1. July 22, 2004 Hot Quarks 2004 Taos Valley, NM Charmonia in the Deconfined Plasma from Lattice QCD Masayuki Asakawa Osaka University Intuitive and Quick Introduction to Lattice QCD and

2. M. Asakawa (Osaka University) Warm Up: Quantum Mechanics 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

3. M. Asakawa (Osaka University) AB Effect and Path Integral 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

4. M. Asakawa (Osaka University) Path Integral in Field Theory 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)

5. M. Asakawa (Osaka University) Discretization, a.k.a. Lattice 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

6. M. Asakawa (Osaka University) QCD on the Lattice 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

7. M. Asakawa (Osaka University) Why Monte Carlo Method? 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 )) if exp(-S p ) or exp(-S p +log(detM q )) is Real Monte Carlo Method

8. M. Asakawa (Osaka University) Why Reweighting? Why Reweighting? This is the case at  =0, non-zero  (SU(2)), but not the case at non-zero  (QCD). Generate Gauge Field Configurations with weight exp(-S p ) or exp(-S p +log(detM q )) if exp(-S p ) or exp(-S p +log(detM q )) is Real Redefine weight and operator With some applicability limit Glasgow Group, Fodor-Katz,...

9. M. Asakawa (Osaka University) Systematic and Statistical Errors Systematic and Statistical Errors Lattice Calculation is carried out with Finite Lattice Spacing a, Finite Lattice Volume V, quark mass m ≠ m phys Necessity of Extrapolations (origin of systematic errors) Monte Carlo Method to evaluate Necessity to Generate Many Configurations (to reduce statistical errors) as in usual integral x f(x) xx

10. M. Asakawa (Osaka University) Things not to do Things not to do van Leeuwen for NA49, QM2002 Fixed lattice spacing a, m ≠ m phys Fodor and Katz (2002)

11. M. Asakawa (Osaka University) Order of Phase Transition at  = 0 Order of Phase Transition at  = 0 3d O(4) scaling tricritical point 3d Ising scaling 2 nd order line: m ud (m s *-m s ) 5/2 near the tricritical point 3d Z(3) Potts Kanaya, QM2002

12. M. Asakawa (Osaka University) Critical Endpoint in Effective Models Critical Endpoint in Effective Models Compilation by Stephanov Yazaki and M.A. NPA (1989) Debut of CEP in QCD

13. M. Asakawa (Osaka University) What is the evidence of strongly int. QGP? What is the evidence of strongly int. QGP? Karsch, 2001 e rises quickly, while p increases slowly Is this due to interesting physics? Models with Massive Quarks and Gluons...etc. Some Hadronization Models before RHIC Ultrarelativistic free gase = 3p e - 3p: interaction measure

14. M. Asakawa (Osaka University) What is the “normal” behavior of e, p, e-3p? Gyulassy, 2004 However, the definition of e and p has ambiguity On the Lattice, e and p are normalized to 0 at T =  = 0 On the other hand, entropy density s has no ambiguity (s ~ deg. of freedom) Even if s approaches the Stefan-Boltzmann value just above T c, p does not approach p SB In addition, e - 3p never vanishes! At  = 0, At T c, p is continuous (from thermodynamics) Thus, p cannot increase rapidly ! Hatsuda and M.A., 1997

15. M. Asakawa (Osaka University) Example Example  /T c = 0.05 Hatsuda and M.A., 1997 Interaction Measure does not measure interaction !

16. M. Asakawa (Osaka University) PLAN of the Second Part PLAN of the Second Part What is Spectral Function? Why Spectral Function? Necessity of MEM (Maximum Entropy Method) Models at Finite T/  MEM Outline Importance of Error Analysis Finite Temperature Results for J/  and  c Error Analysis Statistical Systematic Hatsuda and M.A., PRL 2004

17. M. Asakawa (Osaka University) Spectral Function Spectral Function Pretty important function to understand QCD Dilepton production rate, Real Photon production rate,...etc. holds regardless of states, either in Hadron phase or QGP Definition of Spectral Function

18. M. Asakawa (Osaka University) CERES(NA45) AB data CERES(NA45) AB data

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

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

21. 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?

22. M. Asakawa (Osaka University) Lattice? But there was difficulty... Lattice? But there was difficulty... and are related by What’s measured on 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 !

23. M. Asakawa (Osaka University) Way out ? Way out ? Example of  2 -fitting failure 2 pole fit by QCDPAX (1995)

24. M. Asakawa (Osaka University) Difficulty on Lattice 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

25. M. Asakawa (Osaka University) MEM MEM Maximum Entropy Method successful in crystallography, astrophysics, …etc.

26. M. Asakawa (Osaka University) a method to infer the most statistically probable image given data Principle of MEM 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

27. M. Asakawa (Osaka University) Ingredients of MEM Ingredients of MEM 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

28. M. Asakawa (Osaka University) Error Analysis in MEM (Statistical) 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

29. M. Asakawa (Osaka University) Uniqueness of MEM Solution 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])

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

31. M. Asakawa (Osaka University) Nakahara, Hatsuda, and M.A. Prog. Nucl. Theor. Phys., 2001 Application of MEM to Lattice Data (T=0) Application of MEM to Lattice Data (T=0) Resonance Physics has become possible on Lattice

32. M. Asakawa (Osaka University) Parameters on Lattice at finite T Parameters on Lattice at finite T 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 a → a  a → a  time direction spatial direction Anisotropic Lattice

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

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

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

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

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

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

39. M. Asakawa (Osaka University) Summary and Perspectives (1) Summary and Perspectives (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

40. M. Asakawa (Osaka University) Summary and Perspectives (2) Summary and Perspectives (2) Thermodynamical Quantities : Sudden Increase of Degrees of Freedom at T c but, Hadrons (or Strong Correlations) still exist at least up to T ~ 1.6 T c Hadronization seems to proceed through simple recombination of constituent quarks (RHIC data) charmonia and lighter hadrons But why does the constituent quark picture hold? Further study needed for better understanding of QGP and Hadronic Modes in QGP !