A Schematic Model for QCD. III: Hadronic States

A schematic model for QCD. III: Hadronic states. M. Nuñez V., S. Lerma H. and P. O. Hess, Inst. Cs Nucleares, Univ. Nacional Autónoma de México, Apdo. Postal 70-543, México 04510 D.F. S. Jesgarz Inst. Fis., Univ. de São Paulo, CP 66318, São Paulo, 05315-970, SP, Brasil O. Civitarese and M. Reboiro Dep. Fis., Univ. Nac. de La Plata, c.c. 67 1900, La Plata, Argentina. The hadronic spectrum obtained in the framework of a QCD-inspired schematic model, is presented. The model is the extension of a previous version, whose basic degrees of freedom are constituent quarks and antiquarks, and gluons. The interaction between quarks and gluons is a phenomenological interaction and its parameters are fixed from data. The classification of the states, in terms of quark and antiquark and gluon configurations is based on symmetry consider- ations, and it is independent of the chosen interaction. Following this procedure, nucleon and ∆ resonances are identified, as well as various penta- and hepta-quarks states. The lowest pentaquarks state is predicted at 1.5 GeV and it has negative parity, while the lowest hepta-quarks state has positive parity and its energy is of the order of 2.5 GeV. PACS numbers: 12.90+b, 21.90.+f I. INTRODUCTION In previous papers [1, 2, 3], we have proposed a schematic model aimed at the description of the non-perturbative regime of QCD. The main concepts about the model can be found in [1], together with the applications to the meson spectrum of QCD. Along this line, Ref. [2] deals with the appearance of phase transitions and condensates. A preliminary set of results, about the energy and parity of systems of the type q3(qq¯) (pentaquarks), and q3(qq¯)2 (heptaquarks), was presented in [3]. As it was discussed in [1], this model describes reasonable well the main features of the meson spectrum of QCD. This is a nice result since it shows that, also in the conditions of low energy QCD, e.g.; large coupling constants and non-conservation of particle number, the use of simple models may be very useful. The model of [1] belongs to arXiv:nucl-th/0405053v1 19 May 2004 the class of models [4] which may be solved by algebraic, group theory and symmetry-enforcing techniques, and that describe the interaction between fermions and bosons. Some examples of this class of models can be found in [5], [6], and [7]. Concerning the use of symmetry principles, the work of [8] shows how the ordering of levels of many gluon states may be understood. In [8], the predictive power of the model, based on the microscopical treatment of gluons configurations, was tested by means of the spectrum related to exotic quantum numbers. The purpose of this work is to extend the model of Ref. [1] to describe the main structure of the hadronic spectrum of QCD. The essentials of the model are the following: a)color, flavor and spin degrees of freedom are taken explicitly into account to built the configurations, both in the quark and gluon sectors of the model; b)the quarks and antiquarks are placed in a s-state; c)the interaction of the quarks with the gluons proceeds via gluon pairs coupled to spin and color zero, only. Other possible gluon states are considered as spectators. The present work is organized as follows. Section II gives a brief description of the model. We shall avoid as much as possible the repetition of details which can be found in the already published papers [1, 2, 3], and concentrate in the aspects which are relevant for the description of the hadronic spectrum. In Section III we discuss the classification of many quark-antiquark states, and the classification of many gluon states [8]. In Section IV we show and discuss the hadronic spectrum predicted by the model, as a natural continuation of the study presented in [1], where the meson sector was analyzed. Conclusions are drawn in Section V. II. THE MODEL The basic degrees of freedom of the model are illustrated in the level scheme shown in Figure 1. The quarks occupy two levels, one at negative and the other at positive energy. Antiquarks are depicted as holes in the lower level. If 2 ω 1.6 b [GeV] ω 0.33 f 0 ω -0.33 - f FIG. 1: Schematic representation of the model space. The fermion levels are indicated by their energies ±ωf . The gluon-pairs are represented by the level at the energy ωb. † αi with cαi and c we denote the creation and annihilation operators of quarks, in the upper (i = 1) and lower (i = 2) level, the relation to quark (antiquark) creation and annihilation operators is given by a† c† d c† α = α1, α = α2 aα = cα1, d† α = cα2 , (1) where the index α refers to color, flavor and spin quantum numbers. Since three degrees of color, three degrees of flavor (only u, d and s quarks are taken into account) and two degrees of spin are considered, each level is 18-fold degenerate. The energy of each level is approximated to one third of the nucleon mass and it represents a constituent (effective) mass of the quarks. In this picture, the mass comes mainly from the kinetic energy because quarks are confined to a sphere of radius 1fm. The model describes quarks and antiquarks distributed in an orbital s-level. Orbital excitations can also be accounted for by increasing the number of degrees of freedom. To this system of quarks we add a boson level, which represents pairs of gluons of color and spin zero. Its position is fixed as it was done in Ref. [8]. The coupling between quarks and antiquarks and gluons proceeds via this gluon-pair configurations. Other couplings to different pairs of gluons are neglected, because their contributions, at least for the meson spectrum, were found to be negligible [1]. Nevertheless, gluons of other types are still present in the model but they are considered as spectators. † λf,SM The basic excitations in the fermion sector are qq¯ pairs, which are described by Bλf,SM and B pair operators [1], where λ=0,1 stands for flavor states (0,0) and (1,1), and the spin S can be 0 or 1. These pair of fermions, a sub-class of bi-fermion operators [6, 7], can be mapped onto a boson space [9]. We denote the corresponding operators † by bλf,SM . This is the basic structure of the model presented previously [1]. In order to extend it to the description of baryons, and because we work in the boson space of quark-antiquark pairs, one should add three valence quarks via (λ0,µ0)S0 † (λ,µ) ideal † (λ0,µ0)S0 the operators [D × a ]f,m , where two quark operators aα [9] are coupled as a di-quark Df0m0 = † † (λ0,µ0)S0 [a × a ]f0m0 with flavor, spin (λ0,µ0)S0 either (0,1)0 or (2,0)1. The model Hamiltonian is written as in Ref. [1], with the inclusion of new terms which are needed for the description of baryons, namely: H = 2ωf nf + ωbnb + 3 n V (b† )2 +2b† b + (b )2 (1 − f )b+ X λS nh λS λS λS λS i 2Ω λS n b†(1 − f ) (b† )2 +2b† b + (b )2 + 2Ω h λS λS λS λS io n n b† b n n b† b (0,1)0 D1 b + D2( + ) + (2,0)1 E1 b + E2( + ) , (2) n b† 2 b† b† f where ( λS) = ( λS · λS) is a short hand notation for the scalar product [24]. The factor (1 − 2Ω ) simulates the effect of the terms which would appear in the exact boson mapping of the quark-antiquark pairs [7, 9]. The operators b† and b are boson creation and annihilation operators of the gluon pairs with spin and color zero. The interaction describes scattering and vacuum fluctuation terms of fermion and gluon pairs. The strength VλS is the same for each allowed value of λ and S, due to symmetry reasons, as shown in [1]. These parameters were determined in [1] and they are taken as fixed values in the present calculations. The matrix elements of the meson n part of the Hamiltonian are calculated in a seniority basis. For details, please see Ref. [1]. The operators (λ0,µ0)S0 are number operators of the di-quarks coupled to flavor-spin (λ0,µ0)S0. The interaction does not contain terms which connect states with different hypercharge and isospin. It does not contain flavor mixing terms, either. Therefore, in the comparison with data, the energies should be corrected by the mass formula given in Ref. [10], with parameters deduced as in [11] and turning off flavor mixing terms. The procedure is explained in details in [1]. The parameters Dk are adjusted to the nucleon resonances, while the parameters Ek are adjusted to the ∆ resonances. The two last terms in (2) are needed to describe the interaction between the valence quarks and the meson-like states. The parameters ωf and ωb were fixed at 0.33 GeV and 1.6 GeV, respectively. The parameters VλS were adjusted to the mesonic spectrum and their values are: V00=0.0337 GeV, V01=0.0422 GeV, V10=0.1573 GeV and V11=0.0177 GeV, respectively. This shows the strong effects of the V10 interaction, which acts on pairs coupled to flavor (1,1) and spin 0 (pion-like states ). The parameters Dk and Ek (k=1,2) are adjusted to the nucleon resonances and ∆ resonances respectively.

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