November 23, Fields in the Era of Teraflop-Computing Thermal Properties of Hadrons: below and above the phase transition Masayuki Asakawa Osaka University

M. Asakawa (Osaka University) PLAN QCD Phase Diagram QGP, Color Super Conductors, Critical End Point (CEP) Freezeouts: Experimental Facts and Interpretation Hadrons in Matter and Observables Necessity of MEM (Maximum Entropy Method) MEM Outline Importance of Error Analysis Finite Temperature Results for J/  and  c Error Analysis Statistical Systematic Strongly Interacting Matter at RHIC? CEP Observables

M. Asakawa (Osaka University) What’s Quark Gluon Plasma? Hot Hadron Soup Dense Hadron Soup CEP 2SC, dSC, CFL... Color Super Conductors

M. Asakawa (Osaka University) Lattice Results (Finite T) Pressure Energy Density Entropy Density Jump in Entropy Density Karsch et al. (2000)

M. Asakawa (Osaka University) Critical End Point on the Lattice m ≠ m phys Fodor and Katz (2002) m = m phys Fodor and Katz (2004) N  = 4

M. Asakawa (Osaka University) First suggestion of CEP in QCD Critical End Point in Effective Models Compilation by Stephanov Yazaki and M.A. NPA (1989)

M. Asakawa (Osaka University) Freezeout Points in HI Collisions ChemicalParticle # Thermal Kinetic Particle Distribution Two Freezeouts Experimentally, T ch ＞ T th

M. Asakawa (Osaka University) Chemical Freezeout Data – Fit (  ) Ratio Particle ratios well described by T ch = 160  10 MeV,  B = 24  5 MeV These Numbers are obtained by assuming FREE BE/FD distributions (s 1/2 = 200 A GeV)

M. Asakawa (Osaka University) Hadrons in Matter? How to Observe? initial state pre-equilibrium QGP and hydrodynamic expansion hadronization hadronic phase and freeze-out Fast Time Evolution In Ultrarelativistic Heavy Ion Collisions Experimentally observed hadrons: hadrons at T=0 Hadronic Decays in Medium: Decay products are rescattered e.g., Mass shift is expected NOT to be observed Indeed NO mass shift or width change is observed for  reconstructed from K + K - Furthermore, modified hadrons are quantum mechanically different from hadrons in the vacuum ⇒ discuss later

M. Asakawa (Osaka University) Hadrons and Leptons/Photons Leptons, Photonsinteract only weakly (e.m.) thus, carry information of QGP/hadron phases Hadrons strong interaction thus, only info. of hadron phase (with some exceptions)

M. Asakawa (Osaka University) CERES(NA45) pA data for e + e - production Dileptons yields and spectra: well-described by superposition of leptonic decay of final state hadrons

M. Asakawa (Osaka University) Hadron Modification in HI Collisions? Comparison with Theory (with no hadron modification) Experimental Data Mass Shift ? Broadening ? or Both ? or More Complex Structure ?

M. Asakawa (Osaka University) More CERES(NA45) AB data Also, in heavier systems, the enhancement is observed

M. Asakawa (Osaka University) Why Theoretically Unsettled Mass Shift (Partial Chiral Symmetry Restoration) Spectrum Broadening (Collisional Broadening) Observed Dileptons Sum of All Contributions (Hot and Cooler Phases)

M. Asakawa (Osaka University) Vector Channel SPF Spectral Function and Dilepton Production Dilepton production rate, info. of hadron modification...etc.: encoded in A holds regardless of states, either in Hadron phase or QGP Definition of Spectral Function (SPF) Dilepton production rate

M. Asakawa (Osaka University) Many Body Approach 1 Klingl et al. (1997)    vector dominance +  hole model

M. Asakawa (Osaka University) Many Body Approach 2 Rapp and Wambach (1999) Due to  -hole contribution, non-Lorentzian Lorentzian Assumption ab initio : not justified Neither is Smeared Source (on the lattice)

M. Asakawa (Osaka University) Mass shift or Coll. broadening or  -hole or... QCDSR Assumption for the shape of the spectral function Strongly depends on 4-quark condensates Conventional Many Body Approach Model dependence How is the effect of chiral symmetry restoration taken into account?

M. Asakawa (Osaka University) Lattice? But there was difficulty... and are related by What’s measured on the Lattice is Correlation Function D(  ) However, Measured in Imaginary Time Measured at a Finite Number of discrete points Noisy Data Monte Carlo Method  2 -fitting : inconclusive ! K( ,  ): Known Kernel

M. Asakawa (Osaka University) Way out ? Example of inconclusiveness of  2 -fitting 2 pole fit by QCDPAX (1995) Furthermore, too much freedom in the choice of ansatz

M. Asakawa (Osaka University) Difficulty on Lattice Typical ill-posed problem Problem since Lattice QCD was born Thus, what we have is Inversion Problem continuous d i s c r e t e noisy

M. Asakawa (Osaka University) MEM Maximum Entropy Method successful in crystallography, astrophysics,...etc. a method to infer the most statistically probable image (such as A(  )) given data In MEM, can put and must put statistical errors Reconstructed from X-ray diffraction image Fourier Transformed Available at only Discrete Points Noisy Data Sc 3 C 8 2

M. Asakawa (Osaka University) a method to infer the most statistically probable image (=A(  )) given data Principle of MEM Theoretical Basis: Bayes’ Theorem In MEM, Statistical Error can be put to the Obtained Image MEM In Lattice QCD In MEM, basically Most Probable Spectral Function is calculated H: All definitions and prior knowledge such as D: Lattice Data (Average, Variance, Correlation…etc. ) Bayes' Theorem

M. Asakawa (Osaka University) such as semi-positivity, perturbative asymptotic value, …etc. Default Model given by Shannon-Jaynes Entropy For further details, Y. Nakahara, and T. Hatsuda, and M. A., Prog. Part. Nucl. Phys. 46 (2001) 459 Ingredients of MEM

M. Asakawa (Osaka University) Error Analysis in MEM (Statistical) MEM is based on Bayesian Probability Theory In MEM, Errors can be and must be assigned This procedure is essential in MEM Analysis For example, Error Bars can be put to Gaussian approximation

M. Asakawa (Osaka University) Uniqueness of MEM Solution Unique if it exists ! T. Hatsuda, Y. Nakahara, M.A., Prog. Part. Nucl. Phys. 46 (2001) 459 only one local maximummany dimensional ridge  S - L if  = 0 (usual  2 -fitting) The Maximum of  S - L ∝ log(P[D|AH]P[A|H]) = log(P[A|DH])

M. Asakawa (Osaka University) Result of Mock Data Analysis (1) N(# of data points)-b(noise level) dependence

M. Asakawa (Osaka University) Result of Mock Data Analysis (2) N(# of data points)-b(noise level) dependence

M. Asakawa (Osaka University) Nakahara, Hatsuda, and M.A. Prog. Nucl. Theor. Phys., 2001 Application of MEM to Lattice Data (T=0) Doubler States

M. Asakawa (Osaka University) Parameters on Lattice 1.Lattice Sizes 32 3  32 (T = 2.33T c ) 40 (T = 1.87T c ) 42 (T = 1.78T c ) 44 (T = 1.70T c ) 46 (T = 1.62T c ) 54 (T = 1.38T c ) 72 (T = 1.04T c ) 80 (T = 0.93T c ) 96 (T = 0.78T c ) 2.  = 7.0,  0 = 3.5  = a  / a  = 4.0 (anisotropic) 3. a  = 9.75  fm L  = 1.25 fm 4.Standard Plaquette Action 5.Wilson Fermion 6.Heatbath : Overrelaxation  1  : sweeps between measurements 7.Quenched Approximation 8.Gauge Unfixed 9. p = 0 Projection 10.Machine: CP-PACS

M. Asakawa (Osaka University) Result for V channel (J/  ) J/  ( p  0 ) disappears between 1.62T c and 1.70T c A(  )   2  (  )

M. Asakawa (Osaka University) Result for PS channel (  c )  c ( p  0 ) also disappears between 1.62T c and 1.70T c A(  )   2  (  )

M. Asakawa (Osaka University) Statistical Significance Analysis for J/  Statistical Significance Analysis = Statistical Error Putting T = 1.62T c T = 1.70T c Ave. ±1 

M. Asakawa (Osaka University) Statistical Significance Analysis for  c Statistical Significance Analysis = Statistical Error Putting T = 1.62T c T = 1.70T c

M. Asakawa (Osaka University) Dependence on Data Point Number (1) N   = 46 (T = 1.62T c ) V channel (J/  ) Data Point # Dependence Analysis = Systematic Error Estimate

M. Asakawa (Osaka University) Dependence on Data Point Number (2) N   = 40 (T = 1.87T c ) V channel (J/  ) Data Point # Dependence Analysis = Systematic Error Estimate

M. Asakawa (Osaka University) Also for lighter quark systems Hatsuda, Nakahara, and M.A. (2002) also Bielefeld group for V channel

M. Asakawa (Osaka University) 1. Can J/  produced and escape before QGP is formed ? 2. Can J/  survive as a Coulombic resonance ? 3. Are there competitive non-plasma J/  suppression mechanisms ? 4. Could J/  suppression be compensated in the hadronization stage ? 5. Could enhanced thermal dileptons prevent clear observation of J/  suppression ? J/  suppression as a QGP signature J/  suppression: prototype of QGP signature proposal Debye Screening ⇒ Melting of Heavy Quark Resonances Matsui and Satz, 1986 Check list by Matsui and Satz Now, we know J/  exists above T c

M. Asakawa (Osaka University) What’s Degrees of Freedom above T c ? PHENIX collaboration Jet Quenching working for pions, but not for (anti)protons

M. Asakawa (Osaka University) Recombination of Quarks Jet Quenching and Fragmentation + Recombination (Coalescence) of Quarks Recombination Fragmentation Jet Quenching Effects become visible at lower p T for mesons Duke Group, Texas A&M Group, Oregon Group (2002) GeV/c pQCD ReCo Hydro Mesons Baryons

M. Asakawa (Osaka University) Meson-Baryon Systematics (in R CP ) Baryons Mesons R CP p T (GeV/c)  behaves like meson ? Relative Enhancement of Protons: ⇒ not mass effect

M. Asakawa (Osaka University) Meson-Baryon Systematics (in v 2 ) Bass et al., PRC, 2003

M. Asakawa (Osaka University) Elliptic Flow : v 2 Esumi (QM2002) Target Projectile b:impact parameter Reaction plane Larger Pressure Gradient Reaction Plane More Flow in parallel to Reaction Plane = Elliptic Flow in addition to Radial Flow central region

M. Asakawa (Osaka University) Implication of success of recombination Very good description of meson baryon systematics at intermediate p T Quark Recombination: Q(G)P phase is assumed Gluon Dynamics is not included : Non-perturbative feature of Q(G)P phase just above T c ? The unexpected charge fluctuation at RHIC Actually, explained by Quark Recombination (Constituent Quark Matter) First pointed out by Bialas in a schematic way (2002) Analysis with Correlations due to Gluon Fragmentation, Diquarks,..etc in progress (Nonaka, Muller, Bass, M.A.) v 2 & yields: also sensitive to internal structure of hadrons such as a 0, f 0,  +,...etc. NMFBA (2004) Mysterious observed Charge Fluctuation (STAR, without (questionable) correction) In Naive QGP: D ~ 1 Naive Resonance gas: D ~ 3

M. Asakawa (Osaka University) Critical End Point in QCD CEP How can CEP be seen in Heavy Ion Collisions? Color Super Conductors 2SC, dSC, CFL...

M. Asakawa (Osaka University) CEP = 2nd order phase transition, but... If expansion is adiabatic Divergence of Fluctuation Correlation Length Specific Heat ? CEP = 2nd Order Phase Transition Point How does the system evolve near CEP? What can be observed if CEP exists ? General Belief: Non-Monotonic Behavior as a function of the beam energy Questions Coll. Energy T BB

M. Asakawa (Osaka University) Need to know EOS If the critical region is tiny, hopeless to see, anyway Assume critical region is large Universality: belongs to the same universality class as 3d Ising Model CLUE to EOS T BB T BB

M. Asakawa (Osaka University) EOS Mapping of 3d Ising Model variables to those in QCD Matching of Singular Behavior near CEP and Hadronic Matter/QGP EOS below/above phase transition For Details, C. Nonaka and M.A., nucl-th/ T E =154.7MeV,  BE =367.8MeV CEP crossover 1st order

M. Asakawa (Osaka University) Isentropic Trajectories In each volume element, Entropy(S) and Baryon Number (N B ) are conserved, as long as entropy production can be ignored Isentropic Trajectories (n B /s = const.) without CEP(EOS in usual hydro calculation) with CEP Focusing of Isentropic Trajectories Excluded Volume Approximation + Bag Model EOS

M. Asakawa (Osaka University) Consequences Yes Without Focusing Coll. Energy T BB Non-monotonic Behavior in Observables Only Weakly With Focusing Coll. Energy T BB

M. Asakawa (Osaka University) Correlation Length, Fluctuation,... Furthermore, Correlation Lengths, Fluctuations,...: Hadronic Observables Subject to Final State Interactions fluctuation ~   Time Evolution along given isentropic trajectories Widom’s scaling law r,M: Ising side variables Model H (Hohenberg and Halperin RMP49(77)435)

M. Asakawa (Osaka University) Effect on Freezeouts 1 (chemical) If focusing takes place, are the chemical freezeout points focused? Not Necessarily Hadrons near CEP (generally in hot phase) Quantum Mechanically Different from ones in the vacuum e.g., Csorgo, Gyulassy, and M.A., PRL 99

M. Asakawa (Osaka University) Effect on Freezeouts 2 (Thermal) On the left side of CEP s decreases smoothly, while on the right side of CEP s decreases quickly (1st order PT) On the left side of CEP, thermal freezeout takes place at lower T? Xu and Kaneta, nucl-ex/ (QM2001) Yes, this is indeed the case! At RHIC, radial expansion is faster HBT size: similar Similar or Higher Thermal Freezeout Temperature would be expected, but this is not the case crossover 1st order

M. Asakawa (Osaka University) Summary (1) It seems J/  and  c ( p = 0 ) remain in QGP up to ~1.6T c Sudden Qualitative Change between 1.62T c and 1.70T c ~34 Data Points look sufficient to carry out MEM analysis on the present Lattice and with the current Statistics (This is Lattice and Statistics dependent) Physics behind is still unknown Spectral Functions in QGP Phase were obtained for heavy quark systems at p = 0 on large lattices at several T Both Statistical and Systematic Error Estimates have been carefully carried out

M. Asakawa (Osaka University) Summary (2) Recent RHIC data suggest Realization of Deconfined Phase But No Gluonic Degrees of Freedom look needed Success of Hydrodynamics without Viscosity suggests existence of strongly interacting matter just above T c Focusing Effect near CEP for Isentropic Trajectories ⇒ Weak Non-monotonic behavior of observables as a function of collision energy? Effect on Flow? Low Thermal Freezeout Temperature at RHIC suggests existence of CEP

M. Asakawa (Osaka University) Back Up Slides

M. Asakawa (Osaka University) Spectral Function at Work Dilepton Production Rate at Finite T hold regardless of state, either in Hadron Phase or QGP ! Real Photon Production Rate at Finite T R ratio

M. Asakawa (Osaka University) Chiral Extrapolation Continuum Kernel Lattice Kernel Other Lattice Data or Experimental Value (3) (1) (15) 0.348(27) 0.331(22) 1.88(8) 1.74(8) 1.68(13) 2.44(11) 2.25(10) 1.90(3) 2.20(3) Good Agreement with the Results of Conventional Analyses + Ability to Extract Resonance Masses

M. Asakawa (Osaka University) What Result of Mock Data Analysis tells us General Tendency (always statistical fluctuation exists) The Larger the Number of Data Points and the Lower the Noise Level The result is closer to the original image 40 3  30 lattice  = 6.47, 150 confs. isotropic lattice (T<T c ) How many data points are needed ? may depend on statistics, , K, ,...etc. N , min  or larger

M. Asakawa (Osaka University) Parameters in MEM analysis With Renormalization of Each Composite Operator on Lattice The m-dependence of the result is weak Data Points at are not used Continuum Kernel Small Enough Temporal Lattice Spacing Data at these points can be dominated by such unphysical noise channel PS V m(  )  Default Models used in the Analysis

M. Asakawa (Osaka University) Parameters in MEM Analysis (cont’d) Furthermore, in order to fix resolution, a fixed number of data points (default value = 33 or 34) are used for each case Dependence on the Number of Data Points is also studied (systematic error estimate)

M. Asakawa (Osaka University) Number of Configurations As of November 23, 2004 T  T c T c  T N  T / T c # of Config

M. Asakawa (Osaka University) Polyakov Loop and PL Susceptibility Deconfined Phase Confining Phase Polyakov Loop Susceptibility Deconfined Phase Confining Phase Polyakov Loop 0 in Confining Phase 0 in Deconfined Phase

M. Asakawa (Osaka University) Result of Mock Data Analysis m(default model) dependence

M. Asakawa (Osaka University) Dependence on Data Point Number N   = 54 (T = 1.38T c ) V channel (J/  )

M. Asakawa (Osaka University) Dependence on Data Point Number N  = 54 (T = 1.38T c ) PS channel (  c )

M. Asakawa (Osaka University) Dependence on Data Point Number N   = 46 (T = 1.62T c ) PS channel (  c )

M. Asakawa (Osaka University) Debye Screening in QGP Original Idea of J/  Suppression as a signature of QGP Formation: Debye Screening (Matsui & Satz, 1986) J/y Melting at Debye Screening of Potential between Karsch et al. (2000) Need to start over asking a question “ What is QGP ? ” ?

Transition in the continuum limit Transition gets smoother on finer lattices, imrovement of flavor symmetry ? HYP staggered fermions at Nt=4 -> Hasenfratz, Knechtli, hep-lat/ MILC Coll.