Quantum Fluctuation and scaling in the Polyakov loop -Quark-Meson model Chiral models: predictions...
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Transcript of Quantum Fluctuation and scaling in the Polyakov loop -Quark-Meson model Chiral models: predictions...
Quantum Fluctuation and scaling in the Polyakov loop -Quark-Meson model
Chiral models: predictions under mean field dynamics Role of Quantum and Thermal fluctuations: Functional Renormalization Group (FRG) Approach FRG in QM model at work: O(4) scaling of an order parameter FRG in PQM model Fluctuations of net quark number density beyond MF Work done with: B. Friman, E. Nakano, C. Sasaki, V. Skokov, B. Stokic & B.-J. Schaefer Krzysztof Redlich, University of Wroclaw & CERN
Effective QCD-like models
2 2 25
2
( )
( )3
( ) [( ) ( ) ] ( )
( ) ( [ ], [ ], )q
SS VP
V
JL
I
N
V
L q i m q qq qi q q q
q q q q q
G
UG A A
G
T
D
q
Polyakov loop
2 25
2
40 0
1 1( [ ]) ( ) ( )
2 2
( [ ], [ ], ) ( , )
1( exp[ ( , )])
P
c
QML q i i q
U A A T U
Tr P i d A xN
D g
D i A
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K. Fukushima, C. Ratti & W. Weise, B. Friman & C. Sasaki , ., .....
B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman 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
3
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 then (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
Alford et al.Shuryak et al.Rajagopal et al.
B
broken Sasaki et al.
Inverse compressibility and 1st order transtion
at any spinodal points:
21
|.T
q
qnPV
V compres
Singularity at CEP arethe remnant of that alongthe spinodals
CEP
spinodals
spinodals
C. Sasaki, B. Friman & K.R., Phys.Rev.Lett.99:232301,2007.
Including quantum fluctuations: FRG approach
FRG flow equation (C. Wetterich 93)
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
FRG for quark-meson model
•LO derivative expansion (J. Berges, D. Jungnicket, C. Wetterich) (η small) •Optimized regulators (D. Litim, J.P. Blaizot et al., B. Stokic, V. Skokov et al.)
•Thermodynamic potential: B.J. Schaefer, J. Wambach, B. Friman et al.
,
11 2 1 2( , : ) [3 4 ]q a q
k k o k f cq
n nn nT N N
E E E
Non-linearity through self-consistent determination of disp. rel. 2 2
i iE k M with ' ' ''2 2 2 2
0, 0,2 2k k kk q kM M M g
and 2k k h 0,
'/ |
kk k with
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
FRG-Scaling of an order parameter in QM model
log( ) log( )
log( )t log( )h
t
The order parameter shows scaling. From the one slope 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 3th order when solving FRG flow equation:
see D. Litim analysis for O(4) field Lagrangian
0 5
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
Two type of susceptibility related with order parameter
1. longitudinal
2. transverse
Fluctuations & susceptibilities
/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
Extracting delta from chiral susceptibilities
Within the scaling region and at t=0 the ratio is
independent on h
FRG in QM model consitent
with expected O(4) scaling
( , ) /t h
1, 0
lim ( , ) { 0/
0 0
1
t
t h t
t
0h
1/ 4.9
Lines of constant s/n
Grid
Taylor
Two indep.calculations
FRG at work –global observables
FRG - E. Nakano et al. MF results: see also K. Fukushima
Focusing of Isentrops and their signature
Idea: ratio sensitive to μB
Isentropic trajectories dependent on EoS
In Equilibrium:momentum-dep.of ratio reflects history
Caveats:•Critical slowing down
•Focusing non-unsiversal!!
Asakawa, Bass, Müller, Nonaka
large baryons emitted early, small latertq tq/p p
/p p
Renormalization Group equations in PQM model
Quark densities modify 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
The FRG flow equation has to be solved together with
*( : , 0 a, ) ndTL LL
*
*( : , ) 0 wh, ereTL L
L
* * *0( , : , ) ( , : , ) ( , )kL L T L L T U L L
Fluctuations of an order parameter Mean Field dynamics FRG results
Deconfinement and chiral transition approximately same Within FRG broadening of fluctuations and their
strength: essential modifications compare with MF
Thermodynamics of PQM model in the presence of mesonic fluctuations within FRG approach
Quantitative modification of the phase diagram due to quantum and thermal fluctuations.
Shift of CP to the lower density and higher temperature
Net quark number density fluctuations
Coupling to Polykov loops suppresses fluctuations in broken phase Large influence of quantum fluctuations. Problem with cut-off effects at high T in FRG calulations. !
Probes of chiral trans.22
42
0
( / )| , ( )
( / )q
n q
P Tc N
Tc
T
MF-results
FRG results
PQM
PQM
QM
QM
4th order quark number density fluctuations
Peak structure might appear due to chiral dynamics. In GL-theory
4 2 24 ( ) 3 ( )q qc N N
PQM
QM
2 24( , ) ( , ) ( ),reg ca T b c a T A T T B
2 2 4 4( ) , ( )reg regcc c
c
T Tc c C T T c c D T T
T
Kink-like structure Dicontinuity
MF results
FRG results
QM modelQM model
PQM model
PQM model
Kurtosis as 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
F. Karsch, Ch. Schmidt et al., S. Ejiri et al.
Kur
tosi
s
44,2
2
1
9B c
Rc
Observed quark mass dependence
of kurtosis, remnant of chiral O(4)
dynamics?
Kurtosis of net quark number density in PQM model
Strong dependence on pion mass, remnant of O(4) dynamics
MF results FRG resultsV. Skokov, B. Friman &K.R.
Smooth change with a rather weak dependen- ce on the pion mass
Conclusions The FRG method is very efficient to include
quantum and thermal fluctuations in thermodynamic potential in QM and PQM models
The FRG provide correct scaling of physical obesrvables 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 calculations indicate the remnant of the O(4) dynamics at finite pion mass , however much weaker than that expected in the mean field approach
Experimental Evidence for 1st order transition
2
2
2TT T q
P qP
q
S s sT TV
T n nC
Specific heat for constant pressure: Low energy nuclear collisions
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 smother
Mean Field dynamics FRG results
QM
PQM
<L> QM
PQM
<L>
Two independent methods employed Grid method – exact solution Taylor expansion around minimum:
Solving the flow equation
Flow eqns. for coefficients (truncated at N=3):
Followsminimum
0( ), ,) (kT T
Removing cut-off dependence in FRG
Matching of flow equations k 0k ( , : )PQM
k T ( , )QGPk T
2 * *3
22( 1) 1 2 ( ) 4 1 ( , ) ( , )
12QGP
k k c B c f q qL L
kN n k N n L LnN
We integrate the flow equation bellow from
and switch at to discussed
previously
to k
k ( , : )PQMk T