Determination of Density Dependence of Nuclear Matter Symmetry Energy in HIC’s

1. ISOSPIN PHYSICS AND LIQUID GAS PHASE TRANSITION, CCAST, Beijing, Aug. 19-21, 2005 Determination ofDensity Dependence of Nuclear Matter Symmetry Energy in HIC’s 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. Contents • Nuclear Matter Symmetry Energy • Two-Nucleon Correlation Functions • Light Cluster Production and Coalescence Model • Isospin Transport/Diffusion • Discussions • Summary References: PRL90, 162701 (2003); PRC68, 017601 (2003); PRC68, 014605 (2003); NPA729, 809(2003); PRC69, 054606 (2004); PRL94, 032701 (2005); Nucl-th/0508024.

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

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 SkX~Variation Many-Body Theory BA Brown, PRL85

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

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 σexp • b. In-medium N-N cross section from the Dirac-Brueckner • approach based on Bonn A potential σin-medium • c. Mean-field consistent cross section due to m* • Isospin-dependent Pauli Blocking

9. Two-Nucleon Correlation Functions How to detect the space-time structure of nucleon emission experimentally? 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))

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

11. Light Cluster Production and Coalescence Model 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,… The covariant coalescence model • Depends on constituents’ space-time structure at freeze-out • Neglecting the binding energy effect (T>>Ebinding), 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! Chen,Ko,Li, PRC68; NPA729

12. Dynamical coalescence model

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

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

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

16. Symmetry Energy Effects on t/3He ratio • Stiffer symmetry energy gives smaller t/3He ratio • With increasing kinetic energy, t/3He 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. Lane Potential Chen,Ko,Li, PRL93,2005 MDI ~Finite Range Gogny Interaction

22. Discussions 1. Effects of momentum-dependence of nuclear potential Two-nucleon correlation functions MDI Das, Das Gupta, Gale and Li PRC67, (2003) Pairs with P>500 MeV: n-p CF: 11% The sensitivitybecomes weaker with momentum-dependence 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. 2. Effects of momentum-dependence of nuclear potential t/3He ratio Still sensitive to the stiffness of the symmetry energy

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 Li,Chen, Nucl-th/0508024. Ri(isospin transport/diffusion) Symmetry potential and np collisions

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

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

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

29. 6. Momentum Dependence of Symmetry Potential Di Toro et al. Recent progress: E.N.E. van Dalen, C. Fuchs, A. Faessler, NPA744, (2004); PRL95,(2005) Zhong-yu Ma, Jian Rong, Bao-Qiu Chen, Zhi-Yuan Zhu, Hong-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’?

30. Summary • Two-particle correlation functions and t/3He 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！