Probability distribution of conserved charges and the QCD phase transition K. Redlic h
Probing QCD Phase Diagram with Fluctuations of conserved charges
description
Transcript of Probing QCD Phase Diagram with Fluctuations of conserved charges
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
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
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
B. Friman, V. Skokov &K.R.
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
B. Friman, V. Skokov &K.R.
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