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description
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T
B
Hadronic matter
Quark-Gluon Plasma
Chiral symmetrybroken
Chiral symmetryrestored
LH
C
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
A-A collisions fixed s
x
1st principle calculations: perturbation theory pQCD LGT
, :QCDT , :QCDT
:q T
Renormalization Group and Quark number fluctuations near chiral phase transition
Krzysztof Redlich University of Wroclaw
Based on recent work with: B. Friman, V. Skokov; & F.
Karsch
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Modelling QCD phase diagram
Preserve chiral symmetry
with condensate as an order parameter
K. Fukushima (2004)C. Ratti & W. Weise (07)
( ) ( )L f R fSU N SU N
Preserve center symmetry with Polyakov loop
( )cZ N
as an order parameter
Polyakov loop dynamics Confinement
Nambu & Jona-Lasinio
SpontaneousChiral symmetry
Breaking
Synthesis
40
1( exp[ ( , )])
c
L Tr P i d xAN
PNJL Model
R. Pisarski (2000)
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Sketch of effective chiral models coupled to Polyakov loop
coupling with meson fileds PQM chiral model
Nambu-Jona-Lasinio model PNJL chiral model
1/3
0
i *4
nt ( , ),[ ](( ) )T
q
V
S d d x iq V U Lq q q LA q q
the chiral invariant quark interactions described through:
K. Fukushima; C. Ratti & W. Weise; B. Friman , C. Sasaki ., ….
B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman, V. Skokov, ...
int ( , )V q q
*( , )U L L the invariant Polyakov loop potential (3)Z
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Polyakov loop potential fixed from pure glue Lattice Thermodynamics
First order deconfinement phase transition at
fixed to reproduce pure SU(3) lattice results
* * *3 34 22
*3 4/ ( ) ( ) ( ) ( ) ( ) ( ),GP T U b T b T b T
0 270T MeV
( )kb T
C. Ratti & W. Weise 07
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Thermodynamics of PQM model under MF approximation
Fermion contribution to thermodynamic potential
Entanglement of deconfinement and chiral symmetry
Suppresion of thermodynamicsdue to „statistical confinement”
( )/ 2(3 3( )/)/(ln[1 3( ) ]q q qE T E T
E Tp e e ed ( )q q
Suppresion
suppresion
SBlimit
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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
6
Generic Phase diagram from effective chiral Lagrangians
1st order
Zhang et al, Kitazawa et al., Hatta, Ikeda;Fukushima et al., Ratti et al., Sasaki et al.,Blaschke et al., Hell et al., Roessner et al., ..
0qm (Pisarski-Wilczek) O(4)/O(2) univ.; see LGT , Eijri et al 09
0qm
crossover
Asakawa-Yazaki
CP2nd order, Z(2) (Stephanov et al.)
Hatsuda et al.
Alford et al.Shuryak et al.Rajagopal et al.
B
broken
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Probing CEP with charge fluctuations
Net quark-number fluctuations
where
21 1 1
36 4 6Q q Iq I
P
The CP ( ) and TCP ( ) are the only points where in an equilibrium medium the diverge ( M. Stephanov et al.)
, 0u dm , 0u dm A non-monotonic behavior of charge fluctuations is an excellent probe of the CP
( , )q Q
CP
2 32 ( )q qqN VT ) 4( ( / ) / ( / )n n
q qn P T T
( , )q Q
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Phase diagram in chiral models with CP
at any spinodal points:
21
|.T
q
qnPV
V compres
Singularity at CEP isthe remnant of that along spinodals
CP
spinodals
spinodals
C. Sasaki, B. Friman & K.R.V. Koch et al., I. Mishustin et al., ….
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Including quantum fluctuations: FRG approach
FRG flow equation (C. Wetterich 93)J. Berges, D. Litim, B. Friman, J. Pawlowski, B. J. Schafer, J. Wambach, ….
start at classical action and includequantum fluctuations successively by lowering k
Regulator function suppressesparticle propagation with momentum lower than k
0lim(( ) ), ( /) k kk
T T VV
k k kk R
k-dependentfull propagator
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Renormalization Group equations in PQM model
Quark densities modified by the background gluon fields
Flow equation for the thermodynamic potential density in the PQM model with Quarks Coupled to the Background Gluonic Fields
4
* *2
23 11 2 ( ) 1 2 ( ) 1 ( , ) ( , )
12q
qqk B qk B
dkn E n E L n
E E ELn L L
**
*
1 2 exp( ( )) exp(2 ( ))( , )
1 3 exp(2 ( )) 3 exp(2 ( )) exp(3 ( ))q q
q q q
L LL L
L L
E En
E E E
with fixed such that to minimise quantum potential,
* *0
*( : , ) ( : , ) (, ), ,kL L LT T UL L L
highly non-linear equation due to
*( , )L L
V. Skokov, B. Friman &K.R.
2( ) ( , / )nkE k E k
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FRG at work –O(4) scaling:
Near critical properties obtained from the singular part of the free energy density
cT
Resulting in the well known scaling behavior of
1/ /( , ) ( , )dSF b F bh ht t b
c
c
tT T
T
external field :h Phase transition encoded in the “equation of state”
sF
h
1/ 1/( ) ,hh F z z th
'| | ( | | )st F h t
1/
( ) , 0, 0{
, 0, 0
t tB
ht
h
Bh
coexistence line
pseudo-critical point
* 1 0L L and
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FRG-Scaling of an order parameter in QM model
log( ) log( )
log( )t log( )h
t
The order parameter shows scaling. From the slopes one gets
0.401(1) 4.818(2
0.5 3
0.3836(4
9
6) 4.851( 2)
)
2
MF
LG
R
T
F G
However we have neglected field-dependent wave function renormal. Consequently and . 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
0 5
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Effective critical exponents Approaching from the side of
the symmetric phase, t >0, with small but finite h : from Widom-
Griffiths form of the equation of state
For and
1/ 1/ 1/( ) ,cB h f x x t
0t 0h 0
( )x f x x
t h , thus
( ) , 0, 0{
, 0, 0c
t th
h tht
B
B
Define:
0log( ): {
0log( )
tdR
td t
R
1.53 1MF
0.4 cT
1.48LGT
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Two types of susceptibility related with order parameter
1. longitudinal
2. transverse
Fluctuations of the order parameter
/l h
/t h max ( )t h
max ( )h
Scaling properties
at t=0 and
1/ 1B h
0h
1/m
(ax
)t h
m xx ama( )t t
1/ ( ) 0.49 1.6
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The order parameter in PQM model in FRG approach
For a physical pion mass, model has crossover transition Essential modification due to coupling to Polyakov loop The quantum fluctuations makes transition smoother
Mean Field dynamics FRG results
QM
PQM
<L> QM
PQM
<L>
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Fluctuations of order parameters Mean Field dynamics FRG results
Deconfinement and chiral transition approximately same Within FRG broadening of fluctuations and their
strength: essential modifications compared with MF
d / dTLd / dTL
d /dT
d /dT
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The Phase Diagram and EQS
Mean-field approximation Function Renormalization Group
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)
CEP
CEP
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Probing phase diagram with moments of net quark number fluctuations
Smooth change of and peak-like structure in as in O(4)
Moments probes of the chiral transition
22
( ) 2 3( ) /q qc N VT
MF-results
PQM
PQM
QMFRG-results
4( ) ( / )
( / )
nn
q n
P T
T
QM
PQM
4 2 24 ( ) 3 ( )q qc N N
2 ( )c T 4 ( )c T
2c
1 0
0
q
q
t for negligible
t for cusp structure
4c
(2 ) 0
0q
q
t for cusp structure
t for diverges
2c
QMmodel PQM
model
PQMmodel
QMmodel
SB limit
SB
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Kurtosis of net quark number density in PQM modelV. Skokov, B. Friman &K.R.
For
the assymptotic value
2
24 2
( ) 2 3 3cosh
3qq f q q
c
qP T N m
KT T T
m
N T
due to „confinement” properties
Smooth change with a rather weak dependen- ce 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
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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
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Quark number fluctuations at finite density Strong increase of fluctuations with baryon-chemical potential
In the chiral limit the and daverge at the O(4) critical line at finite chemical potential
3c 4c
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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
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QCD phase boundary & Heavy Ion Data
QCD phase boundary appears near freezeout line
Particle yields and their ratio, well described by the Hadron Resonance Gas
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QCD phase boundary & Heavy Ion Data
Excellent description of LGT EQS by HRG
A. Majumder&B. Muller
LGT by Z. Fodor et al..
R. Hagedorn
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QCD phase boundary & Heavy Ion Data
Is there a memory that the system has passed through the region of the QCD phase transition ?
Consider the net-quark number fluctuations and their higher moments
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STAR DATA ON MOMENTS of FLUCTUATIONS
Mean
B p p
pB p NM N
Variance
B B BN N M
22 ( )BB N Skewness
3 3( ) /BB BS N
4 4( ) / 3B B BN
Kurtosis
Phys. Rev. Lett. 105, 022302 (2010)
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Properties of fluctuations in HRG
Calculate generalized susceptibilities:from Hadron Resonance Gas (HRG) partition function:
then, and
resulting in:
(2)
(1)coth( / )B T
(3)
(2)tanh( / )B T
Compare this HRG model predictions with STAR data at RHIC:
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Comparison of the Hadron Resonance Gas Model with STAR data
Frithjof Karsch & K.R. K.R.
RHIC data follow generic properties expected within HRG model for different ratios of the first four moments of baryon number fluctuations
Can we also quantify the energy dependence of each moment separately using thermal parameters along the chemical freezeout curve?
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Mean, variance, skewness and kurtosis obtained by STAR and rescaled HRG
STAR Au-Au
STAR Au-Au
these data, due to restricted phase space:
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
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LGT and phenomenological HRG model C. Allton et al.,
Smooth change of and peak
in at expected from O(4) universality argument and HRG
, ( )q I T
4qd cT
4
4 4 4
1
( / )q P
dT T
S. Ejiri, F. Karsch & K.R.
( )q I For fluctuations as
expected in the Hadron Resonance Gas
cT T
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To see deviations from HRG results due to deconfinement and chiral transtion one needs to measure higher order fluctuations:
Lattice QCD results model calculations
:
C. Schmidt
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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
The quantum fluctuations modified the mean field results leading to smearing of the chiral cross over transition
The RHIC data on the first four moments of net- proton fluctuations consitent with expectations from HRG: particle indeed produced from thermal source
To observe large fluctuations related with O(4) cross-over, measure higher order fluctuations, N>6