PHYSICAL REVIEW D 101, 076025 (2020) Pentaquark components in low-lying baryon 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 Physics 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 quark in SU(3) flavor symmetry; Q ¼ c, b quark) systems with a general group theory approach in the constituent quark model, 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 baryons 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-spin- flavor-spatial wave function for q3QQ¯ systems in the compact pentaquark picture are constructed systematically for studying hidden charm 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 Æ MeVÞ is as large as Nð1535Þ→Nπ ð . Æ 19 η decades, but theoretical results are still largely inconsistent MeVÞ [12] indicates that N(1535) may couple to the meson 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 strangeness component in Nð1535Þ1=2 is Roper resonance Nð1440Þ, Nð1520Þ, and Nð1535Þ have shown to account for the mass 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 nucleon in Ref. [14]. As for − the lowest positive-parity 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 isospin 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 masses 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 Particle 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<j ij ð Þ 1116 1129 . Σð1193Þ 1193 1163 2.56 C C X λ · λ Σà 1385 HOGE −C i j σ⃗ σ⃗; ð Þ 1385 1372 0.97 hyp ¼ OGE m m i · j ð1Þ Ξð1318Þ 1318 1329 −0.83 i<j i j ΞÃð1530Þ 1533 1510 1.49 Ωð1672Þ 1672 1662 0.62 where Aij and Bij are mass-dependent coupling parameters, ΛCð2286Þ 2286 2272 0.62 taking the form ΣCð2455Þ 2454 2428 1.06 Σà 2520 rffiffiffiffiffiffiffi rffiffiffiffiffiffiffi Cð Þ 2518 2486 1.26 m m Ξ 2470 −0 82 A a ij;Bb u ; Cð Þ 2469 2489 . ij ¼ ij ¼ ð2Þ Ξà 2645 mu mij Cð Þ 2646 2633 0.47 ΩCð2695Þ 2695. 2751 −2.07 à ΩCð2770Þ 2766 2789 −0.84 with mij being the reduced mass of ith and jth quarks, 2mimj ΛBð5620Þ 5620 5599 0.37 defined as mij ¼ m m which corresponds to the relative iþ j ΣBð5811Þ 5811 5781 0.51 à Jacobi coordinates of two-body system in Appendix B. ΣBð5832Þ 5832 5801 0.54 OGE The hyperfine interaction, H includes only one-gluon ΞBð5792Þ 5792 5819 −0.47 hyp à 2 ΞBð5945Þ 5950 5953 −0.05 exchange contribution, where COGE ¼ Cmmu, with mu ΩBð6046Þ 6046 6097 −0.84 being the constituent u quark mass and Cm a constant. 076025-2 PENTAQUARK COMPONENTS IN LOW-LYING BARYON … PHYS. REV. D 101, 076025 (2020) 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].