1. EOS of Asymmetric Nuclear Matter Beijing, Aug. 2005 W. Zuo Institute of Modern Physics, LanZhou, China

2. Collaboration I. Bombaci Pisa University A. Lejeune IPN, Liege Z. H. Li, G.C.Lu IMP, Lanzhou U. Lombardo INFN-LNS, Catania J. F. Mathiot Blaise-Pascal Uni. H.-J. Schulze INFN, Catania C.W.Shen, L.G.Cao INFN-LNS, Catania B. A. Li Arkansa State University

3. Introduction (Motivation) Theoretical approaches BHF approach, TBF Results Symmetry enery, EOS at finite Tempertature, TBF effects Summary Outline

4. Motivations EOS of asymmetric nuclear matter, especially EOS of asymmetric nuclear matter, especially High-density behavior of symmetry energy High-density behavior of symmetry energy ---- New Challenge ! ---- New Challenge ! P. Danielewicz et al., Science 298(2002)1592; B.A.Li, PRL88(2002)192701 P. Danielewicz et al., Science 298(2002)1592; B.A.Li, PRL88(2002)192701 M. Di Toro, Phys.Rep. to appear M. Di Toro, Phys.Rep. to appear Nuclear Physics Nuclear Physics 1) The properties of neutron rich nuclei 1) The properties of neutron rich nuclei I. Tanihata, NPA 616 (1997) 560; T. Glasmachet et al., PLB 395 (1997) I. Tanihata, NPA 616 (1997) 560; T. Glasmachet et al., PLB 395 (1997) 2) Strong correlation between the neutron skin thinkness 2) Strong correlation between the neutron skin thinkness and the slope of symmetry energy and the slope of symmetry energy 3) Heavy ion collisions 3) Heavy ion collisions B. A. Li et al., Int. J. Mod. Phys. E7 (1998) 147 B. A. Li et al., Int. J. Mod. Phys. E7 (1998) 147

5. Motivations Implications for astrophysics Implications for astrophysics M.Prakash et al., Phys. Rep. 280(1997)1; M.Prakash et al., Phys. Rep. 280(1997)1; C.J.Pethick, Rev. Mod. Phys. 64(1992)1133; C.J.Pethick, Rev. Mod. Phys. 64(1992)1133; Lect. Notes Phys., 578 (2001) Lect. Notes Phys., 578 (2001) 1) Sturctures of neutron stars 1) Sturctures of neutron stars EOS of ANM is a basic input of the nutron star EOS of ANM is a basic input of the nutron star structure model structure model 2) Chemical Compositions of neutron stars 2) Chemical Compositions of neutron stars determined by symmetry energy determined by symmetry energy 3) Cooling of neutron stars 3) Cooling of neutron stars Fast cooling via direct URCA process Fast cooling via direct URCA process

6. Oyamatsu et al., NPA634(1998)3. Properties of Neutron-rich Nuclei

7. R.J.Furnstahl, NPA706(2002)85. Correlation between symmetry energy and neutron skin thinkness

8. B.A.Li, PRL88(2002)192701. Heavy ion collisions

9. J.M. Lattimer and M. Prakash, Science Vol. 304 (2004) 536-542. Matter in neutron stars

10. Lattimer et al., PRL66(1991)2701. Composition of neutron star matter (n,p,e,μ)

11. Lattimer and Prakash, Science 304(2004)536. Cooling of neutron stars Condition for dURCA ： Proton fraction is determined by symmetry energy Momentum conservation

12. Neutron Star Structure X.R.Zhou et al., PRC69(2004)018801 TOV equation

13. Theoretical Approaches  Skyrme-Hartree-Fock  Relativistic Mean Field Theory, Relativistic Hartree-Fock  Variational Approach  Brueckner-Hartree-Fock Approach  Dirac-Brueckner Approach  Effective Field Theory

14. B.A.Brown, PRL85(2000)5296 Theoretical predictions of symmetry energy 各种理论模型预言的对称能的密度依赖存在很大的分歧！ Greco et al., PRC63(2001)035202

15. Theroetical predictions of symmetry energy Wiringa et al., PRC38(1988)1010. Dieperink et al., PRC67(2003)064307.

16. Bethe-Goldstone Theory  Bethe-Goldstone equation and effective G-matrix → Nucleon-nucleon interaction: ★ Two-body interaction : AV18 (isospin dependent) ★ Effective three-body force → Pauli operator : → Single particle energy : → “Auxiliary” potential : continuous choice

17. Microscopic Three-body Forces  Based on meson exchange approach  Be constructed in a consistent way with the adopted two-body force---------microscopic TBF !  Grange et.al PRC40(1989)1040 Z-diagram

18. Schematic Comparison between Dirac-BHF & the Microscopic TBF  Leading relativistic correct in Dirac BHF approach   Brown et. al., Comments Nucl. Phys. 17(1987)39; Serot and Walecka, Int. J Mod. Phys. E6(1997)515.  TBF  Contribution of the TBF to energy per nucleon Meson parameters :

19. Effective Microscopic Three-body Force  Effective three-body force → Defect function:  (r 12 )=  (r 12 ) –  (r 12 ) ★ Short-range nucleon correlations (Ladder correlations) ★ Evaluated self-consistently at each iteration  Effective TBF ---- Density dependent  Effective TBF ---- Isospin dependent for asymmetric nuclear matter

20. TBF effect on the EOS of asymmetric nuclear matter The TBF makes the the EOS much stiffer at high densities

21. Asymmetric nuclear matter at finite temperature W. Zuo, Z.H.Li,A. Li, U.lombardo, NPA745(2004)34. T=0,8,10,12,14,16MeV

22. W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418 TBF is necessary for reproducing the empirical saturation properties of nuclear matter in a non-relativistic microscopic framework. Z-diagram Full TBF Saturation Mechanism  (fm -3 ) E A (MeV) K (MeV) 0.19–15.0210 0.26–18.0230 饱和点性质 :

23. W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418 Z-diagram Full TBF Relativistic effect in Dirac-BHF approach and TBF effect Z-diagram

24. Critical temperature for liquid-gas phase transition Z-diagram Full TBF SHF : 14-20MeV RMT : 14MeV DBHF: 10MeV BHF(2BF): 16MeV BHF(TBF):13MeV BHF(Z-d): 11MeV

25. W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418 Parabolic law W. Zuo, Z.H.Li,A. Li, G.C.Lu, PRC 69(2004)064001 The EOS of ANM is determined by the EOS of SNM and symmetry energy

26. W. Zuo, A. Lejeune, U.Lombardo, J.F.Mothiot, NPA706(2002)418 TBF effect on symmetry energy W. Zuo, Z.H.Li,A. Li, G.C.Lu, PRC 69(2004)064001

27. Isospin splitting of nucleon mean field W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005.

28. Neutron-proton effective mass splitting in neutron-rich matter M* n > M* p W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005. neutrons protons DBHF: m n * > m p * Z. Y. Ma et al., PLB 604 (2004)170 Skyrme-like interactions: m p * < m n * or m n * < m p *

29. Isosping splitting of k-mass and e-mass W. Zuo, L.G. Gao, B.A. Li et al., Phys. Rev. C72 (2005)014005. neutrons protons Neutron-proton effective masses is determined by the isospin splitting of k-mass.

30. Microscopic origin of the isospin splitting Neutron-proton effective masses is controlled by the tensor component of the NN interaction

31. Proton fraction in β-stable neutron star matter A. Lejeune, U.Lombardo, W. Zuo, Phys.Lett. B477(2000)45

32. Kaon condensation in neutron stars Variational BHF + 3BF RMT W. Zuo. A. Li, Z.H.Li, U. Lombardo, PRC70(2004)055802. Critical condition Kaon Condensed Phase

33. TBF effect on the 1S0 neutron and proton gap in neutron star matter W. Zuo et al., Phys.Lett. B595(2004)44

34. Summary  The TBF provides a repulsive contribution to the EOS and improves remarkably the predicted saturation properties.  The TBF from the Z-diagram provides the saturation mechanism and gives the main relativistic effect in DBHF approach.  The empirical parabolic law for the EOS of ANM can be extended to the highest asymmetry and to the finite-temperature case.  The TBF leads to a strong enhancement of symmetry energy and the proton fraction in β-stable matter at high density.  The neutron-proton effective mass splitting is  The neutron-proton effective mass splitting is determined by the splitting of the k-mass.  The neutron-proton effective mass splitting is essentially controlled by the nature of the NN interaction.  The TBF suppresses strongly the proton superfluidity in the 1S0 channel induced by the two-body NN interaction. m* n > m* p

35. Thank you !

36. Superfluidity in β-stable matter The superfludity in a homogeneous Fermi system is discribed by the pairing gap which is determined by the standard BCS gap equation → Realistic Nucleon-nucleon interaction: → Energy spectrum: → Single-particle energy :

37. Two main effects are missing from the BCS approach screening of the pairing interaction due to the surrounding nucleons (polarization effect) medium corrections of the single-particle spectrum Up to now all investigations have predicted a reduction of the superfluidity gap in the channel due to the above effects.. D.J.Dean, M. H. Jensen, Rev. Mod. Phys. 75(2003)607 U.lombardo, H.J.Schulze, Lecture Notes in Physics, vol. 578, 2001.

38. Screening pairing suppression

39. U. Lombardo, P. Schuck, W. Zuo, PRC64 (2001) 021301R