1. July 2006@National Tsing Hua University, Taiwan An Introduction to High Temperature QCD and Maximum Entropy Method Masayuki Asakawa Department of Physics, Osaka University Maximum Entropy Method Reviews: M. Jarrell and J. E. Gubernatis, Phys. Rept. 269 (1996) 133 T. Hatsuda, Y. Nakahara, and M.A., Prog. Part. Nucl. Phys. 46 (2001) 459 hep-lat/0011040

2. M. Asakawa (Nuclear Theory Group) Maximum Entropy Method successful in crystallography, astrophysics,...etc. A method to obtain most probable (plausible) image from discrete data Maximum Entropy Method

3. M. Asakawa (Osaka University) Heating Up Forever ? What happens when matter is heated ?

4. M. Asakawa (Osaka University) The Strong Force Elementary Particles in QCD quark (spin 1/2, flavors (u,d,s,...), 3 colors) ⇔ electron gluon (spin 1, mass 0, 8 colors) ⇔ photon Elementary Particles in QED (= Electromagnetism) quarks and gluons have never been isolated ⇒ confined in hadrons (Confinement) Baryon (proton, neutron,...) Meson (pion, kaon,...) Bag-like picture QCD = Quantum ChromodynamicsQED = Quantum Electrodynamics 3 quarks quark-antiquark plus Asymptotic Freedom

5. M. Asakawa (Osaka University) Hot and Dense Hadron Soup Hot Hadron SoupDense Hadron Soup Hadrons : particles which strongly interact with each other (meson, baryon) Soup of Quarks and Gluons (Quark-Gluon Plasma, QGP)

6. M. Asakawa (Osaka University) Phase Diagram 1st order or cross-over ~150MeV (1.5×10 12 K) Color Superconducting Phase quark pairs of spin ↑ and spin ↓

7. M. Asakawa (Osaka University) Finite T Lattice Gauge Theory Results Entropy Density Energy Density Pressure 3 pions (spin 0) quarks ((u,d), 3 colors, spin 1/2, quark-antiquark) gluons (8 colors, spin 1)

8. M. Asakawa (Osaka University) Creating Matter in the Early Universe The History of the Universe Production of Quarks Big Bang 10 -5 s 0.00s 10 13 s 10 2 s 1by 15by 10 -34 s 10 -43 s Production of Atoms Production of Helium Nuclei Production of 1st Gen. Stars Matter-Anti Matter Sym. Breaking Inflation Cosmic Background Radiation Production of Heavy Elements Birth of Solar System Now Super Nova Explosions Confinement of Quarks 150 MeV (1.5×10 12 K)

9. M. Asakawa (Osaka University) How to create such high temperature High Energy Collision of Heavy Ions (=Nuclei) 1 [GeV] = 10 9 [eV] Nuclei Most Dense Matter on the Earth

10. M. Asakawa (Osaka University) RHIC from a Satellite RHIC = Relativistic Heavy Ion Collider

11. M. Asakawa (Osaka University) What’s RHIC ?

12. M. Asakawa (Osaka University) What’s RHIC ? 12:00 o’clock 2:00 o’clock 4:00 o’clock 6:00 o’clock 8:00 o’clock PHOBOS 10:00 o’clock BRAHMS STAR PHENIX RHIC AGS LINAC BOOSTER TANDE MS Pol. Proton Source High Int. Proton Source HEP/NP  g-2 U-line

13. border between Switzerland and France Geneva Airport SPS and LHC at CERN

14. Circumference: ~27km SPS and LHC at CERN

15. M. Asakawa (Osaka University) What happens in High Energy Nucleus Coll.? Almost Baryon (= nucleons ) Free Region Fermi-Landau Picture (Stopping Nuclei, Larger Baryon Density) Bjorken Picture (Passing-Through Nuclei) Baryon: proton, neutron,...etc.

16. M. Asakawa (Osaka University) Time Evolution 10 -23 ~ 10 -22 s ! initial state pre-equilibrium QGP and hydrodynamic expansion hadronization hadronic phase and freeze-out If QGP and Hot/Dense Matter are created in Nucleus Collision, they disappear in a very short time In order to understand what kind of matter was created, need to compare many observables Experimentally observed hadrons: hadrons at T=0

17. M. Asakawa (Osaka University) Example of Nucleus Collisions at RHIC STAR at RHIC Tracks of Charged Particles Need to compare a lot of observables ! Difference in Color = Difference in Momentum Result of a Au+Au Collision

18. 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)

19. M. Asakawa (Osaka University) Why We Started This Business data sum of cocktail sources including the CERES/NA45 More Recently NA60

20. 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

21. 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 ?

22. M. Asakawa (Osaka University) Why We Started This Business data sum of cocktail sources including the CERES/NA45 More Recently NA60

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

24. M. Asakawa (Osaka University) Is Parametrization of SPF at finite T/  Easy? –With increasing T (>0), bound state poles are moving off the real ω axis into complex ω plane ・ The appropriate functional form is “Breit-Wigner type” Sometimes hear statements like Finite T/  Spectral Functions are not always given by “shift + broadening”

25. M. Asakawa (Osaka University) A Good Example (for  meson) Rapp and Wambach (1999) Due to  -hole contribution, non-Lorentzian Lorentzian Assumption ab initio : not justified and many more examples in many fields

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

27. M. Asakawa (Osaka University) A Good Example (for  meson) Rapp and Wambach (1999) Due to  -hole contribution, non-Lorentzian Lorentzian Assumption ab initio : not justified and many more examples in many fields

28. M. Asakawa (Osaka University) Vector Channel SPF Spectral Function and Dilepton Production Dilepton production rate, info. of hadron modification...etc.: encoded in A Definition of Spectral Function (SPF) Dilepton production rate If Smeared Source is used on the Lattice, This Link is Lost

29. M. Asakawa (Osaka University) Lattice? But SPF cannot be measured... and are related by What’s measured on the Lattice is Imaginary Time 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

30. M. Asakawa (Osaka University) Example of Lattice Data Very Roughly Speaking contribution from higher states

31. 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

32. M. Asakawa (Osaka University) Similar Difficulties in Many Areas Analytic Continuation to Imaginary Time is measured Measured at a Finite Number of discrete points Noisy Data Smeared Images due to Finite Resolution are measured Measured by a Finite Number of Pixels Noisy Data Fourier Transformed Images are measured Measured at a Finite Number of data points Noisy Data Lattice Observational Astronomy X-ray Diffraction Measurement in Crystallography

33. M. Asakawa (Nuclear Theory Group) Maximum Entropy Method successful in crystallography, astrophysics,...etc. A method to obtain most probable (plausible) image from discrete data Maximum Entropy Method

34. M. Asakawa (Osaka University) Example of MEM Application Lattice Observational Astronomy X-ray Diffraction Measurement in Crystallography Will be shown shortly Sc 3 C 8 2 Hubble Image of a Nebula (b) H 2 inside C n (c) no H 2 inside C n both observed at SPring-8

35. M. Asakawa (Osaka University) SPring-8 in Japan, a photon factory

36. M. Asakawa (Osaka University) Example of MEM Application Lattice Observational Astronomy X-ray Diffraction Measurement in Crystallography Will be shown shortly Sc 3 C 8 2 Hubble Image of a Nebula (b) H 2 inside C n (c) no H 2 inside C n both observed at SPring-8

38. M. Asakawa (Osaka University) Warm Up: Quantum Mechanics In (Undergraduate) Quantum Mechanics, Transition amplitude is given by The same quantity has another expression, Very roughly speaking, stands for sum over paths

39. M. Asakawa (Osaka University) AB Effect and Path Integral Famous AB(Aharonov-Bohm) effect ( Modern Quantum Mechanics, J.J.Sakurai) Amplitude at the interference point common phases for paths above and below the cylinder source interference point impenetrable wall

40. M. Asakawa (Osaka University) Path Integral in Field Theory Field Theory : Quantum Mechanics with Infinite Degrees of Freedom degree of freedom distributed at each point Sum over PathsSum over Field Distributions (configurations)

41. M. Asakawa (Osaka University) Discretization, a.k.a. Lattice So far, no approximation has been done In order to calculate path integral numerically, Space-Time is discretized Also, let us consider field theory in Euclidean space-time Remember x f(x) xx

42. M. Asakawa (Osaka University) Continuum QCD QCD (Quantum Chromodynamics) SU(3) Gauge Theory (3 colors for quarks, fundamental representation) Lagrangian density This Lagrangian Density is discretized Requirement: Gauge Invariance Simple-minded Discretization of the Quark Part leads to Doubler Problem

43. M. Asakawa (Osaka University) QCD on the Lattice Gauge Fields on the Lattice In order to retain Gauge Invariance, Gauge Fields are distributed between lattice points (=link), while Fermion Fields still live at lattice points continuum gauge field action color field matrix

44. M. Asakawa (Osaka University) Why Monte Carlo Method? Expectation value of QCD observable (on the lattice) Operator with Each Configuration is summed up with weight exp(-S lat ) Each Link has 8 degrees of freedom (SU(3) gluon color) Average over Configurations with Huge Degrees of Freedom! Generate Gauge Field Configurations with weight exp(-S p ) or exp(-S p +log(detM q )) Monte Carlo Method

46. M. Asakawa (Osaka University) Lattice? But SPF cannot be measured... and are related by What’s measured on the Lattice is Imaginary Time 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

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

48. M. Asakawa (Osaka University) MEM Maximum Entropy Method a method to infer the most statistically probable image (such as A(  )) given data, instead of solving the (ill-posed) inversion problem Theoretical Basis: Bayes’ Theorem In Lattice QCD H: All definitions and prior knowledge such as D: Lattice Data (Average, Variance, Correlation…etc. ) Bayes' Theorem

49. M. Asakawa (Osaka University) such as semi-positivity, perturbative asymptotic value, operator renormalization 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  2 -likelihood function

50. M. Asakawa (Osaka University) “Monkey Argument” for (1) Suppose a monkey throws M ( M : large) balls and that i -th cell has a probability p i to receive a ball Then, the probability that a certain image is realized : Note that

51. M. Asakawa (Osaka University) “Monkey Argument” for (2) Next, introduce a small quantum (ball quantum) q and define the finite image f i and the default model m i as Then,

52. M. Asakawa (Osaka University) Axiomatic Approach to Entropy (1) For an image f, we assign f is a better image than g Axiom 1: Locality S(f) is a local functional of f(x) As a result, Lebesgue measure m(x) is introduced Axiom 2: Coordinate Invariance f(x) and m(x) behave as a scalar density and S(f) is also scalar under the coordinate transformation An Alternative Way to Entropy  ( f ( x ), x ): local function J. Skilling, 1989

53. M. Asakawa (Osaka University) Axiom 3: System Independence Let f(x) and g(y) be independent images and define the total image F(x,y) as F(x,y)=f(x)g(y) Axiom 4: Scaling In the absence of constraints, the initial measure m(x) is recovered, i.e., Then, f(x) and g(y) are reconstructed independently, i.e., In addition, S(f) can be normalized to 0 for f(x)=m(x) by adding a constant to S(f) For further details, T. Hatsuda, Y. Nakahara, M.A., Prog. Part. Nucl. Phys. 46 (2001) 459 Axiomatic Approach to Entropy (2) From the four Axioms, S(f) is shown to be given by

54. M. Asakawa (Osaka University) Summary up to This Point What is needed to do MEM Analysis Data (Lattice, X-ray rescattering data,...etc.) Prior Knowledge about Image (Spectral Function, Distribution,...etc.) Kernel (Relation) between Image and Observables  2 -likelihood function

55. 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 Bayes' Theorem The Maximum of  S - L in Image Space: MEM Solution

56. M. Asakawa (Osaka University) Proof of Uniqueness (1) In general, if for real and smooth function, is negative definite, i.e., Then, F has only one maximum if it exists Proof is by contradiction Suppose  space is discretized into N  sections

57. M. Asakawa (Osaka University) Proof of Uniqueness (2) Since it can be shown that  S – L has at most one maximum (local) if it exists On the other hand, Thus, the solution of (minimum of  2 ) is not unique

58. 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

59. M. Asakawa (Osaka University) How to determine  (1) The Maximum of  S - L in Image Space : MEM Solution (Image) Many ways to determine  For example,  is determined so that  S (entropy factor) becomes similar order as L (  2 ) In state-of-the-art MEM,  is also determined by Bayes’ theorem

60. M. Asakawa (Osaka University) How to determine  (2) From the Definition of  S and L can be evaluated numerically by expanding Q(A) around the maximum of Q(A) There are two standard choices for the prior probability for , Laplace’s rule P[  |Hm]= const. or Jeffreys’ rule P[  |Hm]=1/  Practically, either choice is as good as each other

61. M. Asakawa (Osaka University) Definition of Output Image Final output image A out is defined as a weighted average over A and  under the assumption that P[A|DH  m] is sharply peaked around A  (  ) We integrate in [  min,  max ], where  min and  max satisfy with renormalization:

62. M. Asakawa (Osaka University) How to Obtain Maximum of Q=  S-L C ij : Covariance Matrix L and S depend on A (  ) differently In Lattice gauge case, Singular Value Decomposition (SVD) is used

63. M. Asakawa (Osaka University) 1. Take a test input image 2. Transform with an appropriate Kernel 3. Make a mock data by adding noise to 4. Apply MEM to and construct the output image 5. Compare with Mock Data Analysis Dirichlet Kernel

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

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

66. M. Asakawa (Osaka University) Statistical and Systematic Error Analyses in MEM The Larger the Number of Data Points and the Lower the Noise Level The result is closer to the original image Generally, as we saw, Need to do the following: Put Error Bars and Make Sure Observed Structures are Statistically Significant Change the Number of Data Points and Make Sure the Result does not Change Statistical Systematic in any MEM analysis

67. M. Asakawa (Osaka University) Parameters of Actual Calculation (isotropic) 1.Lattice Size 20 3 × 24  = 6.0 3.Naive Unimproved Action 4.Wilson Fermion 5.Quenched Approximation 6. Heatbath : Overrelaxation = 1 : 4 200 sweeps between measurements 7.Gauge Unfixed 10.a = 0.429 GeV -1 = 0.0847 fm 11. Projection 12. Dirichlet Boundary Condition for the Gauge Field 13.Machine SR2201 (32PU) at JAERI

68. M. Asakawa (Osaka University) SPF in V Channel Actually Lattice Artifact

69. M. Asakawa (Osaka University) SPF in PS Channel Actually Lattice Artifact

70. M. Asakawa (Osaka University) Error Analysis in V Channel Perturbative Continuum Value + Renormalization of Composite Operator on Lattice

71. M. Asakawa (Osaka University) m-Dependence of Error in V Channel Perturbative Value / 5

72. M. Asakawa (Osaka University) Mock Data Analysis with various “m” m(default model) dependence

73. M. Asakawa (Osaka University) How to determine m (default model)? So far, we have not discussed the m-dependence of results Mock Data analysis suggests the following: Good Default Model tends to reproduce the input well Thus need to use prior knowledge (perturbative value, operator renormalization etc.) as much as possible Bad Default Model tends to produce fake structures ( ⇒ Ringing) However, if Bad Model is used, Error Bars also tend to increase It looks like Fake Structures are within statistical error bars

74. M. Asakawa (Osaka University) N(# of Data Points) Dependence This must be carefully checked Need to do the following: Put Error Bars and Make Sure Observed Structures are Statistically Significant Change the Number of Data Points and Make Sure the Result does not Change Statistical Systematic in any MEM analysis This is important, in particular, in Finite T Lattice calculation, where # of Data points is often small

75. M. Asakawa (Osaka University) N dependence of Data (isotropic case) 40 3  30 lattice  = 6.47, 150 confs. isotropic lattice (T<T c ) may depend on statistics, , K, ,...etc. N , min  or larger

76. M. Asakawa (Osaka University) Parameters on Lattice (anisotropic case) 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  10 -3 fm L  = 1.25 fm 4.Standard Plaquette Action 5.Wilson Fermion 6.Heatbath : Overrelaxation  1  : 4 1000 sweeps between measurements 7.Quenched Approximation 8.Gauge Unfixed 9. p = 0 Projection 10.Machine: CP-PACS

77. 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)

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

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

80. 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  Both Persistence and Disappearance of the peak are Statistically Significant

81. 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

82. 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

83. M. Asakawa (Osaka University) Back Up Slides

84. M. Asakawa (Osaka University) Big Bang Cosmology Big Bang Weak Electromagnetic Strong Gravity Four Forces

85. M. Asakawa (Osaka University) Correlation among Different Times

86. M. Asakawa (Osaka University) How to Obtain Maximum of Q=  S-L (1) C ij : Covariance Matrix L and S depend on A (  ) differently In Lattice gauge case, Singular Value Decomposition (SVD) is used

87. M. Asakawa (Osaka University) How to Obtain Maximum of Q=  S-L (2) Let us replace the integrals by sums Let us put C = I, for simplicity. Then, Since A l is positive semi-definite,

88. M. Asakawa (Osaka University) How to Obtain Maximum of Q=  S-L (3) Thus, is in N s ( ) dimensional subspace This can be seen by carrying out singular value decomposition of K t This tells us that can be parametrized as Bryan Method (R. K. Bryan, Eur. Biophys. J., 18 (1990) 165)

89. M. Asakawa (Osaka University) Singular Value Decomposition For M×N matrix K t U : M×N matrix satisfying U t U = I V : N×N matrix satisfying V t V = VV t = I  : N×N diagonal matrix with positive semi-definite elements

90. M. Asakawa (Osaka University) How to Obtain Maximum of Q=  S-L (4) Since Note that L depends on (and ) This can be solved by the Newton method with increment with This yields where  : Marquardt-Levenberg parameter (to decrease )  and are restriction of  and V to the singular space : restriction of U to the singular space

91. M. Asakawa (Osaka University) How to Obtain Maximum of Q=  S-L (5) In (2), we assumed C = I, for simplicity If C is not diagonal, C can be diagonalized ( C : real symmetric matrix) with an orthogonal matrix R Using this R, let us redefine K and D as follows: and are used as the kernel and the data, respectively, instead of K and D With this and, L is written as

92. M. Asakawa (Osaka University) Chiral Extrapolation Continuum Kernel Lattice Kernel Other Lattice Data or Experimental Value 0.1570(3) 0.1569(1) 0.1571 0.348(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

93. 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(  )  2 1.15 0.40 Default Models used in the Analysis

94. 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)

95. M. Asakawa (Osaka University) Number of Configurations As of November 23, 2004 T  T c T c  T N  32 40 42 44 46 54 72 80 96 T / T c 2.33 1.87 1.78 1.70 1.62 1.38 1.04 0.93 0.78 # of Config. 141 181 180 182 150 110 194

96. 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

97. M. Asakawa (Osaka University) NA60: dimuon and charm production in p-A and In-In collisions Low mass: excess in central In-In collisions / =1.2, fixed from high p T data Clear excess of data above cocktail  rising with centrality  more important at low p T data sum of cocktail sources including the See next slide for cocktail definition

98. M. Asakawa (Osaka University) NA60: dimuon and charm production in p-A and In-In collisions A simple approach is used to subtract known sources (except the  )  and  : yields fixed to get, after subtraction, a smooth underlying continuum  : set upper limit by “saturating” the yield in the mass region 0.2–0.3 GeV  leads to a lower limit for the excess at low mass Getting the mass shape of the excess

99. M. Asakawa (Osaka University) NA60: dimuon and charm production in p-A and In-In collisions Low mass: comparison with models Excess shape consistent with broadening of the  (Rapp-Wambach) Models predicting a mass shift (Brown-Rho) ruled out These conclusions are also valid as a function of p T (see parallel talk) Predictions for In-In by Rapp et al. (2003) for = 140 Theoretical yields folded with NA60 acceptance and normalized to data in the mass window m  < 0.9 GeV

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

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

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

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

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

105. 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

106. M. Asakawa (Osaka University) Signature Problem (cont’d) Harris and Muller, 1996 And more! Need to Compare Many Observations!

107. M. Asakawa (Osaka University) Summary and Outlook (1) Spectral Functions in QGP Phase were obtained for heavy quark systems at p = 0 on large lattices at several T in the quenched approximation 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 This result is, roughly, in accordance with other lattice calculations (e.g., Bielefeld-BNL) and potential model analyses (e.g., C.Y. Wong) No finite p calculation for J/  yet (Two Spectral Functions, Transverse and Longitudinal, for J/  at finite p ): ← p T dependence of J/  Suppression Non-Quench Calculation Started (Swansea-Dublin) Full QCD Calculation Aarts et al. (hep-lat 0511028) J/  disappears between ~1.5T c and 2T c Important to include possibility of dissociation into DD

108. M. Asakawa (Osaka University) Summary and Outlook (2) ~34 Data Points look sufficient to carry out MEM analysis on the present Lattice and with the current Statistics (But this is Lattice and Statistics dependent) Both Continuum and Thermodynamical (V→\) Limits: Still Needed Both Statistical and Systematic Error Estimates have been carefully carried out This must be carried out in any MEM analysis