T

B

Hadronic matter

Quark-Gluon Plasma

Chiral symmetrybroken

Chiral symmetryrestored

LH

C

Modelling QCD phase diagram

Polyakov loop extended quark-meson model

Including quantum and thermal

fluctuations: FRG-approach

Probing phase diagram with

fluctuations of net baryon number

theory & experiment

A-A collisions fixed s

x

1st principle calculations: perturbation theory pQCD LGT

, :QCDT , :QCDT

:q T

Renormalization Group and Quark number fluctuations near chiral phase transition

Krzysztof Redlich University of Wroclaw

Based on recent work with: B. Friman, V. Skokov; & F.

Karsch

Modelling QCD phase diagram

Preserve chiral symmetry

with condensate as an order parameter

K. Fukushima (2004)C. Ratti & W. Weise (07)

( ) ( )L f R fSU N SU N

Preserve center symmetry with Polyakov loop

( )cZ N

as an order parameter

Polyakov loop dynamics Confinement

Nambu & Jona-Lasinio

SpontaneousChiral symmetry

Breaking

Synthesis

40

1( exp[ ( , )])

c

L Tr P i d xAN

PNJL Model

R. Pisarski (2000)

Sketch of effective chiral models coupled to Polyakov loop

coupling with meson fileds PQM chiral model

Nambu-Jona-Lasinio model PNJL chiral model

1/3

0

i *4

nt ( , ),[ ](( ) )T

q

V

S d d x iq V U Lq q q LA q q

the chiral invariant quark interactions described through:

K. Fukushima; C. Ratti & W. Weise; B. Friman , C. Sasaki ., ….

B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman, V. Skokov, ...

int ( , )V q q

*( , )U L L the invariant Polyakov loop potential (3)Z

Polyakov loop potential fixed from pure glue Lattice Thermodynamics

First order deconfinement phase transition at

fixed to reproduce pure SU(3) lattice results

* * *3 34 22

*3 4/ ( ) ( ) ( ) ( ) ( ) ( ),GP T U b T b T b T

0 270T MeV

( )kb T

C. Ratti & W. Weise 07

Thermodynamics of PQM model under MF approximation

Fermion contribution to thermodynamic potential

Entanglement of deconfinement and chiral symmetry

Suppresion of thermodynamicsdue to „statistical confinement”

( )/ 2(3 3( )/)/(ln[1 3( ) ]q q qE T E T

qq

E Tp e e ed ( )q q

Suppresion

suppresion

SBlimit

The existence and position of CP and transition is model and parameter dependent

Introducing di-quarks and their interactions with quark condensate results in CSC phase and dependently on the strength of interactions to new CP’s

6

Generic Phase diagram from effective chiral Lagrangians

1st order

Zhang et al, Kitazawa et al., Hatta, Ikeda;Fukushima et al., Ratti et al., Sasaki et al.,Blaschke et al., Hell et al., Roessner et al., ..

0qm (Pisarski-Wilczek) O(4)/O(2) univ.; see LGT , Eijri et al 09

0qm

crossover

Asakawa-Yazaki

CP2nd order, Z(2) (Stephanov et al.)

Hatsuda et al.

Alford et al.Shuryak et al.Rajagopal et al.

B

broken

7

Probing CEP with charge fluctuations

Net quark-number fluctuations

where

21 1 1

36 4 6Q q Iq I

P

The CP ( ) and TCP ( ) are the only points where in an equilibrium medium the diverge ( M. Stephanov et al.)

, 0u dm , 0u dm A non-monotonic behavior of charge fluctuations is an excellent probe of the CP

( , )q Q

CP

2 32 ( )q qqN VT ) 4( ( / ) / ( / )n n

q qn P T T

( , )q Q

Phase diagram in chiral models with CP

at any spinodal points:

21

|.T

q

qnPV

V compres

Singularity at CEP isthe remnant of that along spinodals

CP

spinodals

spinodals

C. Sasaki, B. Friman & K.R.V. Koch et al., I. Mishustin et al., ….

Including quantum fluctuations: FRG approach

FRG flow equation (C. Wetterich 93)J. Berges, D. Litim, B. Friman, J. Pawlowski, B. J. Schafer, J. Wambach, ….

start at classical action and includequantum fluctuations successively by lowering k

Regulator function suppressesparticle propagation with momentum lower than k

0lim(( ) ), ( /) k kk

T T VV

k k kk R

k-dependentfull propagator

Renormalization Group equations in PQM model

Quark densities modified by the background gluon fields

Flow equation for the thermodynamic potential density in the PQM model with Quarks Coupled to the Background Gluonic Fields

4

* *2

23 11 2 ( ) 1 2 ( ) 1 ( , ) ( , )

12q

qqk B qk B

dkn E n E L n

E E ELn L L

**

*

1 2 exp( ( )) exp(2 ( ))( , )

1 3 exp(2 ( )) 3 exp(2 ( )) exp(3 ( ))q q

q q q

L LL L

L L

E En

E E E

with fixed such that to minimise quantum potential,

* *0

*( : , ) ( : , ) (, ), ,kL L LT T UL L L

highly non-linear equation due to

*( , )L L

V. Skokov, B. Friman &K.R.

2( ) ( , / )nkE k E k

FRG at work –O(4) scaling:

Near critical properties obtained from the singular part of the free energy density

cT

Resulting in the well known scaling behavior of

1/ /( , ) ( , )dSF b F bh ht t b

c

c

tT T

T

external field :h Phase transition encoded in the “equation of state”

sF

h

1/ 1/( ) ,hh F z z th

'| | ( | | )st F h t

1/

( ) , 0, 0{

, 0, 0

t tB

ht

h

Bh

coexistence line

pseudo-critical point

* 1 0L L and

FRG-Scaling of an order parameter in QM model

log( ) log( )

log( )t log( )h

t

The order parameter shows scaling. From the slopes one gets

0.401(1) 4.818(2

0.5 3

0.3836(4

9

6) 4.851( 2)

)

2

MF

LG

R

T

F G

However we have neglected field-dependent wave function renormal. Consequently and . The 3% difference can be

attributed to truncation of the Taylor expansion at 3rd order when solving FRG flow equation:

see D. Litim analysis for O(4) field Lagrangian

0 5

Effective critical exponents Approaching from the side of

the symmetric phase, t >0, with small but finite h : from Widom-

Griffiths form of the equation of state

For and

1/ 1/ 1/( ) ,cB h f x x t

0t 0h 0

( )x f x x

t h , thus

( ) , 0, 0{

, 0, 0c

t th

h tht

B

B

Define:

0log( ): {

0log( )

tdR

td t

R

1.53 1MF

0.4 cT

1.48LGT

Two types of susceptibility related with order parameter

1. longitudinal

2. transverse

Fluctuations of the order parameter

/l h

/t h max ( )t h

max ( )h

Scaling properties

at t=0 and

1/ 1B h

0h

1/m

(ax

)t h

m xx ama( )t t

1/ ( ) 0.49 1.6

The order parameter in PQM model in FRG approach

For a physical pion mass, model has crossover transition Essential modification due to coupling to Polyakov loop The quantum fluctuations makes transition smoother

Mean Field dynamics FRG results

QM

PQM

<L> QM

PQM

<L>

Fluctuations of order parameters Mean Field dynamics FRG results

Deconfinement and chiral transition approximately same Within FRG broadening of fluctuations and their

strength: essential modifications compared with MF

d / dTLd / dTL

d /dT

d /dT

The Phase Diagram and EQS

Mean-field approximation Function Renormalization Group

Mesonic fluctuations shift the CEP to higher temperature The transition is smoother No focusing of isentropes (see E. Nakano et al. (09))

(c.f., C. Nonaka & M. Asakawa &. B. Muller)

CEP

CEP

Probing phase diagram with moments of net quark number fluctuations

Smooth change of and peak-like structure in as in O(4)

Moments probes of the chiral transition

22

( ) 2 3( ) /q qc N VT

MF-results

PQM

PQM

QMFRG-results

4( ) ( / )

( / )

nn

q n

P T

T

QM

PQM

4 2 24 ( ) 3 ( )q qc N N

2 ( )c T 4 ( )c T

2c

1 0

0

q

q

t for negligible

t for cusp structure

4c

(2 ) 0

0q

q

t for cusp structure

t for diverges

2c

QMmodel PQM

model

PQMmodel

QMmodel

SB limit

SB

Kurtosis of net quark number density in PQM modelV. Skokov, B. Friman &K.R.

For

the assymptotic value

2

24 2

( ) 2 3 3cosh

3qq f q q

c

qP T N m

KT T T

m

N T

due to „confinement” properties

Smooth change with a rather weak dependen- ce on the pion mass

cT T9

4 2/ 9c c

For cT T

4

( )qqP T

T

4 22/ 6 /c c

4 2/c c

Kurtosis as an excellent probe of deconfinement

HRG factorization of pressure:

consequently: in HRG In QGP, Kurtosis=Ratio of cumulants

excellent probe of deconfinement

( , ) ( ) cosh(3 / )Bq qP T F T T

4 2/ 9c c 26 /SB

42

4 2 2

( )/ 3 ( )

( )qq q

qq

Nc c N

N

S. Ejiri, F. Karsch & K.R.

Kur

tosi

s

44,2

2

1

9B c

Rc

F. Karsch, Ch. Schmidt

The measures the quark

content of particles carrying

baryon number

4,2BR

Quark number fluctuations at finite density Strong increase of fluctuations with baryon-chemical potential

In the chiral limit the and daverge at the O(4) critical line at finite chemical potential

3c 4c

Deviations of the ratios of odd and even order cumulants from their asymptotic, low T-value, are increasing with and the cumulant order Properties essential in HIC to discriminate the phase change by measuring baryon number fluctuations !

4 2 3 1/ / 9c c c c /T

4,2 4 2/R c c

Ratio of cumulants at finite density

QCD phase boundary & Heavy Ion Data

QCD phase boundary appears near freezeout line

Particle yields and their ratio, well described by the Hadron Resonance Gas

QCD phase boundary & Heavy Ion Data

Excellent description of LGT EQS by HRG

A. Majumder&B. Muller

LGT by Z. Fodor et al..

R. Hagedorn

QCD phase boundary & Heavy Ion Data

Is there a memory that the system has passed through the region of the QCD phase transition ?

Consider the net-quark number fluctuations and their higher moments

STAR DATA ON MOMENTS of FLUCTUATIONS

Mean

B p p

pB p NM N

Variance

B B BN N M

22 ( )BB N Skewness

3 3( ) /BB BS N

4 4( ) / 3B B BN

Kurtosis

Phys. Rev. Lett. 105, 022302 (2010)

Properties of fluctuations in HRG

Calculate generalized susceptibilities:from Hadron Resonance Gas (HRG) partition function:

then, and

resulting in:

(2)

(1)coth( / )B T

(3)

(2)tanh( / )B T

Compare this HRG model predictions with STAR data at RHIC:

Comparison of the Hadron Resonance Gas Model with STAR data

Frithjof Karsch & K.R. K.R.

RHIC data follow generic properties expected within HRG model for different ratios of the first four moments of baryon number fluctuations

Can we also quantify the energy dependence of each moment separately using thermal parameters along the chemical freezeout curve?

Mean, variance, skewness and kurtosis obtained by STAR and rescaled HRG

STAR Au-Au

STAR Au-Au

these data, due to restricted phase space:

Account effectively

for the above in the HRG model

by rescaling the volume

parameter by the factor 1.8/8.5

200s 8.5

p pM

200s

1.8p p

M

30

LGT and phenomenological HRG model C. Allton et al.,

Smooth change of and peak

in at expected from O(4) universality argument and HRG

, ( )q I T

4qd cT

4

4 4 4

1

( / )q P

dT T

S. Ejiri, F. Karsch & K.R.

( )q I For fluctuations as

expected in the Hadron Resonance Gas

cT T

31

To see deviations from HRG results due to deconfinement and chiral transtion one needs to measure higher order fluctuations:

Lattice QCD results model calculations

:

C. Schmidt

Conclusions The FRG method is very efficient to include quantum and

thermal fluctuations in thermodynamic potential in QM and PQM model

The FRG provides correct scaling of physical observables expected in the O(4) universality class

The quantum fluctuations modified the mean field results leading to smearing of the chiral cross over transition

The RHIC data on the first four moments of net- proton fluctuations consitent with expectations from HRG: particle indeed produced from thermal source

To observe large fluctuations related with O(4) cross-over, measure higher order fluctuations, N>6