1. Determination of Density Dependence of Nuclear Matter Symmetry Energy in HIC’s ISOSPIN PHYSICS AND LIQUID GAS PHASE TRANSITION, CCAST, Beijing, Aug. 19-21, 2005 Lie-Wen Chen (Department of Physics, Shanghai Jiao Tong University) Collaborators: V. Greco, C. M. Ko (Texas A&M University) B. A. Li (Arkansas State University)

2.  Nuclear Matter Symmetry Energy  Two-Nucleon Correlation Functions  Light Cluster Production and Coalescence Model  Isospin Transport/Diffusion  Discussions  Summary Contents References: PRL90, 162701 (2003); PRC68, 017601 (2003); PRC68, 014605 (2003); NPA729, 809(2003); PRC69, 054606 (2004); PRL94, 032701 (2005); Nucl-th/0508024.

3. Neutron Stars … Structures of Radioactive Nuclei, SHE … Isospin Effects in HIC’s … Isospin in Intermediate Energy Nuclear Physics Many-Body Theory Transport Theory General Relativity Nuclear Force EOS for Asymmetric Nuclear Matter Density Dependence of the Nuclear Symmetry Energy HIC’s induced by neutron- rich nuclei (CSR,GSI, RIA,…) Pre-eq. n/p Isospin fractionation Isoscaling in MF n-p differential transverse flowProton differential elliptic flow π-/π+…π-/π+… Isospin diffusion Two-nucleon correlation functions Light clusters (t/ 3 He) Thickness of neutron skin Most uncertain property of an asymmetric nuclear matter

4. Nuclear Matter Symmetry Energy EOS of Asymmetric Nuclear Matter (Parabolic law) Isospin-Independent Part (Skyrme-like) Nuclear Matter Symmetry Energy

5. Density dependence of the symmetry energy from SHF BA Brown, PRL85 SkX~Variation Many-Body Theory

7. Most recent parameterization for studying the properties of neutron stars H. Heiselberg& M. Hjorth-Jensen, Phys. Rep. 328(2000) The symmetry potential acting on a nucleon The neutron and proton symmetry potentials with the stiff ( γ=2 ) and soft ( γ =0.5 ) symmetry energies γ =0.5: L=52.5 MeV and K sym =-78.8 MeV γ=2.0: L=210.0 MeV and K sym =630.0 MeV Phenomenologically parameterizing the nuclear matter symmetry energy

8. Isospin-dependent BUU (IBUU) model Solve the Boltzmann equation using test particle method Isospin-dependent initialization Isospin-dependent mean field Isospin-dependent N-N cross sections a. Experimental free space N-N cross section σ exp b. In-medium N-N cross section from the Dirac-Brueckner approach based on Bonn A potential σ in-medium c. Mean-field consistent cross section due to m* Isospin-dependent Pauli Blocking

9. Two-Nucleon Correlation Functions The two-particle correlation function is obtained by convoluting the emission function g(p,x), i.e., the probability of emitting a particle with momentum p from space-time point x=(r,t), with the relative wave function of the two particle, i.e., The two-particle correlation function is a sensitive probe to the space-time structure of particle emission source by final state interaction and quantum statistical effects ( φ (q,r)) Correlation After Burner: including final-state nuclear and Coulomb interactions (Scott Pratt, NPA 566, 103 (1994)) How to detect the space-time structure of nucleon emission experimentally?

10. Pairs with P>500 MeV: n-n CF: 20% p-p CF: 20% n-p CF: 30% Symmetry Energy Effects on Two-Nucleon Correlation Functions Effects are very small for both isoscalar potential and N-N cross sections Chen,Greco,Ko,Li, PRL90, PRC68, (2003)

11. The covariant coalescence model Chen,Ko,Li, PRC68; NPA729 Butler,Pearson,Sato,Yazaki,Gyulassy,Frankel,Remler,Dove,Scheibl,Heinz,Mattiello,Nagle,Polleri, Biro,Zimanyi,Levai,Csizmadia,Hwa,Yang,Ko,Lin,Voloshin,Molnar,Greco,Fries,Muller,Nonaka,Bass, …  Depends on constituents ’ space-time structure at freeze-out  Neglecting the binding energy effect (T>>E binding ), Coalescence probability: Wigner phase-space density in the rest-frame of the cluster.  Rare process has been assumed (the coalescence process can be treated perturbatively). Higher energy collisions and higher energy cluster production! Light Cluster Production and Coalescence Model

12. Dynamical coalescence model

13. Hulthen wave function Wigner phase-space density for Deuteron Wigner transformation Chen,Ko,Li, NPA729

14. t/ 3 He Wigner phase-space density and root-mean-square radius: Wigner phase-space density for t/ 3 He Assume nucleon wave function in t/ 3 He can be described by the harmonic oscillator wave function, i.e.,

15. Isospin symmetric collisions at E/A≈100 MeV  Deuteron energy spectra reproduced  Low energy tritons slightly underestimated  Inverse slope parameter of 3 He underestimated; probably due to neglect of larger binding effect stronger Coulomb effect wave function Data are taken from INDRA Collaboration (P. Pawlowski, EPJA9) Try Coalescence model at intermediate energies! Chen,Ko,Li, NPA729

16. Symmetry Energy Effects on t/ 3 He ratio  Stiffer symmetry energy gives smaller t/ 3 He ratio  With increasing kinetic energy, t/ 3 He ratio increases for soft symmetry energy but slightly decreases for stiff symmetry energy

17. Isospin Transport/Diffusion How to measure Isospin Transport? PRL84, 1120 (2000) ______________________________________ A+A,B+B,A+B X: isospin tracer

18. E=50 AMeV and b=6 fm

19. _____________ Chen,Ko,Li, PRL94,2005

20. MDI ~Finite Range Gogny Interaction Lane Potential Chen,Ko,Li, PRL93,2005

22. MDI Das, Das Gupta, Gale and Li PRC67, (2003) Two-nucleon correlation functions The sensitivity becomes weaker with momentum-dependence 1. Effects of momentum-dependence of nuclear potential Pairs with P>500 MeV: n-p CF: 11% Discussions The isospin effects on two-particle correlation functions are really observed in recent experimental data !!! R. Ghetti et al., PRC69 (2004) 031605 肖志刚等

23. t/ 3 He ratio Still sensitive to the stiffness of the symmetry energy 2. Effects of momentum-dependence of nuclear potential

24. 3. Effects of in-medium cross sections on isospin transport Li,Chen, Nucl-th/0508024. np cross section is reduced in nuclear medium

25. 3. Effects of in-medium cross sections on isospin transport R i (isospin transport/diffusion) Symmetry potential and np collisions Li,Chen, Nucl-th/0508024.

26. 4. Have We Already Known the Density Dependence of Nuclear Matter Symmetry Energy at Sub-saturated Densities? W. D. Tian, Y. G. Ma, et al., Isoscaling + CQMD _______________ ________________________________ ___________________ arXiv:nucl-ex/0505011 Isocaling+AMD

27. 5. The High Density Behaviors of Nuclear Matter Symmetry Li,Chen,Ko,Yong,Zuo, nucl-th/0504008; Li,Chen,Das, Das Gupta,Gale,Ko,Yong, Zuo, nucl-th/0504069 B. A. Li, PRL88 (2002) 192701

28. nucl-th/0504065, Phys.Rev. C71 (2005) 054907 Other possible observations: Kaons, Σ, … ———————————————————— ————————————————————————— ——————————————

29. 6. Momentum Dependence of Symmetry Potential Recent progress: E.N.E. van Dalen, C. Fuchs, A. Faessler, NPA744, (2004); PRL95,(2005) Zhong-yu MaZhong-yu Ma, Jian Rong, Bao-Qiu Chen, Zhi-Yuan Zhu, Hong-Qiu Song,Jian RongBao-Qiu ChenZhi-Yuan ZhuHong-Qiu Song PLB604, (2004) F. Sammarruca, W. Barredo, P. Krastev, PRC71, (2005) W. Zuo, L.G. Cao, B. A. Li, U. Lombardo, C.W. Shen, PRC72, (2005) L.W. Chen, C.M. Ko, B.A. Li, to be submitted ‘Puzzle’? Di Toro et al.

30. Summary Two-particle correlation functions and t/ 3 He ratio are useful probes of the nuclear symmetry energy The sub-saturated density behavior of the symmetry energy become more and more clear from the isospin diffusion and isoscaling, and n-skin of Pb The high density behavior of the symmetry energy and the momentum dependence of the symmetry potential need much further effort Thank you! 谢谢大家！