1. 0 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. 1 Introduction The purpose of this study is to understand the properties of finite nuclei by using a chiral sigma model with pion mean field within the relativistic mean 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 framework Parity projection SummaryParity projected relativistic Hartree equations Contents

3. 2 Lagrangian Linear Sigma Model

4. 3 Extended Chiral Sigma Model Lagrangian(ECS) Dynamical mass generation term for omega meson J. Boguta, Phys. Lett. 120B(1983)34 Non- linear realization New nucleon field

5. 4 Mean Field Equation Parity mixed single particle wave function Dirac equation Klein-Gordon equations

6. 5  = 0.1414 fm -3 E/A-M = -16.14 MeV K = 650 MeV g  = M / f   g  = m  / f   ~ m  = 777 MeV g  =7.0337 Free parameter m  = 783MeV m  = 139 MeV M = 939 MeV f  = 93MeV Hadron property ECS TM1(RMF) Effective mass : M* = M + g   m*   =  m   + g   ~ Saturation property = (m  2 - m  2 ) / 2f    =  Non-linear coupling Character of the ECS model in nuclear matter Large incompressibility Small LS-force ~ 80 %

7. 6 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)

8. 7 Single Particle Spectrum Large incompressibility It is hard energetically to change a density. The state with large L bounds deeper. Anomalous pushed up 1s-state. N = 18 Without pionWith pion

9. 8 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 the mean 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

10. 9 Parity Projection Single particle wave function Total wave function 0+0+ 0-0- 1h-state 2h-state 1p-1h 2p-2h H. Toki, S. Sugimoto, K. Ikeda, Prog. Theor. Phys. 108 (2002) 903.

11. 10 N. Kaiser, S. Fritsch, W. Weise, Nucl. Phys. A697(2002)255 0-0- 0-0-  2p-2hK = 255 MeV K = 250 + 25 MeV _ Experiment

12. 11 g 7/2 Fermi surface 56 Ni 40 Ca On the other hand, in 40 Ca case the j-upper state is far from Fermi level. In 56 Ni case the j-upper state is Fermi level. 0-0- 0-0- 

13. 12 Hamiltonian density Hamiltonian

14. 13 Field operator for nucleon Creation operator for nucleon in a parity projected state  Parity projected wave function Total energy

15. 14 Parity-projected relativistic mean field equations Nucleon part Variation after projection

16. 15 Meson part We solve these self-consistent equations by using imaginary time step method.

17. 16 Difficulties of relativistic treatment Total energy minimum variation condition gives difficulty to the relativistic treatment, because the relativistic theory involves the negative energy states. Summary We avoid this problem due to elimination of lower component. We however treat 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.