parity partners of the well known nucleon and Δ resonances, but also effectively solved pentaquark in the ½21C½21FS½21F½21S configuration are the long-standing ordering problem of Nð1440Þ, Nð1520Þ, close to the experimental values of 4440 and 4457 MeV, but þ and Nð1535Þ by mixing the q3 and q4q¯ components. still about 100-200 MeV higher than the Pcð4312Þ state. 3 4 P In general, a q state may mix with two or even more q q¯ The higher predicted c masses may result from the states as well as meson-baryon ones. However, the present compact spacial configuration in our pentaquark picture. P work can not give more meaningful information by The observed c may probably be baryon-meson molecular including two or more pentaquark states in the mixture. states or mixtures of compact pentaqark states and mole- A better understanding of Nð1440Þ, Nð1520Þ, and Nð1535Þ cules. For the hidden-bottom pentaquarks, the work pre- may be achieved by studying the helicity amplitude of dicts the mass of the ground states to be 10.9-11.2 GeV, Nð1440Þ, Nð1520Þ and Nð1535Þ with both q3 and q4q¯ state lying below the threshold of a single bottom baryon and BðBÞ , which is consistent with other work [27]. contributions since there are much more sensitive exper- P imental data available. The newly observed c states by the LHCb collaboration have been largely interpreted as hadronic states since there are abundant charmed meson and charmed ¯ IV. q3QQ PENTAQUARK SPECTRUM baryon thresholds available [26]. Within the molecular scenario, the mass spectrum [27–40] and dynamical proper- Motivated by the hidden-charm pentaquark candidates – recently found by the LHCb Collaboration [26] we also ties [27,36 40] have been successfully explained in various calculate the mass spectra of hidden heavy pentaquarks methods. The compact pentaquark interpretation works well 3 [41–43] when the parameters are fixed to both baryons and of q QQ¯ systems. The quark configurations and wave mesons. With the limited experimental results, the of functions of the q3QQ¯ systems are derived in Appendix B. Pc states will keep as an open question in the near future. The spatial wave functions, which are derived in the harmonic oscillator quark-quark interaction and grouped V. SUMMARY in Appendix B according to the permutation symmetry, are employed as complete bases to study the q3QQ¯ systems The masses of low-lying q3 states and ground q4q¯ states described with the color dependent Hamiltonian in Eq. (1). are evaluated, where all model parameters are predeter- The mass spectra of the hidden charm and hidden bottom mined by fitting the theoretical masses to the experimental pentaquarks of the q3 color octet configuration are pre- data for the baryons which are believed to be mainly 3q sented in Tables IX and X separately. states. In the work we have assumed that the Roper It’s noted that the hidden-charm pentaquark mass spectra resonance is the first radial excitation state of nucleon. in this work is slightly higher than the three narrow It is interesting that the theoretical work predicts the þ þ 31 22 31 pentaquarklike states, Pcð4312Þ , Pcð4440Þ , and pentaquark state with the ½ FS½ F½ S configuration þ P 1 1− Pcð4457Þ measured by LHCb. The predicted values of and the quantum numbers IðJ Þ¼2 ð2 Þ has the lowest 4483 and 4495 MeV for the lowest hidden-charm mass, about 1680 MeV. One may make a bold guess that

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TABLE XI. Explicit q3 color-orbital-spin-flavor wave functions.

P SUð6ÞSF l SUð6ÞSF × Oð3Þ wave functions

N Representations O(3) SUð3ÞF octet SUð3ÞF decuplet 0þ P 1þ P 3þ 056 J ¼ 2 J ¼ 2 1 c 0 c 0 pffiffi ψ ϕ Φ χ Φ χ ψ ϕ Φ χ 2 ½111 00sð λ ρ þ ρ λÞ ½111 00S S S 1− P 1− 3− P 1− 3− 170 ;J ¼ 2 ; 2 J ¼ 2 ; 2 1 c 1 1 1 c 1 1 ψ ϕ Φ χ Φ χ ϕ Φ χ − Φ χ pffiffi ψ Φ ϕ χ ϕ χ 2 ½111½ 1mρð λ ρ þ ρ λÞþ 1mλð ρ ρ λ λÞ 2 ½111 Sð 1mλ λ þ 1mρ ρÞ P 1− 3− 5− J ¼ 2 ; 2 ; 2 1 c 1 1 pffiffi ψ χ ϕ Φ ϕ Φ 2 ½111 Sð 1mλ λ þ 1mρ ρÞ 0þ P 1þ P 3þ 256 J ¼ 2 J ¼ 2 1 c 2 c 2 pffiffi ψ ϕ Φ χ Φ χ ψ Φ ϕ χ 2 ½111 00sð λ ρ þ ρ λÞ ½111 S 00S S 0þ P 1þ P 1þ 70 J ¼ 2 J ¼ 2 1 c 2 2 1 c 2 2 pffiffi ψ ϕ Φ χ Φ χ ϕ Φ χ − Φ χ ψ Φ ϕ χ ϕ χ 2 ½111½ 00ρð λ ρ þ ρ λÞþ 00λð ρ ρ λ λÞ 2 ½111 Sð 00λ λ þ 00ρ ρÞ P 3þ J ¼ 2 1 c 2 2 pffiffi ψ χ ϕ Φ ϕ Φ 2 ½111 Sð 00λ λ þ 00ρ ρÞ 1þ P 1þ 3þ 220 J ¼ 2 ; 2 ψ c ϕ2 Φ χ − Φ χ ½111 1mAð ρ ρ λ λÞ 2þ P 3þ 5þ P 1þ 3þ 5þ 7þ 256 J ¼ 2 ; 2 J ¼ 2 ; 2 ; 2 ; 2 1 c 2 c 2 pffiffi ψ ϕ Φ χ Φ χ ψ ϕ Φ χ 2 ½111 2mSð ρ ρ þ λ λÞ ½111 2mS S S 2þ P 3þ 5þ P 3þ 5þ 70 J ¼ 2 ; 2 J ¼ 2 ; 2 1 c 2 2 1 c 2 2 ψ ϕ Φ χ Φ χ ϕ Φ χ − Φ χ pffiffi ψ Φ ϕ χ ϕ χ 2 ½111½ 2mρð λ ρ þ ρ λÞþ 2mλð ρ ρ λ λÞ 2 ½111 Sð 2mλ λ þ 2mρ ρÞ P 1þ 3þ 5þ 7þ J ¼ 2 ; 2 ; 2 ; 2 1 c 2 2 pffiffi ψ χ ϕ Φ ϕ Φ 2 ½111 Sð 2mλ λ þ 2mρ ρÞ this q4q¯ pentaquark state could be the isospin-1=2 narrow Y. Y. acknowledge support from SUT under Grant resonance Nþð1685Þ which can not be accommodated as a No. SUT-PhD/13/2554. S. S. acknowledges support from q3 particle. the Faculty of Science, Burapha University. A. K., Z. Z., The work shows that the ordering problem of the and A. L. acknowledge support from SUT. X. Y. L. Nð1440Þ, Nð1520Þ, and Nð1535Þ may be solved by intro- acknowledges support from Young Science Foundation ducing the q4q¯ contribution. The same calculation leads from the Education Department of Liaoning Province, to that the Nð1895Þ1=2−, Nð1875Þ3=2−, Δð1900Þ1=2−, China (Project No. LQ2019009). and Δð1940Þ3=2− resonances may pair respectively with − − − the Nð1535Þ1=2 , Nð1520Þ3=2 , Δð1620Þ1=2 , and APPENDIX A: EXPLICIT q3 WAVE Δð1700Þ3=2− in the q3 and q4q¯ interpretation. FUNCTIONS The mass spectra of ground hidden heavy pentaquark states 3 3 In this Appendix the q color-orbital-spin-flavor wave q QQ¯ are accurately evaluated using the same predetermined functions with the principle quantum number N ≤ 2 are model parameters. It is found that the hidden charm penta- N0 listed in Table XI, where χi, Φj, and ϕ 0 0 are the spin, quark states with the ½21C½21FS½21F½21S configuration L M y have the lowest masses which are slightly larger than the flavor, and spatial wave functions, respectively. The LHCb results. In this communication, however, the work SUð3ÞF singlet states are excluded since only nucleon Δ cannot draw any conclusion about the nature of Pc states. and resonances are discussed.

ACKNOWLEDGMENTS APPENDIX B: CONSTRUCTION OF PENTAQUARK WAVE FUNCTIONS We are grateful for insightful comments and suggestions FOR q3QQ¯ SYSTEM from Prof. Thomas Gutsche. This work is supported by 3 ¯ Suranaree University of Technology (SUT) and the Office The construction of the q QQ pentaquark state follows 3 ¯ of the Higher Education Commission under the National the rule that q QQ state must be a color singlet and the Research University (NRU) project of Thailand. K. X. and q3QQ¯ wave function should be antisymmetric under any

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3 ¯ permutation between identical quarks. Requiring the q QQ where the ρ and λ stand for the types of ½218 color octet pentaquark to be a color singlet demands that the color part configuration in Eq. (B1). The detailed color wave function 3 ¯ 3 ¯ of the q and QQ must form a ½2221 singlet state, there are for both color singlet and color octet states for the q and QQ two possible color configurations: the color part of the q3 is a are listed in Table XII. 3 ¯ [111] singlet and the QQ¯ is also a singlet and the color part of We construct the spatial wave functions of q QQ systems the q3 is a [21] octet and QQ¯ is also an octet. The pentaquark in the harmonic oscillator potential for the quark-quark state in the q3QQ¯ system with the q3 color singlet configu- interaction. A new set of relative Jacobi coordinates was 3 ¯ ration corresponds to the hadronic molecular pentaquark introduced for the q QQ system, different from the ones in 4 state which is not confined in our Hamiltonian. And the our previous work [22] for q q¯ system, the Hamiltonian for q3QQ¯ system in the compact pentaquark picture takes the q3 the harmonic oscillator potential is written as color octet configuration. Requiring the wave function of the p⃗2 p⃗2 p⃗2 p⃗2 λ ρ σ χ ⃗2 2 2 2 three-quark configuration to be antisymmetric, the spatial- H 3 2 5C λ ρ⃗ σ⃗ χ⃗ q Q ¼ 2m þ 2m þ 2M þ 2u þ ð þ þ þ Þ spin-flavor part of q3 is required to be [21] state by χ conjugation, and directly couples with the spatial-spin- ðB4Þ flavor part of QQ¯ . First we study the total antisymmetric q3 with uχ being the reduced quark mass of the fourth Jacobi wave function for the color octet configuration, 5mM coordinate, defined as uχ ¼ 3m 2M, m and M are the mass of 1 þ ffiffiffi c osf c osf light quark and heavy quark respectively. C is the coupling ψ 3 ¼ p ðψ 21 ψ 21 − ψ 21 ψ 21 ÞðB1Þ ½ A 2 ½ λ ½ ρ ½ ρ ½ λ constant, and the relative Jacobi coordinates and the corre- sponding momenta are defined respectively as with X 1 1 osf o sf ρ⃗ ffiffiffi r⃗−r⃗ λ⃗ ffiffiffi r⃗ r⃗−2r⃗ ψ ¼ bijψ ψ ; ¼ p ð 1 2Þ ¼ p ð 1 þ 2 3Þ ½21ρ;λ ½Xi ½Yj 2 6 i;j¼S;ρ;λ X 1 1 sf s f σ⃗ ffiffiffi r⃗−r⃗ χ⃗ ffiffiffiffiffi 2 r⃗ r⃗ r⃗ −3 r⃗ r⃗ ψ ¼ cijψ ψ ; ¼ p ð 4 5Þ ¼ p ð ð 1 þ 2 þ 3Þ ð 4 þ 5ÞÞ ½Y ½xi ½yj 2 30 i;j S;ρ;λ ¼ 1 1 s s s p⃗ ffiffiffi p⃗−p⃗ p⃗ ffiffiffi p⃗ p⃗−2p⃗ ψ X ¼fψ 3 ; ψ 21 g; ρ ¼ p ð 1 2Þ λ ¼ p ð 1 þ 2 3Þ ½ i ½ S ½ ρ;λ 2 6 f f f f 1 ψ Y ¼fψ 3 ; ψ 111 ; ψ 21 gðB2Þ ½ j ½ S ½ A ½ ρ;λ p⃗σ ¼ pffiffiffiðp⃗4 −p⃗5Þ 2 3 ffiffiffi The total color wave function for q QQ¯ pentaquark state p   5 2Mðp⃗1 þp⃗2 þp⃗3Þ−3mðp⃗4 þp⃗5Þ takes the form, p⃗χ ¼ pffiffiffi 6 3mþ2M 1 X8 Ψc ffiffiffi ψ c q3 ψ c QQ¯ p⃗ r⃗ i 21 ¼ p 21 i ð Þ 21 i ð ÞðB3Þ where i and i are the momenta and coordinate of th quark, ½ j¼ρ;λ 8 ½ j ½ j i the antiquark is assigned the coordinate r⃗5, the fourth and

TABLE XII. q3QQ¯ color wave functions.

Color list q3 color WF ρ type qq¯ q3 color WF λ type 1 1 pffiffi RGB − GRB GBR pffiffi RR¯ GG¯ BB¯ Color singlet 6 ð þ 3 ð þ þ Þ −BGR þ BRG − RBGÞ 1 1 pffiffi RGR − GRR BR¯ pffiffi 2RRG − RGR − GRR Color octet 1 2 ð Þ 6 ð Þ 1 1 pffiffi RGG − GRG BG¯ pffiffi RGG GRG − 2GGR Color octet 2 2 ð Þ 6 ð þ Þ 1 1 pffiffi RBR − BRR −GR¯ pffiffi 2RRB − RBR − BRR Color octet 3 2 ð Þ 6 ð Þ 1 1 1 RBG GBR − BRG − BGR pffiffi RR¯ − GG¯ pffiffiffiffi 2RGB 2GRB − GBR Color octet 4 2 ð þ Þ 2 ð Þ 12 ð þ −RBG − BRG − BGRÞ 1 1 pffiffi GBG − BGG RG¯ pffiffi 2GGB − GBG − BGG Color octet 5 2 ð Þ 6 ð Þ 1 1 1 pffiffiffiffi 2RGB − 2GRB − GBR pffiffi 2BB¯ − RR¯ − GG¯ RBG BRG − BGR − GBR Color octet 6 12 ð 6 ð Þ 2 ð þ Þ þ BGR − BRG þ RBGÞ 1 1 pffiffi RBB − BRB −GB¯ pffiffi RBB BRB − 2BBR Color octet 7 2 ð Þ 6 ð þ Þ 1 1 pffiffi GBB − BGB RB¯ pffiffi GBB BGB − 2BBG Color octet 8 2 ð Þ 6 ð þ Þ

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TABLE XIII. Normalized q3 spatial wave functions with quantum number, N0 ¼ 2n and L0 ¼ M0 ¼ 0. 000 ½3S (0,0,0,0) 1 1 200 3 pffiffi 1; 0; 0; 0 pffiffi 0; 0; 1; 0 ½ S 2 ð Þ, 2 ð Þ pffiffi qffiffi pffiffi 400 5 3 5 ½3S 4 ð2; 0; 0; 0Þ, 8ð1; 0; 1; 0Þ, 4 ð0; 0; 2; 0Þ pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi 600 3 14 18 18 14 ½ S 8 ð3; 0; 0; 0Þ, 8 ð2; 0; 1; 0Þ, 8 ð1; 0; 2; 0Þ, 8 ð0; 0; 3; 0Þ pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi 800 3 42 14 15 14 42 ½ S 16 ð4; 0; 0; 0Þ, 8 ð3; 0; 1; 0Þ, 8 ð2; 0; 2; 0Þ, 8 ð1; 0; 3; 0Þ, 16 ð0; 0; 4; 0Þ pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi pffiffiffiffi 1000 3 33 45 50 50 45 33 ½ S 16 ð5; 0; 0; 0Þ, 16 ð4; 0; 1; 0Þ, 16 ð3; 0; 2; 0Þ, 16 ð2; 0; 3; 0Þ, 16 ð1; 0; 4; 0Þ, 16 ð0; 0; 5; 0Þ pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffi 1200 3 429 594 675 175 675 594 ½ S 64 ð6; 0; 0; 0Þ, 64 ð5; 0; 1; 0Þ, 64 ð4; 0; 2; 0Þ, 32 ð3; 0; 3; 0Þ, 64 ð2; 0; 4; 0Þ, 64 ð1; 0; 5; 0Þ, pffiffiffiffiffiffi 429 64 ð0; 0; 6; 0Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1400 3 1430 2002 2310 2450 2450 2310 ½ S 128 ð7; 0; 0; 0Þ, 128 ð6; 0; 1; 0Þ, 128 ð5; 0; 2; 0Þ, 128 ð4; 0; 3; 0Þ, 128 ð3; 0; 4; 0Þ, 128 ð2; 0; 5; 0Þ, pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 2002 1430 128 ð1; 0; 6; 0Þ, 128 ð0; 0; 7; 0Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffi 1600 3 4862 429 2002 539 2205 539 ½ S 256 ð8; 0; 0; 0Þ, 64 ð7; 0; 1; 0Þ, 128 ð6; 0; 2; 0Þ, 64 ð5; 0; 3; 0Þ, 128 ð4; 0; 4; 0Þ, 64 ð3; 0; 5; 0Þ, pffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi 2002 429 4862 128 ð2; 0; 6; 0Þ, 64 ð1; 0; 7; 0Þ, 256 ð0; 0; 8; 0Þ pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1800 3 4199 5967 1755 1911 7938 7938 ½ S 256 ð9; 0; 0; 0Þ, 256 ð8; 0; 1; 0Þ, 128 ð7; 0; 2; 0Þ, 128 ð6; 0; 3; 0Þ, 256 ð5; 0; 4; 0Þ, 256 ð4; 0; 5; 0Þ, pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1911 1755 5967 4199 128 ð3; 0; 6; 0Þ, 128 ð2; 0; 7; 0Þ, 256 ð1; 0; 8; 0Þ, 256 ð0; 0; 9; 0Þ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 2000 3 58786 20995 99450 6825 28665 29106 ½ S 1024 ð10; 0; 0; 0Þ, 512 ð9; 0; 1; 0Þ, 1024 ð8; 0; 2; 0Þ, 256 ð7; 0; 3; 0Þ, 512 ð6; 0; 4; 0Þ, 512 ð5; 0; 5; 0Þ, pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 28665 6825 99450 20995 58786 512 ð4; 0; 6; 0Þ, 256 ð3; 0; 7; 0Þ, 1024 ð2; 0; 8; 0Þ, 512 ð1; 0; 9; 0Þ, 1024 ð0; 0; 10; 0Þ pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2200 3 52003 74613 88825 98175 103950 106722 ½ S 1024 ð11; 0; 0; 0Þ, 1024 ð10; 0; 1; 0Þ, 1024 ð9; 0; 2; 0Þ, 1024 ð8; 0; 3; 0Þ, 1024 ð7; 0; 4; 0Þ, 1024 ð6; 0; 5; 0Þ, pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi 106722 103950 98175 88825 74613 52003 1024 ð5; 0; 6; 0Þ, 1024 ð4; 0; 7; 0Þ, 1024 ð3; 0; 8; 0Þ, 1024 ð2; 0; 9; 0Þ, 1024 ð1; 0; 10; 0Þ, 1024 ð0; 0; 11; 0Þ fifth quark form the third Jacobi coordinate σ and the centers symmetric spatial wave function while the spatial wave of first three quarks and the last two heavy quarks form the function for other possible permutation symmetries fourth Jacobi coordinate χ. The permutation symmetry of f½21ρ;λ, and ½111A will not be specified here. Note that 3 pentaquarks is simply represented by the q cluster since the we have set M0 ¼ 0 and used the abbreviation, ψ σ⃗ ψ χ⃗ nσ ;lσ ð Þ and nχ ;lχ ð Þ is fully symmetric for any permuta- X C ψ ρ⃗ψ λ⃗ tion between quarks. The total spatial wave function of nρ;lρ;mρ;nλ;lλ;mλ nρlρmρ ð Þ nλlλmλ ð Þ pentaquarks may take the form, fni;li;mig X 3 ½Xy q ½Xy ≡ C ψ n ;l ;n ;l Ψ ψ ⊗ ψ σ⃗ ⊗ ψ χ⃗ nρ;lρ;nλ;lλ ð ρ ρ λ λÞ NLM ¼ N0L0M0 n ;l ð Þ n ;l ð ÞðB5Þ σ σ χ χ n ;l fXi ig which is simply the product of the q3 spatial wave function ≡ Cn ;l ;n ;l ðnρ;lρ;nλ;lλÞðB7Þ shown in Table XIII and the harmonic oscillator wave ρ ρ λ λ ψ σ⃗ ψ χ⃗ fni;lig functions nσ ;lσ ð Þ and nχ ;lχ ð Þ for the Jacobi coordinate σ and χ. ½Xy stands for all possible permutation symmetries The spatial wave functions of pentaquarks with the 3 3 ¯ of the q cluster, where, ½Xy ¼f½3S; ½21ρ;λ; ½111Ag. N, L, q QQ symmetry ½5S are listed in the Table XIV (Up and M are respectively the total principle quantum number, to N ¼ 14 energy level is sufficient for the numerical 3 total angular momentum and magnetic quantum number of ψ q L0 M0 0 calculations), where N0L0M0 ( ¼ ¼ )and the pentaquark (lσ ¼ 0, lχ ¼ 0), with ψ σ⃗ l 0 ψ χ⃗ l 0 nσ ;lσ ð Þ ( σ ¼ ), nχ ;lχ ð Þ ( χ ¼ ) are the spatial q3 N ¼ 2nρ þ lρ þ 2nλ þ lλ þ 2nσ þ lσ þ 2nχ þ lχ ðB6Þ wave functions of the subsystem and the harmonic oscillator wave function for the σ⃗ and χ⃗ coordinates, respectively. Without any limitation for nσ and nχ,all 3 3 ¯ The spatial wave functions of the q subsystem of q QQ degenerate states of each pentaquark energy level up to pentaquarks with the permutation symmetries ½3S are listed N ¼ 14 served as a complete basis. N ≤ 8 states are 0 0 in Table XIII up to N ¼ 22, where lρ, lλ, and are L are listed below, the higher ones follow the rule that limited to 0, 1, and 2 only. To save space, we show only the N ¼ Nq3 þ 2ðnσ þ nχÞ.

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TABLE XIV. q3QQ¯ pentaquark spatial wave functions of symmetric type.

q3QQ¯ q3 Ψ000 ψ 000 ψ 0;0ðσ⃗Þψ 0;0ðχ⃗Þ ½5S ½3S q3QQ¯ q3 q3 q3 Ψ200 ψ 200 ψ 0;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 000 ψ 1;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 000 ψ 0;0ðσ⃗Þψ 1;0ðχ⃗Þ ½5S ½3S ½3S ½3S q3QQ¯ q3 q3 q3 q3 Ψ400 ψ 400 ψ 0;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 200 ψ 1;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 200 ψ 0;0ðσ⃗Þψ 1;0ðχ⃗Þ, ψ 000 ψ 2;0ðσ⃗Þψ 0;0ðχ⃗Þ, ½5S ½3S ½3S ½3S ½3S q3 q3 ψ 000 ψ 1;0ðσ⃗Þψ 1;0ðχ⃗Þ, ψ 000 ψ 0;0ðσ⃗Þψ 1;0ðχ⃗Þ ½3S ½3S q3QQ¯ q3 q3 q3 q3 Ψ600 ψ 600 ψ 0;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 400 ψ 1;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 400 ψ 0;0ðσ⃗Þψ 1;0ðχ⃗Þ, ψ 200 ψ 2;0ðσ⃗Þψ 0;0ðχ⃗Þ, ½5S ½3S ½3S ½3S ½3S q3 q3 q3 q3 ψ 200 ψ 1;0ðσ⃗Þψ 1;0ðχ⃗Þ, ψ 200 ψ 0;0ðσ⃗Þψ 2;0ðχ⃗Þ, ψ 000 ψ 3;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 000 ψ 2;0ðσ⃗Þψ 1;0ðχ⃗Þ, ½3S ½3S ½3S ½3S q3 q3 ψ 000 ψ 1;0ðσ⃗Þψ 2;0ðχ⃗Þ, ψ 000 ψ 0;0ðσ⃗Þψ 3;0ðχ⃗Þ ½3S ½3S q3QQ¯ q3 q3 q3 q3 Ψ800 ψ 800 ψ 0;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 600 ψ 1;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 600 ψ 0;0ðσ⃗Þψ 1;0ðχ⃗Þ, ψ 400 ψ 2;0ðσ⃗Þψ 0;0ðχ⃗Þ, ½5S ½3S ½3S ½3S ½3S q3 q3 q3 q3 ψ 400 ψ 1;0ðσ⃗Þψ 1;0ðχ⃗Þ, ψ 400 ψ 0;0ðσ⃗Þψ 2;0ðχ⃗Þ, ψ 200 ψ 3;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 200 ψ 2;0ðσ⃗Þψ 1;0ðχ⃗Þ, ½3S ½3S ½3S ½3S q3 q3 q3 q3 ψ 200 ψ 1;0ðσ⃗Þψ 2;0ðχ⃗Þ, ψ 200 ψ 0;0ðσ⃗Þψ 3;0ðχ⃗Þ, ψ 000 ψ 4;0ðσ⃗Þψ 0;0ðχ⃗Þ, ψ 000 ψ 3;0ðσ⃗Þψ 1;0ðχ⃗Þ, ½3S ½3S ½3S ½3S q3 q3 q3 ψ 000 ψ 2;0ðσ⃗Þψ 2;0ðχ⃗Þ, ψ 000 ψ 1;0ðσ⃗Þψ 3;0ðχ⃗Þ, ψ 000 ψ 0;0ðσ⃗Þψ 4;0ðχ⃗Þ ½3S ½3S ½3S

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