Compact Sssc Pentaquark States Predicted by a Quark Model
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Compact sssc ̄c pentaquark states predicted by a quark model a < b,c,d,e c d c,d Qi Meng , , Emiko Hiyama , Kadir Utku Can , Philipp Gubler , Makoto Oka , c,d,e a,f,g Atsushi Hosaka and Hongshi Zong aDepartment of Physics, Nanjing University, Nanjing 210093, China bDepartment of Physics, Kyushu University, Fukuoka 819-0395, Japan cNishina Center for Accelerator-Based Science, RIKEN, Wako 351-0198, Japan dAdvanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan eResearch Center for Nuclear Physics, Osaka University, Ibaraki, Osaka 567-0047, Japan fJoint Center for Particle, Nuclear Physics and Cosmology, Nanjing 210093, China gState Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing 100190, China ARTICLEINFO ABSTRACT Keywords: Several compact sssc ̄c pentaquark resonances are predicted in a potential quark model. The Hamil- pentaquark system sssc ̄c tonian is the best available one, which reproduces the masses of the low-lying charmed and strange quark model hadrons well. Full five-body calculations are carried out by the use of the Gaussian expansion method, few-body problem and the relevant baryon-meson thresholds are taken into account explicitly. Employing the real scal- P * * ing method, we predict four sharp resonances, J = 1_2 (E = 5180 MeV, Γ = 20 MeV), 5_2 * + (5645 MeV, 30 MeV), 5_2 (5670 MeV, 50 MeV), and 1_2 (5360 MeV, 80 MeV). These are the candidates of compact pentaquark resonance states from the current best quark model, which should be confirmed either by experiments or lattice QCD calculations. + 1. Introduction to Θ [18] and Pc [19] in the constituent quark model with the scattering channels in the energy region of the observed Observations of candidates of multi-quark hadrons such states taken into account. They, however, did not find states X; Y ; Z Pc as tetraquarks [1,2] and pentaquarks [3,4] gave in the experimentally observed region, but a few narrow states a great impact to hadron physics community and drove many at significantly higher energies. theoretical discussions. There have been various suggestions In these studies, it was found that the coupling to the scat- Pc for their structures, here in particular for ; compact multi- tering states is crucially important; many states that could be quarks [5,6,7,8,9], hadronic molecules [10, 11, 12, 13, 14], found in the absence of the coupling disappear when scat- their admixtures [15] and even baryocharmonium [16]. The X.3872/ tering states are taken into account. It was also found that well established and recently observed narrow pen- the surviving narrow states with higher energies had a spa- Pc taquark states ’s (4312, 4440, 4457) are widely expected tially compact structure with little coupling to any scatter- to emerge as hadronic molecules of long range nature. Yet ing states. Hence the five-body analysis considering the fall- compact multiquark structure with quark dynamics is an im- apart dynamics is essentially important. Yet, other features portant issue to be investigated when the molecular picture which are difficult to implement and were not considered, can not explain high energy production processes [17]. In are those of pion dynamics; the pion exchange force and pion this paper we address this question in a quark model solved emission decays. The former is important for the formation by the latest advanced few-body method. of hadronic molecules with spatially extended structure, the The model we employ is the constituent quark model latter appear as three-body decays. which accommodates important dynamics of quarks; color Knowing these merits and demerits of the five-body method, confinement and color magnetic spin-dependent interactions. sssc ̄c arXiv:1907.00144v2 [nucl-th] 11 Sep 2019 we propose to study the pentaquark state of . Because Hadrons are then made of minimum numbers of valence quarks. of the flavor contents without u; d quarks, the coupling to Incorporating non-relativistic kinetic and potential energies the pion can be expected to be suppressed. Possible meson in its Hamiltonian, the model has successfully explained many exchange is also suppressed due to their heavier flavor con- properties of low-lying conventional hadrons including their tents such as s̄s or s ̄c. Moreover, thresholds of three-body quantum numbers, masses and even interactions. open channels containing strange hadrons appear about 500 For multi-quark states, however, the situation changes MeV above the lowest two-body ones, significantly larger dramatically not only because of more degrees of freedom than 200 MeV for the decays accompanying the pion. This but also due to couplings to fall-apart (scattering) channels. work’s focus on the sssc ̄c system is furthermore promising The latter occurs because multiquarks can be decomposed for future comparisons between the quark model and lattice into more than one color singlet subsets. Considering these QCD calculations [20], since a lattice calculation would have aspects two of the present authors (E. H. and A. H.) and lowered computational costs due to the absence of the light collaborators studied the pentaquark systems corresponding (u and d) valance quarks. In addition, as the quark model < Corresponding author calculation is performed at finite volume, the guidance this Email address: [email protected] (Q. Meng) work provides can be helpful to understand the finite vol- Meng Qi et al.: Preprint submitted to Elsevier Page 1 of 10 Short Title of the Article th Table 1 where mi and pi are the mass and momentum of the i quark, respectively. TG is the kinetic energy of the center-of-mass The two parameter sets of the employed quark-quark interac- a motion. i are the color SU(3) Gell-mann matrices for the tion, AP1 and AL1 [21]. th i quark with color index a. We label the strange quarks, s AP1 AL1 as i = 1; 2; 3, the charm quark, c as i = 4, and the anticharm quark ̄c as i = 5. p 2/3 1 We use the quark-quark interaction potential proposed by mu;d (GeV) 0.277 0.315 Semay and Silvestre-Brac [21, 22], given by ms(GeV) 0.553 0.577 mc (GeV) 1.819 1.836 p Vij.r/ = * + r *Λ 0.4242 0.5069 r ¨ exp.*r2_r2/ ¨ 1.8025 1.8609 2 0 (2) + i ⋅ j; p+1 3m m 3_2r3 (GeV ) 0.3898 0.1653 i j 0 Λ(GeV) 1.1313 0.8321 B 0.3263 0.2204 with 2m m A(GeVB*1) 1.5296 1.6553 i j *B r0.mi; mj/ = A. / : (3) mi + mj Table 2 This potential consists of the color Coulomb potential, The calculated masses (in MeV) of the mesons and baryons the linear confining part, a (color-electric) constant term and relevant for the thresholds to be considered together with the the color-magnetic spin-spin interaction term. The last term comes from a magnetic gluon exchange, where the func- experimental values. tion in the Breit-Fermi interaction is modified by a cutoff Hadron J P Exp. AP1 AL1 parameter r0. Note that r0 depends on the reduced quark masses. The two sets of parameter choices appearing in this * c 0 2984 2984 3007 work, AP1 and AL1, are listed in Table1. J_ 1* 3097 3104 3103 The present Hamiltonian is tested by computing the static * Ds 0 1968 1955 1963 properties of low-lying baryons and mesons. The calculated D< 1* 2112 2107 2102 masses are given in Table2 for the AP1 and AL1 parameters s together with the corresponding experimental values. In one Ω 3_2+ 1672 1673 1675 earlier work, this Hamiltonian was used in a pentaquark sys- + Ωc 1_2 2695 2685 2679 tem (qqqc ̄c) calculation [15]. We choose AP1 in our present < + Ωc 3_2 2766 2759 2752 calculation since it reproduces the relevant thresholds bet- ter. In addition, we have tested the AL1 in our five-body calculation also and have confirmed that the results are not ume lattice spectrum better. A related lattice QCD study is qualitatively modified by this alternative choice. currently underway. This paper is organized as follows. After the introduc- 3. Method tion, the Hamiltonian and the employed computational method are discussed in Secs.2 and3, respectively. In Sec.4, we In this section, we briefly discuss our method of numer- discuss our results and give a summary in Sec.5. ically solving the five-body Schrödinger equation. We de- scribe the five-body wave function with five types of Jacobi C = 1 2 2. Model Hamiltonian coordinates shown in Fig.1. and are configura- tions in which two color-singlet clusters may fall apart along .c/ the inter-cluster coordinates R .C = 1; 2/ . Namely, for C = 1, the color wave function is chosen as the product of color-singlet sss plus c ̄c, which correspond to cΩ and J_ Ω configurations. For C = 2, the color wave function is The Hamiltonian of the non-relativistic quark model is chosen as the product of color-singlet ssc plus s ̄c, which cor- given by < < < < respond to DsΩc, Ds Ωc, DsΩc , and Ds Ωc configurations. In contrast, the other three configurations, C = 3 * 5, do 5 2 É pi not describe color-singlet subsystems, and represent the five H = mi + * TG quarks as always connected by a confining interaction. In 2mi i this sense, we call C = 3 * 5 as the "connected" (confining) 5 8 (1) 3 É É configurations. * .a ⋅ a/V .r / ; 16 i j ij ij i<j=1 a Meng Qi et al.: Preprint submitted to Elsevier Page 2 of 10 Short Title of the Article s c c¯ c¯ s s 2 3 5 c s 2 4 5 c s 2 5 4 c s 2 5 4 c s 2 3 5 c (1) ¯ (2) ¯ s(4) R(5) ¯ ρ ρ s(3) (1) (2) (5) r(1) R(1) s r(2) R(2) s r(3) ρ(3) r(4) R(4) ρ(4) r(5) ρ(5) s R(3) c s 1 4 c s 1 3 s s 1 3 s s 1 3 s s 1 4 C = 1 C = 2 C = 3 C = 4 C = 5 Figure 1: Five sets of the Jacobi coordinate systems.