June 13, Hard Probes 2006 Masayuki ASAKAWA Department of Physics, Osaka University Quarkonium States at Finite Temperature An Introduction to Maximum Entropy Method What to do AND What not to do An Introduction to Maximum Entropy Method What to do AND What not to do

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

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)

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

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

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

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 Direct Inversion: ill-posed !

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

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

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

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

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) Result of Mock Data Analysis N(# of data points)-b(noise level) dependence

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

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  (  ) Lattice Artifact

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  Both Persistence and Disappearance of the peak are Statistically Significant

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) 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 ) J/  disappears between ~1.5T c and 2T c Important to include possibility of dissociation into DD

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