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Modelling Pentaquark and Heptaquark States

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Modelling Pentaquark and Heptaquark States Modelling Pentaquark and Heptaquark States M. Nu˜nez V., S. Lerma H. and P. O. Hess, Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico Apdo. Postal 70-543, M´exico 04510 D.F. S. Jesgarz Instituto de Fisica, Unversidade de S˜ao Paulo, CP 66318, S˜ao Paulo, 05315-970, SP, Brasil O. Civitarese and M. Reboiro Departamento de F´ısica, Universidad Nacional de La Plata, c.c. 67 1900, La Plata, Argentina. 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 π 1 − pentaquark and heptaquark states. The predicted lowest pentaquark state (J = 2 ) lies at the energy of 1.5 GeV and it is associated to the observed Θ+(1540) state. For heptaquarks (J π = 1 + 3 + 2 , 2 ) the model predicts the lowest state at 2.5 GeV. PACS numbers: 12.90+b, 21.90.+f In a series of previous publications [1, 2, 3] a schematic were proposed in Ref. [13], enforcing particle number model for QCD was developed. The model was used to conservation. test the meson spectrum of QCD. In spite of its schematic In what follows we shall classify the basis states and nature the model seems to contain the relevant degrees of solve the Hamiltonian in the framework of the boson ex- freedom, as it was shown in the comparison between cal- pansion method [14, 15]. Finally, we shall compare the culated and experimental meson spectra [2]. This letter results of the calculations with recently published exper- is devoted to the extension of the model to accommodate imental data [4, 5, 6, 7] baryonic features. Particularly, we shall concentrate on The model Hamiltonian is written the appearance of exotic baryonic states, like pentaquark and heptaquark states [4, 5, 6, 7, 8]. H n n The essentials of the model were discussed in detail in = 2ωf f + ωb b arXiv:nucl-th/0405052v1 19 May 2004 n Ref. [2]. It consists of two fermionic levels in the quark + V (b† )2 +2b† b + (b )2 (1 − f )b 2 X λS nh λS λS λS λS i 2Ω (q) and antiquark (¯q) sector and a gluonic (g ) state con- λS taining pairs of gluons. These are the elementary de- n + b†(1 − f ) (b† )2 +2b† b + (b )2 grees of freedom of the model. The interaction among 2Ω h λS λS λS λS io these degrees of freedom is described by excitations of n n b† b pairs of quarks and antiquarks mediated by the exchange + (0,1)0 D1 b + D2( + ) of pairs of gluons. The pairs of quarks are classified n n b† b in a flavor-spin coupling scheme. The pairs of gluons + (2,0)1 E1 b + E2( + ) . (1) are kept in the angular momentum (J), parity (π) and πC charge conjugation (C) state J = 0++. The strength The distance between the fermion levels is 2ωf =0.66 of various channels of the interaction, as well as the con- GeV, ωb=1.6 GeV is the energy of the glue ball, nf stituent masses, are taken from a phenomenological anal- and nb are the number operators for fermion and gluon n m ysis. The model describes meson ((qq¯) (g2) ) states and pairs, respectively, VλS is the strength of the interac- baryonic (q3(qq¯)n(g2)m) states. Among these states we tion in the flavor(λ) and spin (S) channel. The ac- focus on q3(qq¯) states (pentaquarks) and q3(qq¯)2 states tual values λ = 0, 1 refer to flavor (0,0) and (1,1) con- (heptaquarks), where the configurations indicated repre- figurations, while the spin channel is S=0 or 1. The sent the leading terms in an expansion over many quark- adopted values are: V00=0.0337 GeV, V01=0.0422 GeV, antiquark and gluon states. The basis states are classified V10=0.1573 GeV, and V11=0.0177 GeV [2] . The oper- b† b using group theoretical methods [2]. The interaction of ators λS and λS are boson images of quark-antiquark quark-antiquark pairs with gluon pairs is particle non- pairs [2]. The products which appear inside brackets in n conserving. f (1) are scalar products. The factor (1− 2Ω ) results from The above described model belongs to a class of exactly the boson mapping [2]. The mapping is exact for the solvable models of coupled fermion and boson systems [9, channel [λ, S] = [0, 0] and simulates the effect of the bo- 10, 11, 12]. Alternative descriptions of pentaquark states son mapping for the other channels. The operator b† (b) 2 creates (annihilates) gluon pairs with spin-color zero, and the relevant irreps for mesonic states. (More details are n (λ0,µ0)S0 is the number operator of di-quarks coupled to given in Ref. [19]). flavor-spin (λ0,µ0)S0. The parameters D1(2) and E1(2) are adjusted to the nucleon and ∆ resonances. The corre- SU (3) U(4) [q1q2] nq Sq [¯q1q¯2] nq¯ Sq¯ S sponding terms describe the interaction between valence f (0,0), (1,1), (2,2) [8811] [11] 2 0 [88] 2 0 0 quarks and glueballs. The Hamiltonian (1) does not con- (1,1), (3,0), (0,3) [9711] [11] 2 0 [97] 2 1 1 tain terms which distinguish between states with different (1,1), (3,0), (0,3) [8820] [20] 2 1 [88] 2 0 1 hypercharge and isospin. It does not contain flavor mix- (0,0), (1,1), (2,2) [9720] [20] 2 1 [97] 2 1 0, 1, 2 ing terms, either. Therefore, the predicted states have to 1 1 (1,1) [9810] [10] 1 2 [98] 1 2 0, 1 be corrected in the way described in [16] to allow a com- (1,1) [9810] [11] 2 0 [97] 2 1 1 parison with data. The adopted values of D1(2) and E1(2) (1,1) [9810] [11] 2 0 [88] 2 0 0 are: D1 =-1.442GeV, D2=-0.4388GeV, E1=-1.1873GeV (1,1) [9810] [20] 2 1 [97] 2 1 0, 1, 2 and E2=-0.3622GeV. The Hamiltonian contains all rele- (1,1) [9810] [20] 2 1 [88] 2 0 1 vant degrees of freedom requested by QCD. (0,0) [9900] [00] 0 0 [99] 0 0 0 (0,0) [9900] [10] 1 1 [98] 1 1 0, 1 The complete classification of quark-antiquark config- 2 2 urations was given in Ref. [2]. (0,0) [9900] [20] 2 1 [97] 2 1 0, 1, 2 The unperturbed ground state is composed by 18 TABLE I: Flavor irreps coupled to the quark-antiquark con- quarks occupying the lowest fermionic level. The bary- tent of some different U(4) irreps. Shown are the irreps which onic states are described by three quarks in the upper contain, at most, two quarks and two antiquarks. The number n fermionic level to which we add (qq¯) states. The group of quarks (antiquarks) in a given configuration are denoted by chain which describes these states is nq (nq¯). N T [1 ][h] = [h1h2h3][h ] In the boson representation, the states are given by the U(4Ω) ⊃ U( Ω )⊗ U(12) (2) direct product of one-, three-, eight and 24-dimensional 3 harmonic oscillators [2]. For each harmonic oscillator the where Ω=9 accounts for three color and three flavor de- basis states are given by grees of freedom. The irreducible representation (irrep) T N −ν of U(4Ω) is completely antisymmetric, and [h ] is the † λS λS b 2 2 transposed Young diagram of [h] [17]. For N particles, NNλSνλS ( λS ) |νλS αλS > , (5) and due to the antisymmetric irrep [1N ] of U(4Ω), the where N is the number of bosons of type [λ, S], ν irreps of U(Ω/3) and U(12) are complementary and the λS λS is the corresponding seniority and N is a normal- irrep of U(Ω/3) is the color group, which is reduced to NλSνλS ization constant. The seniority is defined as the number SU (3) with the color irrep (λ ,µ ). The U(12) group C C C of uncoupled bosons. The quantity α represents the is further reduced to λS other quantum numbers needed to specify a particular harmonic oscillator. U(12) ⊃ Uf (3) ⊗ U(4) ⊃ SUf (3) ⊗ SUS(2) [p1p2p3p4] (λf ,µf ) S, M , (3) a) Nucleon resonances where (λf ,µf ) is the flavor irrep and [p1p2p3p4] denotes the possible U(4) irreps. The group reduction is done The quality of the model predictions, concerning me- using the methods exposed in Ref. [18]. The basis is son states, was discussed in Ref. [2]. Figure 1 shows the spanned by the states lowest nucleon and ∆ resonances predicted by the model. In the same Figure are shown the calculated penta- and |N, [p1p2p3p4](λC ,µC),ρf (λf ,µf )YTTz,ρSSM > ,(4) heptaquark low lying states. For each state we indicate the spin, parity (J π), and the quark and gluon content where N is the number of particles, Y is the hypercharge (nq +nq¯, ng). The quantity nq +nq¯ is the total number of and (T ,Tz) denotes the isospin and its third component, quarks and antiquarks, which is equal to the number of ρf and ρS are the multiplicities of the flavor and spin valence quarks (0 for mesons, 3 baryons) plus the num- representations. The color labels (λC ,µC ) are related to ber of quarks and antiquarks of the qq¯-pairs, and ng gives the hi via λC = h1 − h2 and µC = h2 − h3.
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