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

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

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

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

Hubei University, Wuhan Relativistic Hartree approximation (RHA) propagators: G (N), ( ) , D ( ) 2018/9/19 Hubei University, Wuhan

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

Introduction of CJT formalism Effective action of composite operators Where: J. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428 2018/9/19 Hubei University, Wuhan

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

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

Hubei University, Wuhan Using imaginary-time formalism The thermodynamical potential 2018/9/19 Hubei University, Wuhan

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

Hubei University, Wuhan From the stationary condition we have 2018/9/19 Hubei University, Wuhan

Hubei University, Wuhan The diagrammatic representation of the gap equations 2018/9/19 Hubei University, Wuhan

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 2018/9/19 Hubei University, Wuhan

Hubei University, Wuhan Approximation 2: Neglect the fluctuations of the vacuum where 2018/9/19 Hubei University, Wuhan

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

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 2018/9/19 Hubei University, Wuhan

Hubei University, Wuhan The thermodynamic functions 2018/9/19 Hubei University, Wuhan

Hubei University, Wuhan The numerical results When T=0, The binding energy curve 2018/9/19 Hubei University, Wuhan

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 2018/9/19 Hubei University, Wuhan

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

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 2018/9/19 Hubei University, Wuhan

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

Hubei University, Wuhan The numerical results When 2018/9/19 Hubei University, Wuhan

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

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

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

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