Probing QCD Phase Diagram with Fluctuations
of conserved charges
HIC
 / T  fixed ~



s NN
QCD phase boundary and
its O(4) „scaling”
Higher order moments of conserved
charges as probe of QCD criticality
in HIC
STAR data & expectations
K. Fukushima
Krzysztof Redlich University of Wroclaw & EMMI/GSI
Particle yields and their ratio, as well as LGT results at
T  Tc are well described by the Hadron Resonance Gas
Partition Function (F. Karsch, A. Tawfik & K.R.).

Budapest-Wuppertal LGT data

 

1

0.006
6
A. Andronic et al., Nucl.Phys.A837:65-86,2010
.  T 
0
Tc
O(4) universality
Tc
2
HRG model
Chiral crosover Temperature from LGT
Tc  154  9 M eV
HotQCD Coll. (QM’12)
Chemical Freezeout LHC (ALICE)
T f  (152,164) M eV
P. Braun-Munzinger (QM’12)

Particle yields and their ratio, as well as LGT results are
T  Tc well described by the Hadron Resonance Gas
S. Ejiri, F. Karsch & K.R.
Partition Function .
(1 )
(3)
(2)
(4)
P  G M  G M  FB
O(4) universality
Tc
Tc

0
 
 1  0.006 6  
T 
2
HRG model
(2)
FB

5
18
c2 
q
2
3
I
c4
 FB
O(4) scaling and critical behavior
Near Tc critical properties obtained from the
singular part of the free energy density

F  Freg  FS
with
T  Tc
  
 

 Tc 
with t 
Tc

FS ( t , h )  b
   
 
h
1/ 
h

t, b
  /
  
pseudo-critical line
 Fh ( z ) ,
F (b
1/
2
Phase transition encoded in
the “equation of state”
 Fs
d
z  th
 1/  
h
1/
F. Karsch et al
h)
O(4) scaling of net-baryon number
fluctuations

The fluctuations are quantified by susceptibilities
 (P / T )
n
B

( B / T )
n
: c n
with
P  Preg  PSingular

From free energy and scaling function one gets

(n)
B


(n)
4
(n)
B


(n)
r
(n)
r
c h
2   n / 2
 c h
2   n
f
(n/2)

f
(n)

(z)
(z)
for   0 and n even
for   0
Resulting in singular structures in n-th order moments
which appear for n  6 at   0 and for
n  3 at   0 since    0 .2 in O(4) univ. class
Kurtosis as an excellent probe of deconfinement
S. Ejiri, F. Karsch & K.R.: Phys.Lett. B633 (2006) 275
1 c4
B
R4 ,2 
 HRG factorization of pressure:
9 c2
P (T ,  q )  F (T ) cosh(3  q / T )
Kurtosis
B

F. Karsch, Ch. Schmidt

consequently: c 4 / c 2  9 in HRG
2
In QGP, SB  6 / 
Kurtosis=Ratio of cumulants
 ( N q ) 
4
c4 / c2 
q
B
The R 4 , 2 measures the quark
content of particles carrying
baryon number
q
 ( N q ) 
2
 3  ( N q ) 
2
with  N q  N q   N q  is an
excellent probe of deconfinement
Kurtosis of net quark number density in PQM model

B. Friman, V. Skokov &K.R.
For T  Tc
the assymptotic value
due to „confinement” properties
Pq q (T )
T
4
9
c4 / c2
2
3m q 
3 q
 3m q 
f 

 K2 
 cosh
2 
2 7  T 
T
 T 
2N
c4 / c2  9

For T  Tc
Pq q (T )
T
4
4

2
1  
1 
7
N f N c[


]




2
2  T 
6T 
180
c4 / c2  6 / 
2
2

Smooth change with a
very weak dependence
on the pion mass
Kurtosis as an excellent probe of deconfinement
Ch. Schmidt at QM’12
1 c4
B
R4 ,2 
9 c2

HRG factorization of pressure:
P (T ,  q )  F (T ) cosh(3  q / T )
Kurtosis
B

F. Karsch, Ch. Schmidt

consequently: c 4 / c 2  9 in HRG
2
In QGP, SB  6 / 
Kurtosis=Ratio of cumulants
 ( N q ) 
4
c4 / c2 
q
B
The R 4 , 2 measures the quark
content of particles carrying
baryon number
q
 ( N q ) 
2
 3  ( N q ) 
2
excellent probe of deconfinement
Ratios of cumulants at finite density: PQM +FRG
B. Friman, V. Skokov &K.R.
HRG
R4 ,2  c4 / c2
HRG
Deviations of the ratios of odd and even order cumulants from
their asymptotic, low T-value, c 4 / c 2  c 3 / c1  9 are increasing
with  / T and the cumulant order
Properties essential in HIC to discriminate the phase change by
measuring baryon number fluctuations !
Comparison of the Hadron Resonance Gas Model
with STAR data
B
(2)

Frithjof Karsch & K.R.

 coth(  B / T )
(1)
B

B
(4)

(2)
B
B
1
RHIC data follow
generic properties
expected within
HRG model for
different ratios of
the first four
moments of baryon
number fluctuations
(3)

(2)
B
 tanh(  B / T )
Error Estimation for Moments Analysis: Xiaofeng Luo arXiv:1109.0593v
STAR data:
Deviations from HRG
 Critical behavior?
 Conservation laws?
A. Bzdak, V. Koch, V.Skokov
ARXIV:1203.4529
Volume fluctuations?

B. Friman, V.Skokov &K.R.
B
S 
2


(2)
B
B
(4)
(3)
,  
2

B
(2)
Ratio of higher order cumulants
B. Friman, V. Skokov &K.R.
Deviations of the ratios from their asymptotic, low T-value,
are increasing with the order of the cumulant
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.
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.
13
Theoretical predictions and STAR data
Deviation from HRG if freeze-out curve close to
Phase Boundary/Cross over line
Lattice QCD
14
L. Chen, BNL workshop, CPOD 2011
STAR Data
Polyakov loop extended
Quark Meson Model
STAR Preliminary
Strong deviations of data from the HRG model (regular part of
QCD 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
 N
k


k
N P(N )
N

Probability quantified by all cumulants
P(N ) 
1
2
2
 dy exp [ i yN   ( iy ) ]
0
Cumulants generating function:  ( y )   V [ p (T , y   )  p (T ,  ) ] 

k
In statistical physics
P(N ) 

ZC (N )
Z GC
N
e
T
k y
k
Probability distribution of the net baryon number

For the net baryon number P(N) is
described as Skellam distribution
B
P(N )   
B

In HRG the means
B, B
N /2
I N (2 B B ) exp[  ( B  B )]
P(N) for net baryon number N entirely
given by measured mean number of
baryons B and antibaryons B
F (T f ,  f , V )
are functions of thermal parameters at
Chemical Freezeout
Influence of criticality on the shape of P(N)
In the first approximation the  B
quantifies the width of P(N)
P ( N )  exp(  N / 2V T  B )
2
3
Shrinking of the probability
distribution already expected due to
deconfinement properties of QCD
HRG
hotQCD Coll.
LGT
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  B in 3d Ising model
universality class
Influence of O(4) criticality on P(N)
Consider Landau model:
   bg 
1
4
| t |
1
2
t 
2
2

1

4
4
t  (T ,  ) 
Scaling properties:
Mean Field   0
O(4) scaling    0 .2 1
n  3  cn
sin g
 
Contribution of a singular part to P(N)
P(N ) 
Z ( N , T ,V )
N
e
K. Morita,
B. Friman, V. Skokov &K.R.
T
Z GC
Got numerically from:
For MF broadening of P(N)
For O(4) narrower P(N)
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  6 /  2 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


Influence of volume fluctuations on cumulants
Probability distribution at fixed V
B
P ( N ,V )   
B
V
N /2
I N (2V
n B n B ) exp[ V ( n B  n B )]
Moments of net charge
N 
 N
k


k
N P(N )
N  
Cumulants
central moment
1 
N  N N 
3 
1
2 
 N 
V
1
V
 ( N ) 
3
4 
1
 ( N ) 
2
V
1
V
2
(  ( N )   3  ( N )  )
4
2
Volume fluctuations and higher order cumulants
in PQM model in FRG approach
General expression for any P(N):
B n ,i
- Bell polynomials
c2   2   1 v2
2
V. Skokov, B. Friman & K.R.