1. Probing the symmetry energy at high densities Outline: Why is the symmetry energy so uncertain at supra-saturation densities? Can the symmetry energy become super-soft or even negative at high densities? --- Some observations by a non-expert of many-body theories Indications of a super-soft symmetry energy at supra-saturation densities from transport model analyses of the FOPI/GSI experimental data on pion production Can neutron stars be stable with a super-soft or negative symmetry energy at supra-saturation densities? --- Some observations and attempts by a non-expert of astrophysics & collaborators: De-Hua Wen and Chang Xu, Texas A&M University-Commerce Lie-Wen Chen, Shanghai Jiao-Tung University Zhigang Xiao and Ming Zhang, Tsinghua University, China Gao-Chan Yong, Institute of Modern Physics, China Bao-An Li

2. E sym (ρ) predicted by microscopic many-body theories Symmetry energy (MeV) Density Effective field theory DBHF RMF BHF Greens function Variational A.E. L. Dieperink et al., Phys. Rev. C68 (2003) 064307

3. Why is the symmetry energy so uncertain especially at high densities? Based on the Fermi gas model (Ch. 6) and properties of nuclear matter (Ch. 8) of the textbook: Structure of the nucleus by M.A. Preston and R.K. Bhaduri Kinetic Isoscalar Isovector Correlation function

4. Nuclear Structure based on Correlated Realistic Nucleon-Nucleon Potentials R. RothR. Roth, T. Neff, H. Hergert, H. Feldmeier, Nucl. Phys. A745 (2004) 3-33T. NeffH. HergertH. Feldmeier Tensor correlations in the Unitary Correlation Operator Method Thomas NeffThomas Neff, Hans Feldmeier, Nucl. Phys. A713 (2003) 311-371Hans Feldmeier Correlation function

5. At saturation density Using Paris potential I. Bombaci and U. Lombardo PRC 44, 1892 (1991) Using the Reid93 interaction PRC68, 064307 (2003)

7. The most important contributions of nuclear force Lecture notes of R. Machleidt at 2005 RIKEN summer school

8. In-medium properties of the short-range tensor force G.E. Brown and Mannque Rho, PLB 237, 3 (1990) G.E. Brown and Mannque Rho, PRL 66, 2720 (1991), Phys Rep. 396, 1 (2004)

9. Gogny central force Correlation function in Fermi gas Gale-Bertsch-Das Gupta’s parameterization of the K-dependent part of the isoscalar potential

10. Can the symmetry energy becomes negative at high densities? Yes, when the short-range repulsive tensor force in isosinglet n-p pairs dominates At high densities, the energy of pure neutron matter becomes lower than symmetric matter leading to negative symmetry energy This

11. In phenomenological models where there is no explicit tensor force, the symmetry energy starts decreasing when the Lane potential becomes negative Gogny-Hartree-Fock C.B. Das, S. Das Gupta, C. Gale and B.A. Li, PRC 67, 034611 (2003). B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004). L.W. Chen, C.M. Ko and B.A. Li, Phys. Rev. C72, 064309 (2005).

12. Why is the symmetry energy so uncertain especially at high densities? Some observations of a non-expert: Poor knowledge on short-range NN correlations in-medium properties of the short-range tensor force -------- Can the symmetry energy becomes super-soft or even negative at high densities? There seems to be NO first principle forbidding it It happens when the repulsive short-range tensor force due to the ρ-meson exchange in the n-p singlet channel dominates. Using a 20% reduction of the ρ-meson mass as required to reproduce the half-lift of 14 C, the symmetry energy becomes negative above about 4ρ 0 The Lane potential U n -U p flips sign when the Esym starts decreasing with density

13. Momentum and density dependence of the symmetry (isovector) potential Lane potential extracted from n/p-nucleus scatterings and (p,n) charge exchange reactions provides only a constraint at ρ 0 : P.E. Hodgson, The Nucleon Optical Model, World Scientific, 1994 G.W. Hoffmann and W.R. Coker, PRL, 29, 227 (1972). G.R. Satchler, Isospin Dependence of Optical Model Potentials, in Isospin in Nuclear Physics, D.H. Wilkinson (ed.), (North-Holland, Amsterdam,1969)

14. Isospin fractionation in heavy-ion reactions low (high) density region is more neutron- rich with stiff (soft) symmetry energy Bao-An Li, Phys. Rev. Lett. 88 (2002) 192701

15. Pion ratio probe of symmetry energy at supra-normal densities GC Coefficients 2

16. Circumstantial evidence for a super-soft symmetry energy at high densities 400 MeV/A FOPI/GSI data on pion production Willy Reisdorf et al., NPA781 (2007) 459 Transport model analysis Z. Xiao et al., PRL 102, 062502 (2009) Au+Au

17. Is the super-soft symmetry energy “unpleasant” or “unphysical”? Unpleasant ! E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, NPA627, 710 (1997); NPA635, 231 (1998). Repeated by several others in other papers Unphysical ! Quoted by several people in a number of papers Crazy!

18. TOV equation P(r+dr) P(r) Gravity Nuclear pressure Why ? The only reason stated is that “ if the symmetry energy is too soft neutron stars will then collapse while they do exist in nature” For npe matter

19. Do we really know gravity at the Fermi distance? ``It's remarkable that gravity, despite being the first to be discovered, is by far the most poorly understood force," says Nima Arkani-Hamed of Harvard University Roland Pease, Nature 411, 986-988 (28 June 2001) Annu. Rev. Nucl. Part. Sci. 2003. 53:77–121

20. Extra dimension at short length or a new Boson? String theorists have published TONS of papers on the extra dimension In terms of the gravitational potential Repulsive Yukawa potential due to the exchange of a new boson proposed in the super-symmetric extension of the Standard Model of the Grand Unification Theory, or the fifth force The neutral spin-1 gauge boson U is a candidate, it can mediate the interaction among dark matter particles, e.g., Pierre Fayet, PLB675, 267 (2009), C. Boehm, D. Hooper, J. Silk, M. Casse and J. Paul, PRL, 92, 101301 (2004). Arkani-Hamed, N., Dimopoulos, S. & Dvali, G. Phys Lett. B 429, 263–272 (1998). J.C. Long et al., Nature 421, 922-925 (2003); Yasunori Fijii, Nature 234, 5-7 (1971); G.W. Gibbons and B.F. Whiting, Nature 291, 636 - 638 (1981) C.D. Hoyle, Nature 421, 899–900 (2003)

21. Experimental constraints on the strength α and range λ of the Yukawa term M.I. Krivoruchenko et al., PRD 79, 125023 (2009) E.G. Adelberger et al., PRL 98, 131104 (2007) D.J. Kapner et al., PRL 98, 021101 (2007) Serge Reynaud et al., Int. J. Mod. Phys. A20, 2294 (2005)

22. Influences of the Yukawa term on Neutron stars It has NO effect on finite nuclei M.I. Krivoruchenko et al, PRD 79, 125023 (2009)

23. Partially constrained EOS for astrophysical studies Danielewicz, Lacey and Lynch, Science 298, 1592 (2002))

24. EOS of MDIx1+WILB

25. Upper limit Lower limit to support neutrons stars with a super-soft symmetry energy at high densities

26. Some thoughts and observations It may not be that crazy to think about a super-soft and/or even negative symmetry energy at supra-saturation densities especially if you are a string theorist! The FOPI/GSI pion data indicates a super-soft symmetry energy at high densities The high-density behavior of the nuclear symmetry energy relies on the short-range correlations and the in-medium properties of the short-range tensor force in the n-p singlet channel. NS can be stable even with a super-soft symmetry energy if one considers the possibilities of extra-dimensions, new bosons and/or a 5 th force as proposed in string theories and super-symmetric extensions of the Standard Model

27. The moment of inertia provides a sensitive probe to determine g 2 /  2

28. Symmetry energy and single nucleon potential used in the IBUU04 transport model ρ C.B. Das, S. Das Gupta, C. Gale and B.A. Li, PRC 67, 034611 (2003). B.A. Li, C.B. Das, S. Das Gupta and C. Gale, PRC 69, 034614; NPA 735, 563 (2004). soft stiff Single nucleon potential within the HF approach using a modified Gogny force: Density ρ/ρ 0 The momentum dependence of the nucleon potential is a result of the non-locality of nuclear effective interactions and the Pauli exclusion principle The x parameter is introduced to mimic various predictions on the symmetry energy by different microscopic nuclear many-body theories using different effective interactions Default: Gogny force

29. W. Reisdorf et al. for the FOPI collaboration, NPA781 (2007) 459 IQMD: Isospin-Dependent Molecular Dynamics C. HartnackC. Hartnack, Rajeev K. Puri, J. Aichelin, J. Konopka,Rajeev K. PuriJ. AichelinJ. Konopka S.A. BassS.A. Bass, H. Stoecker, W. GreinerH. StoeckerW. Greiner Eur.Phys.J. A1 (1998) 151-169 Near-threshold π - /π + ratio as a probe of symmetry energy at supra-normal densities low (high) density region is more neutron-rich with stiff (soft) symmetry energy Need a symmetry energy softer than the above to make the pion production region more neutron-rich!

30. Formation of dense, asymmetric nuclear matterSoft Stiff Soft E sym Stiff E sym density Symmetry energy n/p ratio at supra-normal densities Central density π - / π + probe of dense matter

31. Momentum dependence of the isoscalar potential Compared with variational many-body theory

32. Constraining the radii of NON-ROTATING neutron stars APR: K 0 =269 MeV. The same incompressibility for symmetric nuclear matter of K 0 =211 MeV for x=0, -1, and -2 Bao-An Li and Andrew W. Steiner, Phys. Lett. B642, 436 (2006) Nuclear limits ●.