1. Constraining the EoS and Symmetry Energy from HI collisions Statement of the problem Demonstration: symmetric matter EOS Laboratory constraints on the symmetry energy from nuclear collisions Summary and outlook Statement of the problem Demonstration: symmetric matter EOS Laboratory constraints on the symmetry energy from nuclear collisions Summary and outlook William Lynch, Yingxun Zhang, Dan Coupland, Pawel Danielewicz, Micheal Famiano, Zhuxia Li, Betty Tsang

2. The density dependence of symmetry energy is largely unconstrained. What is “stiff” or “soft” is density dependent The density dependence of symmetry energy is largely unconstrained. What is “stiff” or “soft” is density dependent EOS: symmetric matter and neutron matter E/A ( ,  ) = E/A ( ,0) +  2  S(  )  = (  n -  p )/ (  n +  p ) = (N-Z)/A E/A ( ,  ) = E/A ( ,0) +  2  S(  )  = (  n -  p )/ (  n +  p ) = (N-Z)/A Brown, Phys. Rev. Lett. 85, 5296 (2001) Neutron matter EOS

3. Need for probes sensitive to higher densities (Why the monopole is not sufficient and K nm is somewhat irrelevant.) If the EoS is expanded in a Taylor series about  0, the incompressibility, K nm provides the term proportional to (  -  0 ) 2. Higher order terms influence the EoS at sub-saturation and supra-saturation densities. The solid black, dashed brown and dashed blue EoS’s all have K nm =300 MeV. If the EoS is expanded in a Taylor series about  0, the incompressibility, K nm provides the term proportional to (  -  0 ) 2. Higher order terms influence the EoS at sub-saturation and supra-saturation densities. The solid black, dashed brown and dashed blue EoS’s all have K nm =300 MeV. The curvature K nm of the EOS about  0 can be probed by collective monopole vibrations, i.e. Giant Monopole Resonance. To probe the EoS at 3  0, you need to compress matter to 3  0.

4. Constraining the EOS at high densities by nuclear collisions Two observable consequences of the high pressures that are formed: –Nucleons deflected sideways in the reaction plane. –Nucleons are “squeezed out” above and below the reaction plane.. Two observable consequences of the high pressures that are formed: –Nucleons deflected sideways in the reaction plane. –Nucleons are “squeezed out” above and below the reaction plane.. pressure contours density contours Au+Au collisions E/A = 1 GeV)

5. Danielewicz et al., Science 298,1592 (2002). Note: analysis required additional constraints on m* and  NN. Flow confirms the softening of the EOS at high density. Constraints from kaon production are consistent with the flow constraints and bridge gap to GMR constraints. Note: analysis required additional constraints on m* and  NN. Flow confirms the softening of the EOS at high density. Constraints from kaon production are consistent with the flow constraints and bridge gap to GMR constraints. Constraints from collective flow on EOS at  >2  0. E/A ( ,  ) = E/A ( ,0) +  2  S(  )  = (  n -  p )/ (  n +  p ) = (N-Z)/A  1 The symmetry energy dominates the uncertainty in the n-matter EOS. Both laboratory and astronomical constraints on the density dependence of the symmetry energy at supra- saturation density are urgently needed. The symmetry energy dominates the uncertainty in the n-matter EOS. Both laboratory and astronomical constraints on the density dependence of the symmetry energy at supra- saturation density are urgently needed. Danielewicz et al., Science 298,1592 (2002).

6. Probes of the symmetry energy To maximize sensitivity, reduce systematic errors: –Vary isospin of detected particle –Vary isospin asymmetry  =(N- Z)/A of reaction. Low densities (  <  0 ): –Neutron/proton spectra and flows –Isospin diffusion High densities (  2  0 ) :  –Neutron/proton spectra and flows –  + vs.  - production To maximize sensitivity, reduce systematic errors: –Vary isospin of detected particle –Vary isospin asymmetry  =(N- Z)/A of reaction. Low densities (  <  0 ): –Neutron/proton spectra and flows –Isospin diffusion High densities (  2  0 ) :  –Neutron/proton spectra and flows –  + vs.  - production E/A( ,  ) = E/A( ,0) +  2  S(  ) ;  = (  n -  p )/ (  n +  p ) = (N-Z)/A symmetry energy <0<0 <0<0

7. Why choose to measure Isospin Diffusion, n/p flows and pion production? Supra-saturation and sub-saturation densities are only achieved momentarily Theoretical description must follow the reaction dynamics self-consistently from contact to detection. Theoretical tool: transport theory: –The most accurately predicted observables are those that can be calculated from i.e. flows and other average properties of the events that are not sensitive to fluctuations. Isospin diffusion and n/p ratios: –Depends on quantities that can be more accurately calculated in BUU or QMD transport theory. –May be less sensitive to uncertainties in (1) the production mechanism for complex fragments and (2) secondary decay. Supra-saturation and sub-saturation densities are only achieved momentarily Theoretical description must follow the reaction dynamics self-consistently from contact to detection. Theoretical tool: transport theory: –The most accurately predicted observables are those that can be calculated from i.e. flows and other average properties of the events that are not sensitive to fluctuations. Isospin diffusion and n/p ratios: –Depends on quantities that can be more accurately calculated in BUU or QMD transport theory. –May be less sensitive to uncertainties in (1) the production mechanism for complex fragments and (2) secondary decay.

8. Measurement of n/p spectral ratios: At E/A = 50 MeV, it probes the pressure due to asymmetry term at  0. Expulsion of neutrons from bound neutron-rich system by symmetry energy. At E/A=50 MeV,  0 is the relevant domain. Has been probed by direct measurements of n vs. proton emission rates in central Sn+Sn collisions. Expulsion of neutrons from bound neutron-rich system by symmetry energy. At E/A=50 MeV,  0 is the relevant domain. Has been probed by direct measurements of n vs. proton emission rates in central Sn+Sn collisions. Double ratio removes the sensitivity to neutron efficiency and energy calibration. soft symmetry energy Bao-An Li et al., PRL 78, 1644 (1997). stiff symmetry energy

9. Collide projectiles and targets of differing isospin asymmetry Probe the asymmetry  =(N-Z)/(N+Z) of the projectile spectator during the collision. The use of the isospin transport ratio R i (  ) isolates the diffusion effects: Useful limits for R i for 124 Sn+ 112 Sn collisions: –R i =±1:no diffusion –R i  0: Isospin equilibrium Collide projectiles and targets of differing isospin asymmetry Probe the asymmetry  =(N-Z)/(N+Z) of the projectile spectator during the collision. The use of the isospin transport ratio R i (  ) isolates the diffusion effects: Useful limits for R i for 124 Sn+ 112 Sn collisions: –R i =±1:no diffusion –R i  0: Isospin equilibrium Isospin diffusion in peripheral collisions, also probes symmetry energy at  <  0. P N mixed 124 Sn+ 112 Sn n-rich 124 Sn+ 124 Sn p-rich 112 Sn+ 112 Sn mixed 124 Sn+ 112 Sn n-rich 124 Sn+ 124 Sn p-rich 112 Sn+ 112 Sn  Systems { Example: neutron-rich projectile proton-rich target measure asymmetry after collision

10. What influences isospin diffusion? Isospin diffusion equation: Naive expectations: –D  increases with S(  ) –D  decreases with  np We tested this by performing extensive BUU and QMD calculations with S(  ) for the form: –S(  ) = 12.5·(ρ/ρ 0 ) 2/3 + S int · (ρ/ρ 0 ) γ i Results: –Diffusion sensitive to S(0.4ρ 0 ) –Diffusion increases with S int and decreases with  i –Diffusion increases with  np –Diffusion decreases when mean fields are momentum dependent and neck fragments emerge. –Diffusion decreases with cluster production. Isospin diffusion equation: Naive expectations: –D  increases with S(  ) –D  decreases with  np We tested this by performing extensive BUU and QMD calculations with S(  ) for the form: –S(  ) = 12.5·(ρ/ρ 0 ) 2/3 + S int · (ρ/ρ 0 ) γ i Results: –Diffusion sensitive to S(0.4ρ 0 ) –Diffusion increases with S int and decreases with  i –Diffusion increases with  np –Diffusion decreases when mean fields are momentum dependent and neck fragments emerge. –Diffusion decreases with cluster production.

11. Sensitivity to symmetry energy Stronger density dependence Weaker density dependence Lijun Shi, thesis The asymmetry of the spectators can change due to diffusion, but it also can changed due to pre- equilibrium emission. The use of the isospin transport ratio R i (  ) isolates the diffusion effects: The asymmetry of the spectators can change due to diffusion, but it also can changed due to pre- equilibrium emission. The use of the isospin transport ratio R i (  ) isolates the diffusion effects: Tsang et al., PRL92, 062701 (2004)

12. Probing the asymmetry of the Spectators The main effect of changing the asymmetry of the projectile spectator remnant is to shift the isotopic distributions of the products of its decay This can be described by the isoscaling parameters  and  : Tsang et. al.,PRL 92, 062701 (2004 ) no diffusion Liu et al.PRC 76, 034603 (2007).

13. Determining  R i (  ) In statistical theory, certain observables depend linearly on  : Calculations confirm this Experiments confirm this In statistical theory, certain observables depend linearly on  : Calculations confirm this Experiments confirm this Consider the ratio R i (X), where X = , X 7 or some other observable: If X depends linearly on  2 : Then by direct substitution:

14. Probing the asymmetry of the Spectators The main effect of changing the asymmetry of the projectile spectator remnant is to shift the isotopic distributions of the products of its decay This can be described by the isoscaling parameters  and  : Tsang et. al.,PRL 92, 062701 (2004 ) 1.0 0.33 -0.33 Ri()Ri() Liu et al.PRC 76, 034603 (2007).

15. Quantitative values Reactions: – 124 Sn+ 112 Sn: diffusion – 124 Sn+ 124 Sn: neutron-rich limit – 112 Sn+ 112 Sn: proton-rich limit Exchanging the target and projectile allowed the full rapidity dependence to be measured. Gates were set on the values for R i (  ) near beam rapidity. –R i (  )  0.47  0.05 for 124 Sn+ 112 Sn –R i (  )  -0.44  0.05 for 112 Sn+ 124 Sn Obtained similar values for R i (ln(Y( 7 Li)/ Y( 7 Be)) –Allows exploration of dependence on rapidity Reactions: – 124 Sn+ 112 Sn: diffusion – 124 Sn+ 124 Sn: neutron-rich limit – 112 Sn+ 112 Sn: proton-rich limit Exchanging the target and projectile allowed the full rapidity dependence to be measured. Gates were set on the values for R i (  ) near beam rapidity. –R i (  )  0.47  0.05 for 124 Sn+ 112 Sn –R i (  )  -0.44  0.05 for 112 Sn+ 124 Sn Obtained similar values for R i (ln(Y( 7 Li)/ Y( 7 Be)) –Allows exploration of dependence on rapidity Liu et al., (2006)  v  /v beam Liu et al.PRC 76, 034603 (2007).

16. Comparison to QMD calculations IQMD calculations were performed for  i =0.35-2.0, S int =17.6 MeV. Momentum dependent mean fields with m n */m n =m p */m p =0.7 were used. Symmetry energies: S(  )  12.3·(ρ/ρ 0 ) 2/3 + 17.6· (ρ/ρ 0 ) γ i IQMD calculations were performed for  i =0.35-2.0, S int =17.6 MeV. Momentum dependent mean fields with m n */m n =m p */m p =0.7 were used. Symmetry energies: S(  )  12.3·(ρ/ρ 0 ) 2/3 + 17.6· (ρ/ρ 0 ) γ i Experiment samples a range of impact parameters –b  5.8-7.2 fm. –larger b, smaller  i –smaller b, larger  i Experiment samples a range of impact parameters –b  5.8-7.2 fm. –larger b, smaller  i –smaller b, larger  i mirror nuclei

17. Expansion around  0 :  Symmetry slope L & curvature K sym  Symmetry pressure P sym Diffusion is sensitive to S(0.4  ), which corresponds to a contour in the (S 0, L) plane. fits to IAS masses fits to ImQMD CONSTRAINTS ImQMD fits for variable S 0 ImQMD fits for variable S 0 Bao-An Li et al., Phys. Rep. 464, 113 (2008). Bao-An Li et al., Phys. Rep. 464, 113 (2008). PDR: A. Klimkiewicz, PRC 76, 051603 (2007). PDR: A. Klimkiewicz, PRC 76, 051603 (2007). Danielewicz, Lee, NPA 818, 36 (2009) Danielewicz, Lee, NPA 818, 36 (2009) GDR: Trippa, PRC77, 061304 GDR: Trippa, PRC77, 061304 ImQMD fits for S 0 =30.1 MeV ImQMD fits for S 0 =30.1 MeV Tsang et al., PRL 102, 122701 (2009).

18. Why probe higher densities? Example: EoS  neutron star radius The neutron star radius is not strongly correlated with the symmetry pressure at saturation density. –This portends difficulties in uniquely constraining neutron star radii for constraints at subsaturation density. The neutron star radius is not strongly correlated with the symmetry pressure at saturation density. –This portends difficulties in uniquely constraining neutron star radii for constraints at subsaturation density. The correlation between the pressure at twice saturation density and the neutron star radius is much stronger. –additional measurements at supra- saturation density will lead to stronger constraints. The correlation between the pressure at twice saturation density and the neutron star radius is much stronger. –additional measurements at supra- saturation density will lead to stronger constraints.  Would be advisable to have multiple probes that can sample different densities LL

19. High density probe: pion production Larger values for  n /  p at high density for the soft asymmetry term (x=0) causes stronger emission of negative pions for the soft asymmetry term (x=0) than for the stiff one (x=-1).  - /  + means Y(  - )/Y(  + ) –In delta resonance model, Y(  - )/Y(  + )  (  n, /  p ) 2 –In equilibrium,  (  + )-  (  - )=2(  p -  n ) The density dependence of the asymmetry term changes ratio by about 10% for neutron rich system. Larger values for  n /  p at high density for the soft asymmetry term (x=0) causes stronger emission of negative pions for the soft asymmetry term (x=0) than for the stiff one (x=-1).  - /  + means Y(  - )/Y(  + ) –In delta resonance model, Y(  - )/Y(  + )  (  n, /  p ) 2 –In equilibrium,  (  + )-  (  - )=2(  p -  n ) The density dependence of the asymmetry term changes ratio by about 10% for neutron rich system. soft stiff Li et al., arXiv:nucl-th/0312026 (2003). stiff soft This can be explored with stable or rare isotope beams at the MSU/FRIB and RIKEN/RIBF. –Sensitivity to S(  ) occurs primarily near threshold in A+A This can be explored with stable or rare isotope beams at the MSU/FRIB and RIKEN/RIBF. –Sensitivity to S(  ) occurs primarily near threshold in A+A t (fm/c)

20. Au+Au Preliminary results puzzling ? MSU Riken GSI FRIB S(  ) MeV Isospin diffusion, n-p flow Pion production Future determination of the EoS of neutron-rich matter Xiao, et al., arXiv:0808.0186 (2008) Reisdorf, et al., NPA 781 (2007) 459.

21. Double ratio: pion production Double ratio involves comparison between neutron rich 132 Sn+ 124 Sn and neutron deficient 112 Sn+ 112 Sn reactions. Double ratio maximizes sensitivity to asymmetry term. –Largely removes sensitivity to difference between  - and  + acceptances. Double ratio involves comparison between neutron rich 132 Sn+ 124 Sn and neutron deficient 112 Sn+ 112 Sn reactions. Double ratio maximizes sensitivity to asymmetry term. –Largely removes sensitivity to difference between  - and  + acceptances. Yong et al., Phys. Rev. C 73, 034603 (2006) soft stiff

22. Summary and Outlook Heavy ion collisions provide unique possibilities to probe the EOS of dense asymmetric matter. A number of promising observables to probe the density dependence of the symmetry energy in HI collisions have been identified. –Isospin diffusion, isotope ratios, and n/p spectral ratios provide some constraints at  0,. –  + vs.  - production, neutron/proton spectra and flows may provide constraints at  2  0 and above. The availability of fast stable and rare isotope beams at a variety of energies will allow constraints on the symmetry energy at a range of densities. –Experimental programs are being developed to do such measurements at MSU/FRIB, RIKEN/RIBF and GSI/FAIR Heavy ion collisions provide unique possibilities to probe the EOS of dense asymmetric matter. A number of promising observables to probe the density dependence of the symmetry energy in HI collisions have been identified. –Isospin diffusion, isotope ratios, and n/p spectral ratios provide some constraints at  0,. –  + vs.  - production, neutron/proton spectra and flows may provide constraints at  2  0 and above. The availability of fast stable and rare isotope beams at a variety of energies will allow constraints on the symmetry energy at a range of densities. –Experimental programs are being developed to do such measurements at MSU/FRIB, RIKEN/RIBF and GSI/FAIR