The Potential Of Quark Meson Model With Finite Temperature

Thermal field theory is a combination of quantum field theory and statistical mechanics.This leads to the situation that it is both difficult and very interesting. The motivation that westudy it is that we want to describe relativistic thermal phenomenon such as the quark-gluonplasma. The quark gluon plasma phase is regarded to take place of the hadron phase whensystem is above some critical temperature. Lattice calculations suggest that criticaltemperature for this transition is about150MEV---300MEV. In the plasma phase the quarksand gluons are deconfined, they can move freely through the whole plasma. This phase isrelated to the evolution of the universe soon after the big bang. In the end of last century,experiments at the collider RHIC has re-created it in the laboratory by making gold nucleicollide together head-on. Similar experiments at much higher energy has performed, forexample, the LHC experiments at CERN.However confirming the existence of the quark gluon plasma phase in experiments isvery difficult. Even though a plasma is indeed produced, how to identify it, as yet there is nosimple answer. Of course, there are very crude estimates based on non-equilibrium theory,which suggest that the plasma will survive for a time long enough that it reaches thermalequilibrium before it eventually decays back into ordinary matter. As theoretical aspect isconcerned, so far, it is only equilibrium thermal field theory that is well formulated.At present, lattice QCD simulation is regarded as the most reliable method ininvestigating QCD phase diagram, which it is a nonperturbative method based on the firstprinciple. But it has obvious limit. This method depends heavily on computer and moreserious problem is one do not know how to deal with dynamical fermions. Therefore,although considerable development has been obtained by this method in QCD phase graphwith zero chemical potential, how to do it reasonably with nonzero chemical potential is still achallenge.On the other hand, methods based on the effective theory is an very important way infinite temperature field. This is a realistic option given that solving QCD theory rigorous isextremely difficult. For example, extensive investigations on finite temperature QCD hasbeen done by using the linear (or nonlinear) sigma models, quark meson models, chiralmodels, NJL model and its extension. Important progress has been obtained. Although thesemodels are not true QCD, they preserve most of the QCD symmetry and set a solid base forinvestigating QCD. An important difficulty in field theory at finite temperature is how to approximate reasonably. At present the common used approximations include the mean fieldapproximation, large N approximation and effective potential method. One finds that differentapproximation may affect the quantitative behaviors and even qualitative behaviors. Thereforeeven though for the same model, different approximation may lead to different conclusion.In this thesis we gave the effective potential of the quark meson model at finitetemperature. The quark meson model is an important effective model in studying the nature ofhadron both at zero temperature and finite temperature. This model consists of two parts: oneof them is linear sigma model, which includes light scalar and pseudoscalar mesons and theirinteraction, the other part consists of the kinetic terms of quarks and the interaction betweenquarks and mesons. This model presents different symmetry according to the number of quarkflavors, but it has chiral symmetry and its spontaneous breaking anyway. We appliedoptimized perturbation method to this model and deduced effective potential in the first order.But two loop diagrams appeared in calculation, which corresponded to the second order in.Renormalization for zero temperature part may affect the nature of a phase transitionquanlitatively, so we renormalized the zero temperature part by using the modified minimalsubtraction scheme. Finally we obtained the analytic expressions for the first order effectivepotential, which can be used to calculate critical temperature and other quantities.