Schematic model for QCD. II. Finite temperature regime

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 thermody- namical observables are calculated, such as 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. In Ref. @1 #@ hereon referred to as ~I!# a simple model, representative of QCD, was introduced and applied to the calculation of the spectrum of mesons. It is a Lipkin type model @2# for the quark sector, coupled to a boson level that is occupied by gluon pairs with spin zero. The four param- eters of the model were adjusted in order to reproduce 13 known meson states with spin zero or one. The calculated spectra, for mesons with spin different from the ones used in the fit, were found to be in qualitative agreement with data. As reported in ~I!, the calculated meson states contain many quarks, antiquarks, and gluons. The gluon contributions were found to be of the order of 30%. The model predictions ~I! are free of the so-called multiplicity problem, i.e., that a given state can be described in many ways, which is re- moved due to the action of particle mixing interaction. The model itself resembles the one of Ref. @3#, which treats nucleons coupled to pions. Also, it is related to the work of Ref. @4#, which describes quarks and uses particle conserving interactions. Generally speaking, the model of ~I! belongs to the class of models described in Refs. @5,6#. The gluon part in ~I! is fixed @7# and does not contain any new parameters. The validity of the basic theoretical assumptions, and the applications to low and high temperature regimes, has been studied for mesons with flavor ~0,0! and spin 0 @8#. The aim of these studies was to formulate a manageable, schematic, albeit realistic, model to describe qualitatively QCD at low and high energies. Since the model is algebraic, i.e., all ma- trix elements are analytic, and exactly solvable, it can pro- vide a nonperturbative description based on QCD relevant degrees of freedom, such as quarks, antiquarks, and gluons. This, in turn, allows to test other microscopic many body techniques previously applied to the nonperturbative treat- ment of real QCD @9,10#. Although the proposed model ~I! is probably too simple to describe real QCD, it contains all basic ingredients of real QCD. These are the correct number of degrees of freedom associated with color, flavor, and spin, and the orbital degree of freedom, which is contained in the degeneracy 2V of each of the quark levels. In this work we investigate the behavior of the model, in the finite temperature regime. By starting from the model predictions of the meson spectrum, we calculate the partition function and different thermodynamical quantities, such as the energy density and the heat capacity as a function of temperature. Next, we focus on the calculation of meson pro- duction rates. As we shall show, these production rates are in qualitative agreement with the experiments. Also, we calcu- late absolute production rates for pions and kaons. Finally, we concentrate on the transition from the quark-gluon- plasma ~QGP !@ 11,12# to the hadron gas. The results support the notion that the present model may be taken as a toy model for QCD. The paper is organized as follows: In Sec. II the model is shortly outlined, since the details have been presented in ~I!. In Sec. III we calculate the partition function and give the expressions for the relevant observables. In Sec. IV the model is applied to the description of the QGP. There, we present and discuss the results corresponding to some branching ratios and absolute production rates. Finally, con- clusions are drawn in Sec. V.