Charge fluctuations in heavy ion collisions and
QCD phase diagram
K. Redlich, Uni Wroclaw & EMMI

HIC
crossover

CP or Triple Point
Qarkyonic
Matter

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
Particle yields and their ratio, as well as LGT results at
T  Tc are well described by the Hadron Resonance Gas
Partition Function .
Budapest-Wuppertal LGT data

Tc 

O(4)
universality

1

0.006
6
A. Andronic et al., Nucl.Phys.A837:65-86,2010
.  T 
Tc 0
2
HRG model
Thermal and chemical equilibrium: H. Barz, B. Friman, J Knoll & H.Schultz
P. Braun-Munzinger, J. Stachel, Ch. Wetterich
C. Greiner & J. Noronha-Hostler, ……..
O(4) scaling and critical behavior
Near Tc critical properties obtained from the
singular part of the free energy density

F  Freg  FS

T  Tc
  
with t 
Tc
 Tc 

d
with
2
1/
FS (t, h)  b F (b t, b
Phase transition encoded in
the “equation of state”
Fs
   
 pseudo-critical line
h
 
1/ 
 Fh ( z ) , z  th
1/ 
h
 /
 
h1/
F. Karsch et al
h)
O(4) scaling of net-baryon number
fluctuations

The fluctuations are quantified by susceptibilities
B

(n)
n (P / T 4 )

: cn
n
( B / T )
P  Preg  PSingular
From free energy and scaling function one gets
2 n/2
   c h
( n)
B
( n)
r
2 n
    c h
( n)
B

with
( n)
r
f
( n/2)

( n)

( z) for   0 and n even
f ( z)
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
Effective chiral models and their non-perturbative
thermodynamics: Renormalisation Group Approach
 1/T
S

0
d  d 3 x[iq(     A  4 )q  V int (q, q)  q q  q  U ( L, L* )]
V
U ( L, L )  the Z (3) invariant Polyakov loop potential
*
V int (q, q)  the SU(2)xSU(2)   invariant quark interactions described
through:

Nambu-Jona-Lasinio model
PNJL chiral model
K. Fukushima; C. Ratti & W. Weise; B. Friman , C. Sasaki ., ….
coupling with meson fileds
PQM chiral model
B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman, V. Skokov, ...
 FRG thermodynamics of PQM model:

B. Friman, V. Skokov, B. Stokic & K.R.
Kurtosis of net quark number density in PQM model
V. Skokov, B. Friman &K.R.

For T  Tc
the assymptotic value
due to „confinement” properties
Pq q (T )
T4
9
c4 / c2
2 N f  3mq 
3q
 3mq 

 K2 
 cosh
2 
Nc  T 
T
 T 
2
c4 / c2  9

For T  Tc
Pq q (T )
T
4

1    1    7 2
N f Nc [ 2      
]
2  T  6  T  180
c4 / c2  6 /  2
4
2

Smooth change with a
very weak dependence
on the pion mass
Ratio of cumulants at finite density
R4,2  c4 / c2
HRG
HRG
Deviations of the ratios of odd and even order cumulants from
their asymptotic, low T-value, c4 / c2  c3 / 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

Frithjof Karsch & K.R.
 B(2)
 coth( B / T )
(1)
B
 RHIC data follow
generic properties
expected within
HRG model for
(4)
B
different ratios of

1
 B(2)
the first four
moments of baryon
number fluctuations
 B(3)
 tanh( B / T )
(2)
B
Error Estimation for Moments Analysis: Xiaofeng Luo arXiv:1109.0593v
Ratio of higher order cumulants
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 !
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
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.
10
Theoretical predictions and STAR data
Deviation from HRG if freeze-out curve close to
Phase Boundary/Cross over line
Lattice QCD
11
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!!??
Moments obtained from probability
distributions

Moments obtained from probability
distribution
 N k   N k P( N )
N

Probability quantified by all cumulants
2
1
P( N ) 
dy exp[iyN   (iy)]

2 0
k

(
y
)


V
[
p
(
T
,
y


)

p
(
T
,

)
]


y
Cumulants generating function:
k k
 In statistical physics
N
ZC ( N ) T
P( N ) 
e
ZGC
Probability distribution of the net baryon number

For the net baryon number P(N) is
described as Skellam distribution
B
P( N )   
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
In HRG the means B, B F (Tf ,  f ,V ) are functions of thermal parameters at
Chemical Freezeout
Comparing HRG Model with Preliminary
STAR data: efficiency uncorrected
Data presented at QM’11
Data described by Skellam distribution: No sign for criticality
Observed schrinking of the probability
distribution already expected due to
deconfinement properties of QCD
In the first approximation the  B
quantifies the width of P(N)
P( N )  exp( N 2 / 2VT 3 B )
HRG
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  B in 3d Ising model universality class
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  6 /  2 from the HRG
value as expected at the O(4) pseudo-critical line
Deviation of P(N) from HRG seems to
set in at
s  39GeV