An incomple and possibly biased overview: theoretical studies on symmetry energy Bao-An Li Why is the density-dependence of nuclear symmetry energy still very uncertain? What can we say with some confidence about the symmetry energy near the saturation density? What are the major issues at low densities? What are the new issues besides the unresolved old ones at supra- saturation densities? & collaborators: Jeff Campbell, Michael Gearheart, Joshua Hooker, Ang Li, Weikang Lin, Li Ou, Will Newton and Yuan Tian, Texas A&M University-Commerce Lie-Wen Chen, Shanghai Jiao Tong University Chang Xu, Nanjing University, Nanjing, China Che-Ming Ko and Jun Xu, Texas A&M University, College Station Zhigang Xiao and Ming Zhang, Tsinghua University, China Gao-Chan Yong, Institute of Modern Physics, China

The multifaceted influence of the isospin dependence of strong interaction and symmetry energy in nuclear physics and astrophysics A.W. Steiner, M. Prakash, J.M. Lattimer and P.J. Ellis, Phys. Rep. 411, 325 (2005). Isospin physics Isospin physicsn/p isoscaling isoscaling isotransport isotransport isodiffusion isodiffusion t/ 3 He isofractionation isofractionation K + /K 0 isocorrelation isocorrelation π-/π+π-/π+π-/π+π-/π+ in Terrestrial Labs (QCD)(Effective Field Theory)

E sym (ρ) predicted by microscopic many-body theories Symmetry energy (MeV) Density Effective field theory (Kaiser et al.) Dirac-Brueckner HF Relativistic Mean Field Brueckner HF Greens function Variational many-body A.E. L. Dieperink et al., Phys. Rev. C68 (2003)

The E sym (ρ) from model predictions using popular effective interactions Examples: Density 23 RMF models ρ L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. C72, (2005); C76, (2007).

Relation between symmetry energy and the mean-field potential Lane potential Symmetry energy Effective mass kinetic isoscalar isovector J. Dabrowski, Physics Letters 8, 90 (1964)

Why is the symmetry potential/energy so uncertain? Within an interacting Fermi gas model: Structure of the nucleus, M.A. Preston and R.K. Bhaduri (1975) Short-range tensor force due to rho meson exchange Spin-isospin dependence of 3-body forces NN correlation functions Isospin-dependence of strong interaction: Nucleons having isospin t=1/2 t 3 =1/2 for neutrons t 3 =-1/2 for protons T=0 or 1 for NN pairs Isospin-dependence of NN correlations and the tensor force

S=1, T=0 S 12 =2

Uncertainty of tensor force at short distance Takaharu Otsuka et al., PRL 95, (2005); PRL 97, (2006) Cut-off=0.7 fm for nuclear structure studies Gogny Gogny+tensor

Tensor force contribution to symmetry energy G.E. Brown and R. Machleidt, Phys. Rev. C50, 1731 (1994). PLB 18, 54 (1965) S.-O. Bacnman, G.E. Brown and J.A. Niskanen, Phys. Rep. 124, 1 (1985).

Effects of short-range tensor force on symmetry energy Ang Li and Bao-An Li, 2011

Density dependence of the symmetry energy is the main criterion for distinction between Skyrme parameterizations (87 tested) Group I Group IIGroup III Original Skyrme potentials have tensor components, but they are normally dropped in SKF calcluations J.R. Stone et al., PRC 68, (2003) Page 3

+ MANY other papers starting from the same 3-body force, Reduced to different 2-body force with α=1/3, 2/3, 1, etc D. Vautherin and D.M.Brink, Phys.Rev.C5, 626 (1972) x 0 controls the mixing of different spin-isospin channels Necessary to fit the saturation properties of nuclear matter α controls the in-medium many-body effects

Effects of the 3-body force on the symmetry energy X 0 from to 1.92 are used in the over 140 effective interactions in the literature Chang Xu and Bao-An Li, PRC 81, (2010).

R Subedi et al. Science 320, 1475 (2008) M. Strikman, CERN Courier Jan 27, 2009 Isospin-dependence of Short Range NN Correlations and Tensor Force Two-nucleon knockout by an electron H. Baghdasaryan et al. (CLAS ollaboration) Phys. Rev. Lett. 105, (2010) np pp

Isospin dependence of nucleon-nucleon correlation due to tensor force and its effects on single-particle momentum distribution Ann. Rev. Nucl. Part. Sci., 21, (1971) Fermi sphere Repulsive core only Tensor force + Repulsive core

Effects of isospin-dependence of short-range nucleon-nucleon correlation on symmetry energy Chang Xu and Bao-An Li, arXiv:

Symmetry energy E sym (ρ) and its density slope L(ρ) at arbitrary density based on the Hugenholtz-Van Hove (HVH) theorem theoremtheorem Energy density Kinetic energy Single-particle potential Fermi momentum C. Xu, B.A. Li and L.W. Chen and C.M. Ko, arXiv: , NPA (2011) in press.arXiv:

Symmetry potential at saturation density from global nucleon optical potentials Systematics based on world data accumulated since 1969: (1)Single particle energy levels from pick-up and stripping reaction (2)Neutron and proton scattering on the same target at about the same energy (3)Proton scattering on isotopes of the same element (4)(p,n) charge exchange reactions

Constraining the symmetry energy near saturation density using global nucleon optical potentials C. Xu, B.A. Li and L.W. Chen, PRC 82, (2010).

Theoretical predictions on the correlation between E sym (ρ 0 ) and L(ρ 0 ) Under the condition that they Agree with the EOS of PNM at low densities pewdicted by A.Schwenk and C. Pethick, PRL 95 (2005) Will Newton et al. (2011) Chen et al. Steiner using constraints on optical potentials

Iso Diff. (IBUU04, 2005), L.W. Chen et al., PRL94, (2005) IAS+LDM (2009), Danielewicz and J. Lee, NPA818, 36 (2009) PDR (2007) in 208 Pb Land/GSI, PRC76, (2007) Constraints extracted from data using various models Iso. Diff & double n/p (ImQMD, 2009), M. B. Tsang et al., PRL92, (2009). GOP: global optical potentials (Lane potentials) C. Xu, B.A. Li and L.W. Chen, PRC 82, (2010) PDR (2010) of 68 Ni and 132 Sn, A. Carbone et al., PRC81, (2010). SHF+N-skin of Sn isotopes, L.W. Chen et al., PRC 82, (2010) Isoscaling (2007), D.Shetty et al. PRC76, (2007) DM+N-Skin (2009): M. Centelles et al., PRL102, (2009) TF+Nucl. Mass (1996), Myers and Swiatecki, NPA601, 141 (1996) E sym (ρ 0 )≈ 31 ± 4 MeV L≈ 60 ± 23 MeV

Some basic issues on symmetry energy at low densities neutron +proton uniform matter at density ρ and isospin asymmetry as density decreases Invariance of nuclear interaction under n-p exchange, for uniform matter What is the isospin-dependence of the EOS of clustered matter?

Isospin dependence of the EOS of clustered matter T=2 MeV T=4 MeV n=0.001fm -3 Uniform matter Clustered matter Quantum statistical model S. Typel, G. Ropke, T. Klahn, D. Blaschke and H.H. Wolter, PRC 81, (2010) S-matrix approach J. N. De, S. K. Samaddar, PRC 78, (2008) S.K. Samaddar, J.N. De, X. Vinas and M. Centelles, PRC 80, (2009) The anharmonic behavior depends on whether all mirror nuclei are included in pairs

What “symmetry energies’’ are we talking about for clustered matter? Even if we know the answer, at what densities? Normal or really low? For the sake of getting the traditionally defined symmetry energy and compare it with the one for uniform matter, (1) Isospin-independent “symmetry energy-1” (3) Isospin-dependent “symmetry energy” by forcing the EOS to be quadratic (2) Isospin-independent “symmetry energy-2”

Pairing effects on symmetry energy at low densities E. Khan, J. Margueron, G. Colò, K. Hagino, and H. Sagawa, PRC 82, (2010) Considering nn and pp pairing in T=1 only using 3 kinds of pairing intereactions

Effects of n-p pairing on symmetry energy at low densities Yuan Tian and Bao-An Li (2011) using separable Paris potential To my best knowledge, Nobody has considered both clusters and pairing at low densities simultaneously yet

Super-uncertainty of Symmetry Energy at Supra-saturation Densities Z.G. Xiao, Bao-An Li, L.W. Chen, G. C. Yong and M. Zhang, PRL 102, (2009) Old issues: (1) Model dependence because of the complexity of transport models, inconsistent and diverse input mean-field and in-medium elementary hadron-hadron cross sections (2) Few known probes that are clean and strongly sensitive to the symmetry energy which is a small percent of the total potential energy especially at high densities

A new issue: magnetic effects on the pion ratio In sub-Coulomb barrier U+U collisions, the magnetic field B is on the order of G J. Rafelski and B. Muller, PRL 36, 517 (1976) In off-central Au+Au collisions at RHIC, D. Kharzeev, L. McLerran and H.Warringa, NPA803:227 (2008)

Provides the possibility to study properties of dense matter under strong magnetic field as in neutron stars * G Ou Li and Bao-An Li (2011)

Beam energy and impact parameter dependence of magnetic field created in heavy-ion collisions

No magnetic effect on nucleon observables because the Lorentz force is very small compared to nuclear force Nuclear force over magnetic force

Significant Magnetic effects on pion ratio Empirically, no nuclear mean-field on pions in most transport models and the pion cascade is sufficient to describe available data Theoretically, pion mean-field or dispersion relation is very uncertain Pions are light and moving fast, they thus feel stronger Lorentz force

W. Reisdorf et al., NPA781 (2007) 459 No magnetic field effect considered in neither model calculations nor data analysis (extrapolating to very forward and backward angles to obtain the total multiplicity)

Conclusion and Perspectives Conclusion: No! Perspectives: a lot! Thank you!