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. Hubei University, Wuhan
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
2018/9/19
Hubei University, Wuhan

3. Hubei University, Wuhan
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
2018/9/19
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
2018/9/19
Hubei University, Wuhan

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

6. Hubei University, Wuhan
Resumming all the tadpole diagram
where
zero-point energy from vacuum
S. A. Chin, Ann. of Phys. 108 (1977) 403
2018/9/19
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
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Hubei University, Wuhan

8. Hubei University, Wuhan
Effective potential (translational invariance)
Stationary condition
2018/9/19
Hubei University, Wuhan

9. The application of CJT method in studying nuclear matter
The formulation (QHD-I model)
the bare propagators:
2018/9/19
Hubei University, Wuhan

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

11. Hubei University, Wuhan
where
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Hubei University, Wuhan

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

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

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

16. Hubei University, Wuhan
Approximation 3:
Neglect the momentum dependence of the effective nucleon mass
We have
2018/9/19
Hubei University, Wuhan

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

18. Hubei University, Wuhan
The thermodynamic functions
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Hubei University, Wuhan

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

20. Hubei University, Wuhan
T=0, state equation and effective mass
CJT
155.6
521.5
0.17
-15.7
0.55
485
MFT
103.5
156.9
548
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Hubei University, Wuhan

21. Hubei University, Wuhan
When
The isotherms of pressure
versus density (liquid-
gas phase transition)
The effective nucleon mass
as a function of temperature
2018/9/19
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
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Hubei University, Wuhan

23. Hubei University, Wuhan
(1)
(2)
together with the equations of nucleon self-energy
(3)
(4)
these four equations can be numerically solved
2018/9/19
Hubei University, Wuhan

24. Hubei University, Wuhan
The numerical results
When
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Hubei University, Wuhan

25. Hubei University, Wuhan
When
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Hubei University, Wuhan

26. Hubei University, Wuhan
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Hubei University, Wuhan

27. Hubei University, Wuhan
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.
2018/9/19
Hubei University, Wuhan

28. Hubei University, Wuhan
谢 谢！
2018/9/19
Hubei University, Wuhan