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

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

Relativistic Transport approach Collisions ->  ≠0 Field Interaction Free streaming  It’s not a gradient expansion for small  /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! f(x,p) is a one-body distribution function or a classical field f 0 (p) =Boltzmann -> C[f 0 ]=0 -> ideal hydrodynamics ≠ C[f 0 +  f] ≠ 0 deviation from ideal hydro ( finite  or  /s )

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

valid also at intermediate & high p T out of equilibrium  valid also at intermediate & high p T out of equilibrium  valid also at high  /s  -> LHC -  /s(T), cross-over region  CGC p T non-equilibrium phase (beyond the difference in  x ) :  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) assuming small q transfer -> Fokker-Planck eq. (Beraudo’s talk) A unified framework against a separate modelling with a wide range of validity in  p T + microscopic level Motivation for Transport approach Collisions ->  ≠0 Field Interaction Free streaming

Simulate a fixed shear viscosity  =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 Au+Au 200 GeV Viscosity fixed varying  Usually input of a transport approach are cross-sections and fields, but here we reverse it 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 Knudsen number -1 Comparison for the relaxation of pressure anisotropy P L /P T Huovinen and Molnar, PRC79(2009) In the limit of small  /s (<0.16) transport reproduce viscous hydro at least for the evolution p L /p T Large K small  /s

Viscous Hydrodynamics Asantz used - at p T ~3 GeV !?  f/f≈ 5 - this implies Relax. Time and not Chap.Enskog it violates causality, I 0 Navier-Stokes, but it violates causality, II 0 order needed -> Israel-Stewart Problems related to  f Problems related to  f: dissipative correction to f -> f eq +  f neq just an ansatz  f neq /f at p T > 1.5 GeV is large  f neq  /s implies a RTA approx. (solvable)    (t 0 ) =0 -> discard initial non-equil. (ex. minijets) p T -> 0 no problem except if  /s is large 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: ;see also: Wesp et al., Phys. Rev. C 84, (2011); Fuini III et al. J. Phys. G38, (2011). Shear Viscosity in Box Calculation Green-Kubo correlator Needed very careful tests of convergency vs. N test,  x cell, # time steps ! macroscopicobservables microscopic details η ↔ σ(θ), , M, T …. ? η ↔ σ(θ), , M, T …. ?

for a generic 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 -  m D 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) h(a)=  tr /  tot weights cross section by q 2

 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   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…. x y z pxpx pypy Part II - Impact of  /s(T) on the build-up of v 2 (p T ) vs. beam energy

 /s increases in the cross-over region, realizing the smooth f.o.: small  -> natural f.o.  /s increases in the cross-over region, realizing the smooth f.o.: small  -> natural f.o. Different from hydro that is a sudden cut of expansion at some T f.o. Different from hydro that is a sudden cut of expansion at some T f.o. Terminology about freeze-out No f.o. Freeze-out is a smooth process: scattering rate < expansion rate RHIC B=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%) : v 2 from pure QGP, but at LHC less sensitivity to T-dependence of  /s? Without  /s(T) increase T≤Tc we would have v 2 (LHC) < v 2 (RHIC) First application: f.o. at RHIC & LHC

 /s(T) close to a phase transition I 0 T c cross-over I 0 T c cross-over I 0 T c cross-over P. Kovtun et al.,Phys.Rev.Lett. 94 (2005) L. P. Csernai et al., Phys.Rev.Lett. 97 (2006) R. A. Lacey et al., Phys.Rev.Lett. 98 (2007)  Uncertainty Principle Text book  AdS/CFT suggest a lower bound But do we have signatures of a “U” shape of  /s(T) typical of a phase transition? QGP close to this bound!  pQCD at finite T pQCD

 /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~T α, 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 ( μ B >T) W. Schmidt et al., Phys. Rev. C47, 2782 (1993) Danielewicz et al., AIP1128, 104 (2009) (STAR Collaboration), arXiv: [nucl- ex].

 r-space: standard Glauber model  p-space: Boltzmann-Juttner T max = T c [ p T 2-3GeV ] T max0 = 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 GeV200 GeV2.76 TeV T0T0 290 MeV340 MeV590 MeV  fm/c0.6 fm/c0.3 fm/c Typical hydro condition Discarded in viscous hydro

Impact of  /s(T) vs √s NN  4πη/s=1 during all the evolution of the fireball -> no invariant v 2 (p T ) ->smaller v 2 (p T ) at LHC.  Initial p T distribution relevant (in hydro means      but it is not done!   Notice only with RHIC → almost scaling for 4πη/s=1 LHC data play a key role w/o minijet Plumari, Greco,Csernai, arXiv: %

Impact of  /s(T) vs √s NN  η/s ∝ T 2 too strong T dependence→ a discrepancy about 20%.  Invariant v 2 (p T ) suggests a “U shape” of η/s with mild increase in QGP  Hope: v n, n>3 with an event-by-event analysis will put even stronger constraints Plumari, Greco,Csernai, arXiv:

Enhancement of η/s(T) in the cross-over region affect differently build-up of v 2 (p T ) at RHIC to LHC. At LHC nearly all the v 2 from the QGP phase Scaling of v 2 (p T ) 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 Summary Part II

What about Color Glass condensate initial state? - Kinetic Theory with a Q s saturation scale Part III

Saturation scale pTpT dN/d 2 p T Q sat (s) At RHIC Q 2 ~ 2 GeV 2 At LHC Q 2 ~ 5-8 GeV 2 ? Color Glass Condensate Specific of CGC is a non-equilibrium distribution function !

Unintegrated distribution functions (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, (2007) Hirano and Nara, PRC79, (2009) Hirano and Nara, Nucl. Phys. A743, 305 (2004) Albacete and Dumitru, arXiv: [hep-ph] Albacete et al., arXiv: [nucl-th] (f)KLN spectrum Factorization hypothesis: convolution of the distribution functions of partons in the parent nucleus. fKLN realization of CGC pTpT dN/d 2 p T Q sat (s) p-space x-space  x (fKLN)=0.34  x (Glaub.)=0.29

V 2 from KLN in Hydro Heinz et al., PRC 83, (2011) What does it KLN in hydro? 1) r-space from KLN (larger ex) 2) p-space thermal at t 0 ≈0.8 fm/c - No Q s scale, We’ll call it fKLN-Th

Risultati di Heinz da “hadron production………” Larger  x - > higher  /s to get the same v 2 (p T ) Uncertainty on initial conditions implies uncertainty of a factor 2 on  /s Similar conclusion in Drescher et al., PRC (2011) V 2 from KLN in Hydro What does it KLN in hydro? 1) r-space from KLN (larger ex) 2) p-space thermal at t 0 ≈0.8 fm/c - No Q s scale, We’ll call it fKLN-Th

Higher  /s for KLN leads to small v 3 The value of  /s affects more higher harmonics! Adare et al., [PHENIX Collaboration], PRL 107, (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 p T distribution Using kinetic theory at finite  /s we 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 GeV – 20-30% Glauber in x & thermal in p  x =0.289, Qs=0

Hydro - likeFull x & p 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 Q s saturation scale (non-equilibrium distribution) Results with kinetic theory M. Ruggieri et al., [nucl-th]

GeV M. Ruggieri et al., [nucl-th] We clearly see that when the non-equilibrium distribution is implemented In the initial stage (1 fm/c) v 2 grows slowly then distribution is thermal and it grows faster. V 2 normalized time evolution What is going on?

 At LHC the larger saturation Q s ( ≈ 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 -likeFull KLN x & p TeV What happens at LHC? preliminary

Development of kinetic at fixed  /s(T) :  Relax. Time Approx can severely underestimate  /s Chapmann-Enskog I°order agree with Green-Kubo Invariant v 2 (p T ) 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):  Initial non-equilibrium distribution implied by CGC damps the v 2 (p T ) compensating the larger  x v 2 (p T ) can be described by 4  /s ≈1 Summary

Outlook for a kinetic theory approach  Include initial state fluctuations to study v n :  more constraints on  /s(T)  Impact of CGC non-equilibrium distribution  Include hadronization:  statistical model like in hydro  coalescence + fragmentation ( going at high p T )  Heavy quark dynamics within the same framework:  Fokker-Planck is the small transfer approximation is it always a reliable approxiamation?

Kinetic Equation for Heavy Quarks reduce 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) 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| >m q

Therefore the transport equation in small momentum transfer can be written as Fokker-Planck equation widely used to study HQ dynamics in the QGP Calculation in a Box at T=0.4 GeV