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Hidden-Bottom and -Charm Hexaquark States in QCD Sum Rules

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Hidden-Bottom and -Charm Hexaquark States in QCD Sum Rules Eur. Phys. J. C (2020) 80:121 https://doi.org/10.1140/epjc/s10052-020-7701-8 Regular Article - Theoretical Physics Hidden-bottom and -charm hexaquark states in QCD sum rules Bing-Dong Wan1,a , Liang Tang2,b, Cong-Feng Qiao1,3,c 1 School of Physics, University of Chinese Academy of Science, Yuquan Road 19A, Beijing 10049, China 2 College of Physics, Hebei Normal University, Shijiazhuang 050024, China 3 CAS Center for Excellence in Particle Physics, Beijing 10049, China Received: 2 January 2020 / Accepted: 30 January 2020 / Published online: 14 February 2020 © The Author(s) 2020 Abstract In this paper, we investigate the spectra of the and theoretical progresses can be found in comprehensive prospective hidden-bottom and -charm hexaquark states with reviews like [10–14]. quantum numbers J PC = 0++,0−+,1++ and 1−− in the Facing the observations of tetraquark and pentaquark framework of QCD sum rules. By constructing appropriate states, it is nature to conjecture that there should exist the interpreting currents, the QCD sum rules analyses are per- hidden-charm and -bottom hexaquark states, and it is time formed up to dimension 12 of the condensates. Results indi- to hunt for them. For the hexaquark states, the deuteron is cate that there exist two possible baryonium states in b-quark a typical and well-established dibaryon molecular state with P + sector with masses 11.84 ± 0.22 GeV and 11.72 ± 0.26 GeV J = 1 and binding energy EB = 2.225 MeV [15–17]. for 0++ and 1−−, respectively. The corresponding hidden- The baryonium states composed of a baryon and an anti- charm partners are found lying respectively at 5.19 ± 0.24 baryon is another special class of heaxquark configuration. In ¯ GeV and 4.78 ± 0.23 GeV. Note that these baryonium Refs. [18,19], the c-c structure was introduced to explain states are all above the dibaryon thresholds, which enables the production and decays of Y (4260). And the heavy bary- their dominant decay modes could be measured at BESIII, onium was explored as well in the heavy baryon chiral per- BELLEII, and LHCb detectors. turbation theory [20,21]. The method of QCD sum rules [22–26] has been applied successfully to many hadronic phenoemena, such as the 1 Introduction hadron spectrum and hadron decays. In QCD sum rules, based on the proper interpreting currents corresponding to a Hadrons with more than the minimal quark content (qq¯ or hadron of interest, one can construct the two-point or three- qqq) was proposed by Gell-Mann [1] and Zweig [2] in 1964, point correlation functions, which are respectively used to which were named as multiquark exotic states and do not evaluate the mass and the decay property of the related infringe the Quantum Chromodynamics (QCD). Exploring hadron. Then by matching its operator product expansion the existence and properties of such exotic states is one of the (OPE) to its hadronic saturation, the main function for most intriguing research topics of hadronic physics. In past extracting the mass or decay rate of the hadron is estab- few decades, research on the heavy-flavor exotic states has lished. Utilizing this approach, several significant researches made tremendous developments, that is, many charmonium- for the hexaquark states have been done [27–31]. In Ref. [27], like/bottomonium-like XYZ states have been observed [3– six-quark state d∗(2380) was considering as - structure 7]. In 2015, two hidden-charm pentaquarks Pc(4380) and and the corresponding mass was given. Chen et al. gave an Pc(4450) [8] were observed by the LHCb Collaboration in explicit QCD sum rule investigation for hidden-charm bary- /ψ 0 → /ψ − the J invariant mass spectrum via the b J pK onium states with various relevant local interpreting currents, process. Recently, the LHCb Collaboration reported a new and they found some of these currents can couple to hidden- narrow state Pc(4312), and the previously observed struc- charm baryonium states with the masses around 5.0 GeV ture Pc(4450) appears to be split into two narrower struc- [28]. Reference [29] calculated the mass and coupling con- tures Pc(4440) and Pc(4457) [9]. The recent experimental stant of the scalar hexaquark uuddss. Very recently, the bound system spectrum with J P = 0+ and 2+ in a molec- a e-mail: [email protected] ular picture were investigated in Ref. [30], and the results b e-mail: [email protected] suggest the existence of two bound dibaryon states. c e-mail: [email protected] (corresponding author) 123 121 Page 2 of 10 Eur. Phys. J. C (2020) 80 :121 1−− ¯ Wang [31] studied the scalar-diquark–scalar-diquark–scalar- jμ (x) = abc def [Qd (x)γμ Qc(x)] diquark type hexaquark state with the QCD sum rules, where T T ×[q (x)Cγ5q (x)][q ¯ (x)γ5Cq¯ (x)] . (7) the three diquarks were arranged as ud, uc, and dc, respec- a b e f tively. Here, a ... f denote color indices, C is the charge conjuga- Besides the above mentioned hexaquark configurations, tion matrix, Q represents the heavy quarks, q stands for the the QCD theory allows many other possible hexaquark struc- up quark u, and q for the down quark d. tures, which can couple to hidden-charm baryonium states. The correlation function derived from Eqs. (6) and (7) can Moreover, it should be noted that exploring the hidden- be expressed as the following Lorentz covariance form bottom baryonium states is also significant, which is the main qμqν 2 qμqν 2 μν =− gμν − (q ) + (q ), (8) motivation of this work, and they tend to be measurable in the q2 1 q2 0 LHCb experiment. We investigate the hidden-bottom molec- ¯ where the subscripts 1 and 0 denote the quantum numbers ular states in Q-Q configuration with quantum numbers J PC = 0++,0−+,1++ and 1−− in the framework of QCD of the spin 1 and 0 mesons, respectively. However, since the ( 2) sum rules. Their decay properties, as well as their hidden- leading term 1 q is symmetrical and only contains the ( 2) charm partners, are also analyzed. spin 1 component, we shall focus on calculating the 1 q The rest of the paper is arranged as follows. After the intro- and employ it to perform the QCD sum rule analyses. duction, some primary formulas of the QCD sum rules in our On the phenomenological side, adopting the usual pole calculation are presented in Sect. 2. The numerical analysis plus continuum parametrization of the hadronic the spectral ( 2) and results are given in Sect. 3. In Sect. 4, possible decay density, we express the correlation function q as modes of hidden-bottom baryonium states are investigated. (λX )2 1 ∞ ρ(s) PHEN(q2) = + ds , (9) The last part is left for conclusions and discussion. ( X )2 − 2 π − 2 M q s0 s q ¯ where the superscript X denotes the lowest lying Q-Q X ρ( ) 2 Formalism hexaquark state, M is its mass, and s is the spectral density that contains the contributions from higher excited The starting point of the QCD sum rules is the two-point states and the continuum states above the threshold s0.The λX | | =λX correlation function constructed from two hadronic currents decay constant is defined through 0 j X and | | =λX with the following form: 0 jμ X μ. On the OPE side, the correlation function (q2) can be ( ) = 4 iq·x | { ( ) †( )}| ; written as a dispersion relation form: q i d xe 0 T j x j 0 0 (1) ∞ ρ OPE( ) OPE 2 s 4 iq·x † (q ) = ds , (10) μν(q) = i d xe 0|T { jμ(x) jν (0)}|0 , (2) ( + + )2 s − q2 2m Q 2mq 2mq ρ OPE( ) = [OPE( )]/π where, j(x) and jμ(x) are the relevant hadronic currents with where s Im s is the spectral density of J = 0 and 1, respectively. the OPE side, and contains the contributins of the condensates η up to dimension 12, thus We use the notion Q to represent the Dirac baryon fields 2 of Q without free Lorentz indices. It was shown in Ref. [32] ρ OPE(s) = ρ pert(s) + ρ¯qq(s) + ρG (s) η that, Q may take the following quark structure: 2 3 +ρ¯qGq(s) + ρ¯qq (s) + ρG (s) T η ( ) = [ ( ) γ ( )] ( ), ¯qqqGq ¯ ¯qq2G2 Q x i abc qa x C 5qb x Qc x (3) +ρ (s) + ρ (s) ¯ 2 ¯ 4 where Q = b, c. Therefore, the interpolating currents for +ρ qGq (s) + ρ qq (s). (11) ¯ ++ Q-Q baryounium states with quantum numbers 0 , −+ ++ −− In order to calculate the spectral density of the OPE side, 0 ,1 , and 1 can be respectively constructed as q ( ) Q( ) Eq. (11), the full propagators Sij x and Sij p of a light 0++ ¯ quark (q = u, d or s) and a heavy quark (Q = c or b)are j (x) = abc def [Qd (x)Qc(x)] used: ×[ T ( ) γ ( )][ ¯ ( )γ ¯ T ( )] , qa x C 5qb x qe x 5Cq f x (4) a a iδ jkx/ δ jkmq itjkGαβ αβ αβ −+ Sq (x) = − − (σ x/ + x/σ ) 0 ¯ jk π 2 4 π 2 2 π 2 2 j (x) = abc def [Qd (x)γ5 Qc(x)] 2 x 4 x 32 x δ δ / δ 2 T T jk i jkx jkx ×[q (x)Cγ q (x)][q ¯ (x)γ Cq¯ (x)] , (5) − ¯qq+ mq ¯qq− gs q¯σ · Gq a 5 b e 5 f 12 48 192 ++ 1 δ 2 / ta σαβ ( ) = [ ¯ ( )γμγ ( )] i jkx x jk jμ x abc def Qd x 5 Qc x + mq gs q¯σ · Gq− gs q¯σ · Gq 1152 192 ×[ T ( ) γ ( )][ ¯ ( )γ ¯ T ( )] , a qa x C 5qb x qe x 5Cq f x (6) itjk + (σαβ x/ + x/σαβ )mq gs q¯σ · Gq , (12) 768 123 Eur.
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