Introduction
eta/s in field theory
Quasiparticle systems
On the lower bound of the viscosity to entropy
denstiy ratio
Antal Jakovác
BME Technical University Budapest
A. Jakovac and D. Nogradi, arXiv:0810.4181
A. Jakovac, arXiv:0901.2802
A. Jakovac, PhysRevD.81.045020 [arXiv:0911.3248]
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Transport in plasma
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Transport in plasma
Gas vs. fluid
Inhomogeneous distribution of conserved charge density N
⇒ way to equilibrium
if elementary excitations are weakly interacting
mean free path is large
independent smoothing of charge excess at each point
homogenization before equilibration
ballistic regime (gases)
strongly interacting systems
mean free path is small
induced currents depend on the local environment
⇒ linear response theory J = −D∇N ⇒ Ṅ = D4N
equilibration before homogenization
diffusive regime (fluids)
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Transport in plasma
Diffusion constant
The order of magnitude of the diffusion constant D:
(τ lifetime, ` mean free path, v velocity, σ cross section)
δN
v
δN
=D 2
⇒ D ∼ `v ∼ v 2 τ ∼
τ
`
nσ
D is large in weakly, small in strongly interacting systems
Ṅ = D4N
⇒
D ∼ (g 4 ln g )−1 in PT.
description sensible only in strongly interacting (fluid) systems
Determination of D
in QFT from linear response C (x) = h[Ji (x), Ji (0)]i
C (k = 0, ω)
⇒ D = lim
ω→0
ω
Boltzmann equations
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Transport in plasma
Viscosity
transport coefficient in momentum transport
⇒ viscosity
η ∼ %v 2 τ ∼ τ
4k 2 η
.
damping rate of small perturbations: Γ =
3T s
%v̇ ∼ η4v (Navier-Stokes)
⇒
( energy density)
Typical values of η/s:
water at room temperature ∼ 30
superfluid 4 He at λ-point ∼ 0.8
smallest at the phase transition
point
⇓
what is the smallest value for η/s?
(R.A.Lacey, A. Taranenko, PoSCFRNC2006:021 (2006), [arXiv:nucl-ex/0610029v3])
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Lower bound for η/s
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Lower bound for η/s
argumentation: η ∼ τ, s ∼ n
η
∼ Eτ >
∼~
s
⇒
P. Danielewicz, M. Gyulassy, PRD 31, 53 (1985); P. Kovtun, D.T. Son, A.O. Starinets PRL 94, 111601 (2005).
calculation: for N = 4 SYM theory at Nc 1, λ = g 2 Nc 1
from graviton absorbtion in the dual 5D AdS gravity:
η
1
=
s
4π
(P. Kovtun, D.T. Son, A.O. Starinets JHEP 0310, (2003) 064.)
for weaker coupling we expect larger ratio: indeed, first λ, Nc
corrections are positive (R.C. Myers, M.F. Paulos, A. Sinha, arXiv:0806.2156)
universal for a wide class of theories
(A. Buchel, R.C. Myers, M.F. Paulos, A.
Sinha, Phys.Lett.B669:364-370,2008.; M. Haack, A. Yarom, arXiv:0811.1794)
so far we did not find counterexamples experimentally
⇒
commonly accepted lower bound for η/s
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Lower bound for η/s
MC studies
In principle exact method to measure η/s. . .
measure hT12 (x)T12 (0)i correlator on lattice
discrete time
⇒ Euclidean
we need the spectral function, which is related to the
correlator as
∞
R
d 3 x hT12 (τ, x)T12 (0)i =
R
0
dω
π
C (ω, k = 0)
cosh(ω(β/2−τ ))
.
sinh βτ /2
invert this relation with the prior knowledge C (ω > 0) > 0
Maximal Entropy Method, or ad hoc solutions
too little sensitivity to small ω regime ⇒ large
systematical uncertainties; additional assumptions are needed
best estimates η/s = 0.102 (56) at T = 1.24Tc
(H. B. Meyer, Phys. Rev. D 76, 101701 (2007))
⇒
needs analytic control!
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Lower bound for η/s
RHIC data
Non-central heavy ion collisions have initial anisotropy.
Time evolution of anisotropy: the larger the viscosity, the more
extent the initial anisotropy is washed out
25
ideal
η/s=0.03
η/s=0.08
η/s=0.16
PHOBOS
v2
0.06
0.04
0.02
0
0
ideal
η/s=0.03
η/s=0.08
η/s=0.16
STAR
20
v2 (percent)
0.08
15
10
5
100
200
NPart
300
0
0
400
1
2
pT [GeV]
3
4
(P. Romatschke, U. Romatschke, Phys.Rev.Lett.99:172301,2007.)
upper bound:
η
< 0.16
s ∼
⇒
is there a lower bound?
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Lower bound for η/s
RHIC data: is there a lower bound?
25
ideal
η/s=0.03
η/s=0.08
η/s=0.16
STAR
v2 (percent)
20
15
⇒
10
5
0
0
1
2
pT [GeV]
3
4
P. Romatschke, U. Romatschke, PRL.99:172301,2007.
R.A. Lacey, A. Taranenko, R. Wei, arXiv:0905.4368
Quadratic correction in pT is expected
(D.A. Teaney, arXiv:0905.2433)
η
1
η Statistically <
is not excluded (favored: ≈ 0.9 ± 0.07)
s
4π
s hpT i=0
4πη/s = 1 only at infinite coupling and Nc !
RHIC seriously challenges the lower bound!
invalidate?
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Lower bound for η/s
Theoretical caveats
N = 4 SYM theory is not QCD
higher curvature and dilaton corrections to 5D AdS actions
1
η
<
is conceivable dual theory? unitarity?
⇒
s
4π
Counterexample: N non-interacting species ⇒ η is not
1
η
∼
changed, s ∼ ln N (mixing entropy) ⇒
s
ln N
(A. Cherman, T. D. Cohen, and P. M. Hohler, JHEP 02, 026 (2008), 0708.4201.)
metastable system? what is the case with limited N?
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Lower bound for η/s
Theoretical caveats
Eτ >
∼ ~! First is true only if E > ∆E
∼ ~? In fact ∆E τ >
Condition for (small width) quasiparticle system
⇒ the argumentation is applicable only in quasiparticle systems
in QFT there is always a continuum – effect on η/s?
1.0
pure hadron gas vs. hadron
gas with continuum
⇒
considerable difference
ǐs
0.8
0.6
0.4
0.2
0.0
0.15
0.16
0.17
0.18
0.19
T @GeVD
(J. Noronha-Hostler, J. Noronha and C. Greiner, PRL 103, 172302 (2009), 0811.1571)
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Lower bound for η/s
We need exact statements about
η
from first principles!
s
Result: there is a non-universal lower bound at finite entropy
density; for smal s:
s
η ,
∼
s min
NQ LT 4
where
NQ : number of relevant quantum channels (species)
L: interaction range
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Generic formulae
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Generic formulae
Continuous spectrum: energy levels are not discrete
⇒ represented by a spectral function (density of states, DoS):
X
n
|ni hn| = V
Z
XZ d 4 p
|p, Qi hp, Q| ,
%
(p)
|p,
Qi
hp,
Q|
≡
Q
(2π)4
Q
Q
where Q denotes conserved quantities (quantum channel),
p = (p0 , p) is the total energy-momentum of the state.
QM-based description
use volume normalization to calculate densities
effects of interaction
continuous DoS
temperature dependent DoS
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Transport coefficient
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Transport coefficient
CJ (x) = h[Ji (x), Ji (0)]i ⇒ ηJ = limω→0 CJ (ω)/ω
generic transport coefficient.
insert complete basis of energy-momentum eigenstates
CJ (x) =
1 X −βH
n e
J(x) m hm |J(0)| ni − {x ↔ 0}
Z n,m
translation:
J(x) = e iPx A(0)e −iPx
⇒
hn |J(x)| mi = e i(Pn −Pm )x hn |J(0)| mi
Fourier transformation, p = (p0 , p)
CJ (p) =
1 X −βEn
e
− e −βEm (2π)4 δ(p + Pn − Pm )| hm |J(0)| ni |2
Z n,m
introduce spectral densities
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Transport coefficient
ηJ = β
Z
V2 X
d 4k 2
% (k) e −βk0 | hk, Q|Ji |k, Qi |2 .
Z
(2π)4 Q
Q
current matrix element: hk, Q|Ji |k, Qi = JQ (k)
ki
Vk0
∼ ki /k0 ∼ vi since Ji is a current
in free case JQ (k) is the charge carried by the current: for
electric current J = e charge, for viscosity J ∼ kj momentum
in nonperturbative case JQ (k) can be momentum dependent.
angular averaging
Finally:
Z
1 X
d 4 k k2 −βk0
ηJ = β
e
(JQ (k)%Q (k))2 .
3Z
(2π)4 k02
Q
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Entropy density
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Entropy density
free energy density Z = e −βF = Tr e −βH = V
X Z d 4k
%K (k)e −βk0
(2π)4
K
Volume dependence of the free energy:
for small sizes it is arbitrary
as V → ∞ we recover linear volume dependence
crossover at scale L
volume elements larger than L interact weakly (surface
interaction)
coarse graining scale ⇒ free energy density cen be defined
at scales larger than L
L is also an effective IR cutoff for the interactions.
⇒
we choose a volume V = L3 to define free energy density:
X
T
f = − 3 ln 1 + L3
L
K
Z
d 4k
%K (k) e −βk0
(2π)4
!
.
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Entropy density
The eta/s ratio
with s = −
∂f
and with angular averaging
∂T
Z
β X
d 4 k k2 −βk0
e
(JK (k)%K (k))2
4 k2
3Z
(2π)
0
η
K
!.
=
Z
4
s
X
∂ T
d
k
−
ln 1 + L3
%K (k) e −βk0
∂T L3
(2π)4
K
Is there a lower bound in this formula?
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Entropy density
Generic structure
Structure of η/s for small % is
R
f1 %2
ηJ
∼ R
s
f2 %
2
% ≥ h%i2
⇒
rescaling
−→
2
%
.
h%i
2
we expect η >
∼ s up to rescaling factors.
Quasiparticle vs. non-quasiparticle systems:
large peak in %
⇒
% small everywhere
%2 even larger
⇒
⇒
%2 even smaller
η/s large
⇒
η/s small
in non-quasiparticle systems η/s is naturally small!
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Entropy density
Lower bound – mathematical approach
More exactly:
Z
need a sum rule in each energy channel
dk0
%Q (k0 ) = UQ
2π
minimize η by tuning % with respecting the sum rules and
keeping the entropy density constant.
technically: Lagrange multiplicators
two cases analytical: small/large s. The minimum values:
η F(L3 s)
x
for small x
∼
,
F=
x
4
e
/x
for large x
s min
NQ (LT )
NQ : number of effective quantum channels (species).
⇒
There is a lower bound at finite s, but it is not universal.
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Simplifications
For the concrete model calculations we use simplifications
We use generalized
quasiparticle systems:
Z
d 4k
−βk0
%
(k)
(∓)
ln
1
±
e
.
f =T
QP
(2π)4
we omit the effect of JQ and define a “reduced” viscosity
coefficient as
Z
β
d 4 k (k2 )2 −βk0 2
η̄ =
e
%K (k).
15 (2π)4 k02
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Small width case
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Small width case
Assume that the lowest lying states can be approximated with
Breit-Wigner form:
%(q) =
2Γ
.
(q0 − εq )2 + Γ2
In the small width limit %(q)2 ≈
η̄QP
sQP
 540 T

,



A π4 Γ

 ± 2
T
= 30π
,

Γm



2

 16π m ,
ΓT
2
Γ
2πδ(q0 − εq ). We find:
if εk = k
A± = (1, 8/7)
if εk = m +
if εk =
k2
2m
k2
2m
the first two cases are superfluids
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
.
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Small width case
The value of Γ
in conformal case Γ ∼ T
⇒
ηJ
∼ constant
s
lower limit may come from infinte coupling, 1/4π.
massive case at low temperature: the width is the
consequence of scattering on thermal particles
⇒ abundance is e −M/T (M: energy of scattering state)
ηJ
⇒
∼ Te M/T → ∞.
s
⇒ in the small width quasiparticle case there is a lower bound
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Small width case
Off-shell effects
We consider three examples of off-shell effects which can appear in
physical systems
wave function renormalization
large width
low temperature non-quasiparticle systems
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Wave function renormalization
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Wave function renormalization
In the real case the quasiparticle peak never stands alone,
there is always a continuum at higher energies.
1
0.8
ρ
0.6
0.4
0.2
0
m
m+m’
E
But the complete spectral function must obey the sum rule,
like
Z
1=
dk0
%(k0 ) = peak + continuum.
2π
Therefore the peak must be reduced %QP → Z %QP .
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Wave function renormalization
At small temperatures the continuum contribution for η/s is
much smaller than the QP contribution ⇒
η
η
→Z
s
s
⇒ wave function renormalization directly reduces the η/s ratio.
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
High temperature effects
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
High temperature effects
At high temperatures the quasiparticle peak melts into the
continuum, forming a broad spectral function ⇒ the width can
be larger than the peak position.
A simplified example
%(k0 , k) =
2π
Θ(E1 < k0 < E2 )
E2 − E1
step function, where E1,2 (k) =
q
2 . At small
k 2 + m1,2
temperatures (T < m1 )
η
T
= 6π
s
∆m
⇒ by broadening the distribution the viscosity to entropy
density ratio has no lower bound
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
High temperature effects
Realistic example
Self-energy corrections to a free particle leads to a
Breit-Wigner-like form
%=
(k 2
−
−2k0 Im Σ
− Re Σ)2 + Im Σ2
m2
If we neglect the real part of the self energy we arrive at
%BW (k) =
4γk2 k0
1
.
N (k02 − Ek2 )2 + γk4
N ∼ 1 normalization factor
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
High temperature effects
60
log(η/s)
40
η/s
1000
m/T=0
m/T=1
m/T=5
m/T=10
50
30
20
m/T=0
m/T=1
m/T=5
m/T=10
100
10
10
0
1
5
10
15
20
25
30
2
(γ/T)
0.1
1
10
log((γ/T)2)
for small γ/T the quasiparticle picture holds
for large γ/T the η/s ∼ γ 2 /T 2
turnover depends on the mass, for m → ∞ the ∼ γ 2
behaviour until zero!
⇒ in realistic cases m ∼ T find a lower bound at high
temperature near γ ∼ T .
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
100
Introduction
eta/s in field theory
Quasiparticle systems
System with zero mass excitations
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
System with zero mass excitations
Non-quasiparticle excitations in real models
Simplest interacting field theories at zero temperature has a
spectral function (DoS) like
1
0.8
ρ
0.6
0.4
0.2
0
m
m+m’
E
the Dirac-delta at m represents a stable particle
the continuum at m + m0 represents a multiparticle state.
infinite lifetime
⇒
gas
What could spoil this simple picture?
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
System with zero mass excitations
Zero mass excitations
If there are in the system zero mass particles, then m0 = 0
⇒ cut and the delta-peak melt together
1-loop threshold behavior linear
⇒
% ∼ 1/(E − Ethr )
1
1
0.8
0.8
−→
ρ
0.6
ρ
0.6
0.4
0.4
0.2
0.2
0
m
m+m’
0
E
m
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
E
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
System with zero mass excitations
Zero mass excitations
If there are in the system zero mass particles, then m0 = 0
⇒ cut and the delta-peak melt together
1-loop threshold behavior linear
⇒
% ∼ 1/(E − Ethr )
1
1
0.8
0.8
−→
ρ
0.6
ρ
0.6
0.4
0.4
0.2
0.2
0
m
m+m’
0
E
m
This is not normalizable! ⇒ IR divergences near the
threshold, which smear out the 1/x singularity
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
E
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
System with zero mass excitations
Zero mass excitations
1
0.8
ρ
0.6
0.4
0.2
0
m
E
Interpretation: no single charged particle (electron, quark), it
is always surrounded by soft gauge bosons
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
System with zero mass excitations
Zero mass excitations
0.5
1
ρ
0.4
Tc . w( m/Tc )
−→
0.6
0.3
0.2
0.1
0
0.2
-0.1
0
eq. 15
eq. 8
eq. 13
0.4
0.8
m
0
1
E
2
3
m / Tc
4
5
6
Interpretation: no single charged particle (electron, quark), it
is always surrounded by soft gauge bosons
In QCD: from fitting to MC pressure data one obtains similar
distribution of quasiparticle masses
(T.S.Biro, P.Levai, P.Van, J.Zimanyi, Phys.Rev.C75:034910,2007)
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
System with zero mass excitations
The η/s ratio at low temperatures
%# e −βq0 enhances the lowest lying states
near the threshold:
⇒
power expand
%(q) = Cq0 Θ(q − M)(q 2 − M 2 )w .
C is dimensionfull: [C] = [E ]−2(1+w )
ηJ ∼ C 2 and f ∼ C ⇒ C remains in the ratio.
In the massive and massless case we find
ηJ
∼ CM w T 2+w
s
T →0
and CT 2(1+w ) −→
0
for an integrable threshold (w > −1)
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Outlines
1
Introduction
Transport in plasma
Lower bound for η/s
2
eta/s in field theory
Generic formulae
Transport coefficient
Entropy density
3
Quasiparticle systems
Small width case
Wave function renormalization
High temperature effects
System with zero mass excitations
4
Conclusions
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Conclusions
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Conclusions
1/4π lower bound for η/s is true only for quasiparticle and
conformal theories
in general the lower bound in a given environment depend on
several factors;for small s
η
s
>
∼
s
NQ LT 4
there exist several models where η/s < 1/4π is possible
quasiparticle with multiparticle continuum ⇒ wave
function renormalization
finite temperature, broad spectral function
low temperature systems with zero mass excitation
⇒ there even η/s = is conceivable
in QCD any of these effects can be important
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Hydrodynamics
System with local collective flow: QM description?
Construction of the system: at t = −∞ equilibrium in rest
(ū = (1, 0, 0, 0)), then a modified time evolution corresponding to
the flow u – denote ∆u = u − ū:
H (0) = ūµ P (0)µ = uµ P µ = H + ∆uµ T 0µ
Zt
δH =
−∞
dt 0 ∂0 δH = −
Zt
Z
⇒
δH = −
d 3 x ∆uµ T 0µ
dt 0 d 3 x ∂0 uµ T 0µ + ∆uµ ∂0 T 0µ
−∞
With energy-momentum conservation ∂0 T 0µ = −∂i T iµ and
partial integation
Zt
δH = −
dt 0 d 3 x ∂µ uν T µν
−∞
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Linear response theory (→ the flowing and the original
systems are not too far ⇒ nonrelativistic):
Rt 0 Rt 0 00 3 00
δ hX (t)i = i
dt
dt d x h[X (t), T µν (x 00 )]i ∂µ uν .
−∞
−∞
Hydrodynamical approximation: ∂u ≈ const.
time independent.
⇒ δ hX i
spatial rotational
symmmetry
of the ground state ⇒
δ hπij i ≡ δ Tij − 31 δij T.kk = η2 ∂k v` + ∂` vk − 32 δk` ∂v ,
the coefficient from above (denote C (x) = h[T12 (0), T12 (x)]i
Z0
η=i
Zt
dt
−∞
C (ω, k = 0)
ω→0
ω
dt 0 d 3 x0 C (x 0 ) = lim
−∞
Kubo formula
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
Introduction
eta/s in field theory
Quasiparticle systems
Conclusions
Zero temperature limit of viscosity
The viscosity η and the entropy have a common form
Z
3a5
d 4k
Fn,m = 3
Θ(k0 )Θ(k 2 − σ 2 )e −k0 /T (ak0 )n %m (k),
2π T (2π)4
since
η = N 2 ∆2 F0,2 ,
s = 2F1,1 .
After reducing the integrals
3
n
2
3/2 −σ/T
a Fn,m = C (aσ) (a σT )
e
Z∞
dz e −z
2(wz)5/2
1 + (wz)5
0
where w = 2∆−2/5 T /σ rescaled temperature.
BOTH η and s goes to zero at zero temperature
Non-Perturbative Methods in Quantum Field Theory, March 10-12, 2010, Hévı́z
!m
,