Compact sssc ̄c 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 , 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 [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 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 -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-

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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

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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. The s quarks, labeled as 1 − 3, are to be antisymmetrized, while particles 4 and 5 stand for c quark and ̄c quark, respectively. Scatterings of sss + c ̄c and ssc + s ̄c are described in the coordinate bases C = 1 and 2, respectively.

The five-body Schrödinger equation for the total angular The spatial wave function is expanded by Gaussian basis momentum J and its z-component M is given by functions as Φ(C)(r(C),(C),R(C), s(C)) = (H − E)ΨJM = 0. (4) L  (n , l , m , r(C)) × (n , l , m , (C)) We solve it by using the Gaussian Expansion Method 1 1 1 2 2 2 (8) (C) (C)  (GEM) [23, 24], which was successfully applied to various ×(n3, l3, m3, R ) × (n4, l4, m4, s ) L, types of three-body and four-body systems [25, 26, 27, 28, (n, l, m, r) 29]. The total wave function ΨJM is written as a sum of where is defined as 2 components, each described in terms of one of the Jacobi l −(r∕rn) (n, l, m, r) = Nnlr e Ylm(r̂) (9) coordinate bases, with the Gaussian ranges taken in geometric progression, É É (C) B(C) r = r an−1 (n = 1 − n ). ΨJM = 123 1 n 1 max (10) C 4 5 (5) The index in the total wave function ΨJM given in (C) (C) (C) (C) (C) (C) × S ΦL (r ,  ,R , s ) , Eq.5 is defined as JM ≡ {s, ̄s,, S, n1, n2, n3, n4, l1, l2, l3, l4,L}. (11) C (C) (C) where specifies the set of Jacobi coordinates. 1 , S , C For completeness, we note that the orbital angular momenta Φ( ) and L represent the color-singlet wave functions, spin wave are combined in the order of (((l1, l2), l3), l4)L, where the S functions for total spin and spatial wave functions for total intermediate quantum numbers are suppressed. Within the L orbital angular momentum , respectively. 123 denotes the present calculation settings, they are determined uniquely, s anti-symmetrization operator for the three quarks (1,2,3). so that we can omit them. (C) C = 1−5 The color-singlet total wave functions, 1 , for The dimensions of the basis of Gaussian wave functions, are chosen as n1max, n2max, n4max for the C = 1 − 5 channels are 6. n3max (1) for C = 1 and 2 equals 10, and for C = 3 − 5 are set to 6.  = [(123)1(45)1]1, 1 In the present calculation, we investigate both positive (2) , 1 = [(124)1(35)1]1 and negative parity states. For the negative parity states, (3) we take the total orbital angular momentum as L = 0 and  = [[(12)̄ (34)̄ ]35]1, P − − − 1 3 3 the total spin-parity as J = 1∕2 , 3∕2 , and 5∕2 . The (4) , r  R s C = 1 1 = [(12)3̄ [(34)3̄ 5]3]1 orbital angular momenta of , , , and for and (5) 2 are chosen as (l1, l2, l3, l4) = (0, 0, 0, 0), (1, 0, 0, 1), and  = [(12)̄ [(45) 3] ] . 1 3 1 3 1 (6) (0, 1, 0, 1). For C = 3 − 5, we set all the orbital angular The spin wave functions for the total spin S are given by momenta to 0. For the positive parity states, the total or- bital angular momentum is taken to L = 1 and the total (1) J P = 1∕2+ 3∕2+ 5∕2+ S = [[(12)s3](45)̄s]S , spin-parity to , , and . The orbital (2) angular momenta of r, , R, and s for C = 1 and 2 are cho- = [[(12)s4](35)̄s]S , S sen as (l1, l2, l3, l4) = (0, 0, 1, 0), (0, 0, 0, 1), (1, 0, 0, 0), and (3) (0, 1, 0, 0) C = 3 − 5 (l , l , l , l ) = S = [[(12)s(34)̄s]5]S , , and for are chosen as 1 2 3 4 (0, 0, 1, 0) (4) = [(12) [(34) 5] ] , . S s ̄s  S In diagonalizing the five-body Hamiltonian, we use about (5) J P − − + + = [(12)s[(45)̄s3]]S , 40,000 basis functions for = 1∕2 , 3∕2 , 1∕2 , 3∕2 , S (7) P − + and 15,000 basis functions for J = 5∕2 and 5∕2 . s ̄s  where , , and represent the spins of the subsystem des- It should be noted here that all the obtained eigenval- ignated in each definition. ues are discrete. Namely, as the system is computed in a

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finite volume, even the continuum states corresponding to Table 3 the baryon-meson scattering solutions come out as discrete The calculated masses (in MeV) of the excited mesons and states. In earlier works [18, 19] , the real-scaling (stabiliza- baryons relevant to the thresholds to be considered. tion) method [30] was adopted to distinguish genuine reso- nances from the discretized scattering states. In the present Hadron J P AP1 R(1) R(2) case, we scale the basis functions along and by + ℎc (1P ) 1 3468 multiplying all the range parameters simultaneously with a − c (2S) 0 3607 factor as RN → RN . Then, any continuum state will (1P ) 0+,1+,2+ 3492 fall off towards its threshold, while a compact resonant state c − should not be affected by the boundary at a large distance. (2S) 1 3647 + Ds(1P ) 1 2479 − Ds(2S) 0 2648 4. Results ∗ + + + Ds (1P ) 0 ,1 ,2 2507 ∗ − As a first step, we calculated the spectra without the con- Ds (2S) 1 2708 tributions from scattering states. In these calculations, only Ω¨ 1∕2−,3∕2− 1971 C the connected Jacobi coordinate bases in Fig.1, = 3−5 are ¨¨ + P − Ω 3∕2 2190 included. Solving the Schrödinger equation for J = 1∕2 , − − + + + Ω¨ 1∕2−,3∕2− 3078 3∕2 , 5∕2 , 1∕2 , 3∕2 , and 5∕2 , we obtain the spectra c Ω¨¨ 1∕2+ 3185 shown in Figs.2 and3. It can be seen in these figures that c Ω∗¨ 1∕2−,3∕2− 3078 all eigenvalues are obtained above the lowest meson-baryon c Ω∗¨¨ 3∕2+ 3227 thresholds. Because these calculations include only the con- c nected channels C = 3 − 5 without contributions from the scattering channels C = 1 and 2, all states are stable against fall-apart decay. We call these states compactified states in levels of positive parity are located above all the thresholds the following because they are forces to be so. which consist of only ground state baryons and ground state In Figs.2 and3, the dashed lines indicate the relevant mesons. meson-baryon thresholds, which couple to the shown com- Next, we consider the contributions from scattering states pactified states. The calculated masses of the excited mesons by including the scattering channels C = 1 and 2 in our cal- and baryons corresponding to the thresholds, are given in culation. According to Ref. [19], the coupling of the scatter- ¨ Table3 for the AP1 parameters. Here, Ω stand for the first ing states may cause some of the compactified states to melt ¨¨ excited states with negative parity, while Ω represent the into the continuum spectrum. positive-parity excited states of Ω baryons. As we neglect To investigate the nature of each compactified state, we the spin-orbit interaction in the current Hamiltonian, the p- include the scattering state one by one in the real scaling P + + + ∗ + + wave mesons, c(1P )(J = 0 , 1 , 2 ) and Ds (1P )(0 , 1 , method calculation. Namely, we scale the range parameter + ¨ − − 2 ), are degenerate. The same happens for Ω (1∕2 , 3∕2 ), RN of the Gaussian bases as RN → RN for the scatter- ¨ − − ∗¨ − − Ωc(1∕2 , 3∕2 ), and Ωc (1∕2 , 3∕2 ). Note that the lowest ing channel in the Jacobi coordinates of C = 1 or 2. The and second thresholds in our calculation [Ωc + Ds(1S) and eigenvalues corresponding to scattering states will fall down Ω + c(1S)] are reversed from the experimental data, which towards the respective thresholds with the increasing val- give Ωc + Ds(1S)(4663) and Ω + c(1S)(4656). ues. At the same time, the resonant states will stay at their energy independently from the scaling factor . With this procedure, we can determine the dominant meson-baryon component for each compactified state. For more details and J P = 1∕2− For spin-parity , the lowest energy state ap- examples, see Ref. [19]. pears at 4855 MeV, which is above the hidden charm thresh- Following the above procedure, we now study the cou- Ω+J∕ Ω +D Ω +D∗ old and open charm thresholds c s and c s , pling of each compactified state to specific scattering states. Ω∗ + D∗ but below the open charm threshold c s . The second The results are summarized in Table4 for negative parity state is located at 5044 MeV. states and in Table5 for positive parity states. They show J P = 3∕2− For spin-parity , the lowest eigenvalue is ob- that most of the compactified states have significant cou- tained at 4753 MeV. The second state appears at 4866 MeV pling to some scattering states, and they do not survive as J P = 1∕2− which is slightly higher than the lowest state of , resonances. For instance, one sees that the lowest negative while the third one shows up at 4998 MeV. 3∕2− Ω+ P parity state, 4753 MeV ( ) is mainly an c scattering J = 5∕2− − − For spin-parity , the lowest state is found at state, while 4855 MeV (1∕2 ), 4866 MeV (3∕2 ), and 4873 − 4873 MeV which is higher than all the open charm thresh- MeV (5∕2 ), have a dominant overlap with Ω + J∕ scat- olds and hidden charm thresholds formed by ground states. P tering state. The next group of excited states are similarly J = 1∕2+ 5∕2+ ∗ ∗ ∗ The lowest levels of the and channels assigned to Ωc + Ds, Ωc + Ds, and Ωc + Ds states. are located at 5046 MeV and 5050 MeV, respectively. For P After removing the scattering states, compact resonances J = 3∕2+ P − , the lowest eigenvalue appears at 4929 MeV. remain at 5201 MeV and 5320 MeV for J = 1∕2 , at Compared to the negative parity states, the lowest energy

Meng Qi et al.: Preprint submitted to Elsevier Page 4 of 10 Short Title of the Article E[MeV] JP − JP − JP − 5800 = 1/2 = 3/2 = 5/2 E[MeV] 5762 5737 5700 5683 5660 5635 5617 5600 E[MeV] 5566 5542

5500 5495

Ω*c + Ds*(2S)(5467) 5454 5444 5400 E[MeV] 5375

Ω*c ′′ + Ds*(1S)(5335) 5320 5318 5300 Ω + ψ (2S)(5320) Ω′′ + J/ψ (1S)(5293) 5262 5256

5220 5201 5200 5199 Ω*c ′′ + Ds(1S)(5182) 5193 Ω′′ + ηc(1S)(5174) 5140 Ωc′′ + Ds(1S)(5140) 5121 5100 5108 5094 5088

5044

5000 4998

4900 4866 4873 Ω*c + Ds*(1S)(4867) 4855

4800 Ωc + Ds*(1S)(4792)

Ω + J/ψ (1S)(4777) 4753

4700 Ω*c + Ds(1S)(4715)

Ω + ηc(1S)(4657) Ωc + Ds(1S)(4640) 4600 Figure 2: The calculated energy spectra for quantum numbers J P = 1∕2−, 3∕2−, 5∕2−, including only the connected configurations C = 3−5 are shown in units of MeV. The dashed lines are thresholds, drawn according to the theoretical numbers given in Tables.2 and3.

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E[MeV] 5800 P P P J = 1/2+ J = 3/2+ J = 5/2+

5700

E[MeV] 5600 5582 5570 5550

5500 E[MeV] E[MeV] Ω*c + Ds*(2S)(5467) 5422 5400 5408 5404 5373 5358 D S Ωc + Ds(2S)(5330) 5351 5346 Ω*c ′′+ s*(1 )(5335) 5320 5323 5323 5300 Ω + ψ (2S)(5320) 5309 5315 Ω + ψ (2S)(5320) Ω′′ + J/ψ (1S)(5293) 5294 Ω + ηc(2S)(5280) Ω*c + Ds*(1P)(5266) 5259 5245 D P 5226 5220 Ω*c + s(1 )(5238) 5200 5206 Ω*c ′ + Ds*(1S)(5185) 5184 5197

Ω + χc(1P)(5165) Ωc′′ + Ds(1S)(5140) Ω + hc(1P)(5141) 5116 5100

Ω′ + J/ψ (1S)(5074) 5046 5048 5050 Ωc′ + Ds(1S)(5033) 5000

Ω′ + ηc(1S)(4955) 4929 4900

Ω*c + Ds*(1S)(4867)

4800 Ωc + Ds*(1S)(4792)

Ω + J/ψ (1S)(4777)

Ω*c + Ds(1S)(4715) 4700

Ω + ηc(1S)(4657)

Ωc + Ds(1S)(4640) 4600

Figure 3: Same as in Fig.2, but for J P = 1∕2+, 3∕2+, 5∕2+.

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Table 4 Table 5 Dominant Baryon-Meson components for the various compact- Dominant Baryon-Meson components for the various compact- ified states in Fig.2 for the J P = 1∕2−, 3∕2−, 5∕2− channels. ified states in Fig.3 for the J P = 1∕2+, 3∕2+, 5∕2+ channels.

J P = 1∕2− energy configuration J P = 1∕2+ energy configuration (MeV) (MeV)

4855 Ω + J∕ (1S), Ωc + Ds(1S) 5046 Ω + J∕ (1S), Ωc + Ds(1S) ∗ ¨ ¨ 5044 Ωc + Ds(1S), Ωc + Ds (1S) 5116 Ω + c (1S), Ωc + Ds(1S) ∗ ∗ ¨¨ 5094 Ω + J∕ (1S), Ωc + Ds (1S) 5184 Ωc + Ds(1S) 5140 Ω + J∕ (1S) 5226 Ω¨ + J∕ (1S) ¨¨ ∗ ∗ ∗ ∗ 5193 Ωc + Ds(1S), Ωc + Ds (1S) 5294 Ωc + Ds (1S), Ωc + Ds (1S) ¨¨ ∗ 5201 - 5309 Ω + J∕ (1S), Ωc + Ds (1S) ¨¨ ∗ 5320 - 5320 Ω + J∕ (1S), Ωc + Ds (1S) 5351 Ω + (2S), Ω∗¨ + D∗(1S) J P = 3∕2− energy configuration c s 5358 Ω¨ +  (1S), Ω + D (2S) (MeV) c c s 5373 Ωc + Ds(2S) ∗ 4753 Ω + c (1S), Ωc + Ds(1S) 5408 - ∗ 4866 Ω + J∕ (1S), Ωc + Ds(1S) P + ∗ J = 3∕2 energy configuration 4998 Ω + c (1S), Ωc + Ds(1S) ∗ ∗ (MeV) 5088 Ωc + Ds (1S) ∗ 5108 Ω + J∕ (1S), Ωc + Ds (1S) 4929 Ω + c (1S) ¨¨ ∗¨¨ 5199 Ω + c (1S), Ωc + Ds(1S), 5048 Ω + J∕ (1S) ∗ ∗ Ωc + Ds (1S) 5197 Ω + ℎc (1P ) ∗¨ ∗ 5220 Ω + J∕ (1S) 5206 Ω + ℎc (1P ), Ωc + Ds (1S) ¨¨ ¨ ∗ 5262 Ω + c (1S) 5220 Ω + J∕ (1S), Ωc + Ds (1S), ∗ 5318 - Ωc + Ds(1S) 5245 Ω∗ + D (1P ), Ω∗¨ + D∗(1S) J P = 5∕2− energy configuration c s c s 5315 Ω∗ + D∗(1S) (MeV) c s 5323 Ω¨¨ + J∕ (1S)

4873 Ω + J∕ (1S) 5346 Ω + c (2S) ¨¨ 5121 Ω + J∕ (1S) 5404 Ω + c (2S), Ω + J∕ (1S) 5256 Ω∗ + D∗(1S) c s J P = 5∕2+ energy configuration 5375 Ω¨¨ + J∕ (1S), Ω∗¨¨ + D∗(1S) c s (MeV) 5444 Ω¨¨ + J∕ (1S) ¨¨ ∗¨¨ ∗ 5454 Ω + J∕ (1S), Ωc + Ds (1S) 5050 Ω + J∕ (1S) ∗ ∗ ∗ ∗ 5495 Ω + (2S), Ωc + Ds (2S) 5259 Ω + c (1P ), Ωc + Ds (1S) ∗ ∗ ¨¨ 5542 Ωc + Ds (2S) 5323 Ω + J∕ (1S) ∗ ∗ ∗ ∗ 5566 Ω + (2S), Ωc + Ds (2S) 5422 Ω + (2S), Ωc + Ds (1P ) ∗¨¨ ∗ 5617 Ω + (2S) 5550 Ωc + Ds (1S) ∗ ∗ ∗ ∗ 5635 Ω + (2S), Ωc + Ds (2S) 5570 Ω + (2S), Ωc + Ds (2S) 5660 - 5582 - 5683 Ω + (2S) ∗ ∗ 5737 Ω + (2S), Ωc + Ds (2S) 5762 - compactified state at 5201 MeV can be considered as a seed. With such stabilization plots, we can estimate the width of resonance states [30]. The width of the one around 5180 J P = 3∕2− 5318 MeV for , at 5660 MeV and 5762 MeV MeV is estimated to be 20 MeV. J P = 5∕2− J P = 1∕2+ for , at 5408 MeV for , and at 5582 We note that there are two more possible configurations, J P = 5∕2+ J P = 3∕2+ MeV for . For , all the compacti- namely [[(12)3̄ 5]3(34)3̄ ]1 and [(12)3̄ [(35)14]3]1, for the con- fied states in the low energy region have dominant scattering figuration sets of C = 4 and C = 5, respectively. Our numer- configurations. ical tests indicate that these additional configurations indeed Let us investigate one of the compact resonances in more belong to configuration sets C = 4 and C = 5, and their detail. In Fig.4, we show the stabilization plots using the real respective energies lay higher than that of C = 4 and C = 5 J P = 1∕2− scaling method for the channel. Fig.4(a) shows given in Eq.6. Therefore, these additional configurations are C = 1 the results when only the scattering configurations unlikely to be helpful in generating the lowest energy level 2 and are included. Fig.4(b) shows the results when all con- of this system. In addition, the resonance energies from a C = 1−5 figurations are incorporated in the calculation. As solution incorporating all the possible configurations typi- one can see, around 5201 MeV, there is a clear difference be- cally deviate by at most a few 10 MeV from the energies tween only scattering configurations and full configurations. obtained from a solution with the channels employing only C = 3 − 5 By including the connected configurations , a res- C = 3, 4, 5. onance structure appears at around 5180 MeV, for which the

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5250

5200 ] MeV [ E 5150

Ω′c′ + Ds

5100 1.0 1.2 1.4 1.6 1.8 2.0 2.2 α (a) C = 1 and 2

5250

5200 ] 5180MeV MeV [ E 5150

Ω′c′ + Ds

5100 1.0 1.2 1.4 1.6 1.8 2.0 2.2 α (b) C = 1 − 5

Figure 4: The stabilization plots of the eigenenergies E for J P = 1∕2− with the respect to the scaling factor in two cases: (a) including only scattering configurations C = 1 and 2; (b) including full configurations C = 1 − 5. The Gaussian ranges RN for the coordinates R1 and R2 of C = 1 and 2 configurations are scaled as RN → RN with = 1.0 − 2.2. The red line is threshold. The blue line is the location of 5180 MeV.

Table 6 To obtain more information about the spatial structures P − Resonance structures, their "seed" compactified states and the of the lowest J = 1∕2 compact resonance, we calculated estimated decay widths. the two-body correlation functions of ss and c ̄c for this state. J P energy width (MeV) "seed"(MeV) The correlation functions are defined as (MeV) 2 ss(r1) = ðΨJM ð ds1dR1d1dr̂1 1∕2− 5180 20 5201 Ê 5290 >100 5320 2 − c ̄c(s1) = ðΨJM ð dr1dR1d1dŝ1 (12) 3∕2 5300 >100 5318 Ê 5∕2− 5645 30 5660 5670 50 5762 where r1 and s1 are the relative distances between ss and c ̄c. 1∕2+ 5360 80 5408 dr̂1 and dŝ1 denote the integral of angular parts of r1 and s1, 5∕2+ 5570 >100 5582 respectively. The integral is performed at E = 5180MeV and = 1.28. Fig.5 shows the density distributions of 2 2 r1 ss(r1) and s1 c ̄c(r1) as functions of the distance r = r1 = s1. The peak position of c ̄c is found at about 0.25 fm, which is more compact than charmonia J∕ or c. The With the same method, we studied all other compact res- corresponding ss peak lies at about 0.85fm which is more onance structures with different spin-parity quantum num- extended than the Ω baryon. bers. The results are summarized in Table6.

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2.5 (4) There, however, is a caveat in the current quark model. As neither q ̄q creations nor explicit mesons are introduced, 2.0 cc the model Hamiltonian cannot describe meson exchange in- P ] ss teractions between hadrons. It was pointed out that the c - 1 1.5 (uudc ̄c) pentaquarks observed by LHCb can be realized as ̄ ̄ ∗ molecular-type Σc + D and Σc + D resonances due to the

ρ [ fm 1.0 2 attractive pion (meson) exchange potential. Such resonance r states may not appear in the present calculation because once 0.5 the two color-singlet hadrons are separated in C = 1, or 2 in Fig.1, there is no interaction between them. 0.0 Within these conditions, we predict four sharp resonances: 0 1 2 3 4 P − − J = 1∕2 (E = 5180 MeV, Γ = 20 MeV), 5∕2 (5645 − + r[fm] MeV, 30 MeV), 5∕2 (5670 MeV, 50 MeV), and 1∕2 (5360 Figure 5: Density distributions r 2 (r ) and s 2 (s ) as func- MeV, 80 MeV) for the AP1 potential. They reside rather high 1 ss 1 1 c ̄c 1 up, excited by more than 500 MeV, from the lowest thresh- tions of the distance r = r1 = s1 olds, Ωc +Ds or Ω+J∕ . Nevertheless, they all happen to be compact five-quark states, as is shown in the calculated den- sity distribution of Fig.5, whose coupling to baryon-meson scattering channels are weak. 5. Summary Thus we conclude that the potential quark model predicts P In this paper, we have studied sssc ̄c pentaquarks of J = compact pentaquark resonances through the full five-body − − − + + + 1∕2 , 3∕2 , 5∕2 , 1∕2 , 3∕2 , and 5∕2 . The potential calculation. It would be interesting to observe such sharp quark model is used to analyze the spectrum and resonance and compact pentaquarks in future experiments. Simulta- energies are obtained from the most precise five-body calcu- neous production of two charm and three strange quarks is lation available to date. generally very unlikely, but one may utilize bottom quark de- Ξ0 → (sssc ̄c) + K+ The key findings of our calculation can be summarized cay processes such as b followed by the sssc ̄c J as follows. ( ) → Ω+ ∕ decay. Alternatively, high-energy heavy (1) Our Hamiltonian is taken from Semay and Silvestre- ion collisions are known to produce many strange and charm Brac (SSB) [21, 22]. As is shown in Tables2 and3, the quarks, which can lead to the formation of pentaquark states. J SSB model reproduces the hadron masses within 15 MeV, Resonance states may be observed in the Ω − ∕ (or some which are relevant for the open channel thresholds of the cur- other) correlations in the final states. If these states are ob- rent pentaquark systems. This is very important to guaran- served, it would be a strong indication of the quark dynamics tee the correctness of the quark dynamics in the strange and described by the quark model Hamiltonian, such as the quark charm sectors, and also to compare our results with the real confinement mechanism and spin dependent structures. (observed) spectrum that is influenced strongly by the open Another way of confirming our prediction is to investi- channel thresholds. In this sense this is the best available gate the resonance spectrum from lattice QCD calculations. potential for the present calculation. As these pentaquarks contain only charm and strange quarks, (2) In order to estimate the systematic uncertainties of we expect the reliability of the lattice simulation to be better u d the model, we have compared two sets of parameter choices than for systems with light ( and ) quarks. Such a calcu- of the SSB potential, AP1 and AL1. The main results shown lation is now in progress in our group. above are for the AP1 potential, which fits the observed data better than AL1. However we have found that the pentaquark Acknowledgments resonances for AL1 come out at similar energies as AP1. For − instance, we find a sharp 1∕2 resonance at 5220 MeV with Q.M. is supported in part by the National Natural Sci- a width of 25 MeV (AL1), compared to 5180 MeV with a ence Foundation of China (under Grants No.11535005, No. width of 20 MeV (AP1). 11775118 and No.11690030) and the International Science & (3) For the given Hamiltonian, we have solved the sssc ̄c Technology Cooperation Program of China (under Grant states as precisely as possible. The Gaussian expansion method No.2016YFE0129300). P.G. is supported by Grant-in-Aid (based on the variational principle) is employed for the full for Early-Carrier Scientists No. JP18K13542 and the Lead- five-body system. In our five-body calculation, we include ing Initiative for Excellent Young Researchers (LEADER) all relevant meson-baryon scattering channels (Cf. C = 1, 2 of the Japan Society for the Promotion of Science (JSPS). ∗ ∗ of Fig.1) such as Ω + c, Ω + J∕ , Ωc + Ds , and so on. K.U.C is supported by the Special Postdoctoral Researcher As a result, we need more than 40,000 basis functions for (SPDR) program of RIKEN. This work is supported in part this system. To distinguish two-body scattering states and by Grants-in-Aid for Scientific Research on Innovative Ar- compact resonant states, we have employed the real scaling eas (No. JP18H05407, No. JP18H05236 (E.H and A.H.), method (stabilization method) which has been successfully No. JP19H05159 (M.O.)), and for Scientific Research No. applied in previous studies [18, 19]. JP17K05441(C) (A.H.).

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