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A possible approach to the CEP location Juan Xiong Supervisor: Prof. Jiarong Li IOPP-CCNU 2010-04 Outline Introduction Formulation of the NJL model Boundary of two phases coexistence and the T phase diagram Multiple solutions of the chiral order parameter, quark and pion meson effective mass Summary and discussion Introduction QCD phase diagram (Hadron---QGP) Tricritical point (TCP) and critical end point (CEP) Methods to find the location of CEP Formulation of the Nambu–JonaLasinio (NJL) model The two flavor SU(2) NJL model Lagrangian density is defined as with the quark field, Symmetry: (When the current quark mass. =0 ) m0 0, mu ,d m0 0, SU L (2) SU R (2) SU f (2) NJL model Define the chiral condensate The Schwinger-Dyson equation of quark self-energy in the Hartree approximation has the following form, is the quark propagator in momentum space, NJL model After a direct calculation in imaginary time finite temperature field theory formalism, the chiral condensate reads: The NJL grand potential from the Hartree propagator: In the above formula energy of the quark. is the Hartree quasi-particle are the Fermi-Dirac distribution for the antiquark and quark, respectively. The T phase diagram Generally speaking, there are two ways to obtain the T phase diagram 0 Landau theory of phase transition Boundary of two phases coexistence The QCD chiral phase transition and the liquid-gas phase transition belong to one universal class. As usual, the pressure is defined such that its value is zero in the vacuum state The net quark number density could be calculated with the thermodynamic relation and baryon number density Equilibrium condition: as an order parameter The multiple values of the chiral order parameter The multiple values of the chiral order parameter The quark effective mass The constituent quark mass origins from the quark and antiquark Interaction and is determined by the energy gap in the energy spectrum of quasi-particle excitations The quark effective mass The pion meson mass In the NJL model, meson is the quark-antiquark thermal excitation. Under the standard Hartree Fock approximation (HFA) +random phase approximation (RPA), the full correlation function of pion meson has the form is the polarization tensor of the pion meson which reads In the NJL model, can be easily calculated by solving the equation The pion meson mass The pion meson mass Summary and discussion We analyze the chiral phase transition properties in the real physical world of the two flavor NJL model by the approach of analyzing the liquid-gas phase transition. And obtain the corresponding phase diagram . In the first order phase transition region, the chiral order parameter, the quark effective mass and the thermal excitation pion meson have two physical values. With temperature increasing and quark chemical potential decreasing, two physical values close to each other and coincide at the CEP. We advance a notion which is helpful to find the CEP. In the energy scan in the ultra-relativistic nuclear collision, if the happens on the first order phase transition region, the physical observable quantities related to chiral order parameter could appear the same character.