Charge Fluctuations and transport coefficients near CEP

Charge fluctuations in the presence of spinodal instabilities and their scaling

Shear, and Bulk viscosities and its scaling near CEP

Krzysztof Redlich, Workshop on QCD-CP, INT

08

Use an effective chiral models and scaling theory to study:

Scaling properties:

1 1

2 | |TCPT T-

The strength of the singularity at TCP depends on direction in plane( , )BT m

1| |Cq T PT Tc --µ along 1st order line

any direction not parallel

along 2nd order line 11| |

2 Tq CPT Tc --µ

1/ 2| |Tq CPT Tc --µ

Going beyond the mean field:B.-J. Schaefer & J. Wambach

0.53( 0 0.78)| | mq TCPT Tc - ¹ =>µ -

Z(2) univer. class

1| |TCPT T --

11| |

2 TCPT T --

1/ 2| |TCPT T --

(2 ) 0| |ndcT T --

C.Sasaki, B. Friman & K.R.

FRG: Stokic, Friman & K.R

| |q TCPa b T T qc -= + -

See also Y. Hatta, T. Ikeda (0 ) ( 0.25 (2))( 0.3MF Za = - -

O(4) univer. class

Scaling properties: The strength of the singularity at TCP depends on direction in plane( , )BT m

1| |Cq T PT Tc --µ along 1st order line

any direction not parallel

along 2nd order line 11| |

2 Tq CPT Tc --µ

1/ 2| |Tq CPT Tc --µ

Going beyond the mean field:B.-J. Schaefer & J. Wambach

0.53( 0 0.78)| | mq TCPT Tc - ¹ =>µ -

Z(2) univer. class

C.Sasaki, B. Friman & K.R.

FRG: Stokic, Friman & K.R

| |q TCPa b T T qc -= + -

See also Y. Hatta, T. Ikeda (0 ) ( 0.25 (2))( 0.3MF Za = - -

O(4) univer. class

Quark and isovector fluctuations along the critical line

sensitive probes of TCP/CEPNon-monotonic behavior of the net

quark susceptibility as function of

in LGT or in HIC

NJL-model results: C. Sasaki, B. Friman, K.R.

[ ]cT MeV s:

( , )c cT m s

The nature of the 1st order chiral phase transition

instability of a system:

/ 0

0

/ 0

/

P

P

P

V

V

V¶ ¶¶

>¶

¶

<

=

¶ : stable : unstable : spinodal

A-B: supercooling (symmetric phase)B-C: non-equilibrium stateC-D: superheating (broken phase)

.symc -

brokenc -

Convex anomaly in thermodynamic pressure NJL model results

0qm = 0qm ¹

P- is differentiable at all values of m

P- has a cusp at the point where the dynamical quark mass vanishes

symmetric phase

broken phase

Dynamical Quark Mass and Instabilities

0qm = 0qm ¹

Maxwell constration : two degenerate minima in the thermodynamic potential

Spinodal Lines : no-unique solution of the gap equation for between spinodals

( , )M T fixedm =

m

equilibriumtransition

equilibriumtransition

Entropy per baryon and 1st order phase transition

Smooth evolution of entropy/baryon across the 1st order transition

0qm = 0qm ¹

Phase diagram and spinodals

2-order

TCP

1st

mixed phase - Maxwell construction

0qm = 0qm ¹

Fluctuations in the chiral limit across spinodals

Non-singular behavior of fluctuations

when crossing spinodal line from the side of symmetric phase:

=> directly related with a cusp structure of the pressure

TCP

Critical exponents at 1st order line and CEP

1/ 2 (0.53) 2 /3 (0.70 1/

82 1 0 1

)/ 2 1{ , {

q q

TCm st m

P C PtE

sg g= ¹= =

| |cq

c

gm mc

m--: 1/ 2 (0.53) 2 /3 (0.7

0 1/8

2 1 0 1)

/ 2 1{ , {q q

TCm st m

P C PtE

sg g= ¹= =with

1st

1st

TCP

TCP

B. Friman, C. Sasaki & K.R. , Phys.Rev.D77:034024,2008.

Experimental Evidence for 1st order transition

2

2

2TT T q

P qP

q

S s sT TV

T n nC mc c c

é ù¶æ ö= = - +ê úç ÷¶è ø ê úë û

Specific heat for constant pressure: Low energy nuclear collisions

Net-quark fluctuations on spinodals

at any spinodal points:

21

|.T

q

qnPV

V compresc¶

- = =¶

Singularity at CEP arethe remnant of that alongthe spinodals

CEP

spinodals

spinodals

C. Sasaki, B. Friman & K.R., Phys.Rev.Lett.99:232301,2007.

Minimum of shear viscosity may indicate the location of : L. Csernai, J. Kapusta & L. McLerran (06)

Divergent bulk viscosity at CEP from QCD trace anomaly argument: D. Kharzeev & Tuchin 07

F. Karsch, D. Kharzeev & Tuchin 08

cT

Find viscosities in quasi particle models with dynamically generated mass ,

under relaxation time approximation Derive scaling behavior of viscosities in O(4) and Z(2)

universality class using scaling theory

Transport coefficients near CEP

Our goal: C. Sasaki & K.R. hep-ph 0806.4745

( , ) ( , )bareM T m f Tm m= +

Transport Coefficient near phase transition

L. Csernay, J. Kapusta & L. McLerran 06

R. Lacey et al. 07: data for shear viscosity Harvey B. Meyer 08 LGT results

for bulk viscosity, SU(3) gauge th.

0.134(33) 1.65

0.102(56) 1.24c

c

for T T

for T Ts

h ==

=

Shear viscosity to entropy ratio in LGT

Transport coefficients from kinetic theory:

Energy momentum tensor

Assume small deviations from equilibrium with , consequently

3

3

1[ ]

(2 )

d pT p p f f

Emn m n

p= +ò

22 2 ( , )E p M T m= +ur

0f f fd = -1 exp( ) 1f E pu m- = - ±

urrm

3

3

1[ ]

(2 )

d pT p p f f

Emn m nd d d

p= +ò

To get use Boltzmann equation under relaxation time approximation

with the collision time obtained from the particle

density and cross section

fd

0

0

[ ]pf ff

v f C ft

Ep f fm

m

t

dt

-¶+ Ñ = - -

¶

Þ ¶ = -

r ur;

1f f reln vt s- = < >

derivation of transport coefficients:

from Boltzmann equation: energy conservation: charge conservation: stationary condition : thermodynamics relation:

fd00

0 0T¶ =0

0 0j¶ =/ 0P M¶ ¶ =

/ /T P T P Pe m m= ¶ ¶ - + ¶ ¶

use:

get: ij kij k ijT u WVd hd= - ¶ -

bulk viscosity shear viscosity

10f E p fm

md t -= - ¶

shear viscosity is not modified by thermal change of quasi-particle energy

( , )Th m

43

0 0 0 03 2[ (1 ) (1 )]

15 (2 )

g d p pf f f f

T Et t

ph = ± + ±ò

ur

the above coincides with Hosoya & Kajantie (1985),

however, the bulk viscosity

is modified by thermal change of quasi-particle energy through: and( , )TV m

/E T¶ ¶ /E m¶ ¶

( )

( )

3 2

0 0 0 03

2

2

0 0 0 0

[ (1 ) (1 )3 (2 )

3

(1 ) (1 ) ]

n

E E

g d p Mf f f f

T E

p P PE T

E n

M Pf f f f

E

E

n

T e

e

V t tp

me

t t

m m

= - ± + ±

æ öæ ö¶ ¶æ ö æ öç ÷´ - - - +ç ÷ ç ÷ç ÷ç ÷¶ ¶è ø è ø¶ ¶ ¶¶è øè ø

¶æ ö- ± - ± ç ÷¶è

¶ ¶

ø

òur

for the above coincides with Hosoya & Kajantie (1985)

For the above coincides with Arnold, Dogan & Moore (06)/ 0E m¶ ¶ =/ ( ) 0E T m¶ ¶ ¶ =

Near phase transition the bulk viscosity can be singular through derivatives terms:

21

2

E M

x E x

¶ ¶=

¶ ¶

2

xy

P

x yc ¶

=¶ ¶

with , ( , )x y T m=

1| ( )n T

V

Ps n

C mm mmm

c ce c

¶= -

¶

1| ( ( ) )TT T

V

PnT n sT s

n Ce m mmmm

c m c mcc

¶= + - -

¶

2

| TV V TT

sC T T

Tm

mm

cc

c

æ ö¶æ ö= = -ç ÷ç ÷ ç ÷¶è ø è øwhich all can diverge at the critical points !

critical behavior of bulk viscosity: consider a system where dynamical mass acts as an order

parameter

Mean Field Scaling from Ginzburg-Landau potential

=> 2nd order: a=0 and b>0 TCP at a=b=0

with and from gap equation

2 40( , , ) ( , ) ( , )MT a TM M hM Mb Tm m m=W = W + + -

( , ) | | | |c ca T T Tm a b m m= - + -

132 14 20 0 0|singular

TCPV

M M MM t

C T sa cb m

V× -æ ö¶ ¶

- ´ ´ »ç ÷¶ ¶è ø: : :

sin

2

2| 0nd

gular MV =:

Conclusions: under MF approximation there is no singularity of bulk viscosity at the TCP and 2nd order phase transition. actually : in the chiral limit the bulk viscosity vanishes at the transition point

O(4) scaling of bulk viscosity: use the free energy

that gives

consequently

2( , ) ( )S SF T t f t ha bdm - -= 2, | | / , /c c ct t A t T T T Tm m mº + = - =

Ejiri, Karsch & K.R. 06; Hatta & Ikeda 04

1 , , ,TT Vt t C t M ta a a bmmc c- - -: : : :

3sin 4 1

V

gular M M

C Tta bV + -¶

¶: :

for => 0.24 , 0.38a b= - = sin 0.28 0gular tV + ®:Conclusions: bulk viscosity is non-singular at O(4) critical point

Z(2) scaling of bulk viscosity: use the free energy

that gives

consequently

11/( , ) ( )S SF T h f h t

dbddm

+-=

, , , | | /t h t h c ct A t B x x x xmº + = -

Stephanov, Rajagopal & Shuryak 1998

/ / 1/, , , ,T TT Vh C h M hg bd g bd d

mm mc - -: : :sin 4 1

3/ /gular

V

M Mh

C Tg bd dV + -¶

¶: :

for => 1.25 , 0.31 , 5.2g b d= = =sin 0.54 0gular hV + ®:

Conclusions: bulk viscosity is non-singular at Z(2) critical

point within the relaxation time approximation

Karsch et. al. (08)

Beyond the relaxation time approximation

modes with long wave-length could stay in non-equil.

modified by dynamic critical exponents “z” Onuki, “Phase transition dynamics”, Cambridge Univ. Press (02)

For O(4) :

For Z(2) :

V

sin gular zt n aV - +:

sin gular zt aV*- +:

Singularity in bulk viscosity along O(4) critical line and at the TCP

due to dynamic critical exponents

Large change of

bulk viscosity due to contribution from

temperature derivative of dynamical quark mass, /M T¶ ¶

Bulk Viscosity across the phase transition in the NJL model Cross over region CEP and 1st order line

Bulk viscosity to entropy ratio

Cross over line CEP and 1st order line

Shear viscosity normalized by momentum cutoff

Cross over transition CEP and 1st order line

Shear viscosity per entropy Crossover CEP and 1st order line

Bulk and shear viscosity along the phase boundary

shear viscoisity bulk viscoisity

Conclusions

A non-monotonic change of the net-quark susceptibility probes the existence of CEP However in non-equilibrium: due to spinodal instabilitie the charge fluctuations diverge at 1st order critical line

=> Large fluctuations signals 1st order transition Under relaxation time approximation the bulk

viscosity is finite at CEP and O(4) line Þ Divergence of bulk viscosity controlled by the dynamical, rather then static critical exponents