Effective Model Studies On The Phase Transitions Of Strongly Interacting Matter

Strong interaction is one of the four basic interactions of our nature, in which the physical objects are quarks and gluons. There is no free quarks or gluons in the nature, quarks and gluons are confined in the hadron through complex interactions. An important concept related to the strong interaction is chiral symmetry breaking, which is the origin of most masses. Properties of strongly interacting matters are always the frontier research field of theoretical and experimental physics, many countries and laboratories have put a lot of resources and energies into their researches, of which the phase transitions are very important directions. The strongly interacting matter phase transitions are also important for some astrophysical problems, such as understanding the explosion mechanism of supernova, analyses of the internal structure of compact stars, and also the evolution of the universe. Quantum chromodynamics (QCD) is the basic theory of the strong interaction, and has been proved to be a successful theory. When the energy is high enough, the perturbative theory of QCD can give very good results; however, there is still many problems in the more interesting non-perturbative regime, so we can not solve it directly nowadays, and often need to develop some effective field theory models that meet the based spirits of QCD. In this dissertation, using the widely accepted models, such as the (P)NJL model, the Dyson-Schwinger equations and the three-dimensional quantum electrodynamics, we do some studies on some problems related to the phase transitions of strongly interacting matter.We introduce the related theory of the strong interaction matter and basic research contents in Chapter One, as well as the theoretical and experimental research statuses both at home and abroad. We also explain some important concepts that are related to this work, and then describe the specific contents of the study strongly interacting matter phase transition together with the critical end point briefly. In the subsequent Chapter Two, we discuss the Wigner solution of quark gap equation beyond the chiral limit firstly, for the case of nonzero temperature and finite quark chemical potential. Its effects on the chiral phase transition will also be discussed. People usually think that beyond the chiral limit the quark gap equation only has the Nambu solution, and there is no Wigner solution; however, when we introduce the lowest order contribution of quark condensate into the gluon propagator as a feedback, which is inspired by the results of QCD sum rules, the results show that the Wigner solution and the Nambu solution can coexist. This is very interesting in studying the phase transitions of QCD. Based on this, we also discuss the properties of the Wigner solution at finite tempera-ture and finite chemical potential. Then, using the pressure difference between Nambu phase and Wigner phase (the bag constant) as the order parameter, we obtain a possi-ble QCD phase diagram. In Chapter Three we study the crossover region of the chiral phase transition by several vacuum susceptibilities. We introduce and then explain the definitions and relations between various vacuum susceptibilities. We also discuss the rationality of using susceptibilities to analyse the crossover region and the critical end point. The results show that, at low temperature and high density region vari-ous susceptibilities almost give same critical chemical potential of the first-order phase transition, which may due to they are coupled in mathematical form; however, in the crossover region where the temperature is high enough, different susceptibilities give different results, indicate that there is some uncertainties in this region of the system. Therefore, in our opinion it is difficult to define a single critical line in the crossover region, while to define a critical band may be a more suitable choice. Chapter Four is based on Chapter Two, where the Wigner solution of the gap equation beyond chi-ral limit and its influences on QCD phase transitions are discussed, in a model where the confinement effect is included. We also discuss the relationship between the chiral phase transition and the deconfinement. The numerical results show that, the effect of the Polyakov loop on the thermodynamic potential is much larger than that of the quark condensate, and after considering the coexistence of the Wigner solution and the Nambu solution, the chiral phase transition at finite temperature and zero chemical po-tential case might be first-order and happen earlier than the deconfinement. However, further discussions show that the weight factor of the influence of the quark propaga-tor on the gluon propagator may be crucial for drawing some reliable conclusions, for smaller weight of the influence of the quark condensate to the gluon propagator, the coexistence region of the Wigner solution with the Nambu solution may decrease and even disappear, and the first-order chiral phase transition found above may degenerate to a crossover. However, we found that the changes of Wigner solution do not show obvious impact on the deconfinement phase transition. In Chapter Five we use the sim-ilarity of the three-dimensional quantum electrodynamics and QCD to discuss relevant problems, mostly focus on the properties of the chiral phase transition near the criti-cal number of fermion flavor. For this we briefly introduce the theory and application of three-dimensional quantum electrodynamics, and then numerically solve the corre-sponding fermion-boson coupled Dyson-Schwinger equations for the bare vertex and simplified BC vertex under Landau gauge. The results show that, in the bare vertex approximation the system undergoes a high order continuous phase transition from the Nambu phase to the Wigner phase when closing to the critical number of fermion fla-vors, and if we use the simplified BC vertex, the corresponding chiral phase transition is now a typical second order one. This tells us that the choice of the vertex to the study of chiral phase transition is very important, it is.no doubt that this problem is worth to do further studies. In Chapter Six we introduce the effects of chemical potential into the gluon propagator, and then discuss the QCD phase transition at finite temperature and finite chemical potential, especially the position and properties of the critical end point(CEP). We show how to consider the effect of the chemical potential fisrt, and then solve the gap equation beyond chiral limit, the results show that there is a region that different phases can coexist. We also propose a more general way to calculate the chiral condensate and the chiral susceptibility (then people do not need to introduce a cutoff manually to eliminate the divergence), and then discuss the physics of the first order phase transition and the crossover region, the position of CEP is determined as well. Chapter Seven is some works that are not completed yet, including the influences of different regularization schemes on the results in the NJL model, as well as effects of the chiral chemical potential on the QCD phase transitions and the critical end point. We found that using the two cutoff proper time regularization, the dependence of the results on the choice of parameters are really small, which for example is obviously not the case in the three momentum cutoff scheme, and so this can be regarded as an ad-vantage besides it can mimic the confinement; and the introduction of chiral chemical potential is helpful for people to avoid the sign problem of lattice QCD, and then do the discussions of the phase transitions (especially some information about critical end point) of strongly interacting matter, so of course is worthy of further studies. Chapter Eight is a summary of the whole dissertation, also the outlook for further studies.