PHYSICAL REVIEW D 98, 014021 (2018)
Heavy quark spin multiplet structure of ¯ ð ÞΣð Þ molecular states P Q † ‡ Yuki Shimizu,1,* Yasuhiro Yamaguchi,2, and Masayasu Harada1, 1Department of Physics, Nagoya University, Nagoya 464-8602, Japan 2Theoretical Research Division, Nishina Center, RIKEN, Hirosawa, Wako, Saitama 351-0198, Japan
(Received 24 May 2018; published 16 July 2018)
We study the structure of heavy quark spin (HQS) multiplets for heavy meson-baryon molecular ¯ ð ÞΣð Þ states in a coupled system of P Q , constructing the one-pion exchange potential with S-wave orbital angular momentum. Using the light cloud spin basis, we find that there are four types of HQS multiplets classified by the structure of heavy quark spin and light cloud spin. The multiplets which have attractive potential are determined by the sign of the coupling constant for the heavy meson-pion interactions. Furthermore, the difference in the structure of the light cloud spin gives the restrictions of the decay channel, which implies that the partial decay width has the information for the structure of HQS multiplets. This behavior is more likely to appear in the hidden-bottom sector than in the hidden- charm sector.
DOI: 10.1103/PhysRevD.98.014021
I. INTRODUCTION þ There are many theoretical descriptions for Pc penta- The exotic hadrons are the very interesting research quarks. Among those pictures, the hadronic molecular one subjects in hadron and nuclear physics. In 2015, the Large has been used for several other exotic hadrons, especially Hadron Collider beauty experiment (LHCb) Collaboration near the thresholds. For example, since the mass of ð3872Þ ¯ ð3872Þ announced the observation of two hidden-charm penta- X is close to the DD threshold, X includes þ þ 0 ¯ þ ð4380Þ ð4450Þ Λ → the DD molecule structure [44]. The masses of Pc ð4380Þ quarks, Pc and Pc , in the decay of b − þð4450Þ ¯ Σ J=ψK p [1–3]. Their masses are M4380 ¼ 4380 8 and Pc are slightly below the thresholds of D c and ¯ Σ 28 MeV and M4450 ¼ 4449.8 1.7 2.5 MeV, and D c, respectively. They can be considered as the loosely Γ4380 ¼ 205 18 86 bound state of the heavy meson and heavy baryon. decay widths are MeV and þ P Γ4450 ¼ 39 5 19 MeV. The spin and parity J of Charm quarks are included in Pc pentaquarks. The them are not well determined. The one state is J ¼ 3=2 masses of the heavy quarks, charm and bottom, are much and the other state is J ¼ 5=2, and they have opposite larger than the typical scale of low-energy QCD, Λ ∼ 200 parity. QCD MeV. For the heavy quark region, there is a Before the LHCb observation, some theoretical studies characteristic property in the quark interaction. The spin- of hidden-charm pentaquarks were done [4–7]. After the dependent interaction of the heavy quark is suppressed by 1 LHCb announcement, there were many analyses based on the inverse of the heavy quark mass, =mQ. By this the hadronic molecular state [8–24], compact pentaquark suppression, the heavy quark spin symmetry (HQS) state [25–33], quark-cluster model [34], baryocharmonium appeared in the heavy quark limit [45–49]. As a result, model [35], hadroquarkonia model [36], topologial soliton we can decompose the total spin J⃗to heavy quark spin S⃗ model [37], and meson-baryon molecules coupled with and the other spin j⃗: five-quark states [38]. The kinematical rescattering effects – are also discussed in Refs. [39 43]. J⃗¼ S⃗þ j:⃗ ð1Þ
The total spin is conserved and the heavy quark spin is also conserved in the heavy quark limit because of the sup- *[email protected] † pression of the spin-dependent force. Thus, the other spin [email protected] ‡ [email protected] part is also conserved. This conservation leads to the mass degeneracy of heavy hadrons. Let us consider the heavy Published by the American Physical Society under the terms of meson qQ¯ with a light quark q and a heavy quark Q.For the Creative Commons Attribution 4.0 International license. j ≥ 1=2, there are two degenerate states with total spin: Further distribution of this work must maintain attribution to ’ the author(s) and the published article s title, journal citation, ¼ 1 2 ð Þ and DOI. Funded by SCOAP3. J j = : 2
2470-0010=2018=98(1)=014021(10) 014021-1 Published by the American Physical Society SHIMIZU, YAMAGUCHI, and HARADA PHYS. REV. D 98, 014021 (2018)
These two states are called the HQS doublet. There is only the structure of HQS multiplet. It is convenient to deal the J ¼ 1=2 state for j ¼ 0; hence, it is called the HQS with the corresponding spin structure with appropriate singlet. basis. Thus, we define the light cloud spin (LCS) basis as Such an HQS multiplet structure is seen in the charm and a suitable basis to study the HQS multiplet structure and bottom hadron mass spectrum. For example, the small mass discuss that what types of HQS multiplets can exist under difference is obtained between the heavy-light pseudoscalar the OPEP. (J ¼ 0) and vector (J ¼ 1) mesons, 140 MeV between D This paper is organized as follows. In Sec. II, and D , and 45 MeV between B and B . These mass we construct the one-pion exchange potential in the splittings are much smaller than those in the light quark hadronic molecular (HM) basis. The basis transformation sectors, 600 MeV between π and ρ, and 400 MeV between from the HM basis to the LCS basis is discussed in Sec. III. K and K . This observation indicates that the approximate We show the numerical result in Sec. IV. Finally, we heavy quark spin symmetry is realized in the charm and summarize the work in this paper and discuss the results ¼ 0 bottom quark sectors, and these two mesons with J ,1 in Sec. V. belong to the HQS doublet having the heavy spin S ¼ 1=2 and the other spin j ¼ 1=2. The approximate mass degeneracy is also observed in the II. POTENTIAL Σ ð ¼ ð Þ heavy-light baryons. The mass splitting between c J In this section, we construct the OPEP for P¯ ð ÞΣ 1 2Þ Σ ð ¼ 3 2Þ Σ Σ Q = and c J = ( b and b) is about 65 MeV molecular states based on the heavy quark symmetry and (20 MeV). They are the HQS doublet state with the heavy ¯ ð Þ ¼ 1 2 ¼ 1 the chiral symmetry. The P mesons and pion interaction spin S = and the other spin j . On the other hand, Lagrangian is given by [53–57] the heavy-light baryons Λc and Λb with the light diquark spin 0 are a HQS singlet state. ¯ μ In this paper, we study the structure of HQS multiplets of LHHπ ¼ gTr½HHγμγ5A : ð3Þ QQqqq¯ -type pentaquarks regarding them as molecular ¯ ð ÞΣð Þ ¯ ð Þ states of P Q . Here, P means a HQS doublet meson The heavy meson doublet field H is – ¯ ð Þ ¯ ð Þ Σð Þ with an anti-heavy-quark like D B and D B and Q – stands for a HQS doublet baryon with a heavy-quark like 1 þ =v μ Σ ðΣ Þ Σ ðΣ Þ H ¼ ½Pμγ þ iPγ5 : ð4Þ c b and c b . 2 ¯ ð Þ Σð Þ The HQS doublet structures of the P meson and Q baryon which have one heavy quark are well known. The P and P are pseudoscalar meson and vector meson fields ¯ ð Þ HQS multiplet structure of the P N molecular state with in the HQS doublet. The axial vector current for the pion is a single heavy quark is discussed in Refs. [50–52]. They given by showed that the degeneracy of j 1=2 states can be expanded to the multihadron system. In this paper, we i study the HQS multiplet structure of P -like pentaquarks ¼ ðξ†∂ ξ − ξ∂ ξ†Þ ð Þ c Aμ 2 μ μ ; 5 as a doubly heavy quarks system. The appearance of the ¯ ð ÞΣð Þ pffiffiffi HQS multiplet for P Q molecules is demonstrated by where ξ ¼ expðiπˆ= 2fπÞ. The pion decay constant is fπ ¼ introducing the one-pion exchange potential (OPEP) 92.4 MeV and the pion field πˆ is defined by which is derived from the heavy hadron effective theory respecting the heavy quark symmetry. We focus on the pffiffiffi 0 þ ¯ ð ÞΣð Þ π = 2 π P Q molecules with S-wave orbital angular momen- πˆ ¼ pffiffiffi : ð6Þ tum for simplicity. The effect of the tensor force by the π− −π0= 2 S-D mixing is important for OPEP. However, we do not include the D-wave states complicating the system The coupling constant g is determined as jgj¼0.59 from because the S-wave channel is enough to see the spin the decay of D → Dπ [58]. decomposition to the heavy quark spin and the other spin Σð Þ The Q baryons and pion interaction Lagrangian is ¯ ð ÞΣð Þ of the P Q molecules. Our purpose in this paper is to given by [56,59] ð Þ demonstrate the HQS multiplet of P¯ ð ÞΣ . Thus, we Q 3 study the simple S-wave case in the present study. Since μνρσ ¯ LBBπ ¼ g1ivσϵ Tr½SμAνSρ : ð7Þ the heavy quark spin and the other spin are separately 2 conserved by the heavy quark spin symmetry, the heavy Σ Σ meson-baryon molecular basis is not suitable to discuss The superfield Sμ for Q and Q is represented as
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rffiffiffi 1 The wave functions and OPEPs for each spin state are ¼ Σˆ − ðγ þ Þγ Σˆ ð Þ Sμ Qμ 3 μ vμ 5 Q: 8 0 j ¯ Σ i 1 P Q 1=2− ð Þ ˆ HM B ¯ C Σ ψ − ¼ @ jP Σ i − A ð Þ The heavy baryon fields QðμÞ are defined by 1=2 Q 1=2 ; 15 j ¯ Σ i 0 1 P Q 1=2− ð Þþþ ð Þþ Σ p1ffiffi Σ B QðμÞ 2 QðμÞ C Σˆ ð Þ ¼ @ A ð Þ QðμÞ ð Þþ ð Þ0 : 9 0 1 p1ffiffi Σ Σ 0 − p1ffiffi p1ffiffi 2 QðμÞ QðμÞ 3 6 gg1 B C HM ð Þ¼ B − p1ffiffi 2 p1 ffiffi C ð Þ ð Þ Vπ;1=2− r 2 @ 3 3 3 2 ACπ r ; 16 Σ Σ 1 2 3 2 fπ Q and Qμ are spin = and = baryon fields in the HQS p1ffiffi p1 ffiffi 5 6 3 2 6 doublet.pffiffiffi For the coupling constant g1, we use g1 ¼ ð 8=3Þg4 and g4 ¼ 0.999 estimated in Ref. [59]. The 0 1 coupling g4 is determined by the decay of Σ → Λ π j ¯ Σ i c c P Q 3 2− and its sign follows the quark model estimation. B = C ¯ ψ HM ¼ B j Σ i − C ð Þ We construct the one pion exchange potential using the 3=2− @ P Q 3=2 A; 17 above Lagrangians. At each vertex, we introduce a cutoff ¯ jP Σ i − parameter Λ via the monopole type form factor, Q 3=2
2 2 0 1 5 1 Λ − mπ 0 − pffiffi − pffiffiffiffi ð Þ¼ ð Þ 2 3 2 15 F q 2 2 ; 10 B C Λ þjq⃗j gg1 HM ð Þ¼ B − p1 ffiffi − 1 p5 ffiffi C ð Þ ð Þ Vπ;3=2− r 2 @ 2 3 3 6 5 ACπ r ; 18 fπ − p5ffiffiffiffi p5 ffiffi 1 where mπ is a mass of the exchanging pion, and q⃗is its 2 15 6 5 3 momentum. We use the same cutoff for the P¯ ð ÞP¯ ð Þπ and Σð ÞΣð Þπ Q Q vertices for simplicity, and fix the value of the HM ¯ cutoff at 1000 and 1500 MeV. ψ − ¼ jP Σ i − ð Þ 5=2 Q 5=2 ; 19 In the present analysis, we concentrate on the S-wave ¯ ð ÞΣð Þ P Q molecular states to clarify their HQS multiplet structures. In the hadronic molecule (HM) basis, the spin gg1 HM ð Þ¼− ð Þ ð Þ Vπ;5=2− r 2 Cπ r : 20 structures of molecular states are described by the product 2fπ of meson-baryon spins. Then, the possible spins of the ð Þ P¯ ð ÞΣ states are Q The function CπðrÞ is defined as
1 ¯ ¯ 2 −mπ r − −Λr Λ2 − 2 PΣQ ¼½Qq 0 ⊗ ½Q½d 1 1 2 ¼ ; ð11Þ mπ e e mπ −Λr = 2 CπðrÞ¼ − e : ð21Þ 4π r 2Λ
3 ¯ Σ ¼½¯ ⊗ ½ ½ ¼ ð Þ It should be noted that we subtract the contact terms from P Q Qq 0 Q d 1 3=2 ; 12 2 the potential.
¯ ð Þ ð Þ 1 3 III. HQS MULTIPLET STRUCTURE OF P Σ ¯ ¯ Q P Σ ¼½Qq 1 ⊗ ½Q½d 1 1 2 ¼ ⊕ ; ð13Þ Q = 2 2 We construct the OPEP for the HM base in Sec. II. However, it is inconvenient to see the structure of the HQS 1 3 5 multiplet. In this section, we introduce the light cloud spin ¯ Σ ¼½¯ ⊗ ½ ½ ¼ ⊕ ⊕ ð Þ ¯ P Q Qq 1 Q d 1 3=2 ; 14 (LCS) basis, where the spin structure of the QQqd states is 2 2 2 ½ ¯ divided into the heavy quark spin QQ S and light cloud ½ ½ ¯ spin q d 1 j. It is a natural spin description in the heavy where Q, Q, q, and d stand for a heavy quark, heavy hadron systems, because the heavy quark spin and light Σð Þ antiquark, light quark, and diquark in the Q baryon, cloud spin are separately conserved in the heavy quark ½α α respectively, and the index j of j means the spin of . effective theory.
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Here, we treat the HQS structure of the doubly heavy For the spin 5=2, system in the following manner: The pentaquark as a bound ¯ ð Þ Σð Þ ψ LCS ¼ −1 ψ HM state of P and Q is labeled by the velocity v of the 5=2− U5=2− 5=2− ¯ ð Þ Σð Þ pentaquark. It is natural to assume that both P and Q ¼ðj½¯ ⊗ ½ ½ itriplet Þ ð Þ QQ 1 q d 1 3=2 5 2− ; 30 have the same velocity v. = ¯ ð Þ ð Þ The spin structures of the P Σ molecular states in the LCS −1 HM Q V − ðrÞ¼U − V − U5 2− LCS basis are given by π;5=2 5=2 5=2 = ¼ − gg1 ð Þ ð Þ 2 Cπ r : 31 1 2fπ ½ ¯ ⊗ ½ ½ ¼ ð Þ ð Þ QQ 0 q d 1 1=2 2 singlet ; 22 Here, we call the components labeled by (singlet, doublet, or 3 triplet) of ψ LCS the spin J (singlet, doublet, or triplet) state. ½QQ¯ ⊗ ½q½d ¼ ðsingletÞ; ð23Þ JP 0 1 3=2 2 ψ LCS For instance, the first component of 1=2− in Eq. (26), i.e., j½ ¯ ⊗ ½ ½ isinglet 1 2 1 3 QQ 0 q d 1 1=2 1 2− , is called the spin = singlet state. ½QQ¯ ⊗ ½q½d ¼ ⊕ ðdoubletÞ; ð24Þ = 1 1 1=2 2 2 The transformation matrices determined by the Clebsch- Gordan coefficient of spin reconstruction are given by 1 3 5 ½ ¯ ⊗ ½ ½ ¼ ⊕ ⊕ ð Þ ð Þ 0 1 1 2 1 QQ 1 q d 1 3=2 triplet : 25 − pffiffi pffiffi 2 2 2 2 2 3 6 B C 1 5 2 − ¼ B − pffiffi pffiffi C ð Þ There are four types of HQS multiplets—the spin 1=2 U1=2 @ 2 3 6 3 2 A; 32 singlet, spin 3=2 singlet, spin (1=2, 3=2) doublet, and spin p2ffiffi p2 ffiffi − 1 1 2 3 2 5 2 6 3 2 3 ( = , = , = ) triplet which are classified by the heavy 0 pffiffiffiffi 1 ¼ 0 ¼ 1 2 3 2 1 − p1ffiffi 15 quark spin S , 1 and the light cloud spin j = , = . 2 3 6 B pffiffi C Using unitary transformation matrices, we translate the B 1 1 5 C U3 2− ¼ − pffiffi ; ð Þ basis from the HM basis to the LCS basis. For the spin 1=2, = @ 3 3 3 A 33 pffiffiffiffi pffiffi 15 5 1 ψ LCS ¼ −1 ψ HM 6 3 6 1=2− U1=2− 1=2− 0 1 U5 2− ¼ 1: ð34Þ j½ ¯ ⊗ ½ ½ isinglet = QQ 0 q d 1 1=2 1 2− B = C B ¯ doublet C The potential matrices in the LCS basis are diagonalized ¼ j½QQ ⊗ ½q½d i − ð Þ @ 1 1 1=2 1=2 A; 26 corresponding to the HQS multiplet components. We find ¯ triplet the particular values of the matrix elements of the OPEP; j½QQ ⊗ ½q½d i − 1 1 3=2 1=2 þ1 ½ ¯ ⊗ ½ ½ ½ ¯ ⊗ ½ ½ −2 for QQ 0 q d 1 1=2 and QQ 1 q d 1 1=2, and −1 ¯ ¯ LCS ð Þ¼ HM − for ½QQ 0 ⊗ ½q½d 1 3 2 and ½QQ 1 ⊗ ½q½d 1 3 2. Hence, Vπ;1=2− r U1=2− V1=2− U1=2 = = 0 1 these components play a different role, either an attraction 10 0 B C or a repulsion, depending on the whole sign of the potential. ¼ gg1 B 01 0C ð Þ ð Þ 2 @ ACπ r : 27 fπ 1 IV. NUMERICAL RESULTS 00− 2 Before solving the coupled channel Schrödinger equa- For the spin 3=2, tions under the LCS basis potential, let us discuss the sign assignment of a coupling constant of the heavy meson-pion ψ LCS ¼ −1 ψ HM interaction, jgj¼0.59. In the usual case, its sign is taken as 3=2− U3=2− 3=2− 0 1 a plus following the quark models. However, only the j½ ¯ ⊗ ½ ½ isinglet QQ 0 q d 1 3=2 3=2− absolute value is determined by the decay of D → Dπ B C 1 B ¯ doublet C [58], and the sign of g is not determined. Since the sign ¼ j½QQ 1 ⊗ ½q½d 1 1 2i − ; ð28Þ @ = 3=2 A ambiguity of the D-D -π coupling is absorbed into a j½ ¯ ⊗ ½ ½ itriplet redefinition of the relative phase of the D and D meson QQ 1 q d 1 3=2 3=2− fields, the sign of the off-diagonal component of the −1 LCS ð Þ¼ HM − potential matrix is absorbed by the redefinition. This is Vπ;3=2− r U3=2− V3=2− U3=2 0 1 consistent with the fact that the signs of off-diagonal − 1 00 B 2 C components are irrelevant for the eigenvalues. On the other ¼ gg1 B 010C ð Þ ð Þ 2 @ ACπ r : 29 fπ 1 00− 1 It could be argued that the relative sign of g and g1 is not 2 determined.
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gg1 FIG. 1. Attractive potential VðrÞ¼− 2 Cπ for Λ ¼ 1000 2fπ FIG. 2. Obtained binding energies for Λ ¼ 1000 (purple solid (purple solid curve) and 1500 MeV (green dotted curve), where curve) and 1500 MeV (green dotted curve) with g ¼þ0.59. The g ¼ 0.59, g1 ¼ 0.942, and fπ ¼ 92.4 MeV. ¯ Σ energy is measured from the threshold of Pave Qave. The mass parameter μave is changed from 1 GeV to 100 GeV. hand, signs of diagonal components, which originate from π the sign of the D -D - coupling, affect the eigenvalues. symmetry. We define the spin-averaged mass for the P¯ ð Þ The signs of the diagonal components depend on the sign Σð Þ of the coupling constant. Hence, we examined both signs to mesons and Q baryons as clarify the sign dependence. þ 3 This sign assignment is important in the present study. ¼ MP¯ MP¯ ð Þ MP¯ ave ; 36 For example, the coefficients of the HQS singlet and 4 doublet component are þ1 in the spin 1=2 potential of 2MΣ þ 4MΣ LCS basis in Eq. (27). Thus, these potentials play as a Q Q MΣ ¼ ; ð37Þ repulsive one when we assign g ¼þ0.59, but they are the Qave 6 attractive potentials when we choose g ¼ −0.59. On the other hand, the HQS triplet with the coefficient −1=2 has to deal with the degeneracy of the HQS doublet meson and the attractive potential for g ¼þ0.59 and repulsive poten- baryon, respectively. The masses of the relevant charmed tial for g ¼ −0.59. In other words, the sign of the coupling and bottomed hadrons are shown in Table I. The spin- constant (the interaction models in general) determines averaged reduced mass is defined as which multiplets have the attractive potential. We calculate M ¯ MΣ the cases with both signs of g to study the behavior of the μ ¼ Pave Qave ð Þ ave : 38 ¯ þ attractive multiplets in this section. MPave MΣQave
μ ¼ 1 102 A. Results in case of g =+0.59 When ave . , 1.474, 1.699, and 2.779 GeV, the spin- ¯ ð ÞΣð Þ ¯ ð ÞΣð Þ ð ÞΣð Þ ¼þ0 59 averaged masses of D c , D b , B c , and When we assign as g . , the HQS multiplets ð Þ P − ð ÞΣ which have attractive potential are J ¼ 3=2 singlet and B b are reproduced, respectively. We solve the JP ¼ð1=2−; 3=2−; 5=2−Þ triplet. The potential is written as Schrödinger equation keeping the heavy quark spin sym- μ metry by changing the mass parameter ave from 1 to gg1 VðrÞ¼− CπðrÞ; ð Þ 100 GeV. To obtain the bound state solutions, we use the 2 2 35 fπ Gaussian expansion method [60]. Λ ¼ 1000 and we show in Fig. 1. The results of and 1500 MeV are shown in 3 2 1 2 3 2 First, we show the results obtained by solving the Fig. 2. All four states, spin = singlet and spin ( = , = , 5 2 Schrödinger equation preserving the heavy quark spin = ) triplet, are degenerate because of the heavy quark spin symmetry and their bound state solutions are obtained for μ TABLE I. Masses of relevant charmed and bottomed all range of ave. hadrons [58]. Next, we show the results including the effect of the heavy quark spin symmetry breaking. The breaking is ¯ ¯ D D BBintroduced by the nonzero mass difference between the ¯ ¯ Σ Σ 2 Mass[MeV] 1867.21 2008.56 5279.48 5324.65 HQS multiplet, namely P and P , and Q and Q. To see
Σ Σ Σ Σ 2 c c b b The higher-order terms of the effective Lagrangians also Mass[MeV] 2453.54 2518.13 5813.4 5833.6 break the heavy quark symmetry. However, we employ the leading term of Lagrangians in this study.
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TABLE II. Values of parameters to include the effect of heavy quark spin symmetry breaking in Eqs. (39)–(42).
a½GeV2 b½GeV2 c½GeV2 d½GeV2 w½GeV3 x½GeV3 y½GeV3 z½GeV3 −2.0798 −1.8685 1.9889 2.0814 2.9468 3.1677 −3.1729 −3.0629 the mass dependence of a binding energy, the heavy hadron where μ is a parameter corresponding to the reduced mass masses are parametrized as follows: ¯ ð ÞΣð Þ of P Q state. The eight parameters of a, b, c, d, w, x, y a w and z are fixed to reproduce the eight hadron masses in ¯ ¼ 2μ þ þ ð Þ μ ¼ 1 102 ð2 779Þ MP 2μ ð2μÞ2 ; 39 Table I by taking . . GeV for charm (bottom) sector. The values of these parameters are shown in Table II. Note that the charm (bottom) hadron masses b x M ¯ ¼ 2μ þ þ ; ð Þ are reproduced when we take μ ¼ 1.102 ð2.779Þ GeV, and P 2μ ð2μÞ2 40 the heavy quark spin symmetry is restored as the mass parameter μ increases. The energies obtained by solving c y MΣ ¼ 2μ þ þ ; ð41Þ the Schrödinger equations with the effect of heavy quark Q 2μ ð2μÞ2 spin symmetry breaking are shown in Fig. 3. The labels in Fig. 3, e.g., spin 1=2 triplet, are named as being it at the d z ¼ 2μ þ þ ð Þ heavy quark limit. For instance, the solid line named as spin MΣ 2 ; 42 Q 2μ ð2μÞ 1=2 triplet displays the energy of the state which becomes the spin 1=2 triplet state at the heavy quark limit. We note that the components belonging to the same JP state can be mixed in the finite hadron mass region as shown later, while they are not mixed at the heavy quark limit. The ¯ ð ÞΣð Þ ¯ ð ÞΣð Þ ð ÞΣð Þ ð ÞΣð Þ energies of the D c , D b , B c and B b states are correspond to the values at μ ¼ 1.102, 1.474, 1.699, and 2.779 GeV, respectively. All four states are degenerate and the binding energy is −13.7 MeV for Λ ¼ 1000 MeV and −22.3 MeV for Λ ¼ 1500 MeV in the heavy quark limit. As μ becomes smaller, the degeneracy is solved. At μ ¼ 1.102 GeV, only two (three) states can be bound for Λ ¼ 1000 MeV (1500 MeV). For the spin 1=2 and 3=2 states, each component is completely separated in the heavy quark limit as shown in Eqs. (27) and (29). In the finite heavy hadron mass region, however, the kinetic term with the nonzero mass splitting of the HQS multiplets gives a mixing of the HQS singlet, doublet and triplet components. The percentage of (singlet, doublet, triplet) components in wave functions for Λ ¼ 1000 MeV is shown in Table III.Forμ ≥ 3 GeV, each component is perfectly separated. These ratios are hardly changed even in the case of Λ ¼ 1500 MeV. We can see
¯ ð ÞΣð Þ FIG. 3. Energies of the P Q states with heavy quark spin TABLE III. Percentage of (singlet, doublet, triplet) components Λ ¼ 1000 symmetry breaking effect, obtained for MeV (the in wave functions of the spin 1=2 and 3=2 states in the case of upper figure) and 1500 MeV (the lower figure) with ¼þ0 59 Λ¼1000 ¯ g . and MeV. g ¼þ0.59. These energies are measured from PΣQ threshold. The purple solid, green dotted and yellow dashed-dotted curves μ are the energies of spin (1=2, 3=2, 5=2) triplet states respec- ½GeV Spin 1=2 triplet Spin 3=2 triplet Spin 3=2 singlet tively and the light blue dashed curve is that of spin 3=2 singlet 1(0.8%, 0%, 99.2%)(1.6%, 0%, 98.4%) No bound state state. For the sake of reference, we show the result for the case 2(0%, 0%, 100%)(0.9%, 0%, 99.1%) No bound state of keeping the heavy quark spin symmetry by the red dashed- 3(0%, 0%, 100%)(0%, 0%, 100%)(100%, 0%, 0%) dotted-dotted curve (Common mass).
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gg1 FIG. 4. Attractive potential VðrÞ¼ 2 Cπ for Λ ¼ 1000 (purple fπ solid curve) and 1500 MeV (green dotted curve), where g ¼ −0.59, g1 ¼ 0.942 and fπ ¼ 92.4 MeV.
FIG. 6. Energies obtained with heavy quark spin symmetry breaking for Λ ¼ 1000 MeV (the upper figure) and 1500 MeV (the lower figure) with g ¼ −0.59. These energies are measured ¯ Σ FIG. 5. Obtained binding energies for Λ ¼ 1000 (purple solid from P Q threshold. The purple solid and light blue dashed curve) and 1500 MeV (green dotted curve) with g ¼ −0.59. The curves are the energies of spin (1=2, 3=2) doublet states and the ¯ Σ green dotted curve is that of spin 1=2 singlet state. For the sake of energy is measured from the threshold of Pave Qave. The mass parameter μ is changed from 1 GeV to 100 GeV. reference, we show the result for the case of keeping the heavy quark spin symmetry by the red dashed-dotted-dotted curve. that the effect of heavy quark spin symmetry breaking is small. As in the case of g ¼þ0.59, we show the result where the heavy quark spin symmetry is preserved in Fig. 5, and B. Result in case of g = − 0.59 the result where it is broken in Fig. 6, for g ¼ −0.59. 1 2 In the case of g ¼ −0.59, the attractive multiplets are Figure 6 shows that all three states of spin = singlet 1 2 3 2 JP ¼ 1=2− singlet and JP ¼ð1=2−; 3=2−Þ doublet. The and ( = , = ) doublet are degenerate in the heavy quark −29 5 Λ ¼ potential is written as limit and the binding energy is . MeV for 1000 MeV and −48.1 MeV for Λ ¼ 1500 MeV, which ð Þ¼gg1 ð Þ ð Þ agree with the binding energies in the heavy quark V r 2 Cπ r ; 43 fπ limit shown in Fig. 5. Unlike in the case of g ¼ þ0.59, all states are bound even at μ ¼ 1.102 GeV and it is shown in Fig. 4. This potential is twice deeper than ¯ ð ÞΣð Þ that of g ¼þ0.59, and therefore we expect that the binding corresponding to D c state and their binding energies energy is larger. are a few MeV.
TABLE IV. Percentage of (singlet, doublet, triplet) components in wave functions in the case of g ¼ −0.59 and Λ ¼ 1000 MeV.
μ½GeV Spin 1=2 singlet Spin 1=2 doublet Spin 3=2 doublet 1(3.9%, 96.1%, 0%)(96.5%, 3.4%, 0.1%)(0%, 99.9%, 0.1%) 2(0.2%, 99.8%, 0%)(99.9%, 0.1%, 0%)(0%, 100%, 0%) 3(0%, 100%, 0%)(100%, 0%, 0%)(0%, 100%, 0%)
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The mixing ratio of wave-function components for j ¼ 1=2 give a repulsion. In this study, we also obtain that Λ ¼ 1000 MeV is shown in Table IV. The mixing ratio the states with j ¼ 1=2 and j ¼ 3=2 have a different role as of the minor components induced by the heavy quark shown in Eqs. (27), (29) and (31), namely one is attractive symmetry breaking effect is slightly larger than the case of and the other one is repulsive. Thus, we find that a role of g ¼þ0.59, however it is still small. the interaction is characterized by the light cloud spin in both of the quark model and the hadronic molecular model. V. SUMMARY AND DISCUSSIONS It indicates that the discussion of the HQS multiplet structure can be applied not only to the molecules, but We showed that, in Sec. IV, the sign of the coupling also to the compact multiquark states. In Ref. [38], the constant of the heavy meson-pion interaction determines ccuud¯ potential for JP ¼ 3=2− with ðS; jÞ¼ð1; 3=2Þ is which multiplets have the attractive potential. In the case of stronger than that with ðS; jÞ¼ð0; 3=2Þ. This behavior also g ¼þ0.59, the spin 3=2 HQS singlet and spin (1=2, 3=2, agrees with our results. 5=2) HQS triplet have the attractive potential. On the other Focusing on the light cloud spin structure, there are hand, in the case of g ¼ −0.59, the spin 1=2 HQS singlet constraints of the S-wave decay channel of the spin 3=2 and spin (1=2, 3=2) HQS doublet have the attractive HQS singlet and spin (1=2, 3=2, 5=2) HQS triplet. Since ½ ½ potential. their light cloud spin is given by q d 1 3=2, they cannot ¯ ¯ ð Þ ¯ This classification is explained by the light cloud spin couple to the S-wave ½QQ N and P ΛQ states. Here ½QQ , structure in Eqs. (22)–(25). The light cloud spin of the spin N and ΛQ denote the heavy quarkonium, spin 1=2 nucleon 3=2 singlet and spin (1=2, 3=2, 5=2) triplet is ½q½d 1 3 2 = and HQS singlet heavy baryonlike Λc, respectively. and that of the spin 1=2 singlet and spin (1=2, 3=2) doublet Due to the heavy quark spin symmetry, heavy quark spin ½ ½ 3 is q d 1 1=2. Because the pion exchange interaction is and light cloud spin are independently conserved. ½ ½ 3 2 coupled to the light quark spin, the difference of the Therefore, q d 1 3=2 having light cloud spin = does attractive multiplet comes from the difference of the light not couple to the nucleon of spin 1=2. Moreover, ½ ½ cloud spin structure. Moreover, we find the degeneracy of q d 1 3=2 cannot construct the diquark spin 0 by the spin ½ ½ Λ the HQS singlet and triplet (singlet and doublet) in the case rearrangement. So, q d 1 3=2 cannot couple to Q with of g ¼þ0.59 (−0.59). It is a natural result because the diquark spin 0 as well. As a result, the S-wave decay OPEP does not depend on the heavy quark spin structure. ¯ ¯ ð Þ channels to ½QQ N and P ΛQ from the spin 3=2 HQS In the heavy quark limit, four (three) bound states exist singlet and spin (1=2, 3=2, 5=2) HQS triplet are prohibited ¼þ0 59 −0 59 for g . ( . ). However, the heavy quark sym- in the heavy quark limit. D-wave decay channels do exist; metry is broken for real charm/bottom hadrons, so that all however, we expect that they are small. four (three) bound states may not exist in reality as On the other hand, there are no constraints on the S-wave demonstrated in Sec. IV. But we expect that there exist ¯ ¯ ð Þ decay to ½QQ N and P ΛQ for the spin 1=2 HQS singlet some HQS partners of Pc-like pentaquarks. In particular, and spin (1=2, 3=2) HQS doublet which have the light for the bottom sector, the structure of HQS multiplet is ½ ½ cloud spin of q d 1 1=2 in the view of heavy quark spin more clearly than for charm sector, because the realization symmetry. These restrictions are independent of the model of the heavy quark symmetry is better. We expect the and derived only from heavy quark symmetry. The differ- observation of the bottom pentaquarks to confirm the HQS ence in their S-wave decay channel restrictions should multiplet structure of them. appear in the decay branching ratio of the QQqqq¯ The discussion in the LCS basis can be compared to the pentaquark state. We expect the measurement of the quark model calculations, treating the constituent quarks as ð Þ branching ratio to ½QQ¯ N and P¯ Λ to confirm the heavy degrees of freedom of the system. In Refs. [34,38], the Q quark symmetry in the Pc-like pentaquarks. short-range interaction in the Pc pentaquarks are studied, which is derived based on the quark cluster model. The ACKNOWLEDGMENTS contributions from the color magnetic interaction of ccuud¯ are evaluated, and they find that the ccuud¯ configurations The work of Y. S. is supported in part by a JSPS Grant- having the other spin j ¼ 3=2 are important to produce in-Aid for JSPS Research Fellow No. JP17J06300. The an attraction. On the other hand, the configurations with work of M. H. is supported in part by JSPS KAKENHI Grant No. 16K05345. The work of Y. Y. is supported in part 3We do not consider the tensor force in this study, but it is also by the Special Postdoctoral Researcher (SPDR) and determined by the light cloud spin structure. Not only the pion iTHEMS Programs of RIKEN. interaction, but also the other light meson interactions depend on We are grateful to Sachiko Takeuchi, Atsushi Hosaka, the light cloud spin. and Tetsuo Hyodo for useful comments and discussions.
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