how can we extract spectral functions from lattice data ... fileapril 25, 2003 @int how can we...
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April 25, 2003 @INT
How Can We Extract Spectral FunctionsHow Can We Extract Spectral Functionsfrom Lattice Data ?from Lattice Data ?
Masayuki Asakawa
Kyoto University (MELQCD Collab.)
M. Asakawa Kyoto University
PLANPLAN
Spectral Function
Necessity of MEM (Maximum Entropy Method)• Outline (Review)
Zero and Finite Temperature Result
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SpectralSpectral FunctionFunction
Definition of Spectral Function
( ) ( )
( )
: Boson(Fermion)
A Heisenberg Operator with some quant
0/
0' / 43 '
,
( , )0 0 (1 ) ( )
(2 )( )
0 :
nmn
E TP T
mnn m
A k k e n J m m J n e k PZ
J
ηη µ µη η
η
δπ
−−≡ −
− +
∑ ∓
um #
: Eigenstate with 4-momentum n
mn m n
n P
P P P
µ
µ µ µ= −
Pretty important function to understand QCD
production at T)0
20
4 4 2 2 /
( , )(3 1k T
A k kdN e ed xd q k e
µµα
π
+ −
= −−
holds regardless of states, either in Hadron phase or QGP
Dilepton production rate, etc. Also Real Photon Production
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HadronHadron Modification in HI Collisions? Modification in HI Collisions?
Experimental Data Comparison with Theory
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Why Theoretically UnsettledWhy Theoretically Unsettled
Mass Shift(Partial Chiral Symmetry Restoration)
Spectrum Broadening(Collisional Broadening)
Observed Dileptons
Sum of All Contributions(Hot and Cooler Phases)
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DifficultyDifficulty on Latticeon Lattice
What’s measured on Lattice is Correlation Function ( )D τ
† 3( ) ( , ) (0, 0)D O' x O d xτ τ= ∫and are related by( )D τ ( ) ( , 0)A Aω ω≡
0( ) ( , ) ( )D K A dτ τ ω ω ω
∞= ∫
However,• Measured in Imaginary Time• Measured at a Finite Number of discrete points• Noisy Data Monte Carlo Method
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Difficulty on LatticeDifficulty on Lattice
Thus, Inversion Problem
0( ) ( , ) ( )
( ) ( )
A
A
D K A d
D A
τ τ ω ω ωτ ω
∞=
⇒∫
Typical ill-posed problemProblem since Lattice QCD was born
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WayWay out ?out ?
-fitting• need to assume the form of
(1-pole, 2-poles, 1-pole + continuum…etc.)
• many degrees of freedom many solutions
• resonance mass depends on
2χ
minτ
( )A ω
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WayWay out ? (contout ? (cont’’d)d)Example of χ2-fitting failure
lattice36.0
24 54β =×
QCDPAX, 1995
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MEMMEM
Maximum Entropy Method
successful in crystallography, astrophysics, …etc.
82@Y C
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Principle of MEMPrinciple of MEM
a method to infer the most statisticallyprobable image given data( ( ))A ω=
MEM
In MEM, Statistical Error can be given to the Obtained Image
Theoretical Basis: Bayes’ Theorem
Probability of given
[ ]| ] [[ | ][ ]
[ | ] :
PY X P XP X YPY
P X Y X Y
=
A Theorem that reverses the roles of Cause and Result
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ApplicationApplication of MEM to Lattice QCDof MEM to Lattice QCD
In Lattice QCDD: Lattice Data (Average, Variance, Correlation…etc. )
( ) 0A ω ≥H: All definitions and prior knowledge such as
Bayes Theorem [ | ] [ | ] [ | ]P A DH P D AH P A H∝
( )A ω [ | ]P A DH
Most Probable Spectral Function
that Maximizes Posterior Prob.
( )A ω
In MEM, basically this Most Probable Spectral Function is calculated
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IngredientsIngredients of MEMof MEM
: Prior knowledge about ( ) ( )m Aω ω∈ RDefault Model
exp( )[ | ]
( )( ) ( ) ( ) log( )
,
S
SS
SP A H mZ
AS A m A dm
Z e Aα
αα
ωω ω ω ωω
α
=
= − − = ∈
∫
∫ D R
max at ( ) ( )A mω ω=
such as semi-positivity, perturbative asymptotic value, …etc.
[ | ]P D AH -
2
[ | ] exp( )/ LP D AH L Zχ=
= −likelihood function
given by Shannon-Jaynes Entropy[ | ]P A H
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WhatWhat to Maximize in MEMto Maximize in MEMTherefore, we obtain
[ | ] [ | ] [ | ]
exp( )P A DH P D AH P A H m
S Lα
α∝∝ −
Prob. of A given D and H
The Maximum of S Lα −
Unique if it exists !T. Hatsuda, Y. Nakahara, M.A., Prog. Part. Nucl. Phys. 46 (2001) 459
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MockMock Data AnalysisData Analysis
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
2( ) ( )in inA ω ω ρ ω≡
( )inA ω ( , )K τ ω
( )imockD τ ( )in iD τ
2( ) ( )out outA ω ω ρ ω≡
( )imockD τ
Lattice spacing, const.
diagonal (for simplicity)
/ ( ) / ,
:i in i i
ij
D b a a b
C
σ τ τ= × = =
( )outρ ω ( )inρ ω
( ) ( , ) ( ) , ( , )in inD K A d K e ωττ τ ω ω ω τ ω −= =∫ Dirichlet Kernel
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ResultResult of Mock Data Analysis (1)of Mock Data Analysis (1)
N(# of data points)-b(noise level) dependence
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Result of MockResult of Mock Data Data Analysis (2)Analysis (2)
N(# of data points)-b(noise level) dependence
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ErrorError Analysis in MEMAnalysis in MEM
MEM is based on Bayesian Probability Theory
In MEM, Errors can be and must be assignedThis procedure is essential in MEM Analysis
Error Bars can be put to Average of Spectral Function in
Decay Constants, e.g.,
etc.
2
1
1 2
2 1
2
2
[ , ],1 ( )
4 ( )VVpole
d
Zf d
m
ω
ω
ρρ
ω ω
ρ ω ωω ω
κωρ ω ω
−
=
∫
∫
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SPFSPF in V Channel in V Channel (T=0)(T=0)
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SPFSPF in PS Channel in PS Channel (T=0)(T=0)
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T=0 Result in VT=0 Result in V Channel and Error AnalysisChannel and Error Analysis
Perturbative Continuum Value+
Renormalization of Composite Operatoron Lattice
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FiniteFinite T Calculation (1)T Calculation (1)
How many points are needed in τ direction ?
or larger : needed30N τ
lattice
isotropic lattice
340 306.47β×=
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FiniteFinite T Calculation (2)T Calculation (2)This data suggest
more than ~30 points are needed in τ directionat the highest T.
The highest T : set to ~2.5Tc
In order to have large enough Lσ and Nτ,we employ anisotropic lattice
aτaσ
with / 4a aσ τξ = =
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NumberNumber of Configurationsof Configurations
32, 7 .0, 4 .0N σ β ξ= = = As of April 25, 2003
182~1.6
46 968072544032Nτ
194110150150181141# of Config.~0.78~0.93~1.04~1.4~1.9~2.3T / Tc
T > Tc Tc > T
Fairly Large Statistics in Lattice Standard
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Polyakov Loop and PL SusceptibilityPolyakov Loop and PL Susceptibility
T∝cT∼ 2 cT∼
Deconfined PhaseConfining Phase
= 0 in Confining Phase≠ 0 in Deconfined Phase
Polyakov Loop Susceptibility
Deconfined PhaseConfining Phase
T∝cT∼ 2 cT∼
Polyakov Loop
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ResultResult at T ~ at T ~ 1.9T1.9Tcc (PS Channel)(PS Channel)
2
2, , ,
( )
3 11 ( ) 118 3 2 ( )
(1)
pert
s
c c PS
m
Z aσ τ τ
ρ ω
α µπ π κ κ µ
= + =
∼
O
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Hadronic Correlations above THadronic Correlations above Tc c ??T~1.4Tc
Massive Free Quark Gas
Mass Gap ?
Landau Damping ?But statistically NOT significant
T~1.9Tc mπ/mρ ~ 0.7 (at T = 0)
Similar Result on smaller lattice
and
364 161.5 3c cT T T×
chiral limit
Karsch et al. (01)
vector channel
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IsIs the width statistically significant?the width statistically significant?
PS Channel
Narrow Lowest Peak Broad Lowest Peak
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SummarySummary and and Perspective
Hadronic Spectral Functions in QGP Phase were obtainedon large lattices at several T
It seems there are nontrivial modes in QGP
Sudden Qualitative Change between 1.4Tc and 1.9Tc ?
Physics behind still unknown
Perspective
Further study neededfor better understanding of QGP !