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Transport Theory for the Quark-Gluon Plasma

Transport Theory for the Quark-Gluon Plasma. V. Greco UNIVERSITY of CATANIA INFN-LNS. Quark-Gluon Plasma and Heavy-Ion Collisions – Turin (Italy), 7-12 March 2011. z. y. x. All the observables are in a way or the other related with the evolution of the phase space density :.

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Transport Theory for the Quark-Gluon Plasma

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  1. Transport Theory for the Quark-Gluon Plasma V. Greco UNIVERSITY of CATANIA INFN-LNS Quark-Gluon Plasma and Heavy-Ion Collisions – Turin (Italy), 7-12 March 2011

  2. z y x All the observables are in a way or the other related with the evolution of the phase space density : Hydrodynamics No microscopic descriptions (mean free path  -> 0, h=0) implying f=feq What happens if we drop such assumptions? There is a more “general” transport theory valid also in non-equilibrium? Is there any motivation to look for it? + EoS P(e)

  3. Picking-up four main results at RHIC • Nearly Perfect Fluid,Large Collective Flows: • Hydrodynamics good describes dN/dpT + v2(pT) with mass ordering • BUT VISCOSITY EFFECTS SIGNIFICANT (finite l and f ≠feq) • High Opacity, StrongJet-quenching: • RAA(pT)<<1 flat in pT - Angular correlation triggered by jets pt<4 GeV • STRONG BULK-JET TALK: Hydro+Jet model non applicable at pt<8-10 GeV • Hadronization modified, Coalescence: • B/M anomalous ratio + v2(pT) quark number scaling (QNS) • MICROSCOPIC MECHANISM RELEVANT • Heavy quarks strongly interacting: • small RAA large v2 (hard to get both) pQCD fails: large scattering rates • NO FULL THERMALIZATION ->Transport Regime

  4. Initial Conditions Quark-Gluon Plasma Hadronization BULK (pT~T) Microscopic Mechanism Matters! CGC (x<<1) Gluon saturation Heavy Quarks (mq>>T,LQCD) MINIJETS (pT>>T,LQCD) • pT>> T , intermediate pT • m >> T , heavy quarks • h/s >>0 , high viscosity • Initial time studies of thermalizations • Microscopic mechanism for Hadronization can modify QGP observable • Non-equilibrium + microscopic scale are relevant in all the subfields

  5. Plan for the Lectures • Classical and Quantum Transport Theory - Relation to Hydrodynamics and dissipative effects - density matrix and Wigner Function • Relativistic Quantum Transport Theory - Derivation for NJL dynamics - Application to HIC at RHIC and LHC • Transport Theory for Heavy Quarks - Specific features of Heavy Quarks - Fokker-Planck Equation - Application to c,b dynamics

  6. Classical Transport Theory For a classical relativistic system of N particles f(x,p) is a Lorentz scalar & P0=(p2+m2)1/2 Gives the probability to find a particle in phase-space If one is interested to the collective behavior or to the behavior of a typical particle knowledge of f(x,p) is equivalent to the full solution … to study the correlations among particles one should go to f(x1,x2,p1,p2) and so on… Liouville Theorem: if there are only conservative forces -> phase-space density is a constant o motion Force

  7. Relativistic Vlasov Equation The non-relativistic reduction Liouville -> Vlasov -> No dissipation + Collision= Boltzmann-Vlasov Dissipation Entropy production Allowing for scatterings particles go in and out phase space (d/dt) f(x,p)≠0 Collision term

  8. The Collision Term • It can be derived formally from the reduction of the 2-body distribution • Function in the N-body BBGKY hierarchy. • The usual assumption in the most simple and used case: • Only two-body collisions • f(x1,x2,p1.p2)=f(x1,p1) f(x2,p2) • The collision term describe the change in f (x,p) because: • particle of momentum p scatter with p2 populating the phase space in (p’1,p’2) probability finding 2 particles in p e p2 and space x Sum over all the momenta the kick-out The particle in (x,p) Probability to make the transition Collision Rate

  9. In a more explicit form and covariant version: gain loss At equilibrium in each phase-space region Cgain =Closs When one is close to equilibrium or when the lmfp is very small One can linearize the collision integral in df=f-f0 <<f Relaxation time time between 2 collisions What is the f0(x,p)=0?

  10. Local Equilibrium Solution The necessary and sufficient condition to have C[f]=0 is Noticing that p1+p2=p’1+p’2 such a condition is satisfied by the relativistic extension of the Boltzmann distribution: b=1/T temperature u collective four velocity m chemical potential It is an equilibrium solution also with LOCAL VALUES of T(x), u(x), m(x) The Vlasov part gives the constraint and the relation among T,u,m locally • Main points: • Boltzmann-Vlasov equation gives the right equilibrium distributions • Close to equilibrium there can be many collisions with vanishing net effect

  11. Relation to Hydrodynamics Ideal Hydro General definitions Notice in Hydro appear only p-integrated quantities Inserting Vlasov Eq. Integral of a divergency We can see that ideal Hydro can be satisfied only if f=feq , on the other hand the underlying hypothesis of Hydro is that the mean free path is so small that the f(x,p)is always at equilibrium during the evolution. Similarly ∂mTmn , for f≠feqand one can do the expansion in terms of transport coefficients: shear and bulk viscosity , heat conductivity [P. Romatschke] At the same time f≠feq is associated to the entropy production ->

  12. Entropy Production <-> Thermal Equilibrium Boltzmann-Vlasov Eq. Approach to thermal equlibrium is always associated to entropy production All these results are always valid and do not rely on the relaxation time approx. more generally: DS=0 <-> C[f]=0 Collision integral is associated to entropy production but if a local equilibrium is reached there are many collisions without dissipations!

  13. Does such an approach can make sense for a quantum system? One can account also for the quantum effect of Pauli-Blocking in the collision integral does not allow scattering if the final momenta have occupation number =1 -> Boltzmann-Nordheim Collision integral This can appear quite simplistic, but notice that C[f]=0 now is So one gets the correct quantum equilbrium distribution, but what is F(x,p) for a quantum system?

  14. Quantum Transport Theory In quantum theory the evolution of a system can be described in terms of the density matrix operator: and any expectation value can be calculated as For any operator one can define the Weyl transform of any operator: which has the property (*) The Weyl transform of the density operator is called Wigner function and by (*) fW plays in many respects the same role of the distribution function in statistical mechanics

  15. Properties of the Wigner Function However for pure state fW can be negative so it cannot be a probability On the other hand if we interpret its absolute value as a probabilty it does not violate the uncertainty principle because one can show: So if we go in a phase space smaller than DxDp<h/2 one can never locate a particle In agreement with the uncertainty principle

  16. Quantum Transport Equation One can Wigner transform this or the Schr. Equation After some calculations one gets the following equation This exactly equivalent to the Equation for the denity matrix or the Schr. Eq. NO APPROXIMATION but allows an approximation where h does not appear explicitly and still accounting for quantum evolution when the gradient of the potential are not too strong : This has the same form of the classical transport equation, but it is for example exact for an harmonic potential See : W.B. Case, Am. J. Phys. 76 (2008) 937

  17. Transport Theory in Field Theory One can extend the Wigner function (4x4 matrix): It can be decomposed in 16 indipendent components (Clifford Algebra) For example the vector current In a similar way to what done in Quantum mechanics one can start from the Dirac equation for the fermionic field See : Vasak-Gyulassy- Elze, Ann. Phys. 173(1987) 462 Elze and Heinz, Phys. Rep. 183 (1989) 81 Blaizot and Iancu, Phys. Rep. 359 (2002) 355

  18. Just for simplicity lets consider the case with only a scalar field This is the semiclassical approximation. If one include higher order derivatives gets an expansion in terms of higher order derivatives of the field and of the Wigner function For the NJL s=GYY The validity of such an expansion is based on the assumption ħ∂x∂psFW >>1 Again the point is to have not too large gradients: XF typical length scale of the field PW typical momentum scale of the system A very rough estimate for the QGP XF~ RN ~ 4-5 fm , PW ~ T ~ 1-3 fm-1 -> XF·PW ~ 5-15 >> 1 better for larger and hotter systems

  19. Substituting the semiclassical approximation one gets: There is a real and an imaginary part Which contains the in medium mass-shell Including more terms in the gradient expansion would have brougth terms breaking the mass-shell constraint Decomposing, using both real and imaginary part and taking the trace Vlasov Transport Equation in QFT This substitute the force term mFm(x) of classical transport Quantum effects encoded in the fields while f(x,p) evolution appears as the classical one.

  20. Transport solved on lattice Solved discretizing the space in (h, x, y)a cells Rate of collisions per unit of phase space exact solution t0 3x0 D3x Putting massless partons at equilibrium in a box than the collision rate is See: Z. Xhu, C. Greiner, PRC71(04)

  21. Approaching equilibrium in a box Highly non-equilibrated distributions where the temperature is anisotropy in p-space F.Scardina

  22. Transport vs Viscous Hydrodynamics in 0+1D Knudsen number-1 Huovinen and Molnar, PRC79(2009)

  23. Transport Theory • valid also at intermediate pT out of equilibrium: region of modified hadronization at RHIC • valid also at high h/s -> LHC and/or hadronic phase • Relevant at LHC due to large amount of minijet production • Appropriate for heavy quark dynamics • can follow exotic non-equilibrium phase CGC: A unified framework against a separate modelling with a wider range of validity in h, z, pT + microscopic level.

  24. Applications of transport approach • to the QGP Physics • Collective flows & shear viscosity • dynamics of Heavy Quarks & Quarkonia

  25. First stage of RHIC Parton cascade Hydrodynamics Parton elastic 22 interactions (l=1/sr - P=e/3) No microscopic details (mean free path  -> 0, h=0) + EoS P(e) v2 saturation pattern reproduced

  26. z y Parton Cascade Hydrodynamics x c2s= 0.6 py l=0 2v2/e c2s= 0.1 px Measure of P gradients c2s= 1/3 time Information from non-equilibrium: Elliptic Flow v2/e measures the efficiency of the convertion of the anisotropy from Coordinate to Momentum space Fourier expansion in p-space l=(sr)-1 |viscosity | EoS c2s=dP/de Massless gas e=3P -> c2s=1/3 • More generally one can distinguish: • Short range: collisions -> viscosity • Long range: field interaction -> e ≠ 3P Bhalerao et al., PLB627(2005) D. Molnar & M. Gyulassy, NPA 697 (02)

  27. If v2 is very large P. Kolb More harmonics needed to describe an elliptic deformation -> v4 To balance the minimum v4 >0 require v4 ~ 4% if v2= 20% At RHIC a finite v4 observed for the first time !

  28. Viscosity cannot be neglected Relativistic Navier-Stokes but it violates causality, II0 order expansion needed -> Israel-Stewart tensor based on entropy increase ∂m sm >0 th,tz two parameters appears + df ~ feq reduce the pT validity range P. Romatschke, PRL99 (07)

  29. Transport approach Field Interaction -> e≠3P Collisions -> h≠0 Free streaming C23 better not to show… Discriminate short and long range interaction: Collisions (l≠0) + Medium Interaction ( Ex. Chiral symmetry breaking) r,T decrease

  30. We simulate a constant shear viscosity Cascade code Relativistic Kinetic theory (*) =cell index in the r-space =cell index in the r-space Time-Space dependent cross section evaluated locally The viscosity is kept constant varying s A rough estimate of (T) Neglecting and inserting in (*) At T=200 MeV tr10 mb G. Ferini et al., PLB670 (09) V. Greco at al., PPNP 62 (09)

  31. Two kinetic freeze-out scheme Finite lifetime for the QGP small h/s fluid! • collisions switched off • for <c=0.7 GeV/fm3 • b) /s increases in the cross-over region, faking the smooth transition between the QGP and the hadronic phase No f.o. This gives also automatically a kind of core-corona effect At 4ph/s ~ 8 viscous hydrodynamics is not applicable!

  32. Role of ReCo for h/s estimate Parton Cascade at fixed shear viscosity Hadronic h/s included -> shape for v2(pT) consistent with that needed by coalescence A quantitative estimate needs an EoS with e≠ 3P : cs2(T) < 1/3 -> v2 suppression ~30% -> h/s ~ 0.1 may be in agreement with coalescence Agreement with Hydro at low pT • 4/s >3  too low v2(pT) at pT1.5 GeV/c even with coalescence • 4/s =1 not small enough to get the large v2(pT) at pT2 GeV/c without coalescence

  33. I° Hot Quark Short Reminder from coalescence… Quark Number Scaling Molnar and Voloshin, PRL91 (03) Greco-Ko-Levai, PRC68 (03) Fries-Nonaka-Muller-Bass, PRC68(03) Is it reasonable the v2q ~0.08 needed by Coalescence scaling ? Is it compatible with a fluid h/s ~ 0.1-0.2 ?

  34. Effect of h/s of the hadronic phase Hydro evolution at h/s(QGP) down to thermal f.o. ->~50% Error in the evaluation of h/s Uncertain hadronic h/s is less relevant

  35. Effect of h/s of the hadronic phase at LHC Suppression of v2 respect the ideal 4ph/s=1 LHC – 4ph/s=1 + f.o. RHIC – 4ph/s=1 + f.o. RHIC – 4ph/s=2 +No f.o. At LHC the contamination of mixed and hadronic phase becomes negligible Longer lifetime of QGP -> v2 completely developed in the QGP phase S. Plumari, Scardina, Greco in preparation

  36. Impact of the Mean Field and/or of the Chiral phase transition - Cascade -> Boltzmann-Vlasov Transport - Impact of an NJL mean field dynamics - Toward a transport calculation with a lQCD-EoS

  37. NJL Mean Field gas NJL Fodor, JETP(2006) free gas scalar field interaction Associated Gap Equation Two effects: - e≠ 3p no longer a massless free gas, cs <1/3 - Chiral phase transition

  38. Boltzmann-Vlasov equation for the NJL Numerical solution with d-function test particles Contribution of the NJL mean field Test in a Box with equilibrium f distribution

  39. Simulating a constant h/s with a NJL mean field Massive gas in relaxation time approximation =cell index in the r-space M=0 The viscosity is kept modifying locally the cross-section Theory Code s =10 mb

  40. Dynamical evolution with NJL Au+Au @ 200 AGeV for central collision, b=0 fm.

  41. Does the NJL chiral phase transition affect the elliptic flow of a fluid at fixed h/s? S. Plumari et al., PLB689(2010) • NJL mean field reduce the v2 : attractive field • If h/s is fixed effect of NJL compensated by cross section increase • v2<->h/s not modified by NJL mean field dynamics!

  42. Next step - use a quasiparticle model with a realistic EoS [vs(T)] for a quantitative estimate of h/s to compare with Hydro…

  43. Using the QP-model of Heinz-Levai e P ° A. Bazavov et al. 0903.4379 hep-lat U.Heinz and P. Levai, PRC (1998) WB=0 guarantees Thermodynamicaly consistency M(T), B(T) fitted to lQCD [A. Bazavov et al. 0903.4379]data on e and P NJL QP lQCD-Fodor

  44. Summary for ligth QGP • Transport approach can pave the way for a consistency among known v2,4 properties: • breaking of v2(pT)/ & persistence of v2(pT)/<v2> scaling • v2(pT), v4(pT) at h/s~0.1-0.2 can agree with what needed by coalescence (QNS) • NJL chiral phase transition do not modify v2<->h/s • Signature of h/s(T): large v4/(v2)2 • Next Steps for a quantitative estimate: • Include the effect of an EoS fitted to lQCD • Implement a Coalescence + Fragmentation mechanism

  45. A Nearly Perfect Fluid Tf ~ 120 MeV <bT> ~ 0.5 • No microscopic description (l->0) • Blue shift of dN/dpT hadron spectra • Large v2/e • Mass ordering of v2(pT) Finite viscosity is not negligible For the first time very close to ideal Hydrodynamics

  46. Jet Quenching near Medium away How much modification respect to pp? Jet triggered angular correl. Nuclear Modification Factor • Jet gluon radiation observed: • all hadrons RAA <<1 and flat in pT • photons not quenched -> suppression due to QCD

  47. Surprises… Baryon/Mesons Quenching Au+Au p+p PHENIX, PRL89(2003) p0 suppression: evidence of jet quenching before fragmentation In vacuum p/p ~ 0.3 due to Jet fragmentation Protons not suppressed • Jet quenching should affect both Hadronization has been modified pT < 4-6GeV !?

  48. Hadronization in Heavy-Ion Collisions H Partonspectrum • Initial state: no partons in the vacuum but a thermal ensemble of partons -> Use in mediumquarks • No direct QCD factorization scale for the bulk: dynamics much less violent (t ~ 4 fm/c) More easy to produce baryons! Fragmentation: Baryon • energy needed to create quarks from vacuum • hadrons from higher pT Coal. Meson Coalescence: • partons are already there $ to be close in phase space $ • ph= n pT ,, n = 2,3 baryons from lower momenta (denser) Fragmentation V. Greco et al./ R.J. Fries et al., PRL 90(2003) ReCo pushes out soft physics by factors x2 and x3 !

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