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

qq

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