Abstract:The simplest version of a class of toy models for QCD is presented. It is a Lipkin-type model, for the quark-antiquark sector, and, for the gluon sector, gluon pairs with spin zero are treated as elementary bosons. The model restricts to mesons with spin zero and to few baryonic states. The corresponding energy spectrum is discussed. We show that ground state correlations are essential to describe physical properties of the spectrum at low energies. Phase transitions are described in an effective manner, by using coherent states. The appearance of a Goldstone boson for large values of the interaction strength is discussed, as related to a collective state. The formalism is extended to consider finite temperatures. The partition function is calculated, in an approximate way, showing the convenience of the use of coherent states. The energy density, heat capacity *

A schematic model for QCD, developed in a previous paper, is applied to calculate meson properties in the high temperature (up to 0.5 GeV) regime. It is a Lipkin model for quark-antiquark pairs coupled to gluon pairs of spin zero. The partition function is constructed with the obtained meson spectrum and several thermodynamical observables are calculated, like: the energy density, heat capacity, as well as relative production rates of mesons and absolute production rates for pions and kaons. The model predictions show a qualitative agreement with data. Based on these results we advocate the use of the model as a toy model for QCD.

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 states. The predicted lowest pentaquark state (J π = 1 2 − ) lies at the energy of 1.5 GeV and it is associated to the observed Θ + (1540) state. For heptaquarks (J π = 1 2 + , 3 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 model for QCD was developed. The model was used to test the meson spectrum of QCD. In spite of its schematic nature the model seems to contain the relevant degrees of freedom, as it was shown in the comparison between calculated and experimental meson spectra [2]. This letter is devoted to the extension of the model to accommodate baryonic features. Particularly, we shall concentrate on the appearance of exotic baryonic states, like pentaquark and heptaquark states [4,5,6,7,8].The essentials of the model were discussed in detail in Ref. [2]. It consists of two fermionic levels in the quark (q) and antiquark (q) sector and a gluonic (g 2 ) state containing pairs of gluons. These are the elementary degrees of freedom of the model. The interaction among these degrees of freedom is described by excitations of pairs of quarks and antiquarks mediated by the exchange of pairs of gluons. The pairs of quarks are classified in a flavor-spin coupling scheme. The pairs of gluons are kept in the angular momentum (J), parity (π) and charge conjugation (C) state J πC = 0 ++ . The strength of various channels of the interaction, as well as the constituent masses, are taken from a phenomenological analysis. The model describes meson ((qq) n (g 2 ) m ) states and baryonic (q 3 (qq) n (g 2 ) m ) states. Among these states we focus on q 3 (qq) states (pentaquarks) and q 3 (qq) 2 states (heptaquarks), where the configurations indicated represent the leading terms in an expansion over many quarkantiquark and gluon states. The basis states are classified using group theoretical methods [2]. The interaction of quark-antiquark pairs with gluon pairs is particle nonconserving.The above described model belongs to a class of exactly solvable models of coupled fermion and boson systems [9, were proposed in Ref.[13], enforcing particle number conservation.In what follows we shall classify the basis states and solve the Hamiltonian in the framework of the boson expansion method [14,15]. Finally, we shall compare the results of the calculations with recently published experimental data [4,5,6,7] The model Hamiltonian is writtenThe distance between the fermion levels is 2ω f =0.66 GeV, ω b =1.6 GeV is the energy of the glue ball, n f and n b are the number operators for fermion and gluon pairs, respectively, V λS is the strength of the interaction in the flavor(λ) and spin (S) channel. The actual values λ = 0, 1 refer to...

Coherent states are introduced as test functions to formulate the statistical mechanics of fermions and bosons interacting via schematic forces. Finite temperature solutions to the Lipkin model and to the Schütte-Da Providencia model are obtained by performing the statistical sum à la Hecht, e.g., by using coherent states. Comparison between present and exacts results is discussed.