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Probability distribution of conserved charges and the QCD phase transition K. Redlic h. 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 - PowerPoint PPT Presentation
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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
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