Masayuki ASAKAWA

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Quarkonium States at Finite Temperature
An Introduction to Maximum Entropy Method What to
do AND What not to do
Masayuki ASAKAWA
Department of Physics, Osaka University
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Why We Started This Business
CERES/NA45
More Recently NA60
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Hadron Modification and Dileptons
Mass Shift (Partial Chiral Symmetry Restoration)
Spectrum Broadening (Collisional Broadening)
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Is Parametrization of SPF at finite T/m Easy?
Sometimes hear statements like
Finite T/m Spectral Functions are not always
given by shift broadening
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A Good Example (for r meson)
and many more examples in many fields
Rapp and Wambach (1999)
Due to D-hole contribution, non-Lorentzian

• Lorentzian Assumption ab initio not justified

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Spectral Function and Dilepton Production

• Definition of Spectral Function (SPF)

• Dilepton production rate, info. of hadron
  modification...etc. encoded in A

Dilepton production rate

• If Smeared Source is used on the Lattice, This
  Link is Lost

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Lattice? But SPF cannot be measured ...

• Whats measured on the Lattice is
• Imaginary Time Correlation Function D(t )

K(t,w) Known Kernel
However,
c2-fitting inconclusive !

• Measured in Imaginary Time
• Measured at a Finite Number of discrete points
• Noisy Data Monte Carlo Method

Direct Inversion ill-posed !
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Similar Difficulties in Many Areas

• Lattice

• Analytic Continuation to Imaginary Time is
  measured
• Measured at a Finite Number of discrete points
• Noisy Data

• X-ray Diffraction Measurement in Crystallography

• Fourier Transformed images are measured
• Measured at a Finite Number of data points
• Noisy Data

• Observational Astronomy

• Smeared Images due to Finite Resolution are
  measured
• Measured by a Finite Number of Pixels
• Noisy Data

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Example of MEM Application

• Lattice

Will be shown shortly

• X-ray Diffraction Measurement in Crystallography

• Observational Astronomy

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MEM

• Maximum Entropy Method

• a method to infer the most statistically
  probable image (such as A(w))
• given data, instead of solving the (ill-posed)
  inversion problem

• Theoretical Basis Bayes Theorem

• In Lattice QCD

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Ingredients of MEM

given by Shannon-Jaynes Entropy
For further details, Y. Nakahara, and T. Hatsuda,
and M. A., Prog. Part. Nucl. Phys. 46 (2001) 459
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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

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Result of Mock Data Analysis
N( of data points)-b(noise level) dependence
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Statistical and Systematic Error Analyses in MEM
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

in any MEM analysis
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Parameters on Lattice

• Lattice Sizes 323 32 (T 2.33Tc)
  40 (T 1.87Tc) 42 (T 1.78Tc)
  44 (T 1.70Tc) 46 (T
  1.62Tc) 54 (T 1.38Tc) 72
  (T 1.04Tc) 80 (T 0.93Tc)
  96 (T 0.78Tc)
• b 7.0, x0 3.5 x as/at 4.0
  (anisotropic)
• at 9.75 10-3 fm Ls 1.25 fm
• Standard Plaquette Action

• Wilson Fermion
• Heatbath Overrelaxation 1
  41000 sweeps between measurements
• Quenched Approximation
• Gauge Unfixed
• p 0 Projection
• Machine CP-PACS

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Result for V channel (J/y)
A(w) w2r (w)
J/y (p 0) disappears between 1.62Tc and 1.70Tc
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Result for PS channel (hc)

A(w) w2r (w)
hc (p 0) also disappears between 1.62Tc and
1.70Tc
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Statistical Significance Analysis for J/y
Statistical Significance Analysis Statistical
Error Putting
T 1.62Tc
1s
Ave.
T 1.70Tc
Both Persistence and Disappearance of the peak
are Statistically Significant
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Dependence on Data Point Number (1)
Data Point Dependence Analysis Systematic
Error Estimate
Nt 46 (T 1.62Tc) V channel (J/y)
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Dependence on Data Point Number (2)
Data Point Dependence Analysis Systematic
Error Estimate
Nt 40 (T 1.87Tc) V channel (J/y)
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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

• 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/y yet
• (Two Spectral Functions, Transverse and
  Longitudinal,
• for J/y at finite p ) ? pT dependence of
  J/y Suppression
• Non-Quench Calculation Started (Swansea-Dublin)

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Summary and Outlook (2)

• Both Statistical and Systematic Error Estimates
  have been carefully carried out