Post on 11-Sep-2021
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
A. Khvorostukhin et al. February 17, 2011 1/19
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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