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

July 22, 2004

Hot Quarks 2004Taos Valley, NM

Charmonia in the Deconfined PlasmaCharmonia in the Deconfined Plasma from Lattice QCD from Lattice QCD

Masayuki Asakawa

Osaka University

Intuitive and Quick Introduction to Lattice QCDIntuitive and Quick Introduction to Lattice QCDandand

M. Asakawa (Osaka University)

Warm Up: Quantum MechanicsWarm Up: Quantum Mechanics

In (Undergraduate) Quantum Mechanics, Transition amplitude is given by

, , exp ( )

[ , ] 1

F F I I F F I Iq t q t q iH t t q

q p i

The same quantity has another expression,

, , exp ( , ) (usually)F L

I I

q t

F F I I q tq t q t Dq i L q q dt

Very roughly speaking,

stands for sum over pathsDq


AB Effect and Path IntegralAB Effect and Path Integral

Famous AB(Aharonov-Bohm) effect (Modern Quantum Mechanics, J.J.Sakurai)

(0)

(0)

exp exp

exp exp

F F

I I

F F

I I

t x

above t x above

t x

below t x below

D i L dt ie d

D i L dt ie d

x A s

x A s

Amplitude at the interference point

F F

I I

x x

x xbelow above

B

ie d ie d ie d

ie

A s A s A s

common phases for pathsabove and below the cylindersource

0B

0 0 but B A

interferencepoint

impenetrable wall


Path Integral in Field TheoryPath Integral in Field Theory

Field Theory : Quantum Mechanics with Infinite Degrees of Freedom

degree of freedom distributed at each point

Sum over Paths Sum over Field Distributions (configurations)

( ) ( , )

( ) ( , )

q t x t

p t x t

[ , ] [ ( , ), ( , )] ( )q p i x t y t i x y

, , exp ( , )F L

I I

q t

F F I I q tq t q t Dq i L q q dt

4, , exp ( , )F L

I I

t

F F I I tt t D i d x

L


Discretization, a.k.a. LatticeDiscretization, 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

( )t t i

4exp ( , ) exp exp[ ]L

I

t

M Eti d x iS S L

Remember

0( ) lim

xf x dx

x

f(x)

x


QCD on the LatticeQCD 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

x ˆx

ˆx ˆˆx

( )U x

a

( ) exp[ ( )]

( ) (3)

U x iagA x

U x SU

ˆ ˆ ˆˆ ˆ ˆ( , , ) ( ) ( ) ( ) ( )W x U x U x U x U x

2

6 1ˆˆReTr 1 ( , , )

3px

S W xg

0 2 4,

1( ( ))

4a a

p cont EuclidS S F x d x

continuum gauge field action

color fieldmatrix


Why Monte Carlo Method?Why Monte Carlo Method? Expectation value of QCD observable (on the lattice)

lat p qS S S 1

[ , , ] [ , , ]

det

lat

plat

S

SSq

O U DUD D e O UZ

Z DUD D e DUe M

• Operator with Each Configuration is summed up with weight exp(-Slat)

• Each Link has 8 degrees of freedom (SU(3) gluon color)

Average over Configurations with Huge Degrees of Freedom!

Generate Gauge Field Configurations with weightexp(-Sp) or exp(-Sp+log(detMq)) if exp(-Sp) or exp(-Sp+log(detMq)) is Real

Monte Carlo Method


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 weightexp(-Sp) or exp(-Sp+log(detMq)) if exp(-Sp) or exp(-Sp+log(detMq)) is Real

Redefine weight and operator

With some applicability limit

0 0

( )

( ) ( ) ( )0

0

1e det ( )

1 det ( )e det ( ) e

det ( ) new weight

new operator

g

g g g

S

S S S

O DU M OZ

MDU M O

Z M

R

2

6( , , ),m

g

Glasgow Group, Fodor-Katz,...


Systematic and Statistical ErrorsSystematic and Statistical Errors

Lattice Calculation is carried out with Finite Lattice Spacing a, Finite Lattice Volume V, quark mass m ≠ mphys

0 (continuum)

(infinite volume)

(chiral)phys

a

V

m m

Necessity of Extrapolations (origin of systematic errors)

Monte Carlo Method to evaluate O

Necessity to Generate Many Configurations (to reduce statistical errors)

as in usual integral

0( ) lim

xf x dx

x

f(x)

x


Things not to doThings not to do

van Leeuwen for NA49, QM2002

Fixed lattice spacing a, m ≠ mphys

Fodor and Katz (2002)


Order of Phase Transition at Order of Phase Transition at = 0= 0

3d O(4) scaling

tricritical point

3d Ising scaling

2nd order line: mud (ms*-ms)5/2

near the tricritical point

3d Z(3) Potts

Kanaya, QM2002


Critical Endpoint in Effective ModelsCritical Endpoint in Effective Models

Compilation by Stephanov

Yazaki and M.A. NPA (1989)

Debut of CEP in QCD


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 gas e = 3p

e - 3p: interaction measure


What is the “normal” behavior of e, p, e-3p?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 Tc,p does not approach pSB

In addition, e - 3p never vanishes!

At = 0,

0( ) ( )

( ) ( ) ( )

Tp T s t dt

e T Ts T p T

At Tc, p is continuous (from thermodynamics)

Thus, p cannot increase rapidly ! Hatsuda and M.A., 1997


ExampleExample

4/e T

4( 3 ) /e p T4/p T

/Tc = 0.05

Hatsuda and M.A., 1997

Interaction Measure does not measure interaction !


PLAN of the Second PartPLAN 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


0

20

/4 4 2 2

( , )(

3 1

production at )k T

A k kdN e e T

d xd k k e

Spectral FunctionSpectral Function

0/

' 0 /† 4'3

,

( , )0 0 (1 ) ( )

(2 )( )

0 :

: Boson(Fermion)

A Heisenberg Operator with some quan

n

mn

E TP T

mnn m

A k k en J m m J n e k P

Z

J

tum #

: Eigenstate with 4-momentum n

mn m n

n P

P P P

• 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


CERES(NA45) AB dataCERES(NA45) AB data


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 ?


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)


Mass shift or Coll. broadening or Mass shift or Coll. broadening or -hole 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?


Lattice? But there was difficulty ...Lattice? But there was difficulty ...

† 3( ) ( , ) ' (0,0)D O x O d x

and are related by( )D ( ) ( ,0)A A

0( ) ( , ) ( )D K A d

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 !


Way out ?Way out ?

Example of 2-fitting failure

36.0

24 54 lattice

2 pole fit

by QCDPAX (1995)


Difficulty on LatticeDifficulty on Lattice

0( ) ( , ) ( )

( ) ( )

D K A d

D A

Typical ill-posed problem

Problem since Lattice QCD was born

Thus, what we have is

Inversion Problem

continuousd i s c r e t enoisy


MEMMEM

Maximum Entropy Method

successful in crystallography, astrophysics, …etc.

82@Y C


( ( ))A a method to infer the most statisticallyprobable image given data

Principle of MEMPrinciple of MEM

[ ]| ] [[ | ]

[ ]

[ | ] : Probability of given

P Y X P XP X Y

P Y

P X Y X Y

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

( ) 0A H: All definitions and prior knowledge such as

D: Lattice Data (Average, Variance, Correlation…etc. 　 )

[ | ] [ | ] [ | ]P A DH P D AH P A HBayes' Theorem


Ingredients of MEMIngredients of MEM

( )( ) ( ) ( ) log

( )

exp( )[ | ]

[ ],

S

SS

AA m A

m

SP A H m

Z

S d

Z e dA

R

such as semi-positivity, perturbative asymptotic value, …etc.

( ) ( ): Prior knowledge about m A RDefault Model

[ | ]P D AH 2

[ | ] exp( ) /-

LP D AH L Z

likelihood function

given by Shannon-Jaynes Entropy[ | ]P A H

For further details,Y. Nakahara, and T. Hatsuda, and M. A., Prog. Part. Nucl. Phys. 46 (2001) 459

( ) ( )max at A m


Error Analysis in MEM Error Analysis in MEM (Statistical)(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

2

11 2

2 1

1[ , ], ( )Average of Spectral Function in

II A A d

2 2

2 1

12

22 1

1

1[ ] ( ) ( ) [ | ]

( )

1 ( )

( ) ( ) ( )

( ) ( ) ( )

( )

[ ]

I II

I IA A

Nl

l l

A dA d d A A P A DH m

Q Ad d

A A

A A A

Q A S L

dAdA

A

Gaussian approximation


Uniqueness of MEM SolutionUniqueness of MEM Solution

Unique if it exists !T. Hatsuda, Y. Nakahara, M.A., Prog. Part. Nucl. Phys. 46 (2001) 459

Ar A

r

only one local maximum many 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])


Result of Mock Data Analysis Result of Mock Data Analysis

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


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


Parameters on Lattice at finite TParameters on Lattice at finite T1. 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)

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 : Overrelaxation1 : 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


Result for V channel (J/Result for V channel (J/))

J/ (p 0) disappearsbetween 1.62Tc and 1.70Tc

A() 2()


Result for PS channel (Result for PS channel (cc))

c (p 0) also disappearsbetween 1.62Tc and 1.70Tc

A() 2()


Statistical Significance Analysis for J/Statistical Significance Analysis for J/

Statistical Significance Analysis = Statistical Error Putting

T = 1.62Tc

T = 1.70Tc


Statistical Significance Analysis for Statistical Significance Analysis for cc

Statistical Significance Analysis = Statistical Error Putting

T = 1.62Tc

T = 1.70Tc


Dependence on Data Point Number (1)Dependence on Data Point Number (1)

N= 46 (T = 1.62Tc)V channel (J/)

Data Point # Dependence Analysis = Systematic Error Estimate


Dependence on Data Point Number (2)Dependence on Data Point Number (2)

N= 40 (T = 1.87Tc)V channel (J/)

Data Point # Dependence Analysis = Systematic Error Estimate


Summary and Perspectives (1)Summary and Perspectives (1)

It seems J/ and c (p = 0) remain in QGP up to ~1.6Tc

Sudden Qualitative Change between 1.62Tc and 1.70Tc

~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


Summary and Perspectives (2) Summary and Perspectives (2)

Thermodynamical Quantities : Sudden Increase of Degrees of Freedom at Tc

but,

Hadrons (or Strong Correlations) still exist at least up to T ~ 1.6 Tc

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 neededfor better understanding of QGP and Hadronic Modes in QGP !

Documents

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