basic hadronic SU(3) model generating a critical end point in a hadronic model

1. Modeling of the Parton-Hadron Phase Transition Villasimius 2010 Hot and dense matter and the phase transition in quark-hadron approaches OUTLINE • basic hadronic SU(3) model • generating a critical end point in a hadronic model • revisited • including quark degrees of freedom • phase diagram – the QH model • excluded volume corrections, phase transition J. Steinheimer, V. Dexheimer, H. Stöcker, SWS Goethe University, Frankfurt

2. hadronic model based on non-linear realization of chiral symmetry degrees of freedom SU(3) multiplets: baryons (n,Λ, Σ, Ξ) scalars (, , 0) vectors (ω, ρ, φ) , pseudoscalars, glueball field χ A)SU(3) interaction ~ Tr [ B, M ] B , ( Tr B B ) Tr M B) meson interactions ~ V(M) <> = 0  0 <> =  0  0 C) chiral symmetry  m = mK = 0 explicit breaking ~ Tr [ c  ] ( mq q q )  light pseudoscalars, breaking of SU(3) _ _ _ _ _ _ _  ~ <u u + d d> <  ~ <s s> 0 ~ < u u - d d> _

3. fit parameters to hadron masses  mesons * *         K* p,n ’    Model can reproduce hadron spectra via dynamical mass generation  K 

4. Lagrangian (in mean-field approximation) L = LBS + LBV + LV + LS + LSB baryon-scalars: _ LBS= -  Bi (gi  + gi  + gi  ) Bi baryon-vectors: _ LBV= -  Bi (gi  + gi + gi  ) Bi / / / meson interactions: I LBS= k1 (2 + 2 + 2 )2 + k2/2 (4 + 2 4 + 4 + 6 2 2 ) + k32  - k44 - 4 ln /0 +  4 ln [(2 - 2) / (020)] I LV = g4 (4 + 4 + 4 + β22) explicit symmetry breaking: LSB = c1 + c2

5. important reality check nuclear matter properties at saturation density asymmetry energy E/A (p- n) equation of state E/A () binding energy E/A ~ -15.2 MeV saturation (B)0 ~ .16/fm3 compressibility ~ 223 MeV asymmetry energy ~ 31.9 MeV phenomenology: 200 - 250 MeV 30 - 35 MeV + good description of finite nuclei / hypernuclei SWS, Phys. Rev. C66, 064310

6. Task: self-consistent relativistic mean-field calculation coupled 7 meson/photon fields + equations for nucleons in 1 to 3 dimensions parameter fit to known nuclear binding energies and hadron masses 2d calculation of all measured (~ 800) even-even nuclei error in energy (A  50) ~ 0.21 % (NL3: 0.25 %)  (A  100) ~ 0.14 % (NL3: 0.16 %) relativistic nuclear structure models good charge radii rch~ 0.5 % (+ LS splittings) correct binding energies of hypernuclei SWS, Phys. Rev. C66, 064310 (2002)

7. phase transition compared to lattice simulations heavy states/resonance spectrum is effectively described by single (degenerate) resonance with adjustable couplings Tc ~ 180 MeV µc ~ 110 MeV reproduction of LQCD phase diagram, especially Tc, μc + successful description of nuclear matter saturation phase transition becomes first-order for degenerate baryon octet ~ Nf = 3 with Tc ~ 185 MeV D. Zschiesche et al. JPhysG 34, 1665 (2007)

8. Isentropes, UrQMD and hydro evolution lines of constant entropy per baryon, i.e. perfect fluid expansion E/A = 5, 10, 40, 100, 160 GeV E/A = 160 GeV goes through endpoint J. Steinheimer et al. PRC77, 034901 (2008)

9. Including higher resonances explicitly Add resonances up to 2.2 GeV. Couple them like the lowest-lying baryons P. Rau, J. Steinheimer, SWS, in preparation

10. connect hadronic and quark degrees of freedom order parameter of the phase transition confined phase deconfined phase effective potential for Polyakov loop, fit to lattice data U = ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*)3 – 3 (ΦΦ*)2] a(T) = a0T4 + a1 µ4 + a2 µ2T2 baryonic and quark mass shift δ mB ~ f(Φ) δ mq ~ f(1-Φ) q q quarks couple to mean fields via gσ, gω V. Dexheimer, SWS, PRC 81 045201 (2010) Rattiet al. PRD 73 014019 (2006) Fukushima, PLB 591, 277 (2004) minimize grand canonical potential

11. Phase Diagramfor HQM model µc = 360 MeV Tc = 166 MeV µc. = 1370 MeV ρc ~ 4 ρo hybrid hadron-quark model critical endpoint tuned to lattice results V. Dexheimer, SWS, PRC 81 045201 (2010)

12. Mass-radius relation using Maxwell/Gibbs construction Gibbs construction allows for quarks in the neutron star mixed phase in the inner 2 km core of the star V. Dexheimer, SWS, PRC 81 045201 (2010) R. Negreiros, V. Dexheimer, SWS, PRC, astro-ph:1006.0380 

13. isentropic expansion overlap initial conditions Elab = 5, 10, 40, 100, 160 AGeV averaged Cs significantly higher than 0.2 Csph ~ ¼ Cs,ideal

14. Temperature distribution from UrQMD simulation as initial state for (3d+1) hydro calculation initial temperature distribution Speed of sound - (weighted) average over space-time evolution dip in csissmeared out

15. different approach – hadrons, quarks, Polyakov loop and excluded volume Include modified distribution functions for quarks/antiquarks * * Following the parametrization used in PNJL calculations U = - ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*)3 – 3 (ΦΦ*)2] a(T) = a0T4 + a1 T0T3 + a2 T02T2 , b(T) = b3 T03 T χ = χo (1 - ΦΦ* /2) The switch between the degrees of freedom is triggered by excluded volume corrections thermodynamically consistent - Vq = 0 Vh = v Vm = v / 8 ~ ~ ~ e = e / (1+ Σ viρi ) µi = µ i – vi P Steinheimer,SWS,Stöckerhep-ph/0909.4421 D. H. Rischke et al., Z. Phys. C 51, 485 (1991) J. Cleymans et al., Phys. Scripta 84, 277 (1993)

16. quark, meson, baryon densities at µ = 0 densities of baryon, mesons and quarks natural mixed phase, quarks dominate beyond 1.5 Tc ρ Energy density and pressure compared to lattice simulations

17. Temperature dependence of chiral condensate and Polaykov loop at µ = 0 Interaction measure e – 3p latticedatatakenfromBazavov et al. PRD 80, 014504 (2009) speed of sound shows a pronounced dip around Tc !

18. subtracted condensate and polyakov loop different lattice groups and actions From Borsanyi et al., arxiv:1005:3508 [hep-lat]

19. Lattice comparison of expansion coefficients as function of T expansion coefficients lattice results lattice data from Cheng et al., PRD 79, 074505 (2009) Steinheimer,SWS,Stöcker hep-ph/0909.4421 suppression factor peaks

20. Dependence of chiral condensate on µ, T Lines mark maximum in T derivative Φ σ Separate transitions in scalar field and Polyakov loop variable

21. Dependence of Polyakov loop on µ, T Lines mark maximum in T derivative Φ σ Separate transitions in scalar field and Polyakov loop variable

22. Susceptibilitiy c2 in PNJL and QHM for different quark vector interactions QH At least for µ = 0 – smallquarkvectorrepulsion PNJL gqω = 0 gqω = gnω /3 Steinheimer,SWS,hepph/1005.1176

23. σ Φ

24. UrQMD/Hydro hybrid simulation of a Pb-Pb collision at 40 GeV/A red regions show the areas dominated by quarks

25. simple time evolution including π, K evaporation (E/A = 40 GeV) with evaporation C. Greiner et al., PRD38, 2797 (1988) E/A-mN 300 g2 = 0 If you want it exotic … g2 = 2 200 additional coupling g2 of hyperons to strange scalar field g2 = 4 100 follow star calcs by J. Schaffner et al., PRL89, 171101 (2002) g2 = 6 0 0 0.5 1 1.5 barrier at fs ~ 0.4 – 0.6 fs

26. SUMMARY • general hadronic model as starting point • works well with basic vacuum properties, nuclear matter, nuclei, … • phase diagram with critical end point via resonances • implement EOS in combined molecular dynamics/ hydro simulations • quarks included using effective deconfinement field • „realistic“ phase transition line • implementing excluded volume term, natural switch of d.o.f. If you want to see hadronization, grab your Iphone -> Physics to Go! Part 3

27. Hypernuclei -  single-particle energies Nuclear matter Model and experiment agree well

28. Evolution of the collision system Elab≈ 5-10 AGeV sufficient to overshoot phase border, 100-160 AGeV around endpoint

29. amount of volume scanning the critical endpoint (lattice)

30. Comparison of gross properties of initial conditions Overlap of projectile and target

31. time integrated volume around the critical end point

32. effective volume sampling the critical end point window T = Tc ± 10 MeV µ = µc ± 10 MeV maximum shifts in time