Dyson-Schwinger Equation Approach To QCD Chiral Phase Transition

This thesis is devoted to use the Dyson-Schwinger equation to study the QCD chiral phase transition at finite temperature and chemical potential. At first Ⅰ give a brief introduction to the elementary knowledge in chapter1about the quark model, the QCD and the prevailing viewpoint about the QCD phase diagram.Since the study of the chiral phase transition at finite temperature and density will forefront a strong correlation system, so that the conventional perturbative expansion method is invalidated. So that many nonperturbative methods are generally used and developed, such as the lattice QCD, the QCD sum rule, the chiral perturbation theory and the NJL model etc. Dyson-Schwinger equations is one of them, it has good QCD foundations and provides a continuum tool for the exploration of QCD chiral phase transition. In chapter2some topics about the Dyson-Schwinger equations are intro-duced and an explicit presentation of the DCSB at low energy scale by solving the gap equation is given.After decades years of study, a generally accepted picture is that the chiral phase transition of QCD with two flavors at zero temperature and finite chemical potential is of first order and terminates at a critical end point (the CEP), while at finite temper-ature and vanishing chemical potential the transition becomes a crossover. Detecting the CEP has been an important goal of worldwide experiments in relativistic heavy ion collisions. But the accurate position of this point from theoretical calculations is still under debate. In chapter3I shall to plot the QCD phase diagram and locate the critical end point (CEP) in the framework of the Dyson-Schwinger Equations with a generally used rank-2confining separable model gluon propagator. We first show that the rank-2separable model is validate in the study of the chiral phase transition at fi-nite temperature, then we extrapolate it to both finite temperature and finite chemical potential by replacing ω with ωn=ωn+iμ. In this chapter we use B(0,μ02), i.e. the scalar part of the inverse dressing quark propagator at lowest frequency and zero momentum, as the chiral order parameter, and use it’s derivative with respect to the current quark mass m (the chiral susceptibility) and the bag constant as the criteri-on of the chiral phase transition. Our result indicates that a critical end point exists at (TCEP,μCEP)/Tc=(0.735,1.768). In addition to this, our work also shows that the first order phase transition might not end at one point but experiences a domain in which a meta-stable state and a stable state can coexist. At the end of that chapter, we prove the equality of the differently defined chiral susceptibilities.The critical phenomenon in the neighborhood of CEP is an interesting problem and has attracted many concerns. According to the notion of universality, the critical exponents is only determined by the symmetries of the theory and the spatial dimen-sion. So that many different theories with the same exponent values can be grouped into a single universality class. The universality class of the phase transition at the crit-ical end point is considered to be the same as that of the three-dimensional Ising model. But at the tricritical point, the universality class can be described by a mean field theo-ry. Some works reported that there is the hidden TCP effect in the neighborhood of the CEP. In this chapter we will show the origin of this effect and indicate that this might be a model dependent result by studying the chiral susceptibility in the framework of Dyson-Schwinger Equations. From our numerical result, it is found that as the mass limit to zero, the position of CEP limit to TCP faster and faster while the height of the chiral susceptibility in the crossover domain limit to infinity faster and faster which in-dicate the transition from CEP to TCP is continuous. It is this continuity that causes the so called hidden TCP effect. We assert that the hidden TCP effect and the continuity from CEP to TCP must coexist. Here we note that if the current quark mass is at it’s physical value6.6MeV, the critical region of the CEP is about10MeV and the TCP hardly has any effect on the critical domain of CEP. This property is definitely different from that of [Y. Hatta, T. Ikeda, Phys. Rev. D67,014028(2003)] which indicates the hidden TCP plays an important role on the critical behavior in the neighborhood of CEP. We think that this difference is caused by the fact that in any model there is a range of current quark mass value in which the TCP effect is prominent, if the current quark mass beyond this range the TCP effect would be neglect. The value used in [Y. Hatta, T. Ikeda, Phys. Rev. D67,014028(2003)] just lies in this range and that of our models just lies out of this range.