1

Yoko Ogawa (RCNP/Osaka)

Hiroshi Toki (RCNP/Osaka)

Setsuo Tamenaga (RCNP/Osaka)

Hong Shen (Nankai/China)Atsushi Hosaka (RCNP/Osaka)

Satoru Sugimoto (RIKEN)

Kiyomi Ikeda (RIKEN)

Parity projected relativistic mean field theory

for extended chiral sigma model

2

IntroductionThe purpose of this study is to understand the properties of finite nuclei by using a chiral sigma model with pion mean field within the relativisticmean field theory.

Toki, Sugimoto and Ikeda demonstrate the occurrence of surface pion condensation. Prog. Theor. Phys. 108 (2002) 903.

Chiral symmetry : Linear sigma model in hadron physics

M. Gell-Mann and M. Levy,Nuovo Cimento 16(1960)705.

Spontaneous chiral symmetry breaking

Pion :Mediator of the nuclear force

Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122(1961)345.

H. Yukawa, Proc. Phys.-Math.Soc.Jpn., 17(1935)48.

Application of extended chiral sigma model for finite nuclei(N=Z even-even). Prog. Theor. Phys. 111(2004) 75.

Problem of now frameworkParity projection

Summary

Parity projected relativistic Hartree equations

Contents

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Lagrangian

Linear Sigma Model

4

Extended Chiral Sigma Model Lagrangian(ECS)

Dynamical mass generation term for omega mesonJ. Boguta, Phys. Lett. 120B(1983)34

Non- linear realization

New nucleon field

5

Mean Field Equation

Parity mixed single particle wave function

Dirac equation

Klein-Gordon equations

6

-20

-15

-10

-5

0

5

E /A - M (MeV)

0.250.200.150.100.050.00

ρ ( fm-3)

-400

-200

0

200

400

Potential (MeV)

0.250.200.150.100.05

ρ ( fm-3)

Uv

Us

ρ = 0.1414 fm-3

E/A-M = -16.14 MeV

K = 650 MeV

g = M / f

g = m / f 3~

m = 777 MeV

g =7.0337

Free parameter

m = 783MeVm = 139 MeVM = 939 MeV

f = 93MeV

Hadron property

ECS

TM1(RMF)

Effective mass :M* = M + g m* = m + g

~

Saturation property= (m

2 - m2) / 2f

= 33

Non-linear coupling

Character of the ECS model in nuclear matterLarge incompressibility Small LS-force

~ 80 %

7

Finite Nuclei

TM1(RMF)

ECS model (without pion)

ECS model (with pion)

Y. Ogawa, H. Toki, S. Tamenaga, H. Shen, A. Hosaka, S. Sugimoto and K. Ikeda,Prog. Theor. Phys. Vol. 111, No. 1, 75 (2004)

9.4

9.2

9.0

8.8

8.6

8.4

8.2

8.0

B.E./A (MeV)

80706050403020

A (Mass number)

8

Single Particle Spectrum

-50

-40

-30

-20

-10

0

Single Particle Energy (MeV)

40Ca

44Ti

48Cr

52Fe

56Ni

60Zn

64Ge

68Se32

S36

Ar

0s1/2

1s1/2

0p1/20p3/2

0d3/20d5/2

0f5/2

0f7/2

Extended Chiral Sigma model(without pion)

-50

-40

-30

-20

-10

0

Single Particle Energy (MeV)

40Ca

44Ti

48Cr

52Fe

56Ni

60Zn

64Ge 68

Se32

S 36Ar

Extended Chiral Sigma model(with pion)

~

~0s1/2

~

~

~

~

~

~

0f7/2

0f5/2

1s1/2

0d3/2

0d5/2

0p1/2

0p3/2

Large incompressibilityIt is hard energetically to change a density.The state with large L bounds deeper.

Anomalous pushed up 1s-state.

N = 18Without pion With pion

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The magic number appears at N = 18 instead of N = 20.

Large incompressibility Anomalous pushed up 1s 1/2 state

The effect of Dirac sea

Parity projection

We use the parity mixing intrinsic state in order to treat the pion mean field in themean field theory because of the pseudovector(scalar) character of pion.

We need to restore the parity symmetry and the variation after projection.

The Problem and improvement of framework

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Parity Projection

Single particle wave function

Total wave function

0+

0-

1h-state

2h-state

1p-1h

2p-2h

H. Toki, S. Sugimoto, K. Ikeda, Prog. Theor. Phys. 108 (2002) 903.

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N. Kaiser, S. Fritsch, W. Weise, Nucl. Phys. A697(2002)255

0- 0-

2p-2h K = 255 MeV

K = 250 + 25 MeV_Experiment

12

g 7/2

Fermi surface

Fermi surface 56Ni

40Ca

On the other hand, in 40Ca case the j-upper state is far from Fermi level.

In 56Ni case the j-upper stateis Fermi level.

0- 0-

13

Hamiltonian density

Hamiltonian

14

Field operator for nucleon

Creation operator for nucleon in a parity projected state

Parity projected wave function

Total energy

15

Parity-projected relativistic mean field equations

Nucleon part

Variation after projection

16

Meson part

We solve these self-consistent equations by using imaginary time step method.

17

Difficulties of relativistic treatment Total energy minimum variation condition gives difficulty to the relativistictreatment, because the relativistic theory involves the negative energy states.

Summary

We avoid this problem due to elimination of lower component. We howevertreat the equation which is mathematically equal to the Dirac equation.

K. T. R. Davies, H. Flocard, S. Krieger, M. s. Weiss, Nucl. Phys. A342 (1980)111.

P. G. reinhard, M. Rufa, J. Maruhn, W. Greiner, J. Friedrich, Z. Phys. A323, (1986)13.

We derive the parity projected relativistic Hartree equations.

We show the problem in now framework of ECS model.

Magic number at N = 20 ?

Prediction of 0- state

Large incompressibility. Magic number at N = 20.