Fluctuations & Correlations of conserved charges in PNJL...

Fluctuations & Correlations of conserved charges in

PNJL model

Anirban Lahiri

Bose Institute. Kolkata.

Anirban Lahiri (Bose Institute. Kolkata.) Fluctuations & Correlations of conserved charges in PNJL model 1 / 31

Fluctuations : Some introductary remarks.

1 Fluctuations : Some introductary remarks.

2 PNJL model

Motivation

Our modification

Taylor expansion of pressure

3 Results and Discussion

4 ConclusionAnirban Lahiri (Bose Institute. Kolkata.) Fluctuations & Correlations of conserved charges in PNJL model 2 / 31


Fluctuations and correlations are important characteristics of anyphysical system. They provide essential information about theeffective degrees of freedom and their possible quasi-particle nature.




Fluctuations are closely related to phase transitions.





The most efficient way to address fluctuations of a system created ina heavy-ion collision is via the study of event-by-event fluctuations.





The most efficient way to address fluctuations of a system created ina heavy-ion collision is via the study of event-by-event fluctuations.

In addition, the study of fluctuations may reveal information beyondits thermodynamic properties.



∆Ykick∆Ykick

∆Ycorr

∆Yaccept

∆Ytotal

dN/dY

Y

Charge fluctuations will be able to tell us about the properties of the earlystage of the system, the QGP, if the following criteria are met:

∆Yaccept ≫ ∆Ycorr and ∆Ytotal ≫ ∆Yaccept ≫ ∆Ykick

V. Koch, arXiv : 0810.2520


PNJL model Introduction

Motivation behind PNJL Model

Nambu-Jona-Lasinio (NJL) model : Originally proposed for studyinghadronic d.o.f. Later extended to quark d.o.f.Reproduces chiral symmetry breaking of QCD suscessfully through anon-vanishing chiral condensate.





Polyakov loop model : Originally proposed for pure gauge system.Reproduces confinement-deconfinement transition of QCD.





Polyakov loop model : Originally proposed for pure gauge system.Reproduces confinement-deconfinement transition of QCD.

Polyakov loop-Nambu-Jona-Lasinio (PNJL) model tied together thesetwo aspects of QCD.


PNJL model Our modificartion

h

h h h

hh

V V

VV

V

V

MF

MF

(a) (b) (c)

(d) (e) (f)

t <1

t >1

Here τ = NcGΛ2

2π2 > 1 in order to have chiral symmetry broken.A.A. Osipov et. al. Annals of Physics 322 (2007) 2021.



Thermodynamic Potential I

Ω = U ′[Φ, Φ,T ] + 2gS∑

f=u,d ,s

σ2f −gD

2σuσdσs+3

g1

2(

∑

f=u,d ,s

σf2)2

+3g2∑

f=u,d ,s

σ4f − 6∑

f=u,d ,s

∫ Λ

0

d3p

(2π)3EfΘ(Λ− |~p|)

− 2T∑

f=u,d ,s

∫

∞

0

d3p

(2π)3ln[

1 + 3Φe−(Ef −µf )

T + 3Φe−2(Ef −µf )

T + e−3(Ef −µf )

T

]

− 2T∑

f=u,d ,s

∫

∞

0

d3p

(2π)3ln[

1 + 3Φe−(Ef +µf )

T +Φe−2(Ef +µf )

T + e−3(Ef +µf )

T

]



Thermodynamic Potential II

where, σf = 〈ψf ψf 〉 Ef =√

p2 +M2f with,

Mf = mf − 2gSσf +gD

2σf+1σf+2−2g1σf (σ

2u + σ2d + σ2s )− 4g2σ

3f

A. Bhattacharyya et. al., Phys. Rev. D 82, 014021 (2010).

For the Polyakov loop part we have,

U ′(Φ, Φ,T )

T 4=

U(Φ, Φ,T )

T 4−κ ln[J(Φ, Φ)]

S. K. Ghosh et. al. Phys. Rev. D 77, 094024 (2008).

U(Φ, Φ,T )

T 4= −

b2(T )

2ΦΦ−

b3

6(Φ3 + Φ3) +

b4

4(ΦΦ)

2

and,J[Φ, Φ] = (27/24π2)(1− 6ΦΦ + 4(Φ3 + Φ3)− 3(ΦΦ)

2)

J(Φ, Φ) =⇒ VdM determinant.Anirban Lahiri (Bose Institute. Kolkata.) Fluctuations & Correlations of conserved charges in PNJL model 7 / 31

PNJL model Taylor Expansion of Pressure

P(T , µq, µQ , µS) = −Ω(T , µq, µQ , µS),

p(T , µq, µQ , µS)

T 4=

∑

n=i+j+k

cq,Q,Si ,j ,k (T )(

µqT

)i (µQT

)j(µST

)k

where,

cq,Q,Si ,j ,k (T ) =

1

i !j!k!

∂ i

∂(µq

T)i

∂j

∂(µQ

T)j∂k(P/T 4)

∂(µS

T)k

∣

∣

∣

µq,Q,S=0

µu = µq +2

3µQ , µd = µq −

1

3µQ , µs = µq −

1

3µQ − µS


PNJL model Taylor Expansion of Pressure

For diagonal Taylor coefficients we have used,

cXn =1

n!

∂n(

P/T 4)

∂(

µX

T

)n ; n = i + j

For off-diagonal Taylor coefficients we have used,

cX ,Yi ,j =

1

i !j!

∂ i+j(

P/T 4)

∂(

µX

T

)i∂(

µY

T

)j

Diagonal and off-diagonal susceptibilities are respectively defined as,

χXY =∂2(P/T 4)

∂(µX/T )∂(µY /T )χXX =

∂2(P/T 4)

∂(µX/T )2


Results and Discussion

Pressure consists of two parts; one regular part and one non-analyticpart.

P(T , µu, µd) = Pr (T , µu, µd) + Ps(t, µu, µd)

with t = (T − TC )/TC and µu,d = µu,d/T .

t ≡ t + Aµ2q + Bµ2I

From universal scaling behaviour;

Ps(t, µu, µd) ∼ t2−α

Then second and forth cumulant get contribution like;

(∂2Ps/∂µ2X ) ∼ t1−α + regular and (∂4Ps/∂µ

4X ) ∼ t−α + regular

S. Ejiri et. al. Phys. Lett. B 633 (2006) 275.



Taylor Coefficents for µq.....

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5

c 2q

T/Tc

Model AModel B

lattice data 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45 0.5

0 0.5 1 1.5 2 2.5

c 4q

T/Tc

Model AModel B

lattice data

-0.5-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3 0.4

0 0.5 1 1.5 2 2.5

c 6q

T/Tc

Model AModel B

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.5 1 1.5 2 2.5c 8

q

T/Tc

Model AModel B

Lattice data taken from M. Cheng et. al. Phys. Rev. D 79, 074505 (2009).



Taylor Coefficents for µQ .....

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.5 1 1.5 2 2.5

c 2Q

T/Tc

Model AModel B

lattice data 0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.5 1 1.5 2 2.5

c 4Q

T/Tc

Model AModel B

lattice data

-0.006-0.005-0.004-0.003-0.002-0.001

0 0.001 0.002 0.003 0.004 0.005

0 0.5 1 1.5 2 2.5

c 6Q

T/Tc

Model AModel B

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0 0.5 1 1.5 2 2.5c 8

Q

T/Tc

Model AModel B

Lattice data taken from M. Cheng et. al. Phys. Rev. D 79, 074505 (2009).



Taylor Coefficents for µI .....

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0 0.5 1 1.5 2 2.5

c 2I

T/Tc

Model AModel B

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2 2.5

c 4I

T/Tc

Model AModel B

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1 1.5 2 2.5

c 6I

T/Tc

Model AModel B

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 0.5 1 1.5 2 2.5

c 8I

T/Tc

Model AModel B



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5

c 2X

T/Tc

Model A

X=qX=I

X=SX=Q

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.5 1 1.5 2 2.5

c 2X

T/Tc

Model B

X=qX=I

X=SX=Q

All diagonal Taylor coefficients show charecteristic crossover ⇒ QCDphase transition liberates quarks.S. Gottlieb et al., Phys. Rev. Lett. 59, 2247 (1987); R. V. Gavai et al., Phys. Rev. D 40, 2743 (1989).



Kurtosis I

0

5

10

15

20

25

0.5 1 1.5 2 2.5

χ 4q /χ

2q

T/Tc

Model AModel B

lattice data

Kurtosis is a sensetive probe of deconfinement.At low T kurtosis Rq = (NcB)

2 = 9 and at high T it becomes unity inclassical consideration and if corrected by quantum statistics Rq = (6/π2).



Kurtosis II

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2 2.2

0.5 1 1.5 2 2.5

χ 4Q

/χ2Q

T/Tc

Model AModel B

lattice data

0.5

1

1.5

2

2.5

3

0.5 1 1.5 2 2.5

χ 4S/χ

2S

T/Tc

Model AModel B

lattice data

At low T, RQ is dominated by charge fluctuations in pion sector resultingRQ = 1. At high T, RQ = 2/π2 which is its SB limit.Kurtosis for strange sector shows a peak at Tc . Model shows enhancedfluctuations after Tc and then converges to its SB limit.



Wroblewski parameter

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.5 1 1.5 2 2.5

λ s

T/TC

PNJL(BEP)PNJL(UEP)Lattice data

λs =2〈ss〉

〈uu + dd〉≈

χs2

χu2 + χd

2

=χs2

χu2

λ8qs (Tc) ≈ 0.37 and λ6qs (Tc) ≈ 0.48 with experimental boundλRHICs (Tc) ≈ 0.47± 0.04.



B-S Correlation I

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.5 1 1.5 2 2.5

-cB

S11

T/TC

PNJL (BEP)PNJL (UEP)

lattice data 0

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0.5 1 1.5 2 2.5

-cB

S11

/cB

2

T/TC


lattice data

In the high T phase B and S is highly correlated resulting high value ofcBS11 .Lowest lying baryons do not carry strangeness ⇒ Ratio of the right pannelgoes to zero at low temperature.



B-S Correlation II

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.5 1 1.5 2 2.5

-cB

S11

/cS

2

T/TC


lattice data

CBS = −3〈BS〉 − 〈B〉〈S〉

〈S2〉 − 〈S〉2= −3

χBS

χSS

= −3

2

cBS11

cS2



B-S Correlation III

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.5 1 1.5 2 2.5

cBS

22

T/TC


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.5 1 1.5 2 2.5

-cB

S13

T/TC


0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

0.5 1 1.5 2 2.5

-cB

S31

T/TC




Q-S Correlation I

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.5 1 1.5 2 2.5

cQS

11

T/TC


lattice data 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 1 1.5 2 2.5

cQS

11/c

Q2

T/TC


lattice data

At high T, Q and S are related by strange quasiparticle which leads to highvalue of cQS

11 .Lowest lying charged particle do not carry strangeness ⇒ Ratio in theright pannel vanishes at low T.



Q-S Correlation II

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0.5 1 1.5 2 2.5

cQS

11/c

S2

T/TC


lattice data

CQS = −3〈QS〉 − 〈Q〉〈S〉

〈S2〉 − 〈S〉2= 3

χQS

χSS

=3

2

cQS11

cS2



Q-S Correlation III

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.5 1 1.5 2 2.5

cQS

22

T/TC


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.5 1 1.5 2 2.5

cQS

13

T/TC


0 0.002 0.004 0.006 0.008 0.01

0.012 0.014 0.016 0.018 0.02

0.5 1 1.5 2 2.5

cQS

31

T/TC




B-Q Correlation I

0 0.005

0.01 0.015

0.02 0.025

0.03 0.035

0.04 0.045

0.05

0.5 1 1.5 2 2.5

cBQ

11

T/TC


lattice data

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.5 1 1.5 2 2.5

cBQ

11/c

B2

T/TC


lattice data

0

0.05

0.1

0.15

0.2

0.25

0.5 1 1.5 2 2.5

cBQ

11/c

Q2

T/TC


lattice data



B-Q Correlation II

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.5 1 1.5 2 2.5

cBQ

22

T/TC


0

0.005

0.01

0.015

0.02

0.025

0.5 1 1.5 2 2.5

cBQ

13

T/TC


-0.002-0.001

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007

0.5 1 1.5 2 2.5

cBQ

31

T/TC




Strangeness Carriers I

In QGP phase Bs = −13Ss and Qs =

13Ss

No such direct relation for hadron gas.

CBS = −3χBS

χSS

= 1 +χus + χds

χss= 1 +

cus11cs2

CQS = 3χQS

χSS

= 1−2χus − χds

χss= 1−

1

2

cus11cs2

V. Koch et. al., PRL 95, 182301 (2005); R. V. Gavai and S. Gupta, Phys Rev D 73, 014004 (2006).



Strangeness Carriers II

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.5 1 1.5 2 2.5

CX

S

T/TC

X=Q

X=B

CBS (UEP)CBS (BEP)

CBS (LQCD)CQS (UEP)CQS (BEP)

CQS (LQCD)



Light Flavor I

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.5 1 1.5 2 2.5

-cud

11/c

2u

T/TC

PNJL(BEP)PNJL(UEP)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.5 1 1.5 2 2.5

-cus

11/c

2uT/TC

PNJL(BEP)PNJL(UEP)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0.5 1 1.5 2 2.5

-cus

11/c

2s

T/TC

PNJL(BEP)PNJL(UEP)

χBU

χUU

=1

3(1 +

χud + χus

χuu) =

χBD

χDD

χQU

χUU

=1

3(2−

χud + χus

χuu)

χQD

χDD

= −1

3(1−

2χud − χus

χuu)

R. V. Gavai and S. Gupta, Phys Rev D 73, 014004 (2006).


Conclusion

For all chemical potentials second cumulant is smooth around Tc andhigher order cumulants show peak like structure near Tc due tonon-analytic behaviour.


Conclusion


Kurtosis revealed that after 1.5 Tc system behaves like a free quarkgas.


Conclusion



Strangeness is related to baryon number and charge exactly likestrange quark, above TC .


Conclusion




Light flavor sector also shows that u flavor is carried along withB=1/3 and Q=2/3 and d flavor is carried along with B=1/3 andQ=-1/3.


Conclusion




Light flavor sector also shows that u flavor is carried along withB=1/3 and Q=2/3 and d flavor is carried along with B=1/3 andQ=-1/3.

Non-zero extremely small values of flavor off-diagonal susceptibilitiesgive a conception of quark quasiparticles which are dressed byinteraction.


Conclusion

List of collaborators

Rajarshi Ray (Bose Institute)

Sanjay K. Ghosh (Bose Institute)

Sibaji Raha (Bose Institute)

Abhijit Bhattacharayya (Univ. of Calcutta)

Paramita Deb (Univ. of Calcutta)


Conclusion

Thank You.


Fluctuations & Correlations of conserved charges in PNJL...

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