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. Quantum 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, and transitions from the hadronic phase to the quark-gluon plasma are calculated.
Highlights
Schematic models have been very important in order to understand basic concepts in, e.g., nuclear physicspairing, quadrupole interaction, quantum phase transition from spherical to deformed nuclei, etc.͒
In this work we have introduced the essentials of a toy model for QCD
The model consists of two levels with energies Ϯ f, which describe the fermion degrees of freedom, and gluons are introduced via a level of positive energy which can be filled by gluon pairs with spin zero
Summary
Schematic models have been very important in order to understand basic concepts in, e.g., nuclear physicspairing, quadrupole interaction, quantum phase transition from spherical to deformed nuclei, etc.͒. As it is seen from the figure, the results can be interpreted in terms of equal population of fermion and gluon pairs (V1 Ͻ0.008 GeV), fermionic dominance (0.008ϽV1 Ͻ0.015 GeV), and gluonic dominance (V1Ͼ0.015 GeV) Another conclusion concerns the structure of the vacuum state, i.e., it will contain a finite number of quark-antiquark pairs and gluons, whose ratio depends on the strength of the interaction, which in turn has to be adjusted to experiment in the more general model with open flavor and spin17͔. After the analysis of the properties of the low lying spectrum of the two Hamiltonians4͒, we can proceed to discuss finite temperature effects, which may be relevant for the description of the transition from the hadronic phase to the QGP
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