Anisotropic flow in a Boltzmann kinetic approach at fixed h/s(T)

V. Greco UNIVERSITY of CATANIAINFN-LNS

S. PlumariM. RuggieriF. Scardina

nfQCD 2013 – Kyoto, 2-6 December 2013

Outline

Two main results for HIC: Elliptic flow from Color Glass Condensate (fKLN)

going beyond ex and implementing also the p-space with the Qs saturation scale

Are there hints of h/s(T) ?

Transport Kinetic Theory at fixed h/s : Motivations How to fix locally h/s

Relativistic Boltzmann-Vlasov approach

Collisions -> h≠0Field Interaction (EoS) Free streaming

One can expand over microscopic details (2<->2,2<->3…), but in a hydro language this is irrelevant only the global dissipative effect of C[f] is important!

f(x,p) is the one-body distribution function

- C[feq+df] ≠ 0 deviation from ideal hydro (finite l or h/s)

- We map with C[f] the phase space evolution of a fluid at fixed h/s !

Starting from 1-body distribution function f(x,p) and not from Tμν:- Include off-equilibrium at high and intermediate pT

- Implement non-equilibrium implied by CGC-Qs scale (beyond ex)- Relevant at LHC due to large amount of minijet production - freeze-out self-consistently related with h/s(T)

It’s not a gradient expansion h/s: - valid also at high h/s -> LHC (T>>Tc) or cross-over region (T≈ Tc)

Appropriate for heavy quark dynamics (next week talk)

Motivation for Transport approach

Collisions -> h≠0Field Interaction Free streaming

Simulate a fixed shear viscosityUsually 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

g(a=mD/2T) correct function that fixes the relaxation time for the shear motion

Chapman-Enskog agrees with Green-Kubo!

Chapmann-Enskog

S.Plumari et al., PRC86(2012)

h CE good one !

Chapmann-Enskog

Viscosity from Green-Kubo

Correlator estimate in a Box at fixed T

Simulate a fixed shear viscosity

=cell index in r-space

Space-Time dependent cross section evaluated locally

Transport code

Viscosity fixed varying s

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

g(a=mD/2T) correct function that fix the relaxation time for the shear motion

Chapman-Enskog agrees with Green-Kubo

Chapmann-Enskog

G. Ferini et al., PLB670 (2009)

S.Plumari et al., PRC86(2012)

h CE good one !

Chapmann-Enskog

Transport at fixed h/s vs Viscous Hydro in 1+1D

l

l

LK

Knudsen number-1

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

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

Large K small h/s

Denicol et al. have studied derivation of viscous hydro from Boltzmann kinetic theory: PRD85 (2012) 114047

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

h/s or details of the cross section?

Keep same h/s means:

for mD=0.7 GeV -> factor 2 larger stot

is needed respect to isotropic case

cross section

Keep same h/s means:

h/s or details of the cross section?cross section

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=0.7 GeV -> factor 2 larger stot

is needed respect to isotropic case

4ph/s=1

Keep same h/s means:

h/s or details of the cross section?

for mD=1.4 GeV -> 25% smaller stot

for mD=5.6 GeV -> 40% smaller stot

cross section

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!

Keep same h/s means:

h/s or details of the cross section?cross section

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!

eqfTpp

Pf 2

epd

Differences arises just where in viscous hydro df becomes relevant

No Fine tuning! Employed the relaxation time approximation!

Natural extension from low to high pT

Boltzmann transport describes rise and fall of v2(pT)

Transition between low and high pT in a unified framework!

partons

High

ly n

on-p

QCD

Disappearing of non-pQCD

pQCD limit

S. Plumari and VG, EPIC@LHC, AIP1422(2012)- arXiV:1110.4138 [hep-ph]

s=0.3 and mD=0.7 GeV

pQCD limit, but with s=0.3- No Q2 dependence & - No radiative part

Renormalize s to fix h/s

L ≈ 4-5 GeV, g fixed by h/s

What about Color Glass condensate initial state?

- Kinetic Theory Qs saturation scale

xy z

px

py

Unintegrated distributionfunctions (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]

Factorization hypothesis:convolution of parton distribution functions in the parent nucleus.

fKLN realization of CGC

pT

dN/d

2 pT

Qsat(s)

p-space x-space

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

CGC-KLN ex ≈ 30% larger than Glauber

T. Hirano et al., PLB636(06)

Kharzeev et al., PLB561, 93 (2003)Nardi et al., PLB507, 121 (2001)Drescher et al, PRC75, 034905 (2007)Hirano et al., PRC79, 064904 (2009)Albacete and Dumitru, arXiv:1011.5161…

V2 from KLN in HydroWhat does it KLN in hydro?

1) r-space from KLN (larger ex)2) p-space thermal at t0 ≈0.6-0.9 fm/c - No Qs scale , We’ll call it fKLN-Th

Glauber h/s = 0.08CGC-KLN h/s=0.16

Larger ex - > higher h/s to get the same

v2(pT)

See also:Alver et al., PRC 82, 034913 (2010)Heinz et al., PRC 83, 054910 (2011)

Luzum and RomatschkePRC78(2008) 034915

Thermalization in less than 1 fm/c, in agreement with Greiner et al., NPA806, 287 (2008).Not so surprising: h/s is small -> large effective scattering rate -> fast thermalization.

Implementing KLN pT distribution

Using kinetic theorywe can implement full KLN (x & p space) - ex=0.34, Qs =1.4 GeV

KLN only in x space ( like in Hydro)ex=0.341, Qs=0 -> Th-KLN

AuAu@200 GeV – 20-30%

Glauber in x & thermal in pex=0.289 , Qs=0 -> Th-Glauber

M. Ruggieri et al., Phys.Lett. B727 (2013) 177

Longitudinal and transverse pressure

PL/PT show also a very fast equilibration (isotr≈0.3 fm/c)!

However it is not this that makes a difference for v2: isotropization time very similar for all the cases

t=1/Qs≈0.1 fm/c-> PL/PT > 0Gelis & EpelbaumarXiV:1307.2214

Longitudinal and transverse pressure

For h/s > 0.3 one misses fast isotropization in PL/PT ( ≥ 2 fm/c)For h/s ≈ pQCD no isotropizationSemi-quantitative agreement with Florkowski et al., PRD88 (2013) 034028

our is 3+1D not in relax.time but full integral but no gauge field

≈ pQCD

80% level of isotropization

Hydro - like Full x & pAuAu@200 GeV

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

Results with kinetic theory

M. Ruggieri et al., Phys.Lett. B727 (2013) 177 - 1303.3178 [nucl-th]

We clearly see that when non-equilibrium distribution is implementedin the initial stage (≤ 1 fm/c) v2 grows slowly respect to thermal one

What is going on?

Evolution with Centrality

The difference fKLN , Th-fKLN and Th-Glauber disappears at central collisions (like in hydro for Th-fKLN and Th-Glauber)

In peripheral collisions fKLN would even be lower than Th-Glauber due to non-equilibrium impact

Applying kinetic theory to A+A Collisions….

xy z

px

py

Part II

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

h/s increases in the cross-over region, realizing a smooth f.o. self-consistently dependent on h/s:

Different from hydro that is a sudden cut of expansion at some Tf.o.

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

h/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 (cpT) 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].

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.

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 minijets relevant at LHC for pT>1.5 GeV !

LHC: almost insensitivity to cross-over (≈ 5%) : v2 from pure QGP Without h/s(T) increase T≤Tc we would have

v2(LHC)<v2(RHIC)

w/o minijet

Plumari, Greco,Csernai, arXiv:1304.6566

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

Hope: vn, n>3 with an ev.-by-ev. analysis put even stronger constraints

Plumari, Greco,Csernai, arXiv:1304.6566

However for a definite statement needed hadronization + EoS-lQCD

Development of kinetic at fixed h/s(T) : Verified that up to ≈ 1.5-2 GeV:

v2 <-> h/s , cross section microscopic details irrelevant

For a fluid at h/s <0.1 -> PL /PT > 0.8 at isotr <0.3 fm/c

Invariant v2(pT) from RHIC to LHC: Hints of the fall and rise of h/s(T) from BES?!

Sensitivity of v2 to /s(T) anyway quite weak. η

Studying the CGC (fKLN): Non-equilibrium implied by Qs damps v2(pT)

compensating the larger ex ->

v2(pT) can be described by 4ph/s ≈1 also for fKLN

Summary

Outlookfor the kinetic theory approach

Include initial state fluctuations to study vn:

more constraints on h/s(T) and initial state?

something more or new at pT ≈2-4 GeV ?

pA …

Endeavor already undertaken….

Hadronization: statistical model +Cooper-Frye vs. coalescence + fragm.

Field Dynamics:Realistic EoS: M(T) + Bag mean field dynamics [Done!]

Next step –

Include Initial State Fluctuations(Preliminary results)

Monte Carlo Glauber

s = 0.5 fm

G-Y. Qin, H. Petersen, S.A. Bass and B. Muller, PRC82,064903 (2010)H.Holopainen, H. Niemi and K.J. Eskola, PRC83, 034901 (2011)

v2 and v3 linearly correlated to the corresponding eccentricities ε2 and ε3

Initial State Fluctuations: vn vs εn (Preliminary)

C(2,2)=0.93 C(3,3)=0.684ph/s=1 ≈ 5

v4 and ε4 weak correlated similar to hydro calculations: F.G.Gardim,F.Grassi,M.Luzum and J.Y.Ollitrault NPA904 (2013) 503. Niemi, Denicol, Holopainen and Huovinen PRC87(2013) 054901.

Initial State Fluctuations: vn vs εn (Preliminary)

C(2,2)=0.93 4ph/s=1 ≈ 5 C(3,3)=0.23

General agreement with hydro Niemi et al. PRC87(2013), but:- h/s not constant (include cross-over increase)- 3+1 D not 2+1D- s =0.5 fm not 0.8 fm (if relevant at all!)

Like in viscous hydro the data of vn(pT) at RHIC energies aredescribed with 4πη/s=1 ≈ 5

Data taken from: A. Adare et al. [PHENIX collaboration], Phys.Rev. Lett. 107, 252301 (2011).

Initial State Fluctuations: vn(pT) (Preliminary)

Fluctuations allows to extend the studies on impact of CGC-Qs, h/s(T) and study pA …

Distribution function & occupation number

f >1 need of Bose-Einstein statistics (1+f) terms needed in the collision integral -> possibility of Bose Condensate induced by CGCJ. -P. Blaizot et al., arXiv:1305.2119 [hep-ph]; J. -P. Blaizot et al., Nucl. Phys. A904-905 2013, 829c (2013).

At LHC and low pT it could be there and CYM helps!

f >1 at pT> 0.5 our effect manifest at larger pT

Longitudinal expansion lowers average density by -1

but f(p) goes down slowly at pT<0.5 GeV

One should include also gluon to quark conversion

f ≈ 1/ + slope change

Back-up

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

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

macroscopic observables

microscopic details

η ↔ σ(θ), r, 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 2-3 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 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)=str/stot weights cross section by q2

pT

dN/d

p

along x, q=0

along y, q=90°

dp

pT

dN/d

p

along x, q=0along y, q=90°

dp

Put it very simplistic:

If the momentum dp shift is the samewe can expect flatter distributionare less efficient in build-up v2

What is v2?

smaller v2

What happens at LHC?Hydro -like Full KLN x & p PbPb@2.76 TeV

At LHC the larger saturation Qs ( ≈ 2.5 GeV) scale makes the effect larger: - 4ph/s= 2 not sufficient to get close to the data for Th-KLN, but it is sufficient if one implements both x &p Full fKLN implemention change estimate of h/s by about a factor of 3/2

With h/s(T) T2 we cannot reproduce the data:- RHIC we can recover the spectra by lowering T0by a 30 MeV- LHC impact of minijets too large one would need a T0 ≈340 MeV ≈ RHIC

pT-spectra versus h/s(T)

Minijets starts to affect v2(pT)For pT>1.5 GeVEffect non-negligible

Impact of minijets on v2(pT)

Larger sensitivity on h/s (T) at LHCEffect larger respect to viscous hydro, but this depends also on df

Sensitivity in transport using sameη/s(T)

Hydr

o-Ni

emi

Viscous Hydrodynamics

ffTT eqeq dd An Asantz (Grad)

eqfTpp

Pf 2

epd

- at pT~3 GeV !? df/f≈ 5- this implies RTA and not CE

I0 Navier-Stokes, but it violates causality, II0 order needed -> Israel-Stewart

Problems: 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-eq (ex. minijets)

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

dissipidealTT P

K. Dusling et al., PRC81 (2010)

Transport at fixed h/s vs Viscous Hydro a test in 3+1D

Changing M of partons one gets different EoS – cs(T)

v2/ex (0)decrease with cs

Time scales, trends and value quite similar to hydro evolution

An exact comparison under the same conditions has not been done

Au+Au@200AGeV

stot=15 mb

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

Tmax0 = 340 MeV

T0 0 =1 -> 0=0.6 fm/c

We fix maximum initial T at RHIC 200 AGeV

Then we scale it according to initial e

62 GeV 200 GeV 2.76 TeVT0 290 MeV 340 MeV 580 MeV

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

Typical hydrocondition

Discarded in viscous hydro

Multiplicity & Spectra r-space: standard Glauber conditionp-space: Boltzmann-Juttner Tmax=2(3) Tc [pT<2 GeV ]+ minijet [pT>2-3GeV]

No fine tuning

Risultati di Heinz da “hadron production………”

Larger ex - > higher h/s to get the same

v2(pT)

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

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

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