Heptaquarks with Two Heavy Antiquarks in a Simple Chromomagnetic Model
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Heptaquarks with two heavy antiquarks in a simple chromomagnetic model Aaron Park,1, ∗ Woosung Park,1, † and Su Houng Lee1, ‡ 1Department of Physics and Institute of Physics and Applied Physics, Yonsei University, Seoul 03722, Korea (Dated: May 9, 2018) We investigate the symmetry property and the stability of the heptaquark containing two identical heavy antiquarks using color-spin interaction. We construct the wave function of the heptaquark from the Pauli exclusion principle in the SU(3) breaking case. The stability of the heptaquark against the strong decay into one baryon and two mesons is discussed in a simple chromomagnetic 2 3 2 5 model. We find that q s s¯ with I = 0,S = 2 is the most stable heptaquark configuration that could be probed by reconstructing the Λ + φ + φ invariant mass. PACS numbers: 14.40.Rt,24.10.Pa,25.75.Dw I. INTRODUCTION each other in a compact configuration. However, at the same time, bringing additional quarks into compact con- Multiquark hadrons made of more than three quarks figuration will generate additional kinetic energy com- became a theme of interest since Jaffe predicted their pared to having isolated hadrons. If the additional quarks existence using the bag model [1–3]. Unfortunately, ex- or antiquarks are heavy, the additional kinetic energy can tensive experimental search ruled out the existence of be made small while keeping the enhanced color-spin in- a deeply bound H dibaryon, and the initial excitement teraction among light quarks large, which can be effec- about the finding of Θ+(1540)[4] faded away as further tively understood as additional diquark correlation[41], experimental study could not confirm it [5–14]. compared to separated hadrons. To further probe con- On the other hand, there is a renewed interest in the figurations with heavy quarks in yet another multiquark subject triggered by the discovery of the X(3872) by the configuration, we will consider heptaquarks with two Belle Collaboration in B+ K±π+π+J/ψ [15], which heavy antiquarks. was subsequently confirmed→ by several other experiments Heptaquark composed of five quarks and two anti [16–18]. Also, in the dibaryon sector, a resonance struc- quarks has been studied by only a few researchers [42– ture was finally observed in the I(J P ) = 0(3+) channel by 44]. Ref.[44] suggested that as long as there is a stable the WASA-at-COSY collaboration[19, 20]. Furthermore, meson state composed of two heavy quarks and two light LHCb collaboration has recently observed hidden-charm quarks, there will be a stable heptaquark state within the pentaquark states in the J/ψp invariant mass spectrum chiral soliton model. In fact, within a constituent quark 0 − model, there will be a stable tetraquark state with two in the Λb J/ψK p process [21]. Subsequently, these states were→ studied using many theoretical approaches, heavy quarks or antiquarks [40]. Hence, such configura- such as the QCD sum rules [22–24], the molecular ap- tions are also part of the configurations to be probed in proach [25–28] and the quark model [29, 30]. These ex- this work using a constituent quark model with color-spin perimental findings led to the interest in the study of interaction. multiquark hadron states containing heavy quarks. In To study the possible existence of compact exotic fact, recent lattice calculations show that the H dibaryon hadrons, one first has to inspect the configuration with becomes bound in the massive pion cases [31, 32]. the most attractive color-spin interaction. For example, Multiquark configurations with heavy quarks were the color-spin interaction in a H-dibaryon is more attrac- arXiv:1706.10025v2 [hep-ph] 9 Aug 2017 studied before. Silvestre-Brac and Leandri searched sta- tive than that from the two separate ΛΛ system because ble q6, q5Q and q4QQ′ system in the framework of a the former allows for three most attractive diquark con- pure chromomagnetic Hamiltonian [33–35]. Heavy pen- figuration while the later has two separated diquark con- taquarks with heavy quarks were studied in quark models figurations. The most attractive diquark configuration is with color spin [36, 37] and flavor spin interaction[38, 39]. the maximally antisymmetric configuration in terms of Tetraquark states with two heavy quarks were also found color flavor spin. In terms of two quark configura- ⊗ ⊗ to be stable against strong decay if the heavy quark mass tions, there are four states satisfying the Pauli exclusion was taken to be sufficiently large [40]. principle. We represent them together with the matrix Within the constituent quark model, stable multiquark elements for the invariant appearing in the color spin in- configurations arise from a large attraction in the color- teraction in Table I, which can be obtained from spin interaction when more light quarks can interact with N λ λ σ σ − i j i · j i<j ∗Electronic address: [email protected] X † 4 4 Electronic address: [email protected] = N(N 6)+4I(I +1)+ S(S +1)+2Cc , (1) ‡ 3 − 3 Electronic address: [email protected] 2 II. WHY HEAVY HEPTAQUARK? TABLE I: The classification of two quarks and quark- antiquark color-spin interaction, with λi,σi respectively rep- resenting the color and spin matrix of the i quark. Two quarks In this work, we investigate the stability of the hep- state is determined to satisfy Pauli exclusion principle. We taquark using hyperfine potential. Even if the hyperfine denote antisymmetric and symmetric state as A and S respec- potential is attractive for a given heptaquark configura- tively. In the parenthesis, multiplet state is represented. tion, it cannot form a compact stable state if the addi- tional repulsion from kinetic energy is large. Hence, we qq need to consider which flavor state makes the additional Color A(3)¯ S(6) A(3)¯ S(6) kinetic energy lower. In the remaining part of this paper, Flavor A(3)¯ A(3)¯ S(6) S(6) we consider only hyperfine potential, but we can simply estimate the additional kinetic energy using the following Spin A(1) S(3) S(3) A(1) coordinate system and simple Gaussian spatial function. − · − − 4 8 λiλj σi σj 8 3 3 4 If we label the five light quarks as (i =1 5) and the qq¯ two heavy antiquarks as (i = 6, 7), then we∼ can choose Color (1) (8) (1) (8) the Jacobi coordinate system as follows. Spin A(1) A(1) S(3) S(3) 7 6 − · − 16 − 2 1 1 1 λiλj σi σj 16 2 3 3 m r˙ 2 Mr˙ 2 = M x˙ 2 2 i i − 2 CM 2 i i i=1 i=1 X X 7 where M = mi, M1 = M2 = mu, i=1 X 1 2 2mumQ 5mu(mu + mQ) where N is the total number of quarks, Cc = 4 λ the M3 = M4 = , M5 = color Casimir operator, S the spin and I the isospin of mu + mQ 2(4mu + mQ) 4 10 the system. Using Cc = 3 , 3 for color anti-triplet and 7(mu + mQ)(4mu + mQ) M6 = sextet respectively, one notes, as given in the table, that 10(5mu +2mQ) while the most attractive channel has spin 0, the spin 1 7 state also has an attractive combination. For two anti- 1 rCM = miri (2) quarks, we can use the same table simply by replacing 3¯ M i=1 and 6 with 3 and 6¯ for color and flavor state, respectively. X For the quark antiquark configuration, we can construct a similar table as given in the lower part of Table I. Typ- 1 x1 = (r1 r2) ically, the color-spin interaction is inversely proportional √2 − to the two constituent quark masses involved 1/(m m ). i j 2 1 1 Hence, in forming a multiquark configurations, if the ad- x2 = ( r1 + r2 r3) dition involves a light quark and a light antiquark, the r3 2 2 − addition will just fall apart into a meson state. On the 1 x3 = (r4 r6) other hand, if the addition is composed of a light quark √2 − and a heavy antiquark, it could become energetically fa- 1 x4 = (r5 r7) vorable to be in a compact configuration. To probe such √2 − a possibility systematically, we investigate the symmetry 6 1 1 1 mu mQ property and the stability of the heptaquark containing x5 = ( r1 + r2 + r3 r4 r6) two identical heavy antiquarks in a simple chromomag- r5 3 3 3 − mu + mQ − mu + mQ netic model. 10 1 x = (m r + m r + m r + m r 6 7 {4m + m u 1 u 2 u 3 u 4 This paper is organized as follows. We first explain r u Q 1 why kinetic energy favors compact heptaquarks contain- + mQr6) (mur5 + mQr7) . (3) ing heavy flavors in Sec. II. In Sec. III, we represent − mu + mQ } the color and spin basis functions of the heptaquark configuration. In Sec. IV, we construct the flavor This coordinate system can describe the decay mode of color spin part of the wave function of the heptaquark⊗ the heptaquark consisting of one baryon and two mesons. in order⊗ to satisfy the Pauli exclusion principle. In In this coordinate system, x1, x2 describe the baryon sys- Sec. V, we represent the color-spin interaction part of tem, while x3, x4 represent the relative quark distances the Hamiltonian. In Sec. VI, we calculate the binding for the two mesons, respectively. If we chose a simple potential of the heptaquark and plot the results as a Gaussian form as the spatial function, then we can cal- function of the light/heavy quark mass ratio parameter culate the kinetic energy of the heptaquark as follows.