QGP Shear Viscosity & Electric Conductivity
A. Puglisi - S. Plumari - V.
Greco
UNIVERSITY of CATANIA - INFN-LNS
Mainly based on next weekend arXiV submission
Outline
 Transport Coefficients in kinetic theory:

Green-Kubo and Ohm’s Law

Comparison to Relaxation Time Approximation

Kinetic Transport Theory at fixed h/s [M. Ruggeri talk]
 Shear Viscosity and Electric Conductivity:
 Comparison of sel/T with recent lQCD data
 Ratio (h/s)/(sel/T): disentangling q and g interaction?!
Shear viscosity h -> anisotropic flow vn
Operative definition
Fx
v
 h x
Ayz
y
h 1
@ < p >×l
s 15
Green-Kubo
h=
h/s0
1
d 4 x Pxy (x) Pxy (0)
ò
T
h/s0.16
h/s smoothen fluctuations and
affect more higher harmonics
B.Schenke
Shear Viscosity regulates:
 How the fluid drag itself
in the transverse direction ->
damping of anisotropies vn=<cos(nf)>
 Entropy production
B. Schenke, PRC85(2012)
Electric Conductivity
Green-kubo
s el =
¥
V
dt jx (0) jx (t)
ò
T 0
slQCD
Ohm’s Law
J = s el E
Electric Conductivity sel regulates:
s=0
 Damping of Magnetic Field in HIC t ≈ sel L2
DB = ¶2t B+ s el¶t B
Tuchin ‘13, Sokokov-McLerran ‘13, Kharzeev-Rajagopal ’14
-> Chiral Magnetic Effect, charge asymmetry of directed flow v1
 Damping of Magnetic Fields in the Early Universe
 Soft photons rate
dR g
d 3p
=
aem
s f(w)
2 2 el
ep
Kapusta ’93
 Insight into quark vs gluon scattering rates
Electric Conductivity
Green-kubo
s el =
¥
V
dt jx (0) jx (t)
ò
T 0
Ohm’s Law
J = s el E
Electric Conductivity sel regulates:
 Damping of Magnetic Field in HIC t ≈ sel L2
DB = ¶2t B+ s el¶t B
Tuchin ‘13, Sokokov-McLerran ‘13, Kharzeev-Rajagopal ’14
-> Chiral Magnetic Effect, charge asymmetry of directed flow v1
 Damping of Magnetic Fields in the Early Universe
 Soft photons rate
dR g
d 3p
=
aem
s f(w)
2 2 el
ep
Kapusta ’93
 Insight into quark vs gluon scattering rates
Relativistic Boltzmann Equation
m
* m * p
p
¶
+
m
{ m ¶ m ¶m } f (x, p) = C[f]
Collisions
Free streamingField Interaction
fq,g(x,p) is a one-body distribution function for quark and gluons
1
d 3q
3ò
C22 =
3
(2p)
2E p DN
(2p)
coll 2E q
DtD xD p
3
=g
(2p)
f
3 g
( p) fg (q)vrels p,q®p-k,q+k
2ù
4 space
4
- f(q)f(p) Mgg->gg (pq ® p¢q¢) phase
(2p)
d
(p + q - p¢ - q¢)
ûú
Solved discretizing the
space in (h, x, y)a cells
exact
solution
3
r=15 fm ,
stot=10 mb
8
Collision rate
-1
t0
3x0
10
R (fm )
3
2
d 3p¢
d 3q¢ é
Rate
of
collisions
¢
¢
¢
¢
¢
¢
3 3
f
(
q
)
f
(
p
)
M
(
p
q
®
pq)
ò D(2p)
gg->gg
q 2E¢p (2p)3 2E¢q êë
per unit time and
6
T=0.2 GeV
T=0.3 GeV
T=0.5 GeV
4
2
0
0.5
1
1.5
M (GeV)
2
2.5
3
Transport at fixed shear viscosity
Usually input of a transport approach are cross-sections and fields, but here we reverse
it and start from h/s with aim of creating a more direct link to viscous hydrodynamics
Transport simulation
Relax. Time Approx. (RTA)
h

1 p
1 T
=
=
= cost.
s 15 s tr n 5 s tr n
s tr (  (r ), T )  s tr ,a 
Space-Time dependent cross
section evaluated locally
str is the effective cross section
s tr = ò dW q 2
1 pa 1
15 na h / s
ds
2
= s TOT h(a) £ s TOT
dW
3
a=cell index in the r-space
G. Ferini et al., PLB670 (09)
V. Greco at al., PPNP 62 (09)
1+1D expansion
One maps with C[f] the phase space evolution
of a fluid at fixed h/s !
Convergency to IS Viscous
Hydro for large K
K0 =
1 T0t 0
5 h/s
Huovinen-Molnar, PRC79(2009)
Similar results from BAMPS-Frankfurt
- Convergency for small h/s of Boltzmann
transport at fixed h/s with viscous hydro
- Better agreement with 3rd order viscous
hydro for large h/s
Similar studies by Bazow, Heinz, Strickland
for anisotropic hydordynamics
arXiv:1311.6720 [nucl-th]
El, Xu, Greiner, Phys.Rev. C81 (2010) 041901
Do we really have the wanted shear viscosity h
with the relax. time approx.?
- Check h with the Green-Kubo correlator
Shear Viscosity in Box Calculation
1
h=
T
microscopic scatterings
¥
3
xy
xy
dt
d
x
P
(x,
t)
P
(0, 0)
ò ò
0
V
P xy (x,t)P xy (0, 0) = P xy (0, 0)P xy (0, 0) ×e-t/t
macroscopic
thermodynamics
4 eT
=
15 V
η ↔ σ(θ), , M, T …. ?
S. Plumari et al., arxiv:1208.0481;see also:
Wesp et al., Phys. Rev. C 84, 054911 (2011);
Fuini III et al. J. Phys. G38, 015004 (2011).
F. Reining et al., Phys.Rev. E85 (2012) 026302
Needed very careful tests of convergency
vs. Ntest, xcell, # time steps !
Non Isotropic Cross Section - s(q)
Relaxation Time Approximation
hRTA =
4
4
e
e× t tr =
15
15 h(a) s TOTr
h(a) = 4a(1+ a)éë(2a +1)ln(1+ a-1 ) - 2ùû , a = mD2 / s
h(a)=str/stot weights cross section by q2
Chapmann-Enskog (CE)
hCE =
4
4
e
e× t CE =
15
15 g(a) s TOTr
g(a) correct function that fix the
momentum transfer for shear motion
 CE and RTA can differ by about a factor 2
 Green-Kubo agrees with CE
S. Plumari et al., PRC86(2012)054902
RTA is the one usually employed to make
theoroethical estimates: Gavin NPA(1985); Kapusta,
PRC82(10); Redlich and Sasaki, PRC79(10),
NPA832(10); Khvorostukhin PRC (2010) …
for a generic cross section:
-2
ds
µ ( q 2 (q ) + mD2 )
dW
mD regulates the angular dependence
Green-Kubo in a box - s(q)
Viscosity of a pQCD gluon plasma
Agreement with AMY, JHEP 0305 (2003) 051
close to AMY result JHEP(2003),
but there is a significant simplification:
only direct u & t channels with simplified HTL propagator
 We have checked the Chapmann-Enskog:
- CE good already at I° order ≈ 4-5%
- RTA even with str generally underestimates h
(≈25% for pQCD gluon matter, ±15% for udsg matter)
 We know how to fix locally h/s(T) in the transport approach
Applying kinetic theory to A+A Collisions….
- Impact of h/s(T) on the build-up of v2(pT)
z
y
x
Extend to Higher pT
pT ≈3T
Larger h/s
Hydro  Transport
h/s<<1
Initial off-equilibrium
M. Ruggeri’s talk – this afternoon
Heavy Quarks
S.K. Das talk – tomorrow afternoon
Test in 3+1D: v2/e response for almost ideal case
EoS cs2=1/3 (dN/dy tuned to RHIC)
Integrated v2 vs time
Ideal -Hydro
v2/e
Transport at h/s fixed
v2 =
p 2x - p 2y
p 2x + p 2y
Bhalerao et al., PLB627(2005)
Time rescaled
In the bulk the transport has an hydro v2/e2 response!
Just one tip on what can be studied with a transport at fixed h/s:
impact of power law spectrum at intermediate pT
Non equilibrium at larger pT:
impact of minijets on v2(pT)
J.Y. Ollitrault, Plumari, VG, in preparation
- Mini-jets starts to affect v2(pT) for pT>1.5 GeV
- Effect non-negligible. A flatter spectrum leads to smaller v2
- The physics can be mocked-up by arbitrary df (pT) viscous correction in hydro
V¥
s el = ò dt jx (0) jx (t)
T 0
J = s el E
Electric Conductivity in a Box with boundary condition
Ohm’s Law method
J z = s el E z
d i
pz = fi e E z
dt
Jz/Ez independent on Ez
-> one can define
the conductivity
See also Cassing et al., PRL110 (2013) + Moritz talk this afternoon
Comparing with Green-Kubo correlator
Ohm’s Law
Isotropic
Green-Kubo
RTA with ttr
e*2 p2
s el =
t q,tr rq
2
3T E
e*2 = e 2
Similarly to h for anisotropic cross section
the RTA with str underestimate sel
2 2
2
f
=
å j 9e
j=q,q
i=u,d,s,g
j=u,d,s
Moving to more realistic case for QGP:
- Fitting “thermodynamical” part of transport
coefficient by QP model tuned to lQCD thermodynamics
- Using the Relax. Time Approx. for both h and sel
to follow their relation analytically
Simple QP-model fitting lQCD
Plumari, Alberico, Greco, Ratti, PRD84 (2011)
WB=0 guarantees
Thermodynamicaly consistency
wq,g = k 2 + m 2q,g (T)
g(T) from a fit to e from lQCD
-> good reproduction of P, e-3P, cs
l=2.6
Ts=0.57 Tc
g(T) practically identical to DQPM
Electric Conductivity of the QGP
e*2 p2
s el =
3T E 2
u,d,s
å
t j,tr r j
j=u,d,s
i=u,d,s,g
J=u,d,s
s ijtot (s) = bij
pas2 s
m 2D s + m 2D
bqq=16/9 bqq = 8/9
bgg =9
bqg=2
 Most of the difference with DQPM comes from the fact that our
scattering is anisotropic -> large ttr
 QP -DQPM probably overestimates the conductivity,
what happens for h/s?
Shear Viscosity to Entropy Density
Kapusta ’93
i, j=u,d,s,g
 Also the h/s seems to be over estimated!
 What happens to sel rescaling by a K factor the cross section
to have a minimum of h/s = 0.08
Electric Conductivity of the QGP
u,d,s
e*2 pp22
t qj,tr r j
sselel =
q,tr r
22 tå
3T EE j=u,d,s
s ijtot (s) = bij
Ads/CFT
pas2 s
m 2D s + m 2D
bqq=16/9 bqbarq = 8/9
bgg =9 bqg=2
sel is strongly T- dependent
s el
h
-1
» g (T)
T
s
 Rescaling the cross section we get at the same time h/s and sel/T !
 Of course small h/s tend to give small conductivity
Relation between Shear Viscosity and Conductivity
h
1
p4
=
s 15Ts E 2
s el e*2 p2
= 2
T 3T E 2
u,d,s
å
t j,tr r j »
1 eT
t r » T-2 t r
T rs
t j,tr r j »
1 T
-1
-2
t
r
»
g
(T)T
tr
2
T m(T)
j=u,d,s,g
u,d,s
å
j=u,d,s
So one expects:
s el
h
-1
» g (T)
T
s
Steep rise of sel just above Tc even if the h/s is nearly T independent
h/s to sel /T ratio
Depending on the relative
quark to gluon relaxation time
Practically unknown!
Fixed by the lQCD
thermodynamics
Relaxation times
s ijtot (s) = bij
pas2 s
m 2D s + m 2D
= 28/9
= 9/2
h/s to sel /T ratio
Symbols are dividing lQCD data
h/s for the lowest sel/T
Enhancement of scattering
 The ratio is independent on both K-factor and as(T)
 T->Tc increase by one order of magnitude (sel(T) quite stronger T dependence)
 Sensitive to increase in the qq scattering respect to qg, gg
 Not very sensitive to increase of gg respect to qq
h/s to sel /T ratio
Symbols are dividing lQCD data:
- Highest h/s for lowest sel/T
- Lowest h/s highest sel/T
Warning: we are considering
lQCD quenched, unquenched
and with different actions and Tc
 The ratio is independent on both K-factor and as(T)
 T->Tc increase by one order of magnitude (sel(T) quite stronger T dependence)
 Sensitive to increase in the qq scattering respect to qg, gg
 Not very sensitive to increase of gg respect to qq
h/s to sel /T ratio
AdS/CFT
 AdS/CFT would predict a flat behavior
 Agreement with DQPM confirm the ratio
 There could be even a structure
Summary
Numerical Transport approach:
 Chapmann-Enskog I°order agree with Green-Kubo for h
 Relax. Time Approx. underestimate both h and sel
Electric conductivity:
 New lQCD data on sel appear self-consistently
related to 4ph/s ≈ 1, also sel ≈ g-1(T) h/s
 The ratio (h/s)/(sel/T) is :
- independent on K-factor of as(T) coupling
- sensitive to the relative strength of q /g scattering rates
- T-> Tc steep increase , test for AdS/CFT approach
Width has small impact on thermodynamics?
Both fit to WB-lQCD data
DQPM: E. Braktovskaya et al.,NPA856 (2011) 162
QP: Plumari et al., PRD84 (2011)
DQPM
Chapmann-Enskog vs Green Kubo:massive case
Massive case is relevant in quasiparticle models where Mq,g(T)=g(T)T
Hence we need it to extend the approach to Boltzmann-Vlasov transport
Again good agreement with CE 1st order for s(q)=cost.
Isostropic s – massive particles
z=M/T
Still missing Chapmann-Enskog for
massive & anisotropic cross section
Viscous Hydrodynamics


T   Tideal
 P dissip
Teq  d T   f eq  d f
Asantz used
h pT2
p  p p
feq
df 
f eq »
2
3s t T
e P T
K. Dusling et al., PRC81 (2010)
Problems related to df:
 dissipative correction to f -> feq+dfneq just an ansatz
 dfneq/f at pT> 1.5 GeV is large
 dfneq <-> h/s implies a RTA approx. (solvable)
 P (t0) =0 -> discard initial non-equil. (ex. minijets)
 pT -> 0 no problem except if h/s is large
h/s(T) shear viscosity or details of the cross section?
ds
a s2
µ
dW éq 2 (q ) + m 2 ù2
Dû
ë
cross section
Keep same h/s means:
t h-1 = g( mT )s TOT r
D
s TOT ( m1D ) g(m2D )
=
s TOT ( m2 D ) g(m1D )
 h/s is really the physical parameter determining
v2 at least up to 1.5-2 GeV
 microscopic details become relevant at higher pT
 First time h/s<-> v2 hypothesis is verified!
for mD=1.4 GeV -> 25% smaller stot
for mD=5.6 GeV -> 40% smaller stot
Does the microscopic degrees of freedom
matter once P(e) and h/s is fixed?
h/s(T) shear viscosity or details of the cross section?
ds
a s2
µ
dW éq 2 (q ) + m 2 ù2
Dû
ë
cross section
Keep same h/s means:
t h-1 = g( mT )s TOT r
D
s TOT ( m1D ) g(m2D )
=
s TOT ( m2 D ) g(m1D )
 h/s is really the physical parameter determining
v2 at least up to 1.5-2 GeV
 microscopic details become relevant at higher pT
 First time h/s<-> v2 hypothesis is verified!
for mD=1.4 GeV -> 25% smaller stot
for mD=5.6 GeV -> 40% smaller stot
Does the microscopic degrees of freedom
matter once P(e) and h/s is fixed?
h/s(T) shear viscosity or details of the cross section?
ds
a s2
µ
dW éq 2 (q ) + m 2 ù2
Dû
ë
cross section
Keep same h/s means:
t h-1 = g( mT )s TOT r
D
s TOT ( m1D ) g(m2D )
=
s TOT ( m2 D ) g(m1D )
 h/s is really the physical parameter determining
v2 at least up to 1.5-2 GeV
 microscopic details become relevant at higher pT
 First time h/s<-> v2 hypothesis is verified!
for mD=1.4 GeV -> 25% smaller stot
for mD=5.6 GeV -> 40% smaller stot
Does the microscopic degrees of freedom
matter once P(e) and h/s is fixed?
 r-space: standard Glauber model
 h=y Bjorken boost invariance (flexible)
 p-space: Boltzmann-Juttner Tmax [pT<2 GeV ]+ minijet [pT>2-3GeV]
We fix maximum initial T at RHIC 200 AGeV
Tmax0 = 340 MeV
T0 t0 =1 -> t0=0.6 fm/c
Typical hydro
condition
Then we scale r-profile according to initial e
and with beam energy according to dN/dy
62 GeV
200 GeV
2.76 TeV
T0
290 MeV
340 MeV
590 MeV
t0
0.7 fm/c
0.6 fm/c
0.3 fm/c
No Discarded
fine tuning in viscous hydro
Impact of h/s(T) vs √sNN
w/o minijet P (t0) =0
10-20%
f.o.
Plumari, Greco,Csernai, arXiv:1
 4πη/s=1 during all the evolution of the fireball -> no invariant v2(pT)
-> smaller v2(pT) at LHC.
 Initial pT distribution relevant (in hydro means p(t0)  0, but it is not done!
Impact of h/s(T) vs √sNN
Plumari, Greco,Csernai, arXiv:1
 η/s ∝ T2 too strong T dependence→ a discrepancy about 20%.
 Invariant v2(pT) suggests a “U shape” of η/s with mild increase in QGP
See also, Niemi-Denicol et al., PRL106 (2011)
Viscous correction
Terminology about freeze-out
Freeze-out is a smooth process: scattering rate < expansion rate
 h/s increases in the cross-over
region, realizing the smooth f.o.:
small s -> natural f.o.
 Different from hydro that is a
sudden cut of expansion at
some Tf.o.
No f.o.
s tr »
1 p 1
15 n h / s
Comparison for anisotropic cross section
Similarly to h for anisotropic cross section the RTA with str underestimate