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
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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
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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
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Mean Field Equation
Parity mixed single particle wave function
Dirac equation
Klein-Gordon equations
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-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 %
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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)
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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
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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-
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Hamiltonian density
Hamiltonian
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Field operator for nucleon
Creation operator for nucleon in a parity projected state
Parity projected wave function
Total energy
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Parity-projected relativistic mean field equations
Nucleon part
Variation after projection
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Meson part
We solve these self-consistent equations by using imaginary time step method.
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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.