Shear and bulk viscosities for a pure glue matter

A. Khvorostukhina

in collaboration with V.Toneeva, D.Voskresenskyb

a BLTP, JINR

b MEPHI, Moscow

Scienti�c Council, February 17, 2011

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Contents

1 Introduction

2 Equation of state of glue matter

3 Calculation of viscosity coe�cients

4 Shear viscosity

5 Bulk viscosity

6 Conclusion

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Why is a glue matter? Aims

Motivation:

New lattice data for the pure gluon EoS!

There are lattice data for viscosities!

Simplicity: exactly �rst order ⇒ two-phase model have to

work very well!

Aims:

To improve the phenomenological quasiparticle model to

reproduce the new lattice data

To investigate the behavior of viscosity coe�cients for a gluon

system

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Gluon phase

The system of

interacting gluons⇔ a gas of noninteracting quasiparticles

with an e�ective mass

m2g (T ) =

1

2g2(T ) T 2 T-dependent !

g2(T ) =16π2

11 ln [λ(T − Ts)/Tc)]2

εg (T ) = εidg (T ,mg (T )) + B(T ),

Pg (T ) = P idg (T ,mg (T ))− B(T )

The thermodynamical identity (the condition of thermodynamical

consistency)

TdP

dT− P(T ) = ε(T ) ⇒ dB(T )

dT= −

εidg − 3P idg

mg

dmg

dTM.I. Gorenstein, S.N. Yang, Phys. Rev. D 52, 5206 (1995)

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Glueball phase

Gluodynamics: at T < Tc the matter consists of glueballs

Two lowest-lying scalar 0++ and tensor 2++ glueballs

mgb(T ) ≈ m̃gb(T )− 2T +√4T 2 − Γ2gb(T )

N. Ishii et al., Phys. Rev. D 66, 094506 (2002)

m̃gb(T ) = m0gb = const

Γgb(T ) = bgb(T − Tgb) Θ(T − Tgb)F. Buisseret, EPJ C 68, 473 (2010)

Thermodynamic consistency � in the same way as above for gluons.

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Full model and parameters

The Gibbs conditions at the transition point

T gc = T gb

c ≡ Tc

Pg (Tc) = Pgb(Tc)

Tc = 265 MeV in agreement with the lattice results

Fit the new lattice data M. Panero, Phys. Rev. Lett. 103, 232001 (2009)

Ts/Tc = 0.5853, λ = 3.3

B0 � from intersection with glueballs to reproduce Tc -value

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Pressure and energy density

1 2 30,0

0,2

0,4

0,6

0,8

1,0 QP model Panero, 2009 Karsch, 1996

P

/PS

B

T/Tc

1 2 30,0

0,2

0,4

0,6

0,8

1,0

ε/ε S

B

T/Tc

0,6 0,8 1,00,00

0,02

0,04

0,6 0,8 1,00,0

0,1

0,2

The pressure and the energy density normalized to those in the Stefan-Boltzmann limit.

�lled squares � the old Karsch's lattice results G. Boyd et al., Nucl. Phys. B 469, 419 (1996)

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Interaction measure and entropy density

1,0 1,5 2,0 2,5 3,00,0

0,5

1,0

1,5

2,0

2,5

3,0 Panero, 2009 Karsch, 1996 QP model

(ε

-3P

)/T4

T/Tc

1,0 1,5 2,00,0

0,2

0,4

0,6

0,8

1,0

(ε+P

)/(ε S

B+P

SP)

T/Tc

0,6 0,8 1,00,0

0,3

0,6

0,6 0,8 1,00,0

0,1

0,2

The trace anomaly and the entalpy/entropy.

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Energy-momentum tensor of quasiparticles

Tµν =∑a

∫dΓ

pµa p

νa

EaFa

pµa = (Ea(~pa),~pa), Ea(~p) =√~p 2 + m2

a[Fa]

F loc.eq.a (pa, xa) =

[ep

µa uµ/T − 1

]−1Using Boltzmann equation and varying only Fa,

δTµν = −∑a

∫dΓ

{τapµa p

νa

E 2a

pκa∂κFa

}loc.eq.

where τa(~p) � the relaxation time of the given species

By de�nition

δTij = −ζ δij ~∇ · ~u − η(∂iuj + ∂jui −

2

3δij ∂ku

k

), i , j , k = 1, 2, 3

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Expression for shear viscosity

Taking derivatives ∂µFloc.eq.a , by straightforward calculations we

�nd expression

η =1

15T

∑a

∫dΓ τa

~p 4a

E 2a

F eqa (1 + F eq

a )

Simplifying,

τa(~p) = τ̃a = const

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Relaxation time

Glueballs: isotropic cross section σgb = 30 mb, τ−1gb = ngb(T )σgb

Resummation of the hard thermal loops:

τ̃−1 ∼ g2T ln(1/g) R.D. Pisarski, PRL 63, 1129 (1989)

τ̃−1g = Ncg2T

4πln2c

g2A. Peshier, W. Cassing, PRL 94, 172301 (2005)

Two values of c-parameter:

c = 14.4 (Peshier ,Cassing)

c = 11.44 − the limit case τ̃g−1 → 0, T → Tc + 0

τ̃−1BKR = aη/(32π2)T g4 log(aηπ/g2) , aη = 6.8

M. Bluhm et al., Nucl. Phys. A 830, 737c (2009)

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Shear viscosity

0,8 1 2 100,01

0,1

1

163*8 243*8

η/

s

T/Tc

τ̃BKR

τ̃g , c = 11.44

τ̃g , c = 14.4

The horizontal dotted line � η/s = 1/4π bound (AdS/CFT).

Triangles and squares � the lattice data, S. Sakai, A. Nakamura, PoS LAT2007, 221 (2007)

Filled circles � the lattice data, H.B. Meyer, Phys. Rev. D 76, 101701 (2007)

The shaded region � perturbative result

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Remarks on shear viscosity

τ̃g :

perturbative regime is not achieved up to very high T

predictions of our QP model are in a reasonable agreement

with the lattice results and do not contradict perturbative

estimates

τ̃BKR:

for T & 10Tc shear viscosity calculations demonstrate a

noticeable growth exceeding lattice data and even a

perturbative estimate (η/s)pert ≈ 0.8− 1.0

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Expression for bulk viscosity

ζ = − 1

3T

∑a

∫dΓ τa

~p 2a

EaF eqa (1 + F eq

a )Qa,

where the EoS-dependent Qa factor is given by

Qa = −{~p 2a

3Ea− c2s

[Ea − T

∂Ea

∂T

]}c2s = ∂P

∂ε � the speed of sound squared.

The Landau-Lifshitz condition

δT 00 =∑a

τ̃a

∫dΓEaQa = 0

is satis�ed in our QP model.

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Alternative expressions for bulk viscosity

P. Chakraborty, J.I. Kapusta, arXiv: 1006.0257

ζChK =∑a

da

T

∫d3p

(2π)3τ̄aF

eqa (1 + F eq

a )Q2a

M. Bluhm et al., Nucl. Phys. A 830, 737c (2009)

ζBKR =∑a

da

3T

∫d3p

(2π)3τaEa

F eqa (1+ F eq

a )Qa

[m2

a(T )− Tdm2

a(T )

dT

]The QP interaction contributes to the energy-momentum tensor ⇒the second term Tdm2

a(T )/dT in the square bracket.

A perturbative estimate gives

(ζ/s)pert ≈ 0.02α2s , 0.06 ≤ αs ≤ 0.3

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Bulk viscosity

0,8 1 2 1010-4

10-3

10-2

10-1

100

ζ/s

T/Tc

1,0 1,1 1,20,01

0,1

1

τ̃g , c = 11.44

τ̃g , c = 14.4

ζBKR

ζChK

Empty squares � the lattice data, S. Sakai, A. Nakamura, PoS LAT2007, 221 (2007)

Filled circles � the lattice data, H. B. Meyer, Phys. Rev. Lett. 100, 162001 (2008)

The dotted line � perturbative result

Thin short-dashed curve � rough approximation of lattice data

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Remarks on bulk viscosity

Values of ζ/s in our model noticeably underestimate the

corresponding values on the approximating short-dashed curve.

Nevertheless the behavior qualitatively is similar to that given

by the approximating curve.

ζChK yields a strong T suppression at T & 1.5Tc , as

compared to that given by our result.

for T > 1.9Tc ζBKR becomes invalid providing negative values.

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Conclusions

For thermodynamic characteristics the quasiparticle model

results are in good agreement with the latest lattice data.

With the chosen value of the relaxation time the shear

viscosity to entropy density ratio η/s �ts rather well the scant

lattice data.

Although the calculated ζ/s ratio essentially underestimates

the upper limits given by the corresponding lattice data, its

temperature dependence is well described.

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Thank you for attention!

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