A Status Review on Symmetry Energy around Saturation Density: Theory Lie-Wen Chen (陈列文) (INPAC/Department of Physics, Shanghai Jiao Tong University. lwchen@sjtu.edu.cn) Collaborators： Wei-Zhou Jiang (SEU) Che Ming Ko and Jun Xu (TAMU) Bao-An Li and Chang Xu (TAMU-Commerce) De-Hua Wen (SCUT) Zhi-Gang Xiao and Ming Zhang (Tsinghua) Gao-Chan Yong (IMP) Xin Wang, Bao-Jun Cai, Rong Chen, Peng-Cheng Chu, Kai-Jia Sun, Zhen Zhang, Hao Zheng (SJTU) “Topical Workshop on Nuclear Symmetry Energy and Astrophysics”, Xi’an, December 17-19, 2010

Outline The nuclear symmetry energy Theoretical tools of determining the symmetry energy around normal density (1) Many-body approaches (2) Transport theory Symmetry energy around normal density from: (1) Heavy ion collisions (2) Nuclear structures (3) Global nucleon optical potential Symmetry energy around normal density and: (1) Nuclear effective interactions (2) Neutron stars Summary and outlook p. 1

The nuclear symmetry energy p. 2

The Nuclear Symmetry Energy Liquid-drop model (Isospin) Symmetry energy term Symmetry energy including surface diffusion effects (ys=Sv/Ss) Sv ~ the nuclear matter symmetry energy at normal density W. D. Myers, W.J. Swiatecki, P. Danielewicz, P. Van Isacker, A. E. L. Dieperink,…… p. 3

The Nuclear Matter Symmetry Energy EOS of Isospin Asymmetric Nuclear Matter (Parabolic law) Symmetric Nuclear Matter (relatively well-determined) Symmetry energy term (poorly known) Isospin asymmetry The Nuclear Symmetry Energy p. 4

Astrophysics and Cosmology The Symmetry Energy The multifaceted influence of the nuclear symmetry energy A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005). Nuclear Physics on the Earth Symmetry Energy Astrophysics and Cosmology in Heaven The symmetry energy is also related to some issues of fundamental physics: 1. The precision tests of the SM through atomic parity violation observables (Sil et al., PRC05) 2. Possible time variation of the gravitational constant (Jofre et al. PRL06; Krastev/Li, PRC07) 3. Non-Newtonian gravity proposed in grand unification theories (Wen/Li/Chen, PRL09) p. 5

QCD phase diagram in 3D: density ,temperature, and isospin

Theoretical tools of determining the symmetry energy around normal density (1) Many-body approaches (2) Transport theory p. 7

Nuclear Matter EOS: Many-Body Approaches The nuclear EOS cannot be measured experimentally, its determination thus depends on theoretical approaches Microscopic Many-Body Approaches Non-relativistic Brueckner-Bethe-Goldstone (BBG) Theory Relativistic Dirac-Brueckner-Hartree-Fock (DBHF) approach Self-Consistent Green’s Function (SCGF) Theory Variational Many-Body (VMB) approach Green’s Function Monte Carlo Calculation Vlowk + Renormalization Group …… Effective Field Theory Density Functional Theory (DFT) Chiral Perturbation Theory (ChPT) Phenomenological Approaches Relativistic mean-field (RMF) theory Quark Meson Coupling (QMC) Model Relativistic Hartree-Fock (RHF) Non-relativistic Hartree-Fock (Skyrme-Hartree-Fock) Thomas-Fermi (TF) approximations p. 8

Esym: Many-Body Approaches Z.H. Li et al., PRC74, 047304(2006) Dieperink et al., PRC68, 064307(2003) BHF p. 9 10

Esym: Many-Body Approaches Chen/Ko/Li, PRC72, 064309(2005) Chen/Ko/Li, PRC76, 054316(2007) p. 10 11

W.Z. Jiang, B.A. Li, and L.W. Chen, PLB653, 184 (2007) Esym: Many-Body Approaches RMF with chiral limits W.Z. Jiang, B.A. Li, and L.W. Chen, PLB653, 184 (2007) p. 11 12

Resonant Fermi Gases with a Large Effective Range (PRL,2005) Pure Nuetron Matter: Many-Body Approaches Resonant Fermi Gases with a Large Effective Range (PRL,2005) J. Piekarewicz, JPG37, 064038 (2010) Chiral 3NF (PRC,2010) Quantum MC (PRC,2010) Quantum MC (PRL,2008) p. 12 13

Theoretical tools of determining the symmetry energy around normal density (1) Many-body approaches (2) Transport theory p. 13

Transport Theory Central collisions Transport Models Ni + Au, E/A = 45 MeV/A Transport Models for HIC’s at intermediate energies: N-body approaches CMD, QMD,IQMD,IDQMD, ImQMD,ImIQMD,AMD,FMD One-body approaches BUU/VUU, BNV, LV, IBL Relativistic covariant approaches RVUU/RBUU,RQMD… Central collisions Broad applications of transport models in astrophysics, plasma physics, electron transport in semiconductor and nanostructures, particle and nuclear physics, …… p. 14

Transport model for HIC’s Isospin-dependent BUU (IBUU) model Solve the Boltzmann equation using test particle method Isospin-dependent initialization Isospin- (momentum-) dependent mean field potential 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 EOS p. 15

Transport model: IBUU04 Isospin- and momentum-dependent potential (MDI) Das/Das Gupta/Gale/Li, PRC67,034611 (2003) Chen/Ko/Li, PRL94,032701 (2005) Li/Chen, PRC72, 064611 (2005) p. 16

Relativistic VUU/BUU Model Che Ming Ko, Ulrich Mosel, …… Relativistic Vlasov Equation + Collision Term… Wigner transform, Dirac + Fields Equation drift mean field Non-relativistic Boltzmann-Uehling-Uhlenbeck “Lorentz Force”→ Vector Fields pure relativistic term Collision term: p. 17

中能重离子碰撞输运模型 QMD模型 关于QMD的“推导”文献J. Aichelin, Phys. Rep.202, 233 (1991)有详细的讨论，其基本出发点同样是多体薛定谔方程，约化密度矩阵以及对BBGKY系列的截断，但一开始就引入相空间的概念，即对密度矩阵作富里叶变换得到所谓的Wigner密度(或称Wigner表示)．从而直接将多体薛定谔方程表示为类似于经典输运方程的形式，这里我们给出有关的主要公式. AMD/FMD考虑了波函数的反对称化 p. 18

中能重离子碰撞输运模型 QMD模型 r p. 19

中能重离子碰撞输运模型 QMD模型 p. 20

中能重离子碰撞输运模型 QMD模型 - p. 21

Symmetry energy around normal density from: (1) Heavy ion collisions (2) Nuclear structures (3) Global nucleon optical potential p. 22

Symmetry energy probes Promising Probes of the Esym(ρ) (an incomplete list !) Pigmy/Giant resonances Nucleon optical potential B.A. Li/L.W. Chen/C.M. Ko Phys. Rep. 464, 113(2008) p. 23

Esym: Isospin Diffusion in HIC’s Isospin Diffusion/Transport ______________________________________ How to measure Isospin Diffusion? PRL84, 1120 (2000) A+A,B+B,A+B X: isospin tracer p. 24

(1) Esym: Isospin Diffusion in HIC’s Symmetry energy, isospin diffusion, in-medium cross section Chen/Ko/Li, PRL94,032701 (2005) Chen/Ko/Li, PRC72,064309 (2005) Li/ Chen, PRC72, 064611(2005) Isospin Diffusion Data  Esym(ρ0)=31.6 MeV L=88±25 MeV p. 25

Esym: Isoscaling in HIC’s Isoscaling observed in many reactions M.B. Tsang et al. PRL86, 5023 (2001) p. 26

(2) Esym: Isoscaling in HIC’s Consistent with isospin diffusion data! Constraining Symmetry Energy by Isocaling: TAMU Data Shetty/Yennello/Souliotis, PRC75,034602(2007); PRC76, 024606 (2007) Isoscaling Data  Esym(ρ0)=31.6 MeV L=65 MeV Consistent with isospin diffusion data! p. 27

(3) Esym: Isospin diffusion and double n/p ratio in HIC’s ImQMD: n/p ratios and two isospin diffusion measurements Tsang/Zhang/Danielewicz/Famiano/Li/Lynch/Steiner, PRL 102, 122701 (2009) ImQMD: Isospin Diffusion and double n/p ratio  Esym(ρ0)~28 - 34 MeV? L=38 - 103 MeV p. 28

Symmetry energy around normal density from: (1) Heavy ion collisions (2) Nuclear structures (3) Global nucleon optical potential p. 29

(4) Esym: Nuclear Mass in Thomas-Fermi Model Myers/Swiatecki, NPA 601, 141 (1996) Thomas-Fermi Model analysis of 1654 ground state mass of nuclei with N,Z≥8 Thomas-Fermi Model + Nuclear Mass  Esym(ρ0)=32 .65 MeV L=49.9 MeV p. 30

Esym: Neutron Skin of Heavy Nuclei Chen/Ko/Li, PRC72,064309 (2005) Good linear Correlation: S-L Oyamatsu et al., NPA634, 3 (1998); Brown, PRL85,5296 (2000); Horowitz/Piekarewicz, PRL86, 5647 (2001); Furnstahl, NPA706, 85 (2002); Yoshida/Sagawa, PRC73, 044320 (2006) p. 31 32

(5) Esym: Droplet Model Analysis on Neutron Skin N-Skin data measured in antiprotonic atoms Droplet Model + N-skin  Esym(ρ0)=28 - 35 MeV, L=55 ± 25 MeV p. 32

(6) Esym: Skyrme-HF Analysis on Neutron Skin Chen/Ko/Li/Xu PRC82, 024321(2010) N-Skin data of Sn isotopes Esym(ρ0)=30 MeV Neutron skin constraints on L and Esym(ρ0) are insensitive to the variations of other macroscopic quantities. p. 31

Esym: Pygmy Dipole Resonances Electric dipole strength in atomic nuclei By Deniz Savran p. 34

(7)、(8) Esym: Pygmy Dipole Resonances Pygmy Dipole Resonances of 130,132Sn  Esym(ρ0)=32 ± 1.8 MeV L=43.125 ± 15 MeV Pygmy Dipole Resonances of 68Ni and 132Sn  Esym(ρ0)=32.3 ± 1.3 MeV, L=64.8 ± 15.7 MeV p. 35

(9) Esym: IAS+LDM Danielewicz/Lee, NPA 818, 36 (2009) Esym from Isobaric Analog States + Liquid Drop model with surface symmetry energy IAS+Liquid Drop Model with Surface Esym  Esym(ρ0)=32.5 ± 1 MeV L=94.5 ± 16.5 MeV p. 36

Esym from Liquid Drop model with surface symmetry energy (10) Esym: LDM Liu/Wang/Li/Zhang, PRC (2010), in press [arXiv:1011.3865] Esym from Liquid Drop model with surface symmetry energy Liquid Drop Model with Surface Esym  Esym(ρ0)=31.1 ± 1.7 MeV L=66.0 ± 13 MeV p. 37

Symmetry energy around normal density from: (1) Heavy ion collisions (2) Nuclear structures (3) Global nucleon optical potential p. 38

Esym: Global nucleon optical potential Xu/Li/Chen, PRC82, 054607 (2010) Single particle energy at Fermi surface = particle chemical potential Energy density Hugenholtz-Von Hove (HVH) Theorem Symmetry potential (Lane potential) p. 39

Esym: Global nucleon optical potential Xu/Li/Chen, PRC82, 054607 (2010) =26.11 MeV p. 40

(11) Esym: Global nucleon optical potential Xu/Li/Chen, PRC82, 054607 (2010) p. 41

Esym around normal density 11 constraints on Esym (ρ0) and L from nuclear reactions and structures Esym(ρ0)=27 - 36 MeV L=30 - 90 MeV More accurate data are needed to obtain more stringent constraints! p. 42

Symmetry energy around normal density and: (1) Nuclear effective interactions (2) Neutron stars p. 43

Symmetry energy and Nuclear Effective Interaction Chen/Ko/Li, PRC76, 054316(2007) Chen/Ko/Li, PRC72,064309 (2005) Esym(ρ0)= 31.5 ±4.5 MeV and L=55 ± 25 MeV: only 55/118 Esym(ρ0)= 31.5 ±4.5 MeV and L=55 ± 25 MeV: only 8/23 p. 44

Symmetry energy around normal density and: (1) Nuclear effective interactions (2) Neutron stars p. 45

The Nuclear Symmetry Energy and Neutron Stars Lattimer/Prakash, Science 304, 536 (2004) core-crust transition Neutron star has solid crust over liquid core. Rotational glitches: small changes in period from sudden unpinning of superfluid vortices. Evidence for solid crust. 1.4% of Vela moment of inertia glitches. Needs to know the transition density to calculate the fractional moment of inertia of the crust Link et al., PRL83,3362 (99) p. 46

Locating the inner edge of neutron star crust Parabolic Law fails! Xu/Chen/Li/Ma, PRC79, 035802 (2009) Kazuhiro Oyamatsu, Kei Iida Phys. Rev. C75 (2007) 015801 Parabolic Approximation has been assumed !!! pasta Significantly less than their fiducial values: ρt=0.07-0.08 fm-3 and Pt=0.65 MeV/fm3 Xu/Chen/Li/Ma, ApJ 697, 1547 (2009) p. 47

Constraints on M-R relation of NS Xu/Chen/Li/Ma, ApJ 697, 1547 (2009) Lattimer Prakash (Empirical estimate Link et al., PRL83,3362(99)) (Isospin Diff) p. 48

Summary and outlook p. 49

Summary and Outlook Analyzing the data from heavy ion collisions, nuclear structures (mass,neutron skin, PDR…), and nucleon global optical potential already put important constraints on the symmetry energy around the saturation density: Esym(ρ0) =31.5±4.5 MeV and L=60±30 MeV More accurate data are needed to obtain more stringent constraints! Symmetry energy is important for understanding the effective nuclear interactions and many properties of neutron stars. p. 50

谢 谢！ Thanks!

Discussions: Esym around normal density Are important the higher-order characteristic parameters of the symmetry energy defined at normal density? What’s the values of these parameters? How can we determine the Esym(ρ0) more accurately? Any more novel observables/approaches to extract the symmetry energy? Model independent detrmination? p. 33

Discussions: Esym around normal density p. 33

Discussions: Esym around normal density p. 33

Discussions: Esym around normal density p. 33

Discussions: Esym around normal density p. 33

Discussions: Esym around normal density L.G. Cao, Z.Y. Ma, CPL, 2008 HIC+N-Skin Trippa, PRC, 2008 M. Liu et al., PRC, 2010 Esym(ρ0)=30±5 MeV L=58±18 MeV p. 33

Radioactive Ion Beam Facilities Cooling Storage Ring (CSR) Facility at HIRFL in China up to 500 MeV/A for 238U http://www.impcas.ac.cn/zhuye/en/htm/247.htm. Radioactive Ion Beam Factory (RIBF) at RIKEN in Japan http://www.riken.jp/engn/index.html FAIR-NUSTAR/GSI in Germany up to 2 GeV/A for 132Sn http://www.gsi.de/fair/index_e.html. SPIRAL2/GANIL in France http://ganinfo.in2p3.fr/research/developments /spiral2 Facility of Rare Isotope Beams (FRIB) in the USA up to 200(400) MeV/A for 132Sn http://www.nscl.msu.edu/ria/index.php The Korean Rare Isotope Accelerator (KoRIA) up to 250 MeV/A for 132Sn, up to 109 pps …… p. 13

The Skyrme HF Energy Density Functional Standard Skyrme Interaction: There are more than 120 sets of Skyrme- like Interactions in the literature _________ Agrawal/Shlomo/Kim Au PRC72, 014310 (2005) Yoshida/Sagawa PRC73, 044320 (2006) Chen/Ko/Li/Xu PRC82, 024321(2010) 9 Skyrme parameters: 9 macroscopic nuclear properties: p. 29

The Skyrme HF Energy Density Functional Chen/Cai/Ko/Li/Shen/Xu, PRC80, 014322 (2009): Modified Skyrme-Like (MSL) Model Chen/Ko/Li/Xu, arXiv:1004.4672

The Skyrme HF with MSL0 Chen/Ko/Li/Xu, arXiv:1004.4672

Correlations between Nuetron-Skin thickness and macroscopic Nuclear Properties For heavy nuclei 208Pb and 120Sn: Δrnp is strongly correlated with L, moderately with Esym(ρ0), a little bit with m*s,0 For medium-heavy nucleus 48Ca: Δrnp correlation with Esym is much weaker; It further depends on GV and W0 Chen/Ko/Li/Xu PRC82, 024321(2010) Important Terms p. 30

Esym at very low densities: Clustering effects Horowitz and Schwenk, Nucl. Phys. A 776 (2006) 55 S. Kowalski, et al., PRC 75 (2007) 014601. 64

Esym at very low densities: Clustering effects J. B. Natowitz et al., PRL104,202501 (2010) 65