Elliptic flow from kinetic theory at fixed Elliptic flow from kinetic theory at fixed /s(T)/s(T)

V. Greco V. Greco UNIVERSITY of CATANIAUNIVERSITY of CATANIAINFN-LNSINFN-LNS

S. PlumariA. PuglisiM. RuggieriF. Scardina

Padova, 22 May 2013 - ALICE PHYSICS WEEK

OutlineOutline

Two main results for HICTwo main results for HIC: Are there signatures of a phase transition in Are there signatures of a phase transition in /s(T) ?/s(T) ? Elliptic flow in Color Glass Condensate (fKLN) Elliptic flow in Color Glass Condensate (fKLN) beyond beyond x x !! implementing both x & p space…implementing both x & p space…

Transport Kinetic Theory at fixed Transport Kinetic Theory at fixed /s :/s : MotivationsMotivations How to fix locally How to fix locally /s: /s: Green-Kubo vs Chapmann-Enskog Green-Kubo vs Chapmann-Enskog && Relax Time Relax Time

Approx.Approx.

Relativistic Transport Relativistic Transport approachapproach

Collisions -> Collisions -> ≠0≠0Field Interaction Free streaming

It’s not a gradient expansion for small It’s not a gradient expansion for small /s/s

One can include an expansion over microspic details, but in a hydro languagethis is irrelevant only the global dissipative effect of C[f] is important!

f(x,p) is a one-body distribution function or a classical field

f0(p) =Boltzmann -> C[f0]=0 -> ideal hydrodynamics

C[f0+f] ≠ ≠ 0 deviation from ideal hydro (finite or /s)

Highly non-equilibrated distributions

This is a problem that cannot be treated in hydrodynamicsthat assumes locally an equilibrium distribution!

A simple example of kinetic theory applicationParticles in a box

Going to equilibriumE/N=6 GeV -> T=2 GeV

valid also at intermediate & high pvalid also at intermediate & high pTT out of equilibrium out of equilibrium valid also at high valid also at high /s/s-> -> LHC - LHC - /s(T), cross-over region /s(T), cross-over region CGC pCGC pTT non-equilibrium phase non-equilibrium phase (beyond the difference in (beyond the difference in xx)): : Relevant at LHC due to large amount of minijet production Relevant at LHC due to large amount of minijet production Appropriate for heavy quark dynamics Appropriate for heavy quark dynamics assuming small q transfer -> Fokker-Planck eq. (Bassuming small q transfer -> Fokker-Planck eq. (Beeraudo’s talk)raudo’s talk)

A unified framework against a separate modelling with a wide range of validity in pT + microscopic level

Motivation for Transport approachMotivation for Transport approach

Collisions -> Collisions -> ≠0≠0Field Interaction Free streaming

Simulate a fixed shear viscositySimulate a fixed shear viscosity

snpTr trtr /

1151)),(( ,

=cell index in the r-space

Space-Time dependent Space-Time dependent cross section evaluated cross section evaluated locallylocally

V. Greco at al., PPNP 62 (09)G. Ferini et al., PLB670 (09)

Relax. Time Approx. (RTA) Transport simulation

Au+Au 200 GeV

Viscosity fixed varying

Usually input of a transport approach are cross-sections and fields, but here we reverseit and start from/s with aim of creating a more direct link to viscous hydrodynamics

tr is the effective cross section

Transport vs Viscous Hydrodynamics in 1+1D

LK

sTK/5

1

Knudsen number-1

Comparison for the relaxation of pressure anisotropy PL/PT Huovinen and Molnar, PRC79(2009)

In the limit of small /s (<0.16) transport reproduce viscous hydro at least for the evolution pL/pT

Large K small /s

Viscous Hydrodynamics

ffTT eqeq Asantz used

eqfTpp

Pf 2

- at pT~3 GeV !? f/f≈ 5- this implies Relax. Time and not Chap.Enskog

I0 Navier-Stokes, but it violates it violates causality, causality, IIII00 order needed -> Israel-Stewart order needed -> Israel-Stewart

Problems related to Problems related to ff:dissipative correction to f -> feq+fneq just an ansatz

fneq/f at pT> 1.5 GeV is large

fneq <-> /s implies a RTA approx. (solvable)

(t0) =0 -> discard initial non-equil. (ex. minijets)

pT -> 0 no problem except if /s is large

dissipidealTT

K. Dusling et al., PRC81 (2010)

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

Part I

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).

Shear Viscosity in Box CalculationShear Viscosity in Box CalculationGreen-Kubo correlator

Needed very careful tests of convergencyvs. Ntest, xcell, # time steps !

macroscopic macroscopic observablesobservables

microscopic details

η ↔ σ(θ), η ↔ σ(θ), , M, T , M, T ….…. ? ?

for a generic cross section:

Non Isotropic Cross Section - Non Isotropic Cross Section -

CE and RTA can differ by about a factor 2-3 Green-Kubo agrees with CE

Green-Kubo in a box -

mD regulates the angular dependence

g(a) correct function that fix the momentum transfer for shear motion

RTA is the one usually emplyed to make theroethical 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)=tr/tot weights cross section by q2

We know how to fix locally /s(T)

We have checked the Chapmann-Enskog: - CE good already at I° order ≈ 5% (≈ 3% at II° order)

- RTA even with tr severely underestimates

Applying kinetic theory to A+A Collisions….

xy z

px

py

Part II

- Impact of /s(T) on the build-up of v2(pT) vs. beam energy

/s increases in the cross-over /s increases in the cross-over region, realizing the smooth f.o.: region, realizing the smooth f.o.: small small -> natural f.o. -> natural f.o.

DDifferent from hydro that is a ifferent from hydro that is a sudden cut of expansion at sudden cut of expansion at some Tsome Tf.o.f.o.

Terminology about freeze-out Terminology about freeze-out

No f.o.

Freeze-out is a smooth process: scattering rate < expansion rate

RHICB=7.5 fm

RHIC

LHC

RHIC

LHC

RHIC: /s increase in the cross-over region equivalent to double /s in the QGP LHC: almost insensitivity to cross-over (≈ 5%) : v2 from pure QGP,

but at LHC less sensitivity to T-dependence of /s? Without /s(T) increase T≤Tc we would have v2(LHC) < v2(RHIC)

First application: f.o. at RHIC & LHCFirst application: f.o. at RHIC & LHC

/s(T) close to a phase transition/s(T) close to a phase transition

I0 order

@ T c

cross-over I0 order

@ T c

cross-over

I0 order

@ T c

cross-over

P. Kovtun et al.,Phys.Rev.Lett. 94 (2005) 111601.L. P. Csernai et al., Phys.Rev.Lett. 97 (2006) 152303.R. A. Lacey et al., Phys.Rev.Lett. 98 (2007) 092301.

Uncertainty Principle15/1/1 stE

Text book

AdS/CFT suggest a lower bound

But do we have signatures of a “U” shapeof /s(T) typical of a phase transition?

QGP close to this bound!

pQCD at finite T

pQCD

/s(T) for QCD matter/s(T) for QCD matter lQCD some results for quenched approx. (large error bars) A. Nakamura and S. Sakai, PRL 94(2005) H. B. Meyer, Phys. Rev. D76 (2007)

Quasi-Particle models seem to suggest a η/s~Tα, α ~ 1 – 1.5. S.Plumari et al., PRD84 (2011) M. Bluhm , Redlich, PRD (2011)

Chiral perturbation theory (pT) M. Prakash et al. , Phys. Rept. 227 (1993) J.-W. Chen et al., Phys. Rev. D76 (2007)

Intermediate Energies – IE ( μB>T) W. Schmidt et al., Phys. Rev. C47, 2782 (1993) Danielewicz et al., AIP1128, 104 (2009)

(STAR Collaboration),arXiv:1206.5528 [nucl-ex].

r-space: standard Glauber modelp-space: Boltzmann-Juttner Tmax=1.7-3.5 Tc [pT<2 GeV ]+ minijet [pT>2-3GeV]

Tmax0 = 340 MeV

T=1 ->=0.6 fm/c

We fix maximum initial T at RHIC 200 AGeV

Then we scale it according to initial

62 GeV 200 GeV 2.76 TeVT0 290 MeV 340 MeV 590 MeV

0 0.7 fm/c 0.6 fm/c 0.3 fm/c

Typical hydrocondition

Discarded in viscous hydro

Impact of Impact of /s(T) vs √s/s(T) vs √sNNNN

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 ≠but it is not done! Notice only with RHIC → almost scaling for 4 /s=1 LHC data play a key roleπη

w/o minijet

Plumari, Greco,Csernai, arXiv:1304.6566

10-20%

Impact of Impact of /s(T) vs √s/s(T) vs √sNNNN

/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

Hope: vn, n>3 with an event-by-event analysis will put even stronger

constraints

Plumari, Greco,Csernai, arXiv:1304.6566

Enhancement of /s(T) in the cross-over region affect differentlyη build-up of v2(pT) at RHIC to LHC.

At LHC nearly all the v2 from the QGP phase

Scaling of v2(pT) from Beam Energy Scan indicate a 'U' shape of /s(T): ηa first signature of /s(T) behavior typical of a η phase transition

Summary Part II

What about Color Glass condensate initial state?

- Kinetic Theory with a Qs saturation scale

Part III

Saturation scale

pT

dN/d

2 pT

Qsat(s)

At RHIC Q2 ~ 2 GeV2

At LHC Q2 ~ 5-8 GeV2 ?

Color Glass CondensateColor Glass Condensate

3/12

222 ),()()( A

RQxxgQsQ ssat

Specific of CGC is a non-equilibrium distribution function !

Unintegrateddistributionfunctions(uGDFs)

Saturation scale Qs depends on:1.) position in transverse plane;2.) gluon rapidity.

Nardi et al., Nucl. Phys. A747, 609 (2005)Kharzeev et al., Phys. Lett. B561, 93 (2003)Nardi et al., Phys. Lett. B507, 121 (2001)Drescher and Nara, PRC75, 034905 (2007)Hirano and Nara, PRC79, 064904 (2009)Hirano and Nara, Nucl. Phys. A743, 305 (2004)Albacete and Dumitru, arXiv:1011.5161[hep-ph]Albacete et al., arXiv:1106.0978 [nucl-th]

(f)KLN spectrumFactorization hypothesis:convolution of the distributionfunctions of partonsin the parent nucleus.

fKLN realization of CGC fKLN realization of CGC

pT

dN/d

2 pT

Qsat(s)

p-space x-space

x(fKLN)=0.34x(Glaub.)=0.29

V2 from KLN in Hydro

Heinz et al., PRC 83, 054910 (2011)

What does it KLN in hydro?1) r-space from KLN (larger ex)2) p-space thermal at t0 ≈0.8 fm/c - No Qs scale , We’ll call it fKLN-Th

Risultati di Heinz da “hadron production………”

Larger x - > higher /s to get the same v2(pT)

Uncertainty on initial conditions implies uncertainty of a factor 2 on /s

Similar conclusion inDrescher et al., PRC (2011)

V2 from KLN in HydroWhat does it KLN in hydro?1) r-space from KLN (larger ex)2) p-space thermal at t0 ≈0.8 fm/c - No Qs scale , We’ll call it fKLN-Th

Higher /s for KLN leads to small v3

The value of /s affects more higher harmonics!

Adare et al., [PHENIX Collaboration], PRL 107, 252301 (2011)

Can we discard KLN or CGC?!Well at least before one should implement both x and p space

Thermalization in less than 1 fm/c, in agreement with Greiner et al., NPA806, 287 (2008).Not so surprising: /s is small large scattering rates which naturally lead to fast thermalization.

Implementing KLN pImplementing KLN pTT distribution distribution

Using kinetic theory at finite /swe can implement full KLN (x & p space) - x=0.34, Qs =1.44 GeV

KLN only in x space ( like in Hydro)x=0.34, Qs=0

AuAu@200 GeV – 20-30%

Glauber in x & thermal in px=0.289 , Qs=0

Hydro - like Full x & pAuAu@200 GeV

When we implement KLN and Glauber like in Hydro we get the same When we implement full KLN we get close to the data with 4/s =1 : larger x compensated by Qs saturation scale (non-equilibrium distribution)

Results with kinetic theoryResults with kinetic theory

M. Ruggieri et al., 1303.3178 [nucl-th]

AuAu@200 GeV

M. Ruggieri et al., 1303.3178 [nucl-th]

We clearly see that when the non-equilibrium distribution is implementedIn the initial stage (1 fm/c) v2 grows slowly then distribution is thermaland it grows faster.

V2 normalized time evolution

What is going on?

At LHC the larger saturation Qs ( ≈ 2.4 GeV) scale makes the effect larger: - 4/s= 2 not sufficient to get close to the data for Th-KLN - 4/s=1 it is enough if one implements both x &p Full fKLN implemention change the estimate of /s by about a factor of 3

Hydro -like Full KLN x & p PbPb@2.76 TeV

What happens at LHC?

preliminary

Development of kinetic at fixed Development of kinetic at fixed /s(T) :/s(T) :Relax. Time Approx can severely underestimate /s

Chapmann-Enskog I°order agree with Green-Kubo

Invariant vInvariant v22(p(pTT) from RHIC to LHC:) from RHIC to LHC:It is a signature of the fall and rise of /s(T) which a signature of the phase transition

Studying the CGC (fKLN):Studying the CGC (fKLN):Initial non-equilibrium distribution implied by CGC

damps the v2(pT) compensating the larger x

v2(pT) can be described by 4/s ≈1

SummarySummary

OutlookOutlookfor a kinetic theory approachfor a kinetic theory approach

Include initial state fluctuations to study vn:

more constraints on /s(T) Impact of CGC non-equilibrium distribution

Include hadronization: statistical model like in hydro coalescence + fragmentation (going at high pT)

Heavy quark dynamics within the same framework: Fokker-Planck is the small transfer approximation is it always a reliable approxiamation?

Kinetic Equation for Heavy Quarksreduce to Fokker-Planck

(small momentum transfer)

Kinetic equation for small momentum transfer

The collision integral can be formally simplified in term of rate of collision w(p.k)

)( 2121)(, ppppv gqQrel

Using w(p,k) one can rewrite the C[f] in a form more suitable for an expansion in

See: B. Svetitsky, PRD 37 (1988) 2484

Small transfer momentum |k| << |p| due to MQ >>mq

Therefore the transport equation in small momentum transfer can be written as

ii pppA )()( jiij pppppB )()()(

Fokker-Planck equationFokker-Planck equationwwidely used to study HQ idely used to study HQ ddynamics in the QGPynamics in the QGP

ijij

ii

DpBpApA

)()(

Calculation in a Box at T=0.4 GeV