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

23/4/19 Hubei University, Wuhan 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

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Motivation Mean field theory (MFT) in the study of n

uclear matter

QHD model, RHA, HF Develop non-perturbative calculation meth

od beyond MFT in studying nuclear matter

CJT resummation scheme

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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

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Relativistic Hartree approximation (RHA)

propagators: G (N), ( ) , D ( )

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Resumming all the tadpole diagram

vacMFTRHA

vacwhere

zero-point energy from vacuum

S. A. Chin, Ann. of Phys. 108 (1977) 403

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Introduction of CJT formalism Effective action of composite operators

)( ),( G

Where: )()( xx

)()(),( yxTyxG

J. Cornwall, R. Jackiw and E. Tomboulis, Phys. Rev. D10 (1974) 2428

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Effective potential (translational invariance)

Stationary condition

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The application of CJT method in studying nuclear matter The formulation (QHD-I model)

the bare propagators:

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Using imaginary-time formalism

The thermodynamical potential

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where

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From the stationary condition

we have

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The diagrammatic representation of the gap equations

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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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Approximation 2:

Neglect the fluctuations of the vacuum

where

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Approximation 3:

Neglect the momentum dependence of the effective nucleon mass

We have

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Analysis of the thermodynamic consistency

The stationary condition requires

0G

G is a function of , so we have

which ensures the thermodynamic consistency

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The thermodynamic functions

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The numerical results When T=0,

The binding energy curve ,6.1552 g 5.5212 g

0

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T=0, state equation and effective mass

CJT 155.6 521.5 0.17 -15.7 0.55 485

MFT 103.5 156.9 0.17 -15.7 0.55 548

2g

2g )( 3

0fm )(/ MeVM MM /* )(MeVK

0

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When 0T

The effective nucleon mass as a function of temperature

The isotherms of pressureversus density (liquid-gas phase transition)

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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

0

M0

M

these requirements will generate two coupled equations

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together with the equations of nucleon self-energy

these four equations can be numerically solved

(1)

(2)

(3)

(4)

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The numerical resultsWhen 00 T ,

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When 00 T ,

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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.

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谢 谢！