1. Chiho Nonaka QM2009 Nagoya University Chiho NONAKA March 31, 2009@Quark Matter 2009, Knoxville, TN In collaboration with Asakawa, Bass, and Mueller

2. Chiho Nonaka QM2009 Where is the QCD Critical Point? Lattice QCD, Effective theories…. QCP search in heavy ion collisions -Energy scan -Experiments and phenomenology Stephanov,hep-lat/0701002 energy scan focusing t Stephanov, Rajagopal, and Shuryak, PRL81 (1998) 4816

3. Chiho Nonaka QM2009 Towards Quantitative Analyses Realistic dynamical model 3D Hydro + UrQMD (hadron base event generator) Equation of state with QCD critical point Physical observables Signals of QCP should survive after freezeout process. Fluctuations Hadron ratios

4. Chiho Nonaka QM2009 Schematic sketch t freeze-outhadronizationthermalizationcollision 3D Hydro+UrQMD Model Nonaka and Bass PRC75:014902(2007) 3D Hydro + UrQMD  Success of hydrodynamic models at RHIC: --P T spectra, rapidity distribution, elliptic flow  Limitation of Hydro Ex. Elliptic flow at forward/backward rapidity, HBT  Extension of Hydro --Viscosity effect of hadron phase --Final state interactions Viscosity in QGP phase Full 3-d Hydrodynamics EoS :1st order phase transition QGP + excluded volume model Cooper-Frye formula UrQMD t fm/c final state interactions Monte Carlo Hadronization TCTC T SW T C :critical temperature  T SW : Hydro  UrQMD hydrodynamical expansion

5. Chiho Nonaka QM2009 Highlights of 3D Hydro+UrQMD Nonaka and Bass PRC75:014902(2007)

6. Chiho Nonaka QM2009 Schematic sketch t freeze-outhadronizationthermalizationcollision 3D Hydro+UrQMD Model C.N. and Bass PRC75:014902(2007) 3D Hydro + UrQMD  Success of hydrodynamic models at RHIC: --P T spectra, rapidity distribution, elliptic flow  Limitation of Hydro Ex. Elliptic flow at forward/backward rapidity, HBT  Extension of Hydro --Viscosity effect of hadron phase --Final state interactions Viscosity in QGP phase Full 3-d Hydrodynamics EoS :1st order phase transition QGP + excluded volume model Cooper-Frye formula UrQMD t fm/c final state interactions Monte Carlo Hadronization TCTC T SW T C :critical temperature  T SW : Hydro  UrQMD hydrodynamical expansion Asakawa, Bass, Mueller,C.N.PRL101:122302(2008), C.N., Asakawa, Phys.Rev.C71:044904(2005) with QCD critical point

7. Chiho Nonaka QM2009 EOS with QCD Critical Point Singular part near QCD critical point + Non-singular part Singular part - Mapping ： (r, h) 3-d Ising Model  (T,  B ) QCD  r h T QGP Hadronic 3d Ising Model Same Universality Class QCD C.N. and Asakawa, PRC71,044904(2005)

8. Chiho Nonaka QM2009 EOS of 3-d Ising Model Parametric Representation of EOS Guida and Zinn-Justin NPB486(97)626 h : external magnetic field

9. Chiho Nonaka QM2009 Mapping ： 3D Ising Model  QCD h axis ? Critical Region? Lattice QCD Effective theory Experiments r h  T QCD 3d Ising model h h? r tangential line inputs in our model No Universality

10. Chiho Nonaka QM2009 Focusing Effect With QCD critical point Bag Model + Excluded Volume Approximation (No Critical Point) FocusedNot Focused Isentropic Trajectories on QCD phase diagram

11. Chiho Nonaka QM2009 Signal of QCP Signal of QCD critical point should survive even after freezeout process. Fluctuations: conserved values ex. charge, baryon number Hadron ratios: fixed at chemical freezeout temperature T ch freeze-outhadronizationthermalizationcollisionhydrodynamical expansion

12. Chiho Nonaka QM2009 Fluctuation in 1d Hydro. Model H (Halperin RMP49(77)435) fluctuation QCP critical slowing down evolution rate fluctuation : clear signal? static dynamical

13. Chiho Nonaka QM2009 Signal of QCP Signal of QCD critical point should survive even after freezeout process. Fluctuations: conserved values ex. charge, baryon number Hadron ratios: fixed at chemical freezeout temperature T ch Key: T ch depends on transverse momentum freeze-outhadronizationthermalizationcollisionhydrodynamical expansion

14. Chiho Nonaka QM2009 Hadron Production on n B /s const. line UrQMD : no QCD critical point T  Isentropic trajectory higher P T particles lower P T particles T ch depends on transverse velocity. Hadron ratios depend on transverse velocity.

15. Chiho Nonaka QM2009 Demonstration SPS Location of QCP (  B,T)=(550,159) Critical Region parameter Chemical freezeout point (  B,T)=(406,145) from statistical model QCP CF Search of the QCD critical point from experiments

16. Chiho Nonaka QM2009 Isentropic Trajectories FO, CO QCP ratio Hadronization occurs between the phase boundary and chemical freezeout point CrossOver First Order

17. Chiho Nonaka QM2009 Signature of QCP decreases (FO,CO) increases (QCP) with QCP steeper spectra at high P T ~P T

18. Chiho Nonaka QM2009 QCD Critical Point? NA49, PRC73,044910(2006) steeper spectra at high P T NA49

19. Chiho Nonaka QM2009 3D Hydro+UrQMD with QCP Initial Conditions Energy density Baryon number density Parameters Flow EOS: QCP, Bag Model Switching temperature  0 =0.5   =1.5  0 =0.6 fm/c T SW =150 [MeV] v T =0 v L =  Bjorken’s solution );  max GeV/fm 3 n Bmax fm -3 2.00.15 E E MeVMeV,550159==T  QCP: 2 0 1 longitudinal direction: H(  )longitudinal direction: H(  ) T c =170 MeV at  B = 0 MeV

20. Chiho Nonaka QM2009 3D Hydro + UrQMD Hydro with Bag Model Hydro with QCD critical point T [MeV]  B [MeV] 1212 UrQMD Switching temperature: 150 MeV UrQMD QCP  B ~430 MeV  B ~250 MeV T [MeV]  B [MeV]

21. Chiho Nonaka QM2009 P T Spectra At T SW Because of focusing effect QCP Bag Model QCD critical point

22. Chiho Nonaka QM2009 Summary Towards quantitative analyses EoS with QCP Focusing effects in isentropic trajectories 3D Hydro + UrQMD model P T spectra, hadron ratios Focusing Signal of QCD critical point: p/p ratio as a function of transverse momentum Signal of QCD critical point: p/p ratio as a function of transverse momentum

23. Chiho Nonaka QM2009 Back Up

24. Chiho Nonaka QM2009 p Spectra work in progress  Narrow collision energy window is needed.  NA49: anti-p spectra Careful choice for initial condition of hydro calculation

25. Chiho Nonaka QM2009 Critical Region ) QCP CF Entropy density -0.8 0.8 hadron phase QGP phase BB T QCD critical point (  B,T)=(550,159) CF

26. Chiho Nonaka QM2009 Correlation Length r h Widom’s scaling low Large fluctuation

27. Chiho Nonaka QM2009 Fluctuation in 1d evolution Model H (Halperin RMP49(77)435) Critical slowing down Evolution rate fluctuation

28. Chiho Nonaka QM2009 Recent Status of Dynamical Models hydrodynamical expansion freeze-outhadronizationthermalizationcollision initial conditions parametrizationparametrization Glauber type Glauber type equation of states 1 st order phase1 st order phase transition transition freezeout process Sudden freezeoutSudden freezeout Chemical equilibriumChemical equilibrium Color Glass Condensate Flux Tube Initial fluctuation Lattice QCD Partial chemical equilibriumPartial chemical equilibrium Viscosity effect ofViscosity effect of hadron phase hadron phase Final state interactionsFinal state interactions Viscous Hydrodynamics

29. Chiho Nonaka QM2009 Equation of State QCD critical point entropy density trajectories (  B,T)=(550,159)

30. Chiho Nonaka QM2009 Focusing Effect  T r h Guida and Zinn-Justin, NPB486(97)626 Critical exponent Critical slowing down dynamical critical exponent along h is dominant. Berdnikov and Rajagopal,PRC61,105017(99) 3d Ising Model Same Universality Class QCD

31. Chiho Nonaka QM2009 Entropy density Dimensionless parameter: Sc Choice of parameters Singular Part + Non-singular Part Hadron Phase (excluded volume model) QGP phase Critical domain Thermodynamical inequalities Critical exponent near CEP keeps correctly. crit  Bcrit

32. Chiho Nonaka QM2009 Gibbs Free Energy Entropy Density for Singular Part Singular Part of EOS mapping Free energy: Guida and Zinn-Justin NPB486(97)626 crit  Bcrit

33. Chiho Nonaka QM2009 Thermodynamical Quantities 1st order i Baryon number density i Pressure i Energy Density QCP 1st order crossover

34. Chiho Nonaka QM2009 34 Slowing out of Equilibrium B. Berdnikov and K. Rajagopal, Phys. Rev. D61 (2000) 105017 Berdnikov and Rajagopal’s Schematic Argument along r = const. line Correlation length longer than  eq h  faster (shorter) expansion r h slower (longer) expansion Effect of Focusing on  ? Focusing Time evolution : Bjorken’s scaling solution along n B /s    fm, T 0 = 200 MeV   eq

35. Chiho Nonaka QM2009 35 Correlation Length (I) Widom’s scaling low  eq r h depends on n /s.  Max. Trajectories pass through the region where  is large. (focusing) eq B n B /s non-critical component of the EOS

36. Chiho Nonaka QM2009 36 Correlation Length (II)  time evolution (1-d) Model H (Halperin RMP49(77)435)  is larger than  at T f. Differences among  s on n /s are small. In 3-d, the difference between  and  becomes large due to transverse expansion. eq B Critical slowing down

37. Chiho Nonaka QM2009 QCP Search in Heavy Ion Collisions t energy scan focusing t M. Stephanov, K. Rajagopal, and E.Shuryak, PRL81 (1998) 4816