1. The nuclear equation of state is soft C. Hartnack and J. Aichelin Subatech/University of Nantes H. Oeschler Technical University of Darmstadt SQM, march 2006 Why do we want to know it in Astrophysics? in Nuclear physics? How we can measure it? Why it is soft? Is this a robust statement?

2. The nuclear equation of state E/A as a fct of the density (nuclear physics) Pressure as a fct of density (astrophysics) T=0 hard soft

3. Not theoretically accessible yet (Brückner-Hartree-Fock limited to ρ < ρ 0 ) Importance for nuclear/hadron physics: Modifies considerably the energy which is available for particle production Changes the mass of hadrons considerably Makes that hadrons are not a gas of noninteracting particles (and questions therefore the statistical approach to the hadronization) Is responsible for collective effects like v 1 or v 2

4. Life of a type II supernova Protonneutron star has about the same density as nuclei Nuclear equation of state influence many astrophysical processes Double pulsar rotation (astro-ph 0506566) Binary mergers (astro-ph 0512126) Neutron star formation:

5. “Neutron” Star Composition in 2005 Σ,Λ,Ξ,Δ CFL 2SC,... strange quark matter K—K— (F. Weber, Prog. Part. Nucl. Phys. 54 (2005) 193-288 )

6. The present simulations of the stellar core collapse modeling suffer from The complexity of 2D and 3D simulations Because several 2D/3D phenomena influence the shock expansion like Stellar core rotation Convection Shock instabilities and The incompletely known nuclear equation of state Because different EOS’s show significant differences (Janka, astro-ph/0405289)

7. Source of information: heavy ion reactions at energies with Ebeam< 2 AGeV Three propositions: Proposition I: Volume oscillations induced by heavy ion collisions Giant Monopol Resonance A particle with spin = 0 excites the heavy nucleus, the nucleus vibrates and the vibration frequency is proportional to the compressibility modulus Most recent result (nucl-th 0312020) Compressibility modulus K= 248 +/- 6 MeV Density change is tiny Δρ / ρ < 0.01

8. Proposition II In-plane (v 1 ) and elliptic (v 2 ) flow in heavy ion reactions View perpendicular to the reaction plane In plane flow v 1 at beam/targ y Elliptic flow v 2 at midrapidity In the overlap zone the density increases, at its surface the density gradient increases and therefore the pressure. This pressure creates the flow (Frankfurt)

9. How can one relate the flow to the compressibility modulus K? Only way: Simulation of heavy ion reactions on the computer Using different values of K comparison with the experimental data should allow to determine K

10. The IQMD model ● Semiclassical microscopic model on an event by event basis ● Full time-evolution of each event allows for a view inside the reaction ● Includes nucleons, deltas, pions with their isospin degrees of freedom ● Nuclear eos, Coulomb, asymmetry, Yukawa and momentum dep. pot. ● Virtual treatment of strange particles allows for high statistics

11. How is IQMD related with an EOS? IQMD uses potentials between nucleons Therefore it is able to describe non equilibrium processes The potential uses 4 free parameters a,b,c,L These parameters are adjusted by the following procedure: We calculate in infinite nuclear matter the expectation value of the Hamiltonian for T=0: E = = a( ρ / ρ 0 ) + b( ρ / ρ 0 ) c a,b,c are adjusted to reproduce –16 AMeV at ρ = ρ 0 and a compressibility modulus K (curvature) at ρ 0 L, the width of the Gaussian wavefunctions is chosen to fit best the nuclear surface Having adjusted a,b,c at nuclear matter properties we can use the potential now also in out of equilibrium processes  IQMD

12. Time-evolution: the basic scales: 0 fm/c: start of the reaction 4 fm/c: raise of resonance prod. 8 fm/c: max. central  (nuc.) 10 fm/c max central  12 fm/c: max number of  14 fm/c:  dominate over  16-20 fm/c: nucleon spectra become`thermal' 20 fm/c:  number stabilizes

13. Different in medium cross sections K=210 Different ranges of V NN same EOS No conclusive results yet The flow is a tiny effect which depends very sensitive on many things which are not completely under control in the transport theories Different EOS,same range of V NN Andronic nucl-ex 0411024

14. Proposition III : K + production in heavy ion reactions Why do K + may measure the nuclear equation of state ? HI reactions around 1AGeV: - heavy nuclei get considerably compressed ( ρ >> ρ 0 ) - sqrt(s) too low to produce a K + in first chance NN collisions N’s which produce K + had collisions before by these collisions they test the medium. In these collsions Δ ’s are produced, the main source of K + Higher densities  shorter mean free path  more Δ ’s collide before they disintegrate Different EOS  different density profiles  different K + yield Light nuclei may serve as benchmark.

15. INDEED Strong collectivity: Mult/A in AuAu >Mult/A in CC because ρ (CC) << ρ (AuAu) But: Sufficient sensitive to determine the equation of state? First observation: yield for CC too high (CC = superposition of pp) Why? R AA increases by a factor of 5

16. The K + mass shift and its consequences for HI reactions Increase of the K + mass (nucl-th 0404088) Selfconsistent (IQMD) Scattering length Mass shift around 8% for Au+Au 1.5 AGeV

17. Mass increases higher threshold yield decreases CC: well reproduced Result independent of EOS Au+Au: Small differences between soft and hard EOS But: sufficient to determine EOS?

18. To enhance sensitivity: consider σ (AuAu)/ σ (CC) K + production in central Au+Au as compared to CC collisions shows a strong dependence on the Compressibility modulus Only a soft equation of state (K around 210 MeV) is compatible with the data This result is robust: If one varies input parameters which are not precisely known N-Delta cross sections KN-potential Δ lifetimes the conclusion does not change Completely independent programs give the same results

19. Conclusions confirmed by a completely independent observable: Number of K + per A part This variable is robust as well

20. CONCLUSIONS The simulations of heavy ion experiments agree quantitatively with experiments only if the K + change their mass by interactions with the hadronic environment (predicted theoretically). The ratio σ K+ (AuAu)/ σ K+ (CC) is sensitive to the compressibility modulus of the hadronic equation of state Data agree with simulations for a soft EOS (K = 240 MeV) Result is robust with respect to little known input parameters and confirmed by impact parameter dependence of the K + yield. Thus it seems that this long standing problem has been solved. But remember: what we observe is a hadronic system (10-20% π, Δ ) at at high excitation energy out of equilibrium which we describe with a transport theory whose Hamiltonian gives this EOS in infinite matter This is the best we can ever do.

21. Where are the K + produced ? -- around R = 4fm (mid-central) Corresponds to a density of 1.5 ρ / ρ ° More central  more K + rescatting Shortest way (Perp to the reac. Plane) Longest way (in the reac. Plane) Mean free path not really small

22. What tell heavy ion reactions about a possible K - condensate? Pons et al (astro-ph/0008389): « The effects of kaon condensation on metastable stars is dramatic » « A unique signature for kaon condensation will be difficult to identify » Also here accelerator experiments may give interesting results

23. KN interactions are part of the chiral SU(3) Lagrangien in the mean field approximation

24. But not at all for the K - Reason :strong interaction with the baryons

25. We have forgotten that K’s interact with the medium Simplest approach relativistic mean field calculation

26. KN potentials consequences for K + in HI coll Mass shift around 10% for Au+Au 1.5 AGeV K + are « heavier », K - « lighter » in the nucl. environment

27. Influence of the scalar and vector part of the interaction Vector or scalar only changes the cross section dramatically Change of the total yield much easier to observe than the (small) change of the in-plane flow

28. These excitation functions are all but trivial The importance of the different channels varies with - beam energy - size of the system

29. pA are normally a good test ground but not here: low momenta: Fermi motion and potential compensate: σ (pA) = N σ (pp) High momenta: KN potential negligible

30. pA is not sensitive to -- KN potential -- KN cross section Overall well described  basic process understood but this time we cannot learn much from pA

31. The Δ N  K + cross section is unknown Two different approaches conjectured --Tsushima --Isospin scaled NN  K + cross section (here named Giessen) The two give 60% difference in the K + yield

32. How reliable are transport model predicitons? Trento workshop + many homework assignments in order to clarify the discrepancies between the results. Barz Bratkovskaya (HSD) Cassing (HSD) Chen (C. M. Ko) Danielewisz Fuchs (QMD) Gaitanos Hartnack (IQMD) Larinov (Mosel) Reiter (URQMD)

33. After several iterations: differences almost exclusively due to different inputs - different (unknown) cross sections (especially in the Δ chanel) - treatment of Δ resonance in medium Stopping and transverse momentum ok For large systems

34. Pions make more problems but agree fairly for the same input.

35. In the standard versions we see differences in the pion yield of more than 70%. Why?: all codes need a Δ lifetime as input Why problem?: Δ has a width; in order to populate the Δ we need wavepackets with a large width in energy. But in simulation programs we have sharp energies Commonly used descriptions: 1) Kitazoe: 1/ τ ~ Phasespace at the given energy 2) Wigner (phase shift) τ ~ 2 d δ (E)/dE Γ rex >> Γ wave fct 3) τ = 1/ 120 MeV Dramatic consequences for π ’s in HIC

36. Fortunatelly the different approaches for the Δ lifetime do not change the slope of the K + spectra but they change the yield at low pt.

37. No potential KN potential K + : form of dn/dy very similar but differences in yield due to different cross sections. Γ Δ =const Soft EOS

38. p T slopes in good agreement. Not trivial : phase space of the NN( Δ )  N Λ K + collisions. + Fermi motion + KN collisions

39. The calculations show a squeeze due to the KN potential in heavy systems

40. Good news: even if yields vary by 30% all calculations (even using different cross sections, τ Δ etc) point towards a soft nuclear equation of state This is the first time that we have solid information on the nuclear equation of state.

41. The result of a soft equation of state is very robust KN potential +/- π lifetime BB  K + cross section

42. What’s about the K - ? K - are much more complicated than K + Resonances in the K - N channel - which may disappear in the medium ( Λ (1405) Koch, Weise) More cross sections Λ + π N+ K - because the s quark can be transfered to baryons. These cross sections diverge close to the threshold. (not included in Chen) Complicated in medium properties Due to the coupling to the resonances (Lutz, Kolomeitsev, Tolos….) Still quasi particles?

43. There remains work to be done K - rapidity distributions

44. Present status (or on what all agree): - Large difference between pp and AA is due to the production channel Λ + π  N+ K - absent in pp - This cross section dominates the final yield - It couples the K - to the K + yield - K - have a steady state equilibrium due to the hugh difference of Λ + π  N+ K - and N+ K -  Λ + π cross sec. close to threshold Influence of the K - and K + potential on the final K - yield: K + on/off : more or less Λ to create the K - K - on/off : varies the threshold Both yield a factor of two: K - on /K + on yield the same result as K - off / K + off

45. Can one see the K potentials and cross sections directly? Yes, When leaving the nucleus K + gain and K - loose energy therefore spectra distorted in a specific way σ (K +,p T ) / σ ( K -,p T ) at small p T is sensitive to the K potentials Collisions change the slope: Slope mesures the KN cross sections

46. Influence of the KN collisions Changes the slope remarkably Influence of the KN potential Changes the yield at small E cm Slope of the K + spectra « measures » the number of KN collisions

47. SPS and RHIC energies Simulation programs still in development (EPOS) and URQMD Strategy on which most agree: Start with pp pA to have a known environment AA For RHIC: to early, even the elementary degrees of freedom are still discussed (parton casc, strings, CCC, hadronic rescatt.) In the non strange sector (data available) reasonable agreement

48. The more strange the particles the more differ the predicitons Does this allow to find the right process or are there parameters to fix pp yields differ by a factor of 4 for Ω 3 for anti ΛΣ 3 for Ξ 1.5 for K, ΛΣ

49. Pb +Pb 160 AGeV: One sees quite different reaction scenarios Hijing/ HSD much more transparent than EPOS/URQMD Hugh difference in the energy deposit (but almost not visible in the pion yield).

50. For the (multi) strange sector the differences become enormous due to the différent reaction mechanisms Ω (AA)/ Ω (pp) Hijing 58 Epos 1000 Urqmd1.3 1190 Urqmd2.1 4222 Λ yields differ by a factor 2 Ξ yield by a factor of 28 Λ /anti Λ ratios by 4

51. Conclusions: Theory predict without doubt that hadrons change their properties in a hadronic environment. Models are reliable for ρ < ρ ° Baryons and Mesons react differently Transport theories have seen a lot of progress K + and K - production are coupled (NK  πΛ ) The present knowledge of elem. σ requires effective masses Data are only compatible with a soft hadronic « EOS » this result seems to be robust. to progress further: new transport theories for spectral functions more data on pp, np, pA to determine input σ

52. Conclusions: At 1 AGeV: lot of progress due to intense collaborations data are only compatible with a soft nuclear EOS KN potential and cross section may be seen in spectra to progress further: new transport theories with spectral functions more data on pp, np, pA to determine input σ At 200 AGeV (SPS): Comparison of simulation codes have just started presently hugh differences in the strangeness sector extrapolation pp/AA yields very different results to progress further: pp,pA  strangeness dn/dy(dx) data urgently needed determination of observables sensitive to the proposed mechanisms At RHIC: Ideas and manpower needed, but we have no choice

53. Thanks to E. Kolomeitsev and C. Hartnack who made the figures Barz Gaitanos Bleicher (URQMD) Hartnack(IQMD) Bratkovskaya (HSD) Larinov (Mosel) Cassing (HSD) Reiter(URQMD) Chen (C. M. Ko) Topor Pop (Hijing) Danielewisz Werner(Epos) Fuchs (QMD)

54. The present level of data theory comparison