how can we extract spectral functions from lattice data ... fileapril 25, 2003 @int how can we...

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


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


HadronHadron Modification in HI Collisions? Modification in HI Collisions?

Experimental Data Comparison with Theory


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)


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


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


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


WayWay out ? (contout ? (cont’’d)d)Example of χ2-fitting failure

lattice36.0

24 54β =×

QCDPAX, 1995


MEMMEM

Maximum Entropy Method

successful in crystallography, astrophysics, …etc.

82@Y C


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


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


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


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


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


ResultResult of Mock Data Analysis (1)of Mock Data Analysis (1)

N(# of data points)-b(noise level) dependence


Result of MockResult of Mock Data Data Analysis (2)Analysis (2)

N(# of data points)-b(noise level) dependence


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

−

=

∫

∫


SPFSPF in V Channel in V Channel (T=0)(T=0)


SPFSPF in PS Channel in PS Channel (T=0)(T=0)


T=0 Result in VT=0 Result in V Channel and Error AnalysisChannel and Error Analysis

Perturbative Continuum Value+

Renormalization of Composite Operatoron Lattice


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×=


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 = =


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


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


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


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


IsIs the width statistically significant?the width statistically significant?

PS Channel

Narrow Lowest Peak Broad Lowest Peak


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 !

how can we extract spectral functions from lattice data ... fileapril 25, 2003 @int how can we...

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