PHYSICAL REVIEW D 98, 014021 (2018) Heavy quark spin multiplet structure of ¯ ðÞΣðÞ 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 ¯ ðÞΣðÞ 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 ¯ à ð3872Þ announced the observation of two hidden-charm penta- X is close to the DD threshold, X includes þ þ 0 ¯ à þ ð4380Þ ð4450Þ Λ → the DD molecule structure [44]. The masses of Pc ð4380Þ quarks, Pc and Pc , in the decay of b − þð4450Þ ¯ Σà J=ψK p [1–3]. Their masses are M4380 ¼ 4380 Æ 8 Æ and Pc are slightly below the thresholds of D c and ¯ ÃΣ 28 MeV and M4450 ¼ 4449.8 Æ 1.7 Æ 2.5 MeV, and D c, respectively. They can be considered as the loosely Γ4380 ¼ 205 Æ 18 Æ 86 bound state of the heavy meson and heavy baryon. decay widths are MeV and þ P Γ4450 ¼ 39 Æ 5 Æ 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 Æ 1 2 ð Þ and DOI. Funded by SCOAP3. JÆ 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 à à 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 à 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 ðÞ heavy-light baryons. The mass splitting between c J In this section, we construct the OPEP for P¯ ðÞΣ 1 2Þ ΣÃ ð ¼ 3 2Þ Σ Σà 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 ¯ ðÞ 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 ¯ ðÞΣðÞ ¯ ðÞ states of P Q . Here, P means a HQS doublet meson The heavy meson doublet field H is – ¯ ð Þ ¯ Ãð ÃÞ ΣðÞ 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 à μ Σ ðΣ Þ ΣÃðΣÃÞ H ¼ ½Pμγ þ iPγ5: ð4Þ c b and c b . 2 ¯ ðÞ ΣðÞ The HQS doublet structures of the P meson and Q baryon which have one heavy quark are well known. The P and Pà are pseudoscalar meson and vector meson fields ¯ ðÞ 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 Æ 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 ¯ ðÞΣðÞ 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 ¯ ðÞΣðÞ 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 Dà → Dπ [58]. decomposition to the heavy quark spin and the other spin ΣðÞ The Q baryons and pion interaction Lagrangian is ¯ ðÞΣðÞ of the P Q molecules. Our purpose in this paper is to given by [56,59] ðÞ demonstrate the HQS multiplet of P¯ ðÞΣ . 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 Σà meson-baryon molecular basis is not suitable to discuss The superfield Sμ for Q and Q is represented as 014021-2 HEAVY QUARK SPIN MULTIPLET STRUCTURE OF … PHYS.