Charge fluctuations in heavy ion collisions and QCD phase diagram K. Redlich, Uni Wroclaw & EMMI

QCD phase boundary, its O(4) „scaling” & relation to freezeout in HIC

Moments and probablility distributions of conserved charges as probes of the O(4) criticality in QCD

Xu Nu -STAR data & expectations

with: P. Braun-Munzinger, B. Friman F. Karsch, V. Skokov

crossover

Qarkyonic Matter

HIC

CP or Triple Point

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

are well described by the Hadron Resonance Gas Partition Function .

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

cT T

Budapest-Wuppertal LGT data

Thermal and chemical equilibrium: H. Barz, B. Friman, J Knoll & H.Schultz P. Braun-Munzinger, J. Stachel, Ch. Wetterich C. Greiner & J. Noronha-Hostler, ……..

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

Effective chiral models and their non-perturbative thermodynamics: Renormalisation Group Approach

coupling with meson fileds PQM chiral model FRG thermodynamics of PQM 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 SU(2)xSU(2) 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

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

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 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

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

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

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

Ratio of higher order cumulants

Fluctuations of 6th and 8th order moments exhibit strong variations from HRG predictions: Their negative values near chiral transition to be seen in heavy ion collisions at LHC & RHIC

10

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

Theoretical predictions and STAR data 11

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!!??

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

Comparing HRG Model with Preliminary STAR data: efficiency uncorrected

Data described by Skellam distribution: No sign for criticality

Data presented at QM’11

Observed schrinking 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

LGT

hotQCD Coll.Similar: Budapest-Wuppertal Coll

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

Conclusions:

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

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

Observed deviations of the from the HRG

value as expected at the O(4) pseudo-critical line Deviation of P(N) from HRG seems to

set in at

39s GeV

6 2/