Probing QCD Phase Diagram with Fluctuations of conserved charges Krzysztof Redlich University of Wroclaw & EMMI/GSI QCD phase boundary and its O(4) „scaling”

Probing QCD Phase Diagram with Fluctuations of conserved charges

Krzysztof Redlich University of Wroclaw & EMMI/GSI

QCD phase boundary and

its O(4) „scaling” Higher order moments of conserved

charges as probe of QCD criticality

in HIC STAR data & expectations

HIC

/ ~ NNT fixed s

K. Fukushima

Particle yields and their ratio, as well as LGT results at

are well described by the Hadron Resonance Gas Partition Function (F. Karsch, A. Tawfik & K.R.).

A. Andronic et al., Nucl.Phys.A837:65-86,2010. O(4) universality

cT T

Budapest-Wuppertal LGT data

Chiral crosover Temperature from LGT Chemical Freezeout LHC (ALICE) HotQCD Coll. (QM’12) P. Braun-Munzinger (QM’12)

2

00.0061 6c

c

T

T T

HRG model

(152,164)fT MeV154 9cT MeV

Particle yields and their ratio, as well as LGT results are

well described by the Hadron Resonance Gas

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

(3) (4)) )1 (( 2M BM BFGP G F

2 4(2) 5 2

18 3I

BqcF c

cT T

O(4) universality 2

00.0061 6c

c

T

T T

HRG model

O(4) scaling and critical behavior

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

cT

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

2

c

c c

T T

Tt

T

Phase transition encoded in

the “equation of state” sF

h

11/

/( ) ,hF z z th

h

pseudo-critical line

reg SF F F

with1/h

with

F. Karsch et al

O(4) scaling of net-baryon number fluctuations

The fluctuations are quantified by susceptibilities

Resulting in singular structures in n-th order moments which appear for and for

since in O(4) univ. class

From free energy and scaling function one gets

( ) ( ) 2 /2 ( /2) 0( )n nB rn nc h f z for and n even

4( ) ( / )

:( / )

nn

B nnB

P Tc

T

( ) ( ) 2 ( ) ( 0)n nn nB r c h f z for

6 0n at 3 0n at 0.2

reg SingularP P P with

Kurtosis as an excellent probe of deconfinement

HRG factorization of pressure:

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

with is an

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.: Phys.Lett. B633 (2006) 275

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

q q qN N N

Kurtosis of net quark number density in PQM model

For

the assymptotic value

2

24 2

( ) 2 3 3cosh

2

3

7qq f q q qP T N m m

KT T T T

due to „confinement” properties

Smooth change with a very weak dependence 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

4 2 2

2

1 1 7[ ]2 6 180f cN N

T T

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

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

Ch. Schmidt at QM’12

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

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

Ratios of cumulants at finite density: PQM +FRG

HRG HRG


Comparison of the Hadron Resonance Gas Model with STAR data

Frithjof Karsch & K.R.

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

(2)

(1)coth( / )B

BB

T

(3)

(2)tanh( / )B

BB

T

(4)

(2)1B

B

Error Estimation for Moments Analysis: Xiaofeng Luo arXiv:1109.0593v

STAR data:

Deviations from HRGCritical behavior?Conservation laws? A. Bzdak, V. Koch, V.Skokov ARXIV:1203.4529

Volume fluctuations? B. Friman, V.Skokov &K.R.

4)2

(

(2)B

B

(3)

( )2

2,B

B

S

Deviations of the ratios from their asymptotic, low T-value, are increasing with the order of the cumulant

Ratio of higher order cumulants B. Friman, V. Skokov &K.R.

13

The range of negative fluctuations near chiral cross-over: PNJL model results with quantum fluctuations being included : These properties are due to O(4) scaling , thus should be also there in QCD.

Range of deviations from HRG

Fluctuations of the 6th and 8th order moments exhibit strong deviations from the HRG predictions:

=> Such deviations are to be expected in HIC already at LHC and RHIC energies


Theoretical predictions and STAR data 14

Deviation from HRG if freeze-out curve close toPhase Boundary/Cross over line

Lattice QCDPolyakov loop extendedQuark Meson Model

STAR Data

STAR Preliminary

L. Chen, BNL workshop, CPOD 2011

Strong deviations of data from the HRG model (regular part ofQCD partition function) results: Remnant of O(4) criticality!!??

STAR DATA Presented at QM’12

Lizhu Chen for STAR Coll.

V. Skokov

Moments obtained from probability distributions

Moments obtained from probability distribution

Probability quantified by all cumulants

In statistical physics

2

0

( ) (1

[ ]2

)expdy iP yNN iy

[ ( , ) ( , ]( ) ) k

kkV p T y p T yy

( )k k

N

N N P N

Cumulants generating function:

)(

()

NC T

GC

Z N

ZP eN

Probability distribution of the net baryon number

For the net baryon number P(N) is described as Skellam distribution

P(N) for net baryon number N entirely given by measured mean number of baryons and antibaryons BB

/2

( ) (2 )exp[ ( )]N

N

BBN BI B BP

B

In HRG the means are functions of thermal parameters at Chemical Freezeout

(, , , )f fTB F VB

Shrinking of the probability distribution already expected due to deconfinement properties of QCD

HRG

In the first approximation the quantifies the width of P(N)

B

2 3( ) exp( / 2 )BP N N VT

LGThotQCD Coll.Similar: Budapest-Wuppertal Coll

At the critical point (CP) the width of P(N) should be larger than that expected in the HRG due to divergence of in 3d Ising model universality class

B

Influence of criticality on the shape of P(N)

Influence of O(4) criticality on P(N)

Consider Landau model:

Scaling properties:

Mean Field 0

O(4) scaling 0.21

2 2 41 1

2 4bg t

1| |

4t ( , )t T

sin3 gnn c

Contribution of a singular part to P(N)( , ,

( )) N

T

GC

Z N T V

ZP eN

Got numerically from:

For MF broadening of P(N) For O(4) narrower P(N)

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

Conclusions:

Hadron resonance gas provides reference for O(4) critical behavior in HIC and LGT results

Probability distributions and higher order cumulants are excellent probes of O(4) criticality in HIC

Observed deviations of the from the HRG

as expected at the O(4) pseudo-critical line Shrinking of P(N) from the HRG follows

expectations of the O(4) criticality

6 2/

Influence of volume fluctuations on cumulants

V/2

( , ) (2 )exp[ ( )]B

N

BBN B

Bn n n n

BP N IV V V

( )N

k k

N

N N P N

N N N

Probability distribution at fixed V

Moments of net charge

Cumulants

1

1N

V

22

1( )N

V

33

1( )N

V

24 24

1( ( ) 3 ( ) )N N

V

central moment

Volume fluctuations and higher order cumulants in PQM model in FRG approach

General expression for any P(N): - Bell polynomials ,n iB

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

22

2 2 1c v

Documents

Probing QCD Phase Diagram with Fluctuations of conserved charges Krzysztof Redlich University of Wroclaw & EMMI/GSI QCD phase boundary and its O(4) „scaling”