november 23, 2004@quantum fields in the era of teraflop-computing thermal properties of hadrons:...

November 23, 2004 @Quantum Fields in the Era of Teraflop-Computing

Thermal Properties of Hadrons:

below and above the phase transition

Masayuki Asakawa

Osaka University

M. Asakawa (Osaka University)

PLANQCD Phase Diagram

• QGP, Color Super Conductors, Critical End Point (CEP)

Freezeouts: Experimental Facts and Interpretation

Hadrons in Matter and Observables

Necessity of MEM (Maximum Entropy Method)

• MEM Outline

• Importance of Error Analysis

Finite Temperature Results for J/ and c

• Error Analysis

Statistical

Systematic

Strongly Interacting Matter at RHIC?

CEP Observables


What’s Quark Gluon Plasma?

Hot Hadron Soup

Dense Hadron Soup

CEP

CEP

2SC, dSC, CFL...

Color Super Conductors


Lattice Results (Finite T)

Pressure

Energy Density

Entropy Density

Jump in Entropy Density

Karsch et al. (2000)


Critical End Point on the Lattice

m ≠ mphys Fodor and Katz (2002) m = mphys Fodor and Katz (2004)

N = 4


First suggestionof CEP in QCD

Critical End Point in Effective Models

Compilation by Stephanov

Yazaki and M.A. NPA (1989)


Freezeout Points in HI Collisions

Chemical Particle #

Thermal

Kinetic

Particle Distribution

Two Freezeouts

Experimentally,Tch ＞ Tth


Chemical FreezeoutD

ata

– F

it (

)

Rat

io

Particle ratios well described by Tch = 16010 MeV, B = 24 5 MeV

These Numbers are obtainedby assuming FREE BE/FD distributions

STAR @RHIC 　 (s1/2 = 200 A GeV)


Hadrons in Matter? How to Observe?

initial state

pre-equilibrium

QGP andhydrodynamic expansion

hadronization

hadronic phaseand freeze-out

Fast Time Evolution In Ultrarelativistic Heavy Ion Collisions

Experimentally observed hadrons: hadrons at T=0

• Hadronic Decays in Medium: Decay products are rescattered e.g., Mass shift is expected NOT to be observed

• Indeed NO mass shift or width change is observed for reconstructed from K+K-

• Furthermore, modified hadrons are quantum mechanically different from hadrons in the vacuum discuss later⇒


Hadrons and Leptons/Photons

• Leptons, Photons interact only weakly (e.m.)

thus, carry information of QGP/hadron phases

• Hadrons strong interaction

thus, only info. of hadron phase (with some exceptions)


CERES(NA45) pA data for e+e- production

Dileptons yields and spectra: well-described by superposition of leptonic decay of final state hadrons


Hadron Modification in HI Collisions?

Comparison with Theory (with no hadron modification)

Experimental Data

Mass Shift ? Broadening ? or Both ? or More Complex Structure ?


More CERES(NA45) AB data

Also, in heavier systems,the enhancement is observed


Why Theoretically Unsettled

Mass Shift(Partial Chiral Symmetry Restoration)

Spectrum Broadening(Collisional Broadening)

Observed Dileptons

Sum of All Contributions(Hot and Cooler Phases)


Vector Channel SPF

u dJ e u u e d d

0

20

/4 4 2 2

( , )(

3 1

production at )k T

A k kdN e e T

d xd k k e

Spectral Function and Dilepton Production

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

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

holds regardless of states, either in Hadron phase or QGP

Definition of Spectral Function (SPF)

Dilepton production rate


Many Body Approach 1

Klingl et al. (1997)

vector dominance + hole model


Many Body Approach 2

Rapp and Wambach (1999)

Due to -hole contribution, non-Lorentzian

• Lorentzian Assumption ab initio : not justified • Neither is Smeared Source (on the lattice)


Mass shift or Coll. broadening 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 ...

† 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 the 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 !

K(,): Known Kernel


Way out ?

Example of inconclusiveness of 2-fitting

36.0

24 54 lattice

2 pole fit

by QCDPAX (1995)

Furthermore, too much freedom in the choice of ansatz


Difficulty on Lattice

Typical ill-posed problem

Problem since Lattice QCD was born

Thus, what we have is

Inversion Problem

0( ) ( , ) ( )

( ) ( )

D K A d

D A

continuousd i s c r e t enoisy


MEM Maximum Entropy Method• successful in crystallography, astrophysics, ...etc.

• a method to infer the most statistically probable image (such as A()) given data

• In MEM, can put and must put statistical errors

82@Y C

Reconstructed from X-ray diffraction image

• Fourier Transformed• Available at only Discrete Points• Noisy Data

Sc3C82


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

Principle 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


( )( ) ( ) ( ) 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

Ingredients of MEM


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

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 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 (1)

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


Result of Mock Data Analysis (2)

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)

Doubler States


Parameters on Lattice

1. 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


Result for V channel (J/)

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

A() 2()


Result for PS channel (c)

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

A() 2()


Statistical Significance Analysis for J/

Statistical Significance Analysis = Statistical Error Putting

T = 1.62Tc

T = 1.70Tc

Ave.

±1


Statistical Significance Analysis for c

Statistical Significance Analysis = Statistical Error Putting

T = 1.62Tc

T = 1.70Tc


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)


Data Point # Dependence Analysis = Systematic Error Estimate


Also for lighter quark systems

Hatsuda, Nakahara, and M.A. (2002)also Bielefeld group for V channel


1. Can J/ produced and escape before QGP is formed ?

2. Can J/ survive as a Coulombic resonance ?

3. Are there competitive non-plasma J/ suppression mechanisms ?

4. Could J/ suppression be compensated in the hadronization stage ?

5. Could enhanced thermal dileptons prevent clear observation of

J/ suppression ?

J/suppression as a QGP signature

J/suppression: prototype of QGP signature proposal

Debye Screening Melting of Heavy Quark⇒ Resonances

Matsui and Satz, 1986

• Check list by Matsui and Satz

Now, we know J/ exists above Tc


What’s Degrees of Freedom above Tc ?

PHENIX collaboration

Jet Quenching

working for pions, but not for (anti)protons


Recombination of Quarks

Jet Quenching and Fragmentation +Recombination (Coalescence) of Quarks

Recombination

Fragmentation

Jet Quenching Effects becomevisible at lower pT for mesons

Duke Group, Texas A&M Group,Oregon Group (2002)

0 1 2 3 4 5 6 7 8 9 10 11 12 GeV/c

pQCD

ReCoHydro

MesonsBaryons


Meson-Baryon Systematics (in RCP)

Baryons Mesons

RC

P

pT (GeV/c)

behaves like meson ?

Relative Enhancement of Protons: ⇒ not mass effect


Meson-Baryon Systematics (in v2)

Bass et al., PRC, 2003


Elliptic Flow : v2

Esumi (QM2002)

Target

Projectile

b:impact parameter

React

ion

plan

e

Larger Pressure Gradient

Reaction Plane

More Flow in parallel to Reaction Plane

= Elliptic Flow in addition to Radial Flow

central region

0 1 0 2 021 2 cos( ) 2 cos 2( )i i

iT

dN dNN v v

dyd dyd d p


Implication of success of recombination

Very good description of meson baryon systematics at intermediate pT

Quark Recombination: Q(G)P phase is assumed

Gluon Dynamics is not included : Non-perturbative feature of Q(G)P phase just above Tc ?

The unexpected charge fluctuation at RHIC

Actually, explained by Quark Recombination (Constituent Quark Matter)

• First pointed out by Bialas in a schematic way (2002)

• Analysis with Correlations due to Gluon Fragmentation, Diquarks, ..etc in progress (Nonaka, Muller, Bass, M.A.)

v2 & yields: also sensitive to internal structure of hadrons such as a0, f0, +,...etc. NMFBA (2004)

Mysterious observed Charge Fluctuation (STAR, without (questionable) correction)

242.8

ch

QD

N

In Naive QGP: D ~ 1

Naive Resonance gas: D ~ 3


Critical End Point in QCD

CEP

How can CEP be seen in Heavy Ion Collisions?

Color Super Conductors

2SC, dSC, CFL...


CEP = 2nd order phase transition, but...

If expansion is adiabatic

Divergence of Fluctuation Correlation Length Specific Heat ?

CEP = 2nd Order Phase Transition Point

• How does the system evolve near CEP?

• What can be observed if CEP exists ?

General Belief: Non-Monotonic Behavior as a function of the beam energy

QuestionsColl. Energy

T

B


Need to know EOS

If the critical region is tiny,hopeless to see, anyway

Assume critical region is large

Universality: belongs to the same universality class as 3d Ising Model

CLUE to EOS

T

B

T

B


EOS

• Mapping of 3d Ising Model variables to those in QCD• Matching of Singular Behavior near CEP and Hadronic Matter/QGP EOS below/above phase transition

For Details, C. Nonaka and M.A., nucl-th/0410078

TE=154.7MeV, BE=367.8MeV

CEP

crossover

1st order


Isentropic Trajectories• In each volume element, Entropy(S) and Baryon Number (NB) are conserved, as long as entropy production can be ignored

Isentropic Trajectories (nB/s = const.)

without CEP(EOS in usual hydro calculation)with CEP

Focusing of Isentropic Trajectories

Excluded Volume Approximation + Bag Model EOS


Consequences

Yes

Without Focusing

Coll. EnergyT

B

Non-monotonic Behavior in Observables

Only Weakly

With Focusing

Coll. EnergyT

B


Correlation Length, Fluctuation,...Furthermore, Correlation Lengths, Fluctuations,...: Hadronic Observables

Subject to Final State Interactions

fluctuation ~

Time Evolution along givenisentropic trajectories

2 2 /1( , )2

eq

rr M f M g

M

Widom’s scaling law

r,M: Ising side variables

00

( ) ( ) , 1zA

m m m

1( ) ( ) ( )

( )eq

dm m m

d

Model H (Hohenberg and Halperin RMP49(77)435)3z


Effect on Freezeouts 1 (chemical)

If focusing takes place,are the chemical freezeout points focused?

Not Necessarily

Hadrons near CEP (generally in hot phase)Quantum Mechanically Different from onesin the vacuum

e.g., Csorgo, Gyulassy, and M.A., PRL 99


Effect on Freezeouts 2 (Thermal)

On the left side of CEPs decreases smoothly,

while on the right side of CEPs decreases quickly (1st order PT)

On the left side of CEP,thermal freezeout takes placeat lower T?

Xu and Kaneta, nucl-ex/0104021(QM2001)

Yes, this is indeed the case!

• At RHIC, radial expansion is faster

• HBT size: similar

Similar or Higher Thermal Freezeout Temperaturewould be expected, but this is not the case

crossover

1st order


Summary (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 (2)

Recent RHIC data suggest Realization of Deconfined Phase• But No Gluonic Degrees of Freedom look needed

Success of Hydrodynamics without Viscosity suggests existence of strongly interacting matter just above Tc

Focusing Effect near CEP for Isentropic Trajectories

⇒ 　 Weak Non-monotonic behavior of observables as a function of collision energy?

Effect on Flow?

Low Thermal Freezeout Temperature at RHIC suggests existence of CEP


Back Up Slides


hadrons24

( )/ ( ) ( )R e e e e A ss

mm

ps s mm+ - + - + -º ® ® = -

Spectral Function at Work

Dilepton Production Rate at Finite T

production at T)0

20

4 4 2 2 /

( , )(3 1k T

A k kdN e ed xd k k e

mma

p

+ -

= --

r

hold regardless of state, either in Hadron Phase or QGP !

real production at T)4 3 2 /

( , )( 23 1T

A qdNd xd q e

mm

w

w wg aw

p

== -

-

r Real Photon Production Rate at Finite T

R ratio


Chiral Extrapolation

Continuum Kernel Lattice Kernel

Other Lattice Data orExperimental Value

　　 0.1570(3) 0.1569(1) 0.1571

0.348(15) 0.348(27) 0.331(22)

1.88(8) 1.74(8) 1.68(13)

2.44(11) 2.25(10) 1.90(3)

2.20(3)

ckmar

'/m mrr

'/m mrp

( (1450))r( (1700))r

Good Agreement with the Results of Conventional Analyses + Ability to Extract Resonance Masses


What Result of Mock Data Analysis tells us

General Tendency (always statistical fluctuation exists)

The Larger the Number of Data Pointsand the Lower the Noise Level

The result is closer to the original image

403 30 lattice = 6.47, 150 confs.isotropic lattice (T<Tc)

How many data points are needed ?

may depend on statistics, , K, ,...etc.

N, min or larger


Parameters in MEM analysis

With Renormalization of Each Composite Operator on LatticeThe m-dependence of the result is weak

Data Points at are not used 　　　　　　

/ 0, ,3, 3, , 1a N N

Continuum Kernel Small Enough Temporal Lattice Spacing

Data at these points can be dominated by such unphysical noise

channel PS V

m()2 1.15 0.40

Default Models used in the Analysis


Parameters in MEM Analysis (cont’d)

Furthermore, in order to fix resolution, a fixed number of data points (default value = 33 or 34) are used for each case

Dependence on the Number of Data Points is also studied (systematic error estimate)


Number of Configurations

32, 7.0, 4.0 N As of November 23, 2004

T Tc Tc T

N 32 40 42 44 46 54 72 80 96

T / Tc 2.33 1.87 1.78 1.70 1.62 1.38 1.04 0.93 0.78

# of Config.

141 181 180 180 182 150 150 110 194


Polyakov Loop and PL Susceptibility

TcT 2 cT

Deconfined Phase Confining Phase

Polyakov Loop Susceptibility

Deconfined Phase Confining Phase

TcT 2 cT

Polyakov Loop

0 in Confining Phase 0 in Deconfined Phase


Result of Mock Data Analysis

250.001

Nb

==

m(default model) dependence


Dependence on Data Point Number




N= 54 (T = 1.38Tc)

PS channel (c)



N= 46 (T = 1.62Tc)

PS channel (c)


Debye Screening in QGP

Original Idea of J/Suppression as a signature of QGP Formation: Debye Screening (Matsui & Satz, 1986)

J/ Melting at 1.1~ cT TDebye Screening of Potentialbetween cc

3, 0.7 PSf

V

NN

N

Karsch et al. (2000)

Need to start over asking a question“ What is QGP ? ” ?

Transition in the continuum limit

Transition gets smoother on finer lattices,imrovement of flavor symmetry ? HYP staggered fermions at Nt=4 -> Hasenfratz, Knechtli, hep-lat/0202019

MILC Coll.

november 23, 2004@quantum fields in the era of teraflop-computing thermal properties of hadrons:...

Documents

gev slide

later slide

cooler phases slide

t ch t th slide

interpretation hadrons

modified hadrons

thermal properties of

cep observables