Constraints on the Symmetry Energy from Heavy Ion Collisions Hermann Wolter Ludwig-Maximilians-Universität München 44th Karpacz Winter School of Theoretical Physics, and 1st ESF CompStar Workshop, „The Complex Physics of Compact Stars“, Ladek Zdroj, Poland,

Outline: - the symmetry energy and its role for neutron stars - knowledge of the symmetry energy - Investigation in heavy ion collisions - below saturation density: Fermi energies, diffusion, fragmentation - high densities: relativistic energies; flow, particle production - summary Punchline: - we identify several observables in heavy ion collisions which are sensitive to the symmetry energy - however, the situation is not yet at a stage (experimentally and theoretically) to fix the symmetry energy Collaborators: M. Di Toro, M. Colonna, LNS, Catania Theo Gaitanos, U. Giessen C. Fuchs, U. Tübingen; S. Typel, GANIL Vaia Prassa, G. Lalazissis, U. Thessaloniki

Schematic Phase Diagram of Strongly Interacting Matter Liquid-gas coexistence Quark-hadron coexistence SIS

Schematic Phase Diagram of Strongly Interacting Matter Liquid-gas coexistence Quark-hadron coexistence Z/N 1 0 SIS neutron stars

Symmetry Energy: Bethe-Weizsäcker Massenformel E sym    MeV)     Asy-stiff Asy-soft Asy-superstiff High density: Neutron stars Around normal density: Structure, neutron skins heavy ion collisions in the Fermi energy regime

Theoretical Description of Nuclear Matter V ij Non-relativistic: Hamiltonian H = S T i + S V ij, ; V nucleon-nucleon interaction Relativistic: Hadronic Lagrangian y, nucleon, resonances s,w, p,.... mesons phenomenological microscopic (fitted to nucl. matter) (based on realistic NN interactions non-relativistic Skyrme-type Brueckner-HF (BHF) (Schrödinger) Relativistic Walecka-type Dirac-Brueckner HF (DB) (Quantumhadrodyn.) Density functional theory

Decomposition of DB self energy Density (and momentum) dependent coupling coeff.     Dirac-Brueckner (DB)Density dep. RMF (alternative: non-linear model (NL) meson self interactions)

No  f  1.5 f  FREE f    2.5 fm 2 f  5 f  FREE PRC65(2002) RMF Symmetry Energy: 28÷36 MeV NL NLρ NLρδ

The Nuclear Symmetry Energy in different Models The symmetry energy as the difference between symmetric and neutron matter: stiff soft iso-stiff iso-soft empirical iso-EOS‘s cross at about microscopic iso-EOS`s soft at low densities but stiff at high densities C. Fuchs, H.H. Wolter, EPJA 30(2006)5,(WCI book)

Uncertainities in optical potentials Isoscalar PotentialIsovector (Lane) Potential data

GSI SIS LNS, GANIL, MSU Incident energy of Heavy Ion Collision: Low energy (Fermi regime): Fragmentation, liquid-gas phase transition, Deep inelastic High energy (relativistic): Compression, particle production, temperature. Modificaion of hadron properties

Transport description of heavy ion collisions: For Wigner transform of the one-body density: f(r,p;t) Vlasov eq.; mean field 2-body hard collisions Simulation with Test Particles: effective mass Kinetic momentum Field tensor Relativistic BUU eq. loss termgain term Fluctuations from higher order corr.; stochastic treatment

Data: Famiano et al. PRL 06 Calc.: Danielewicz, et al. 07 soft stiff SMF simulations, V.Baran 07 Central Collisions at Fermi energies: Investigation of ratio of emitted pre-equilibrium neutrons over protons 124 Sn Sn 112 Sn Sn

Peripheral collision at Fermi energies: Schematic picture of reaction phases and possible observables pre-equilibrium emission: Gas asymmetry Proton/neutron ratios Double ratios binary events: asymmetry of PLF/TLF transport ratios ternary events: asymmetry of IMF Velocity corr. isospin diffusion/transpo rt Isospin current due to density and isospin gradients: drift coefficients diffusion coeffients Differences in tranport coefficients simply connected to symmetry energy asy- stiff asy- soft Density range in peripheral collisions Opposite effects on drift and diffusion for asy-stiff/soft

Isospin Transport through Neck: Imbalance (or Rami, transport) ratio: (i = proj/targ. rapidity) (also for other isospin sens.quantities) Limiting values: R=0 complete equilibration R=+-1, complete trasnparency Discussed extensively in the literature, and experimental data (MSU) e.g. L.W.Chen, C.M.Ko, B.A.Li, PRL 94, (2005) V. Baran, M. Colonna, e al., PRC 72 (2005)  Momentum dependence important

Isospin Transport through Neck: exp. MSU Asymmetry of IMF in symm. Sn+Sn collisions

Asymmetry of IMF in symm. Sn+Sn collisions  IMF MD MI Stiff-soft, 124 Stiff-soft, 112 Stiff-soft, 124 Stiff-soft, 112 Asymmetry of IMF in peripheral collision rather sensitive to symmetry energy, esp. for 1.MD interactions 2.when considered as ratio relative to asymmetry of residue 3.Effects of the order of 30%,  sensitive variable!

Results from Flow Analysis (P. Danielewicz, R. Lynch,R.Lacey, Science) Flow and elliptic flow described in a model which allows to vary the stiffness (incompressibility K), and has a momentum dependence Deduced limits for the EOS (pressure vs. density) for symmetric nm (left). The neutron EOS (i.e. the symmetry energy) is still uncertain, thus two areas are given for two different assumptions. v 2 : Elliptic flowv 1 : Sideward flow

Asymmetric matter: Differential directed and elliptic flow 132 Sn AGeV b=6fm p n    differential directed flow   differential elliptic flow Difference at high p t first stage Dynamical boosting of the vector contribution T. Gaitanos, M. Di Toro, et al., PLB562(2003) Proton-neutron differential flow and analogously for elliptic flow

Pion production: Au+Au, semicentral Equilibrium production (box results) Finite nucleus simulation: ~ 5 (NLρ) to 10 (NLρδ)

++ ++ -- -- ++ ++ -- -- W.Reisdorf et al. NPA781 (2007) 459 Transverse Pion Flows Simulations: V.Prassa Sept.07 Antiflow: Decoupling of the Pion/Nucleon flows OK general trend. but: - smaller flow for both  - and  + -not much dependent on Iso-EoS

Kaon Production: A good way to determine the symmetric EOS (C. Fuchs, A.Faessler, et al., PRL 86(01)1974) Also useful for Isovector EoS? -charge dependent thresholds - in-medium effective masses -Mean field effects Main production mechanism: NN  BYK, pN  YK

Effect of Medium-Effects on Pion (left) and Kaon (right) Ratios Inelastic cross section K-potential (isospin independent) K-potential (isospin dependent)

Astrophysical Implications of Iso-Vector EOS Neutron Star Structure Constraints on the Equation-of-state - from neutron stars: maximum mass gravitational mass vs. baryonic mass direct URCA process mass-radius relation - from heavy ion collisions: flow constraint kaon producton Equations of State tested: Klähn, Blaschke, Typel, Faessler, Fuchs, Gaitanos,Gregorian, Trümper, Weber, Wolter, Phys. Rev. C74 (2006)

Neutron star masses and cooling and iso-vector EOS Tolman-Oppenheimer-Volkov equation to determine mass of neutron star Proton fraction and direct URCA Onset of direct URCA Forbidden by Direct URCA constraint Typical neutron stars Heaviest observed neutron star (now retracted)

Direct Urca Cooling limit Mass-Radius Relations Gravitational vs. Baryon Mass Heavy Ion Collision obsevables Constraints of different EOS‘s on neutron star and heavy ion observables Maximum mass

Summary: While the Eos of symmetric NM is fairly well determined, the isovector EoS is still rather uncertain (but important for exotic nuclei, neutron stars and supernovae) Can be investigated in HIC both at low densities (Fermi energy regime, fragmentation) and high densities (relativistic collisions, flow, particle production) Data to compare with are still relatively scarce; it appears that the Iso-EoS is rather stiff. Effects scale with the asymmetry – thus reactions with RB are very important Additional information can be obtained by cross comparison with neutron star observations