the CJT Calculation Beyond MFT and its Results in Nuclear Matter

1. the CJT Calculation Beyond MFT and its Results in Nuclear Matter 舒 崧 湖北大学物理学与电子技术院 二零零六年十月二十九日于桂林

2. Outline • Motivation • General introduction of CJT formalism • The application of CJT method in studying nuclear matter • The mean field approximation in the CJT calculation and the comparison to the usual MFT results • The beyond MFT calculation and the results in the CJT calculation • Summary and outlook Hubei University, Wuhan

3. Motivation • Mean field theory (MFT) in the study of nuclear matter QHD model, RHA, HF • Develop non-perturbative calculation method beyond MFT in studying nuclear matter CJT resummation scheme Hubei University, Wuhan

4. MFT in studying nuclear matter • Quantum hadrodynamics (QHD) simple Walecka model (QHD-I) MFT: B.D.Serot, J.D.Walecka, Adv. Nucl. Phys.,Vol.16 (1986) 1 Hubei University, Wuhan

5. Relativistic Hartree approximation (RHA) propagators: G (N), ( ) , D ( ) Hubei University, Wuhan

6. Resumming all the tadpole diagram where zero-point energy from vacuum S. A. Chin, Ann. of Phys. 108 (1977) 403 Hubei University, Wuhan

7. Introduction of CJT formalism • Effective action of composite operators Where: J. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428 Hubei University, Wuhan

8. Effective potential (translational invariance) Stationary condition Hubei University, Wuhan

9. The application of CJT method in studying nuclear matter • The formulation (QHD-I model) the bare propagators: Hubei University, Wuhan

10. Using imaginary-time formalism The thermodynamical potential Hubei University, Wuhan

11. where Hubei University, Wuhan

12. From the stationary condition we have Hubei University, Wuhan

13. The diagrammatic representation of the gap equations Hubei University, Wuhan

14. The mean field approximation in the CJT calculation • Solving the gap equations according to the mean field requirements at the first step • Approximation 1: replace the full meson propagators by the bare ones which means we have neglected the medium effect of the mesons Hubei University, Wuhan

15. Approximation 2: Neglect the fluctuations of the vacuum where Hubei University, Wuhan

16. Approximation 3: Neglect the momentum dependence of the effective nucleon mass We have Hubei University, Wuhan

17. Analysis of the thermodynamic consistency The stationary condition requires G is a function of , so we have which ensures the thermodynamic consistency Hubei University, Wuhan

18. The thermodynamic functions Hubei University, Wuhan

19. The numerical results • When T=0, The binding energy curve Hubei University, Wuhan

20. T=0, state equation and effective mass Hubei University, Wuhan

21. When The isotherms of pressure versus density (liquid- gas phase transition) The effective nucleon mass as a function of temperature Hubei University, Wuhan

22. The beyond MFT calculation and the results in the CJT calculation • The medium effects of mesons will be included in a thermodynamic consistent way The effective mass of and meson will be determined by minimizing the thermodynamic potential with respect to the effective mass these requirements will generate two coupled equations Hubei University, Wuhan

23. (1) (2) together with the equations of nucleon self-energy (3) (4) these four equations can be numerically solved Hubei University, Wuhan

24. The numerical results • When Hubei University, Wuhan

25. When Hubei University, Wuhan

26. Hubei University, Wuhan

27. Summary and outlook • The CJT method has been applied to study the nuclear matter. • We have reproduced the MFT results in the CJT calculation after taking certain approximation. • The beyond mean field results are obtained by self-consistently calculating the meson effective mass. The temperature and density dependences of the effective mass has been displayed. • The thermodynamic consistency has been preserved in our calculation. Hubei University, Wuhan

28. 谢 谢！ Hubei University, Wuhan