Probability distribution of conserved charges and the QCD phase transition K. Redlich

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) crossover criticality

STAR data & expectiations

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

crossover

Qarkyonic Matter ?

HIC

CP or Triple Point ?

Relating freezeout and QCD critical line at small baryon

chemical potential Particle yields and their ratios

as well as LGT thermodynamics

at are well described

by the HRG Partition Function QCD crossover line -

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

Chemical freezeout and QCD crtitical line

cT T

3

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

The HRG partition function provides and excellent approximation of the QCD thermodynamics at

, ( )q I T

4

1

( / )

nqn n

Pd

T T

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

cT TSee also: C. Ratti et al., P. Huovinen et al., B. Muller et al.,…….

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

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

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

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 PQM -RG

HRG HRG

See talk of Vladimir Skokov

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

Ratio of higher order cumulants at finite densitySee talk of Vladimir Skokov

Properties of fluctuations in HRG

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

and

resulting in:

Compare these HRG model predictions with STAR data at RHIC:

mesons baryonsHRGP P P 4 cosh( ) / )/ (baryons BP T F m TT

baryons

4( ) ( ( , ) / )

( / )B

nB

nn

B

P T T

T

and

(

(2

4)

)1B

B

(3)

(1)1B

B

(2)

(1)coth( / )B

BB

T

(3)

(2)tanh( / )B

BB

T

(2)

(1)

2B

B

B

BM

(3)

(2

2)B

BB BS

( )

( )2

2

4

BB

BB

Coparison 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

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

STAR Au-Au

STAR Au-Au

these data, due to transverse momentum cut.

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

Moments obtained from probability distributions

Moments obtained form the probability distribution

Probability quantified by all moments

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

Moments generating function:

)(

()

NC T

GC

Z N

ZP eN

Probability quantified by

:mean numbers of charged 1,2 and 3 particles & their antiparticles

[ , ] 0i S i Se He H S H

( )( , , ) [ ]SH SGCSZ T V Tr e ( , ) [ ]C H

S SZ T V Tr e

/SS

S TC

S

CS

GZ e Z

2

0

1( , ) ( , )

2i GCS

SSZ T V d e Z T iT

conservation on the average exact conservation

,n nS S

The probability distribution for net baryon number N is governed in HRG by the Skellam distribution

( )P N

The probability distribution for net proton number N is entirely given in terms of (measurable) mean number of protons b and anti-protons b

P(N) in the Hadron Resonance Gas for net- proton number

Probability distribution obtained form measured particle spectra: ( ) ( , )

t tp pP N f p p STAR data

Mean proton and anti-proton calculated in the HRG with

taken at freezeut curve and Volume obtained form the net-mean proton number

( , )BT

broken lines( )P N

Comparing HRG Model with Preliminary STAR data: efficiency uncorrected

Data consistent with Skellam distribution: No sign for criticality

Data presented at QM’11, H.G. Ritter, GSI Colloquium

7.7s GeV 11s GeV

HRG energy dependence versus STAR dataData presented at QM’11, H.G. Ritter, GSI Colloquium

arXiv:1106.2926Xiaofeng Luo for the STAR Collaboration

Data consitent with Skellam distribution HRG shows broader distribution

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

Data versus UrQMD

UrQMD provide much too narower probability distribution of net proton number

V. Skokov et al..

Centrality dependence of O(4) criticality?

Weak dependence of chemcialFreezeout on centrality

Decrease of chemical potentil

In the chiral limit decreaseof volume reduce criticlity

Decrease of centrality shifts the chemical line out of chiral O(4) cross over

Influence of volume on criticality requires detaile knowlege of Finite Size Scaling Function for free energy in O(4) universality class

J. Engels

NNs Centrality dependence at 62 and 200 GeVPreliminary STAR dataH.G. Ritter, GSI Colloquium

Published STAR data-efficiency uncorected

HRG shows increasing broadening of distributions with centrality

Centrality dependence of HRG model deviations from data:

Thanks to Xu Nu

Criticality increases with centrality: seen in the probability distributions and moments compared to data

O(4) criticality Hadronic freezout

HRG freezeout

O(4) criticality

The separation between chemical freezeout line and the O(4) chiral line might appear in HIC at collision energies larger than 39 GeV??

schematic diagram Based on the relation between HRG model and presently aveliable data we could conclude that:

Probability distributions and their energy dependence for

The maxima of have very similar values for all ( )P N NNs

thus const. for all p pN N

7 2.8NNGeV s TeV

0.4 0.8tGeV p GeV

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

In the HRG the P(N) is uniqually determined by

the measure multiplicities of charged particles. This avoids ambiguity in determining freezeout parameters and volume of the system

Deviation of P(N) form HRG sets in at and increases with centrality:

This might indicated remnants of O(4) criticality

in STAR data on net proton probability distribution in central Au-Au collisions at

39s GeV

39s GeV