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

Elliptic flow from kinetic theory at fixed h /s(T). V. Greco UNIVERSITY of CATANIA INFN-LNS. S. Plumari A. Puglisi M. Ruggieri F. Scardina. Padova, 22 May 2013 - ALICE PHYSICS WEEK. Outline. Transport Kinetic Theory at fixed h /s : Motivations How to fix locally h /s:

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

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  1. Elliptic flow from kinetic theory at fixed h/s(T) V. Greco UNIVERSITY of CATANIA INFN-LNS S. Plumari A. Puglisi M. Ruggieri F. Scardina Padova, 22 May 2013 - ALICE PHYSICS WEEK

  2. Outline • Transport Kinetic Theory at fixed h/s : • Motivations • How to fix locally h/s: • Green-Kubo vsChapmann-Enskog& Relax Time Approx. What is η ↔ σ(θ), r, M, T, … ? • Two main results for HIC: • Are there signatures of a phase transition in h/s(T) ? • Elliptic flow in Color Glass Condensate (fKLN) beyond ex ! • implementing both x & p space…

  3. Relativistic Transport approach Collisions -> h≠0 Field Interaction Free streaming f(x,p) is a one-body distribution function or a classical field f0(p) =Boltzmann -> C[f0]=0 -> ideal hydrodynamics C[f0+df] ≠ 0 deviation from ideal hydro (finite l or h/s) • It’s not a gradient expansion for small h/s One can include an expansion over microspic details, but in a hydro language this is irrelevant only the global dissipative effect of C[f] is important!

  4. A simple example of kinetic theory application Particles in a box Highly non-equilibrateddistributions Going to equilibrium E/N=6 GeV -> T=2 GeV This is a problem that cannot be treated in hydrodynamics that assumes locally an equilibrium distribution!

  5. Motivation for Transport approach Collisions -> h≠0 Free streaming Field Interaction • valid also at intermediate & high pT out of equilibrium • valid also at high h/s-> LHC - h/s(T), cross-over region • CGC pTnon-equilibrium phase (beyond the difference in ex): • Relevant at LHC due to large amount of minijet production • Appropriate for heavy quark dynamics • assuming small q transfer -> Fokker-Planck eq. (Beraudo’s talk) A unifiedframeworkagainst a separate modelling with a wide range of validity in h, z,pT + microscopiclevel

  6. Simulate a 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 Transportsimulation Relax. Time Approx. (RTA) Space-Time dependent cross section evaluated locally str is the effective cross section =cellindex in the r-space Viscosityfixedvaryings G. Ferini et al., PLB670 (09) Au+Au 200 GeV V. Greco at al., PPNP 62 (09)

  7. Transport vs ViscousHydrodynamics in 1+1D Comparison for the relaxation of pressure anisotropy PL/PT Huovinen and Molnar, PRC79(2009) Knudsen number-1 Large K small h/s In the limit of small h/s (<0.16) transport reproduce viscous hydro at least for the evolution pL/pT

  8. Viscous Hydrodynamics I0Navier-Stokes, butitviolatescausality, II0orderneeded -> Israel-Stewart Asantzused K. Dusling et al., PRC81 (2010) - this implies Relax. Time and not Chap.Enskog - at pT~3 GeV !? df/f≈ 5 • Problemsrelated to df: • dissipative correction to f -> feq+dfneq just an ansatz • dfneq/fatpT> 1.5 GeVis large • dfneq<-> h/simplies a RTA approx. (solvable) • Pmn(t0) =0 -> discardinitial non-equil. (ex. minijets) • pT-> 0 no problemexceptifh/sis large

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

  10. Shear Viscosity in Box Calculation Green-Kubo correlator microscopic details macroscopic observables η ↔ σ(θ), r, M, T …. ? Needed very careful tests of convergency vs. Ntest, Dxcell, # time steps ! S. Plumari et al., arxiv:1208.0481;see also: Wespet al., Phys. Rev. C 84, 054911 (2011); FuiniIII et al. J. Phys. G38, 015004 (2011).

  11. Non Isotropic Cross Section - s(q) Relaxation Time Approximation 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) … for a generic cross section: h(a)=str/stot weights cross section by q2 Chapmann-Enskog (CE) mD regulates the angular dependence Green-Kubo in a box - s(q) g(a) correct function that fix the momentum transfer for shear motion • CE and RTA can differ by about a factor 2-3 • Green-Kubo agrees with CE S. Plumari et al., PRC86(2012)054902

  12. We know how to fix locally h/s(T) • We have checked the Chapmann-Enskog: • - CE good already at I° order ≈ 5% (≈ 3% at II° order) • - RTA even with str severely underestimates h

  13. z y x py px Part II Applying kinetic theory to A+A Collisions…. - Impact of h/s(T) on the build-up of v2(pT) vs. beam energy

  14. Terminology about freeze-out Freeze-out is a smooth process: scattering rate < expansion rate • /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. RHIC B=7.5 fm

  15. First application: f.o. at RHIC & LHC LHC LHC RHIC RHIC • RHIC: h/s increase in the cross-over region equivalent to double h/s in the QGP • LHC: almost insensitivity to cross-over (≈ 5%) : v2 from pure QGP, • but at LHC less sensitivity to T-dependence of h/s? • Without h/s(T) increase T≤Tc we would have v2(LHC) < v2(RHIC)

  16. h/s(T) close to a phase transition 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. Text book • Uncertainty Principle I0 order @ Tc • AdS/CFT suggest a lower bound cross-over I0 order QGP close to this bound! @ Tc pQCD cross-over • pQCD at finite T I0 order @ Tc But do we have signatures of a “U” shape of h/s(T) typical of a phase transition? cross-over

  17. 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].

  18. Initial Conditions • r-space: standard Glauber model • p-space: Boltzmann-JuttnerTmax=1.7-3.5 Tc[pT<2 GeV ]+ minijet[pT>2-3GeV] No fine tuning Discarded in viscous hydro 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 it according to initial e

  19. Impact of h/s(T) vs √sNN w/o minijet 10-20% Plumari, Greco,Csernai, arXiv:1304.6566 • 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 pmn(t0) ≠ 0, but it is not done!) • Notice only with RHIC → almost scaling for 4πη/s=1 LHC data play a key role

  20. Impact of h/s(T) vs √sNN Plumari, Greco,Csernai, arXiv:1304.6566 • 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 • Hope: vn, n>3 with an event-by-event analysis will put even stronger constraints

  21. Summary Part II • Enhancement of η/s(T) in the cross-over region affect differently • 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 η/s(T): a first signature of η/s(T) behavior typical of a phase transition

  22. Part III What about Color Glass condensate initial state? - Kinetic Theory with a Qs saturation scale

  23. dN/d2pT pT Qsat(s) Color Glass Condensate Saturation scale Specific of CGC is a non-equilibrium distribution function ! At RHIC Q2~ 2 GeV2 At LHC Q2~ 5-8 GeV2 ?

  24. dN/d2pT pT Qsat(s) fKLN realization of CGC (f)KLN spectrum Factorization hypothesis:convolution of the distributionfunctions of partonsin the parent nucleus. Unintegrateddistributionfunctions(uGDFs) x-space p-space 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] ex(fKLN)=0.34 ex(Glaub.)=0.29

  25. V2 from KLN in Hydro 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 Heinz et al., PRC 83, 054910 (2011)

  26. V2 from KLN in Hydro 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 Larger ex - > higher h/s to get the same v2(pT) Risultati di Heinz da “hadron production………” Uncertainty on initial conditions implies uncertainty of a factor 2 on h/s Similar conclusion in Drescher et al., PRC (2011)

  27. Higher h/s for KLN leads to small v3 Adare et al., [PHENIX Collaboration], PRL 107, 252301 (2011) The value of h/s affects more higher harmonics! Can we discard KLN or CGC?! Well at least before one should implement both x and p space

  28. Implementing KLN pT distribution AuAu@200 GeV – 20-30% Using kinetic theory at finite h/s we can implement full KLN (x & p space) - ex=0.34, Qs =1.44 GeV KLN only in x space ( like in Hydro) ex=0.34, Qs=0 Glauber in x & thermal in p ex=0.289 , Qs=0 Thermalization in less than 1 fm/c, in agreement with Greiner et al., NPA806, 287 (2008). Not so surprising: h/s is small large scattering rates which naturally lead to fast thermalization.

  29. Results with kinetic theory AuAu@200 GeV Hydro - like Full x & p M. Ruggieri et al., 1303.3178 [nucl-th] • 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 4ph/s =1 : • larger ex compensated by Qs saturation scale (non-equilibrium distribution)

  30. What is going on? V2 normalized time evolution AuAu@200 GeV M. Ruggieri et al., 1303.3178 [nucl-th] We clearly see that when the non-equilibrium distribution is implemented In the initial stage (1 fm/c) v2 grows slowly then distribution is thermal and it grows faster.

  31. What happens at LHC? PbPb@2.76 TeV Hydro -like Full KLN x & p preliminary • At LHC the larger saturation Qs ( ≈ 2.4 GeV) scale makes the effect larger: • - 4ph/s= 2 not sufficient to get close to the data for Th-KLN • - 4ph/s=1 it is enough if one implements both x &p • Full fKLN implemention change the estimate of h/s by about a factor of 3

  32. Summary • Development of kinetic at fixed h/s(T) : • Relax. Time Approx can severely underestimate h/s • Chapmann-Enskog I°order agree with Green-Kubo • Invariant v2(pT) from RHIC to LHC: • It is a signature of the fall and rise of h/s(T) which a • signature of the phase transition • Studying the CGC (fKLN): • Initial non-equilibrium distribution implied by CGC • damps the v2(pT) compensating the larger ex • v2(pT) can be described by 4ph/s ≈1

  33. Outlook for a kinetic theory approach • Include initial state fluctuations to study vn: • more constraints on h/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?

  34. Kinetic Equation for HeavyQuarks reduce to Fokker-Planck (small momentum transfer)

  35. Kinetic equation for small momentum transfer The collisionintegral can be formallysimplified in term of rate of collisionw(p.k) Using w(p,k) one can rewrite the C[f] in a form more suitable for an expansion in Small transfer momentum |k| << |p| due to MQ >>mq See: B. Svetitsky, PRD 37 (1988) 2484

  36. Therefore the transportequation in small momentum transfer can be writtenas Calculation in a Box at T=0.4 GeV Fokker-Planck equation widely used to study HQ dynamics in the QGP

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