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Probing QCD phase diagram with fluctuations
of conserved charges
Krzysztof Redlich University of Wroclaw
Modelling QCD phase diagram
Polyakov loop extended quark-meson
model: Including quantum and thermal

fluctuations: FRG-approach
Probing phase diagram with
fluctuations of net baryon number
theory & experiment (STAR data)
Essential role of higher order
moments and their Ratios
Based on recent work with:
B. Friman, V. Skokov; & F. Karsch
T
LHC

A-A collisions
fixed s
Quark-Gluon
Plasma
Chiral
symmetry
restored
Hadronic matter
Chiral symmetry
broken
x
B
1st principle calculations:
 , T   QCD :  perturbation theory
pQCD
 , T   QCD :
q  T
:
LGT
Sketch of effective chiral models coupled to Polyakov loop
 1/T
S

d  d 3 x[iq(     A  4 )q  V int (q, q)  q q  q  U ( L, L* )]
0
V
U ( L, L )  the Z (3) invariant Polyakov loop potential
*
V int (q, q)  the chiral 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, ...
Thermodynamics of PQM model under MF approximation

Fermion contribution to thermodynamic potential
 q q   d p(ln[1  3( e
3
 ( Eq   )/ T
Entanglement of deconfinement
and chiral symmetry
 2( Eq   )/T
e
)e
3( Eq   )/ T
] (q  q)
Suppresion of thermodynamics
due to „statistical confinement”
SB
limit
suppresion
  

Suppresion
Generic Phase diagram from effective chiral Lagrangians
mqq  00
m
(Pisarski-Wilczek)
O(4)/O(2) univ.; see LGT , Eijri et al 09
crossover
CP
Asakawa-Yazaki

2nd order, Z(3)
(Stephanov et al.)
1st order
  broken
Hatsuda et al.
Alford et al.
Shuryak et al.
Rajagopal et al.
Zhang et al, Kitazawa et al., Hatta, Ikeda;  B
Fukushima et al., Ratti et al., Sasaki et al.,
Blaschke et al., Hell et al., Wambach et al., ..

The existence and
position of CP and
transition is model and
parameter dependent
Introducing di-quarks
and their interactions
with quark condensate
results in CSC phase
and dependently on the
strength of interactions
to new CP’s
4
Phase diagram in chiral models with CP
2 (P / T 4 )
  
(  / T )2
fluctuations of net baryon-number
at any spinodal points:
P
1
V
|T 

V
  compres.
nq 2
spinodals
CP
spinodals
Singularity at CEP is
the remnant of that along
spinodals
C. Sasaki, B. Friman & K.R.
V. Koch et al., I. Mishustin et al., ….
Including quantum fluctuations: FRG approach
k-dependent
full propagator
start at classical action and include
quantum fluctuations successively by lowering k
FRG flow equation (C. Wetterich 93)
J. Berges, D. Litim, B. Friman, J. Pawlowski,
B. J. Schafer, J. Wambach, ….
Regulator function suppresses
particle propagation with
momentum lower than k
(T , V )  lim( k  (T / V ) k )
k 0
k
k  k Rk
FRG at work –O(4) scaling: L  L*  1 and   0

Near Tc critical properties obtained from the
singular part of the free energy density
T  Tc
F  Freg  FS

with
1/
FS (t , h)  b F (b t , b
 /
t
h)
Tc
h : external field
Phase transition encoded in the “equation of state”
   h1/ Fh ( z ) , z  th 1/ 
Fs
   

h

d
  | t | Fs ' (h | t |  )
Resulting in the well known scaling behavior of   
   {

B(t ) , h  0, t  0
Bh1/
, t  0, h  0
coexistence line
pseudo-critical point
FRG-Scaling of an order parameter in QM model
log( )

t

log( )
log(t )
log(h)
The order parameter shows scaling. From the slopes one gets


MF
0.5
3
FRG
0.401(1)
4.818(29)
LGT
0.3836(46) 4.851(22)
However we have neglected field-dependent
wave function renormal. Consequently   0
and   5 . The 3% difference can be
attributed to truncation of the Taylor expansion
at 3rd order when solving FRG flow equation:
see D. Litim analysis for O(4) field Lagrangian

The Phase Diagram and EQS
Mean-field approximation
Function Renormalization Group
CEP
CEP



Mesonic fluctuations shift the CEP to higher temperature
The transition is smoother
No focusing of isentropes (see E. Nakano et al. (09))
(c.f., C. Nonaka & M. Asakawa &. B. Muller)
Probing phase diagram with moments of net quark
number fluctuations
n
4

(
P
/
T
)
(n)

 Moments  q
probes of the chiral transition
n
(  / T )
4
2
2
( 2)
2
3
c

(

N
)


3

(

N
)

c2  q  ( Nq )  /VT
4
q
q
c2
SB
QM
QM PQM
PQM
model
model
FRG-results
PQM
model
QM
QM
model
PQM PQM
SB limit
MF-results

Smooth change of c2 (T ) and peak-like structure in c4 (T ) as in O(4)
 t1
c2  
 t 

for q  0 cusp  structure
 t
c4  
 t  (2 ) for q  0 diverges
for q  0 cusp  structure
for q  0 negligible
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

c4 / c2  6 /  2

Smooth change with a
rather weak dependence on the pion mass
Kurtosis as an excellent probe of deconfinement
S. Ejiri, F. Karsch & K.R.
R4,2
B
1 c4

9 c2

HRG factorization of pressure:
P (T , q )  F (T ) cosh(3q / T )
Kurtosis
B

F. Karsch, Ch. Schmidt

consequently: c4 / c2  9 in HRG
2
In QGP, SB  6 / 
Kurtosis=Ratio of cumulants
c4q / c2q 
B
The R4,2
measures the quark
content of particles carrying
baryon number
 ( N q )4 
 ( N q )2 
 3  ( N q )2 
excellent probe of deconfinement
Ratio of cumulants at finite density
R4,2  c4 / c2
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 !
QCD phase boundary & Heavy Ion Data

Particle yields and their
ratio, well described by
the Hadron Resonance
Gas

QCD phase boundary
appears near freezeout
line
QCD phase boundary & Heavy Ion Data

Excellent description
of LGT EQS by HRG
Karsch, Tawfik &K.R.
A. Majumder&B. Muller
LGT by Z. Fodor et al..
R. Hagedorn
Coparison of the Hadron Resonance Gas Model
with STAR data

LGT results for CP see also:
R. Gavai &S. Gupta
S. Ejiri et al.
Frithjof Karsch & K.R.
 (2)
 coth(B / T )
(1)

 (3)
 tanh(B / T )
(2)


RHIC data follow generic properties expected within HRG model
for different ratios of the first four moments of baryon number
fluctuations
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 !
Ratio of higher order cumulants at finite density
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 !
Lines of vanishing fluctuations for different moments


CP
CP
Broad temperature
range of negative
values of different
moments with n>4
which increases with
their orders.
Strong deviations from
Hadron Resonance
Gas result due to
collective effects
related with chiral
symmetry restoration
and deconfinement
Conclusions



The FRG method is very efficient to include
quantum and thermal fluctuations in thermodynamic
potential in QM and PQM model
The FRG provides correct scaling of physical
observables expected in the O(4) universality class
To observe fluctuations related with O(4) crossover in HIC measure higher order fluctuations,
N>6 relative to N=2 moments
Quark number fluctuations at finite density


Strong increase of fluctuations with baryon-chemical potential
In the chiral limit the c3 and c4 daverge at the O(4) critical line
at finite chemical potential