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

Anisotropic flow in a B oltzmann kinetic approach at fixed h /s(T). V. Greco UNIVERSITY of CATANIA INFN-LNS. S. Plumari M. Ruggieri F. Scardina. nfQCD 2013 – Kyoto, 2-6 December 2013. Outline. Transport Kinetic Theory at fixed h /s : M otivations How to fix locally h /s .

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

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  1. Anisotropic flow in a Boltzmann kinetic approach at fixed h/s(T) V. Greco UNIVERSITY of CATANIA INFN-LNS S. Plumari M. Ruggieri F. Scardina nfQCD 2013 – Kyoto, 2-6 December 2013

  2. Outline • Transport Kinetic Theory at fixed h/s : • Motivations • How to fix locally h/s • 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) ?

  3. Relativistic Boltzmann-Vlasovapproach Collisions -> h≠0 Field Interaction (EoS) Free streaming 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 ! 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!

  4. Motivation for Transport approach Collisions -> h≠0 Free streaming Field Interaction • Starting from 1-body distribution function f(x,p) and not from Tμν: • Include off-equilibriumat high and intermediate pT • Implementnon-equilibriumimplied by CGC-Qs scale (beyondex) • Relevantat LHC due to large amount of minijet production • freeze-out self-consistently related with h/s(T) • It’snot a gradientexpansionh/s: • - validalsoat high h/s-> LHC (T>>Tc) or cross-over region(T≈ Tc) • Appropriate for heavy quark dynamics(next week talk)

  5. Simulate a fixedshearviscosity 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 Chapmann-Enskog g(a=mD/2T) correct function that fixes the relaxation time for the shear motion Chapmann-Enskog Viscosity from Green-Kubo Correlator estimate in a Box at fixed T hCE good one ! S.Plumari et al., PRC86(2012) Chapman-Enskog agrees with Green-Kubo!

  6. Simulate a fixedshearviscosity 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 Chapmann-Enskog Transportcode Space-Time dependent cross sectionevaluatedlocally g(a=mD/2T) correct function that fix the relaxation time for the shear motion =cellindexin r-space Viscosityfixedvaryings Chapman-Enskog agrees with Green-Kubo G. Ferini et al., PLB670 (2009) Chapmann-Enskog hCE good one ! S.Plumari et al., PRC86(2012)

  7. Transportatfixedh/s vs ViscousHydroin 1+1D Comparison for the relaxation of pressure anisotropy PL/PT Huovinen and Molnar, PRC79(2009) Large K small h/s Knudsen number-1 In the limit of small h/s (<0.16) transport converge to viscous hydro at least for the evolution pL/pT Denicol et al. have studied derivation of viscous hydro from Boltzmann kinetic theory: PRD85 (2012) 114047

  8. Test in 3+1D: v2/eresponse for almostideal case EoS cs2=1/3 (dN/dy tuned to RHIC) Integrated v2vs time Ideal -Hydro Transportath/sfixed v2/e Time rescaled Bhalerao et al., PLB627(2005) In the bulk the transport has an hydro v2/e2 response!

  9. h/s or details of the cross section? cross section Keep same h/s means: for mD=0.7 GeV -> factor 2 largerstot is needed respect to isotropic case

  10. h/s or details of the cross section? cross section 4ph/s=1 Keep same h/s means: • 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 largerstot is needed respect to isotropic case

  11. h/s or details of the cross section? cross section Keep same h/s means: • 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=1.4 GeV -> 25% smaller stot for mD=5.6 GeV -> 40% smaller stot

  12. h/s or details of the cross section? cross section Keep same h/s means: • 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! Differences arises just where in viscous hydro df becomes relevant

  13. Natural extension from low to high pT Renormalize s to fix h/s L ≈ 4-5 GeV, g fixed by h/s Disappearing of non-pQCD Highly non-pQCD pQCD limit, but with as=0.3 - No Q2 dependence & - No radiative part pQCD limit partons as=0.3 and mD=0.7 GeV Boltzmann transport describes rise and fall of v2(pT) Transition between low and high pT in a unified framework! No Fine tuning! Employed the relaxation time approximation! S. Plumari and VG, EPIC@LHC, AIP1422(2012)- arXiV:1110.4138 [hep-ph]

  14. z y x py px What about Color Glass condensate initial state? - Kinetic Theory Qs saturation scale

  15. dN/d2pT pT Qsat(s) fKLN realization of CGC Factorization hypothesis:convolution of parton distribution functions in the parent nucleus. 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 … Unintegrateddistributionfunctions (uGDFs) x-space p-space Saturation scale Qs depends on:1.) position in transverse plane;2.) gluon rapidity. CGC-KLN ex ≈ 30% larger than Glauber Nardiet al., Nucl. Phys. A747, 609 (2005)Kharzeevet al., Phys. Lett. B561, 93 (2003)Nardiet 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 T. Hirano et al., PLB636(06)

  16. V2 from KLN in Hydro What does it KLN in hydro? 1) r-space from KLN (larger ex) 2) p-space thermal at t0 ≈0.6-0.9 fm/c - No Qs scale , We’ll call it fKLN-Th Larger ex - > higher h/s to get the same v2(pT) Glauber h/s = 0.08 CGC-KLN  h/s=0.16 Luzum and Romatschke PRC78(2008) 034915 See also: Alver et al., PRC 82, 034913 (2010) Heinz et al., PRC 83, 054910 (2011)

  17. Implementing KLN pT distribution AuAu@200 GeV – 20-30% Using kinetic theory we 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 Glauber in x & thermal in p ex=0.289 , Qs=0 -> Th-Glauber M. Ruggieriet al., Phys.Lett. B727 (2013) 177 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.

  18. Longitudinal and transverse pressure t=1/Qs≈0.1 fm/c -> PL/PT > 0 Gelis& Epelbaum arXiV:1307.2214 • PL/PT show also a very fast equilibration (tisotr≈0.3 fm/c)! • However it is not this that makes a difference for v2: • isotropization time very similar for all the cases

  19. Longitudinal and transverse pressure 80% level of isotropization ≈ pQCD • For h/s > 0.3 one misses fast isotropization in PL/PT (t ≥ 2 fm/c) • For h/s ≈ pQCD no isotropization • Semi-quantitative agreement with Florkowski et al., PRD88 (2013) 034028 • our is 3+1D not in relax.time but full integral but no gauge field

  20. Results with kinetic theory AuAu@200 GeV Hydro - like Full x & p M. Ruggieriet al., Phys.Lett. B727 (2013) 177 - 1303.3178 [nucl-th] • 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)

  21. What is going on? We clearly see that when non-equilibrium distribution is implemented in the initial stage (≤ 1 fm/c) v2 grows slowly respect to thermal one

  22. 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

  23. 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

  24. Terminologyaboutfreeze-out Freeze-out is a smooth process: scattering rate < expansion rate • /sincreases in the cross-over region, realizinga smoothf.o. self-consistentlydependent on h/s: • Different from hydrothatis a suddencut of expansionat some Tf.o.

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

  26. Impact of h/s(T) vs √sNN w/o minijet 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 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)

  27. 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 mildincrease in QGP • Hope: vn, n>3 with an ev.-by-ev. analysis put even stronger constraints Howeverfor a definite statement neededhadronization+ EoS-lQCD

  28. Summary • 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 tisotr<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 η/s(T) anyway quite weak. • 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

  29. Outlook for the kinetic theory approach • Hadronization: • statistical model +Cooper-Frye vs. coalescence + fragm. • Field Dynamics: • Realistic EoS: M(T) + Bag mean field dynamics [Done!] • 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….

  30. 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)

  31. Initial State Fluctuations: vnvsεn (Preliminary) 4ph/s=1 ≈ 5 C(3,3)=0.68 C(2,2)=0.93 • v2and v3 linearly correlated to the corresponding eccentricities ε2and ε3

  32. Initial State Fluctuations: vnvsεn (Preliminary) 4ph/s=1 ≈ 5 C(3,3)=0.23 C(2,2)=0.93 • v4and ε4 weak correlated similar to hydro calculations: • F.G.Gardim,F.Grassi,M.LuzumandJ.Y.Ollitrault NPA904 (2013) 503. • Niemi, Denicol, Holopainen and Huovinen PRC87(2013) 054901. • 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!)

  33. Initial State Fluctuations: vn(pT) (Preliminary) Data taken from: A. Adare et al. [PHENIX collaboration], Phys.Rev. Lett. 107, 252301 (2011). Like in viscous hydro the data of vn(pT) at RHIC energies are described with 4πη/s=1 ≈ 5 Fluctuations allows to extend the studies on impact of CGC-Qs, h/s(T) and study pA …

  34. 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 CGC • J. -P. Blaizot et al., arXiv:1305.2119 [hep-ph]; J. - • P. Blaizot et al., Nucl. Phys. A904-905 2013, 829c (2013). f ≈ 1/t + slope change • f >1 at pT> 0.5 our effect manifest at larger pT • Longitudinal expansion lowers average density by t-1 • but f(p) goes down slowly at pT<0.5 GeV • One should include also gluon to quark conversion At LHC and low pT it could be there and CYM helps!

  35. Back-up

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

  37. 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).

  38. 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

  39. What is v2? along x, q=0 dN/dp dN/dp dp along y, q=90° along x, q=0 along y, q=90° smaller v2 dp pT pT Put it very simplistic: If the momentum dp shift is the same we can expect flatter distribution are less efficient in build-up v2

  40. What happens at LHC? PbPb@2.76 TeV Full KLN x & p Hydro -like • 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 fKLNimplemention change estimate of h/s by about a factor of 3/2

  41. pT-spectra versus h/s(T) • 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

  42. Impact of minijets on v2(pT) Minijets starts to affect v2(pT) For pT>1.5 GeV Effect non-negligible

  43. Sensitivity in transport using sameη/s(T) Hydro-Niemi • Larger sensitivity on h/s (T) at LHC • Effect larger respect to viscous hydro, but this depends also on df

  44. Viscous Hydrodynamics I0 Navier-Stokes, butitviolatescausality, II0orderneeded -> Israel-Stewart An Asantz (Grad) K. Dusling et al., PRC81 (2010) - this implies RTA and not CE - at pT~3 GeV !? df/f≈ 5 • Problems: • 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-eq (ex. minijets) • pT-> 0 no problemexceptifh/sis large

  45. Transportatfixedh/s vs ViscousHydroa test in 3+1D Changing M of partons one gets different EoS – cs(T) Au+Au@200AGeV 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 stot=15 mb

  46. Initial Conditions • r-space: standard Glauber model • p-space: Boltzmann-JuttnerTmax=1.7-3.5 Tc[pT<2 GeV ]+ minijet[pT>2-3GeV] 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

  47. Multiplicity & Spectra • r-space: standard Glauber condition • p-space: Boltzmann-JuttnerTmax=2(3) Tc[pT<2 GeV ]+ minijet[pT>2-3GeV] No fine tuning

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