QGP Shear Viscosity & Electric Conductivity

A. Puglisi - S. Plumari - V. GrecoUNIVERSITY of CATANIA - INFN-LNS

Mainly based on next weekend arXiV submission

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]

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)

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

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

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

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 !

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]

Do we really have the wanted shear viscosity hwith the relax. time approx.?- Check h with the Green-Kubo correlator

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 …. ?

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

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

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….

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

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

- 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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?

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?

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?

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

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.

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)

Viscous correction

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

Comparison for anisotropic cross section

Similarly to h for anisotropic cross section the RTA with str underestimate sel