The Study About The Thermodynamic Properties Of Hadronic Matter And The Non-perturbative CJT Method

The study of the hadronic matter has an important role in low energy nuclear physics, high energy heavy ion collision experiments and astrophysics. As we know, hadrons are composed of quarks. In principle, the properties of hadrons should be described by the fundamental quark level dynamic theory of quantum chromodynam-ics(QCD). However the difficulties of nbn-perturbative calculation in QCD and the complexity of strong interactions between hadrons at low energy make this proposal extremely hard to be fulfilled. At present, the usually method is to construct effective field theory(EFT) to bridge QCD theory to hadronic physics. The EFT should respect the QCD symmetries like Lorentz covariance, parity conservation, time-reversal and charge-conjugation invariance, isospin symmetry and spontaneously broken chi-ral symmetry, etc. There are many representations of low energy EFT. Among them the quantum hadrodynamics(QHD) is a very successful low energy EFT. Our work is based on hadronic level, mainly using QHD to discuss thermodynamics of hadronic matter. Meanwhile we have made some discussions on non-perturbative calculation technics.The theory basis in studying thermodynamics in field theory is finite temperature field theory. We have illustrated its imaginary time formalism and real time formalism. Our discussion is based on imaginary time formalism. As studying strong interacting hadronic matter, the non-perturbative calculation method is very important. We have mainly introduced mean field theory(MFT) and the CJT resumation technics. MFT is well used and very successful in studying nuclear matter. Its main idea is to treat the meson fields as the classic fields, which are determined by calculating their expectation value through many body theory or by the thermodynamic equilibrium condition. MFT result can also be obtained by self-consistently calculating the proper nucleon self-energy in Hartree or Hartree-Fock approximation. CJT method is also a non-perturbative approach in field theory. It is developed from usual technics in generating functional formalism with external sources. It is called CJT for it had been first studied by Cornwall, Jackiw and Tomboulis. The non-perturbative effects have been included by self-consistently resuming certain classof infinite higher order corrections. We have illustrated its general formalism and further discussed its application in 4 theory.The nuclear matter is a nuclear many-particle system which is mainly composed of nucleons with mesons mediating the interactions. The concept of nuclear matter is first generalized from the study of the finite nuclei. In the early years the reliable treatment in studying finite nuclei is the Brueckner-Hartree-Fock(BHF) method. By determining the interactions between nucleons through nucleon-nucleon scattering experiments, it had been used to describe the finite nuclei and nuclear matter very well. Later years it had been generalized to relativistic situation in which it was called Dirac-Brueckner-Hartree-Fock(DBHF). Meanwhile Walecka and others had developed the relativistic quantum dynamic theory based on hadronic level which was called quantum hadrodynamics(QHD). By using the mean field approach they had made very successful descriptions of the finite nuclei and nuclear matter. With the development of finite temperature field theory, it had been also generalized to study the thermodynamics of nuclear matter. When studying the nuclear matter at low temperature, one finds a liquid-gas type phase transition. One part of work is to discuss the phase structure of the nuclear matter. Particularly we have discussed the effective nucleon mass under the condition of this phase transition. It is found that the effective nucleon mass changes non-monotonically with temperature and density when transition occurs. We further determined the effective nucleon masses of the liquid phase and gas phase in coexistent area. Interestingly, other recent works show that at high temperature there is also a phase transition in nuclear matter in which the effective nucleon mass also has non-monotonically changing characteristic.In study nuclear matter, the QHD-I model in mean field approximation is very successful in many aspects. However, in some aspects as nucleon effective mass dropping too fast and a larger compression modulus, these results are not consistent with the empirical observations in nuclear matter. In later years, Zimanyi and Moszkowski proposed derivative coupling model (ZM) to overcome these shortcomings. The main point in ZM model was that they had introduced nonlinear coupling constant into the QHD-I model. In our work we have expanded the nonlinear coupling constant and analysis how the effective mass has been modified in order by order. In this way we find that to introduce this nonlinear coupling constat is equivalent to addtwo opposite potentials: attractive and repulsive potential. When these two potentials compete to infinite order, the effective nucleon mass has been changed to drop smoothly with temperature or density. Furthermore we find that the mechanics of high temperature phase transition like in QHD-I model has been suppressed in ZM model.In recent experiments of relativistic heavy ion collisions, large amount of pions have been created. These attract lots of interests in studying a kind of pion matter. Usually one can use statistic virial expansion, chiral perturbation theory or effective hadronic models to investigate the pion matter. In our work we have used the linear sigma model to study the thermodynamics of the pion matter through the CJT formalism. We have introduced a pion isospin chemical potential into the linear sigma model. In the chiral limit, we mainly discussed the thermodynamic pictures of the chiral phase transition in the pion matter. The effective meson mass and order parameter have been analyzed at finite temperature and density. We find that the chiral phase transition is like a liquid-gas phase transition from low density to high density. We further determine the Maxwell construction and give the — T phase diagram of chiral phase transition. In studying pion matter, we find that Bose-Einstein condensation(BEC) will happen in this system. We have studied how it happens in this system and also given its — T phase diagram. When BEC and chiral phase transition are associated together we have give the united — T phase diagram of them. We find that chiral phase transition can only happen in the non-BEC area, while BEC can take place in not only chiral symmetry broken phase but also chiral symmetry restored phase.In study of the matter of strong interaction, whether in strong coupling quark matter or in strong interacted hadronic matter, one relies on the non-perturbative calculation. At present the usual approach in nuclear matter is MFT while that in strong coupling quark matter is lattice QCD. With the study of strong interaction deepened, one should develop more systematically non-perturbative method in field theory. In this direction, the CJT formalism is relatively a well-developed non-perturbative calculation in field theory at present. Recently this approach have been attracted much attention. This approach has been already applied to study the non-perturbative QCD calculation and the results are found very consistent with thelattice calculation. However in nuclear matter there are no parallel investigations. The last part of our work is to apply CJT approach to study the nuclear matter. As the first step, we have neglected the medium effects to the mesons, the momentum dependent of effective nucleon mass and the fluctuations of the vacuum, then we have obtained the final results which are found to be very consistent with MFT. In the whole calculation, the thermodynamic consistency has been preserved.