Modeling Pentaquark and Heptaquark States
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PHYSICAL REVIEW C 70, 025201 (2004) Modeling pentaquark and heptaquark states M. Nuñez V., S. Lerma H., and P. O. Hess Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70-543, México 04510 D.F. S. Jesgarz Instituto de Fisica, Unversidade de São Paulo, CP 66318, São Paulo 05315-970, SP, Brasil O. Civitarese and M. Reboiro Departamento de Física, Universidad Nacional de La Plata, c.c. 67 1900, La Plata, Argentina (Received 29 January 2004; published 23 August 2004) A schematic model for hadronic states, based on constituent quarks and antiquarks and gluon pairs, is discussed. The phenomenological interaction between quarks and gluons is QCD motivated. The obtained hadronic spectrum leads to the identification of nucleon and ⌬ resonances and to pentaquark and heptaquark ͑ 1 −͒ states. The predicted lowest pentaquark state J = 2 lies at the energy of 1.5 GeV and it is associated to the ⌰+͑ ͒ ͑ 1 + 3 +͒ observed 1540 state. For heptaquarks J = 2 , 2 the model predicts the lowest state at 2.5 GeV. DOI: 10.1103/PhysRevC.70.025201 PACS number(s): 24.85.ϩp, 12.39.Ϫx, 14.20.Ϫc I. INTRODUCTION In what follows we shall classify the basis states and solve the Hamiltonian in the framework of the boson expansion In a series of previous publications [1–3] a schematic method [14,15]. Finally, we shall compare the results of the model for QCD was developed. The model was used to test calculations with recently published experimental data [4–7]. the meson spectrum of QCD. In spite of its schematic nature The model Hamiltonian is written the model seems to contain the relevant degrees of freedom (color, flavor, and spin), as it was shown in the comparison ͓ͭ͑ † ͒2 † ͑ ͒2͔ H =2 fnf + bnb + ͚ VS bS +2bSbS + bS between calculated and experimental meson spectra [2]. This S letter is devoted to the extension of the model to accommo- n n date baryonic features. Particularly, we shall concentrate on ϫͩ f ͪ †ͩ f ͓ͪ͑ † ͒2 † ͑ ͒2͔ͮ 1− b + b 1− b +2b b + b the appearance of exotic baryonic states, like pentaquark and 2⍀ 2⍀ S S S S heptaquark states [4–8]. † † + n͑ ͒ ͑D n + D ͑b + b͒͒ + n͑ ͒ ͑E n + E ͑b + b͒͒. The essentials of the model were discussed in detail in 0,1 0 1 b 2 2,0 1 1 b 2 Ref. [2]. It consists of two fermionic levels in the quark (q) ͑1͒ and antiquark (¯q) sector and a gluonic (g2) state containing pairs of gluons. These are the elementary building blocks of The distance between the fermion levels is 2 f =0.66 GeV, the model. The interaction among these parts is described by b =1.6 GeV is the energy of the glue ball, nf and nb are excitations of pairs of quarks and antiquarks mediated by the the number operators for fermion and gluon pairs, respec- ͑ exchange of pairs of gluons. The pairs of quarks are classi- tively, VS is the strength of the interaction in the flavor ) fied in a flavor-spin coupling scheme. The pairs of gluons are and spin (S) channel. The actual values =0,1 refer to flavor kept in the angular momentum (J), parity (), and charge (0,0) and (1,1) configurations, while the spin channel is S conjugation (C) state J C =0++. The strength of various chan- =0 or 1. The adopted values are: V00=0.0337 GeV, V01 nels of the interaction, as well as the constituent masses, are =0.0422 GeV, V10=0.1573 GeV, and V11=0.0177 GeV [2]. † taken from a phenomenological analysis. The model de- The operators bS and bS are boson images of quark- scribes meson (͑qq¯͒n͑g2͒m) states and baryonic antiquark pairs [2]. The products which appear inside brack- ͑ ⍀͒ (q3͑qq¯͒n͑g2͒m) states. Among these states we focus on q3͑qq¯͒ ets in (1) are scalar products. The factor 1−nf /2 results states (pentaquarks) and q3͑qq¯͒2 states (heptaquarks), where from the boson mapping [2]. The mapping is exact for the ͓ ͔ ͓ ͔ the configurations indicated represent the leading terms in an channel ,S = 0,0 and simulates the effect of the boson †͑ ͒ expansion over many quark-antiquark and gluon states. The mapping for the other channels. The operator b b creates (annihilates) gluon pairs with spin-color zero, and n͑ ͒ is basis states are classified using group theoretical methods 0, 0 S0 [2]. The interaction of quark-antiquark pairs with gluon pairs the number operator of di-quarks coupled to flavor-spin ͑ ͒ is particle nonconserving. 0 , 0 S0. The parameters D1͑2͒ and E1͑2͒ are adjusted to The above described model belongs to a class of exactly reproduce the nucleon and ⌬ resonances. The corresponding solvable models of coupled fermion and boson systems terms describe the interaction between valence quarks and [9–12]. Alternative descriptions of pentaquark states, which mesons. The Hamiltonian (1) does not contain terms which use also an algebraic approach but which include orbital ex- distinguish between states with different hypercharge and citations, and enforce particle number conservation, have isospin. It does not contain flavor mixing terms, either. been proposed in Ref. [13]. Therefore, the predicted states have to be corrected in 0556-2813/2004/70(2)/025201(4)/$22.5070 025201-1 ©2004 The American Physical Society NUÑEZ, LERMA, AND HESS PHYSICAL REVIEW C 70, 025201 (2004) the way described in Ref. [16] to allow a comparison TABLE I. Flavor irreps coupled to the quark-antiquark content ͑ ͒ with data. The adopted values of D1͑2͒ and E1͑2͒ are: D1 of some different U 4 irreps. Shown are the irreps which contain, =−1.442 GeV, D =−0.4388 GeV, E =−1.1873 GeV, and at most, two quarks and two antiquarks. The number of quarks 2 1 ͑ ͒ (antiquarks) in a given configuration are denoted by nq n¯q . E2 =−0.3622 GeV. The Hamiltonian contains all relevant de- grees of freedom requested by QCD. ͑ ͒ ͑ ͓͒ ͔ ͓ ͔ The complete classification of quark-antiquark configura- SUf 3 U 4 q1q2 nq Sq ¯q1¯q2 n¯q S¯q S tions was given in Ref. [2]. (0,0), (1,1), (2,2) ͓8811͔͓11͔ 20 ͓88͔ 20 0 The unperturbed ground state is composed by 18 quarks (1,1), (3,0), (0,3) ͓9711͔͓11͔ 20 ͓97͔ 21 1 occupying the lowest fermionic level. The baryonic states are ͓ ͔͓͔ ͓ ͔ described by three quarks in the upper fermionic level to (1,1), (3,0), (0,3) 8820 20 21 88 20 1 ͓ ͔͓͔ ͓ ͔ which we add ͑qq¯͒n states. The group chain which describes (0,0), (1,1), (2,2) 9720 20 21 97 2 1 0, 1, 2 ͓ ͔͓͔ 1 ͓ ͔ 1 these states is (1,1) 9810 10 1 2 98 1 2 0, 1 (1,1) ͓9810͔͓11͔ 20 ͓97͔ 21 1 ͓1N͔͓h͔ = ͓h h h ͔͓hT͔, 1 2 3 (1,1) ͓9810͔͓11͔ 20 ͓88͔ 20 0 ͓ ͔͓͔ ͓ ͔ ⍀ (1,1) 9810 20 21 97 2 1 0, 1, 2 U͑4⍀͒ ʛ Uͩ ͪ U͑12͒, ͑2͒ (1,1) ͓9810͔͓20͔ 21 ͓88͔ 20 1 3 (0,0) ͓9900͔͓00͔ 00 ͓99͔ 00 0 where ⍀=9 accounts for three color and three flavor degrees (0,0) ͓9900͔͓10͔ 1 1 ͓98͔ 1 1 0, 1 ͑ ⍀͒ 2 2 of freedom. The irreducible representation (irrep) of U 4 (0,0) ͓9900͔͓20͔ 21 ͓97͔ 2 1 0, 1, 2 is completely antisymmetric, and ͓hT͔ is the transposed Young diagram of ͓h͔ [17]. For N particles, and due to the ͓ N͔ ͑ ⍀͒ ͑⍀ ͒ ␣ antisymmetric irrep 1 of U 4 , the irreps of U /3 and bosons. The quantity S represents the other quantum num- U͑12͒ are complementary and the irrep of U͑⍀/3͒ is the bers needed to specify a particular harmonic oscillator. ͑ ͒ color group, which is reduced to SUC 3 with the color irrep Concerning the width of the states, in the present version ͑ ͒ ͑ ͒ C , C . The U 12 group is further reduced to of the model it cannot be calculated. For it, one has to con- ͉ ␣ ͘ struct the explicit form of S S and calculate the overlap ͑ ͒ ʛ ͑ ͒ ͑ ͒ ʛ ͑ ͒ ͑ ͒ U 12 Uf 3 U 4 SUf 3 SUS 2 of the initial state with the states of the exit channel, which can be seen as a cluster component of the initial state. In ͓ ͔͑ ͒ ͑ ͒ p1p2p3p4 f, f S,M, 3 spite of this limitation we can advance some qualitative ar- ͑ ͒ ͓ ͔ guments. We can determine the average numbers of quark- where f , f is the flavor irrep and p1p2p3p4 denotes the possible U͑4͒ irreps. The group reduction is done using the antiquark pairs coupled to a definite flavor and spin and also the average number of gluon pairs coupled to color and spin methods exposed in Ref. [18]. The basis is spanned by the zero. If these average numbers, as obtained for the initial states state, differ with respect to the sum of the average numbers ͉ ͓ ͔͑ ͒ ͑ ͒ ͘ ͑ ͒ N, p1p2p3p4 C, C , f f, f YTTz, SSM , 4 of the particles in the decay channel, then the width of the state may be small. If these average numbers are similar, the where N is the number of particles, Y is the hypercharge, and appearance of a broad width may be expected.