QGP Shear Viscosity & Electric Conductivity
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QGP Shear Viscosity & Electric Conductivity
A. Puglisi - S. Plumari - V. GrecoUNIVERSITY of CATANIA - INFN-LNS
Mainly based on next weekend arXiV submission
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Outline
Shear Viscosity and Electric Conductivity: Comparison of sel/T with recent lQCD data Ratio (h/s)/(sel/T): disentangling q and g
interaction?!
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]
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Shear viscosity h -> anisotropic flow vn
Shear Viscosity regulates:
How the fluid drag itself in the transverse direction -> damping of anisotropies vn=<cos(nf)>
Entropy production
yv
AF x
yz
x
h
h/s0 h/s0.16 h/s smoothen fluctuations and affect more higher harmonics
Green-Kubo
Operative definition
B.Schenke
B. Schenke, PRC85(2012)
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Electric Conductivity sel regulates:
Damping of Magnetic Field in HIC t ≈ sel L2
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 Kapusta ’93
Insight into quark vs gluon scattering rates
Electric ConductivityGreen-kubo Ohm’s Law
s=0
slQCD
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Electric Conductivity sel regulates:
Damping of Magnetic Field in HIC t ≈ sel L2
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 Kapusta ’93
Insight into quark vs gluon scattering rates
Electric ConductivityGreen-kubo Ohm’s Law
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Relativistic Boltzmann Equation
CollisionsField Interaction Free streamingfq,g(x,p) is a one-body distribution function for quark and gluons
Collision rate
Rate of collisionsper unit time and
phase space
Solved discretizing the space in (h, x, y)a cells
t0 3x0
exact solution
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Transport at fixed shear viscosity
snpTr trtr /
1151)),(( , h
ssa
aa
a=cell index in the r-space
Space-Time dependent cross section evaluated locally
V. Greco at al., PPNP 62 (09)G. Ferini et al., PLB670 (09)
Relax. Time Approx. (RTA) Transport simulation
Usually input of a transport approach are cross-sections and fields, but here we reverseit and start from h/s with aim of creating a more direct link to viscous hydrodynamics
str is the effective cross section
Huovinen-Molnar, PRC79(2009)
Convergency to IS ViscousHydro for large K
1+1D expansion
One maps with C[f] the phase space evolution of a fluid at fixed h/s !
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El, Xu, Greiner, Phys.Rev. C81 (2010) 041901
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, Stricklandfor anisotropic hydordynamicsarXiv:1311.6720 [nucl-th]
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Do we really have the wanted shear viscosity hwith the relax. time approx.?- Check h with the Green-Kubo correlator
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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
Shear Viscosity in Box Calculation
Needed very careful tests of convergencyvs. Ntest, xcell, # time steps !
macroscopic thermodynamics
microscopic scatterings
η ↔ σ(θ), , M, T …. ?
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for a generic cross section:
Non Isotropic Cross Section - s(q)
Chapmann-Enskog (CE)
CE and RTA can differ by about a factor 2 Green-Kubo agrees with CE
Green-Kubo in a box - s(q)
mD regulates the angular dependence
Relaxation Time Approximation
g(a) correct function that fix the momentum transfer for shear motion
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) …
S. Plumari et al., PRC86(2012)054902
h(a)=str/stot weights cross section by q2
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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
Viscosity of a pQCD gluon plasma
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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
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Applying kinetic theory to A+A Collisions….
xy z
px
py
- Impact of h/s(T) on the build-up of v2(pT)
Hydro Transport
Extend to Higher pT
Larger h/s
Initial off-equilibrium
pT ≈3T
h/s<<1M. Ruggeri’s talk – this afternoon
Heavy QuarksS.K. Das talk – tomorrow afternoon
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Bhalerao et al., PLB627(2005) v 2/e
Time rescaled
Ideal -Hydro
In the bulk the transport has an hydro v2/e2 response!
Test in 3+1D: v2/e response for almost ideal case EoS cs
2=1/3 (dN/dy tuned to RHIC)
Transport at h/s fixed
Integrated v2 vs time
Just one tip on what can be studied with a transport at fixed h/s:
impact of power law spectrum at intermediate pT
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- 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
Non equilibrium at larger pT:impact of minijets on v2(pT)
minijets
J.Y. Ollitrault, Plumari, VG, in preparation
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Electric Conductivity in a Box with boundary condition
Ohm’s Law method
See also Cassing et al., PRL110 (2013) + Moritz talk this afternoon
Jz/Ez independent on Ez -> one can define the conductivity
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Comparing with Green-Kubo correlator
Green-Kubo
Ohm’s Law
RTA with ttr
Similarly to h for anisotropic cross section the RTA with str underestimate sel
Isotropic
i=u,d,s,gj=u,d,s
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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
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WB=0 guarantees Thermodynamicaly consistency
Simple QP-model fitting lQCD
g(T) from a fit to e from lQCD -> good reproduction of P, e-3P, cs
Plumari, Alberico, Greco, Ratti, PRD84 (2011)
l=2.6 Ts=0.57 Tc
g(T) practically identical to DQPM
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Electric Conductivity of the QGP
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?
i=u,d,s,gJ=u,d,s
bqq=16/9 bqq = 8/9 bgg =9 bqg=2
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Shear Viscosity to Entropy Density
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
i, j=u,d,s,g
Kapusta ’93
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Electric Conductivity of the QGP
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
sel is strongly T- dependent
bqq=16/9 bqbarq = 8/9 bgg =9 bqg=2 Ads/CFT
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Relation between Shear Viscosity and Conductivity
So one expects:
Steep rise of sel just above Tc even if the h/s is nearly T independent
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h/s to sel /T ratio
Fixed by the lQCDthermodynamics
Depending on the relative quark to gluon relaxation time
Practically unknown!
= 28/9= 9/2
Relaxation times
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h/s to sel /T ratio
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
Symbols are dividing lQCD datah/s for the lowest sel/T
Enhancement of scattering
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h/s to sel /T ratio
Symbols are dividing lQCD data:
- Highest h/s for lowest sel/T
- Lowest h/s highest sel/T
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
Overestimate
Underestimate
Warning: we are consideringlQCD quenched, unquenchedand with different actions and Tc
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h/s to sel /T ratio
AdS/CFT would predict a flat behavior Agreement with DQPM confirm the ratio There could be even a structure
AdS/CFT
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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
Summary
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Width has small impact on thermodynamics?
DQPM: E. Braktovskaya et al.,NPA856 (2011) 162 QP: Plumari et al., PRD84 (2011)
Both fit to WB-lQCD data
DQPM
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Chapmann-Enskog vs Green Kubo:massive case
Massive case is relevant in quasiparticle models where Mq,g(T)=g(T)THence we need it to extend the approach to Boltzmann-Vlasov transport
Again good agreement with CE 1st order for s(q)=cost.
Still missing Chapmann-Enskog for massive & anisotropic cross section
z=M/T
Isostropic s – massive particles
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Viscous Hydrodynamics
ffTT eqeq dd
Asantz used
eqfTpp
Pf 2
epd
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
dissipidealTT P
K. Dusling et al., PRC81 (2010)
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h/s(T) shear viscosity or details of the cross section?
Keep same h/s means:
for mD=1.4 GeV -> 25% smaller stot
for mD=5.6 GeV -> 40% smaller stot
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!
cross section
Does the microscopic degrees of freedom matter once P(e) and h/s is fixed?
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h/s(T) shear viscosity or details of the cross section?
Keep same h/s means:
for mD=1.4 GeV -> 25% smaller stot
for mD=5.6 GeV -> 40% smaller stot
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!
cross section
Does the microscopic degrees of freedom matter once P(e) and h/s is fixed?
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h/s(T) shear viscosity or details of the cross section?
Keep same h/s means:
for mD=1.4 GeV -> 25% smaller stot
for mD=5.6 GeV -> 40% smaller stot
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!
cross section
Does the microscopic degrees of freedom matter once P(e) and h/s is fixed?
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Standard Initial Conditions r-space: standard Glauber modelh=y Bjorken boost invariance (flexible)p-space: Boltzmann-Juttner Tmax [pT<2 GeV ]+ minijet [pT>2-3GeV]
Tmax0 = 340 MeV
T0 t0 =1 -> t0=0.6 fm/c
We fix maximum initial T at RHIC 200 AGeV
Then we scale r-profile according to initial e
62 GeV 200 GeV 2.76 TeVT0 290 MeV 340 MeV 590 MeV
t 0 0.7 fm/c 0.6 fm/c 0.3 fm/c
Typical hydrocondition
Discarded in viscous hydro
and with beam energy according to dN/dy
No fine tuning
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Impact of h/s(T) vs √sNN
4 /s=1 during all the evolution of the fireball -> no invariant vπη 2(pT)
-> smaller v2(pT) at LHC.
Initial pT distribution relevant (in hydro means p(t0) ≠ 0, but it is not done!
w/o minijet P (t0) =0
Plumari, Greco,Csernai, arXiv:1304.6566
10-20%
f.o.
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Impact of h/s(T) vs √sNN
/s Tη ∝ 2 too strong T dependence→ a discrepancy about 20%. Invariant v2(pT) suggests a “U shape” of /s with η mild increase in QGP
Plumari, Greco,Csernai, arXiv:1304.6566
See also, Niemi-Denicol et al., PRL106 (2011)
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Viscous correction
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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.
Terminology about freeze-out
No f.o.
Freeze-out is a smooth process: scattering rate < expansion rate
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Comparison for anisotropic cross section
Similarly to h for anisotropic cross section the RTA with str underestimate sel