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

x y z Hydrodynamics No microscopic descriptions (mean free path -> 0,  =0) implying f=f eq + EoS P(  ) All the observables are in a way or the other related with the evolution of the phase space density : 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?

Picking-up four main results at RHIC Nearly Perfect Fluid,Large Collective Flows  Nearly Perfect Fluid, Large Collective Flows::  Hydrodynamics good describes dN/dp T + v 2 (p T ) with mass ordering  BUT VISCOSITY EFFECTS SIGNIFICANT (finite  and f ≠ f eq )  High Opacity, Strong Jet-quenching  High Opacity, Strong Jet-quenching::  R AA (p T ) <<1 flat in p T - Angular correlation triggered by jets p t <4 GeV  STRONG BULK-JET TALK: Hydro+Jet model non applicable at p t <8-10 GeV Hadronization modified, Coalescence:  Hadronization modified, Coalescence::  B/M anomalous ratio + v 2 (p T ) quark number scaling (QNS)  MICROSCOPIC MECHANISM RELEVANT Heavy quarks strongly interacting  Heavy quarks strongly interacting::  small R AA large v 2 (hard to get both) pQCD fails: large scattering rates  NO FULL THERMALIZATION ->Transport Regime

BULK (p T ~T) MINIJETS (p T >>T,  QCD ) CGC (x<<1) Gluon saturation Heavy Quarks (m q >>T,  QCD ) MicroscopicMechanismMatters! Initial ConditionsQuark-Gluon PlasmaHadronization  p T >> T, intermediate p T  m >> T, heavy quarks   /s >>0, high viscosity  Initial time studies of thermalizations Hadronization  Microscopic mechanism for Hadronization can modify QGP observable Non-equilibrium + microscopic scale are relevant in all the subfields

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

For a classical relativistic system of N particles Gives the probability to find a particle in phase-space f(x,p) is a Lorentz scalar & P 0 =(p 2 +m 2 ) 1/2 Classical Transport Theory 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(x 1,x 2,p 1,p 2 ) and so on… Liouville Theorem: if there are only conservative forces -> phase-space density is a constant o motion Force

The non-relativistic reduction Relativistic Vlasov Equation 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

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: 1)Only two-body collisions 2)f(x 1,x 2,p 1. p 2 )=f(x 1,p 1 ) f(x 2,p 2 ) The collision term describe the change in f (x,p) because: a)particle of momentum p scatter with p 2 populating the phase space in (p’ 1,p’ 2 ) probability finding 2 particles in p e p 2 and space x Probability to make the transition Sum over all the momenta the kick-out The particle in (x,p) Collision Rate

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

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

Relation to Hydrodynamics Ideal Hydro Inserting Vlasov Eq. Integral of a divergency We can see that ideal Hydro can be satisfied only if f=f eq, 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 ∂  T , for f≠f eq and 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 -> General definitions Notice in Hydro appear only p-integrated quantities

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

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?

Quantum Transport Theory In quantum theory the evolution of a system can be described in terms of the density matrix operator: 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 f W plays in many respects the same role of the distribution function in statistical mechanics and any expectation value can be calculated as and by (*) (*)

Properties of the Wigner Function However for pure state f W 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  x  p<h/2 one can never locate a particle In agreement with the uncertainty principle

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

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

Just for simplicity lets consider the case with only a scalar field For the NJL  G  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 The validity of such an expansion is based on the assumption ħ∂ x ∂ p  F W >>1 Again the point is to have not too large gradients: X F typical length scale of the field P W typical momentum scale of the system A very rough estimate for the QGP X F ~ R N ~ 4-5 fm, P W ~ T ~ 1-3 fm -1 -> X F ·P W ~ 5-15 >> 1 better for larger and hotter systems

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 This substitute the force term mF  (x) of classical transport Vlasov Transport Equation in QFT Quantum effects encoded in the fields while f(x,p) evolution appears as the classical one.

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

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

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

Transport Theory valid also at intermediate p T out of equilibrium:  valid also at intermediate p T out of equilibrium: region of modified hadronization at RHIC region of modified hadronization at RHIC  valid also at high  /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  p T + microscopic level.

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

Hydrodynamics No microscopic details (mean free path -> 0,  =0) + EoS P(  ) Parton cascade v 2 saturation pattern reproduced First stage of RHIC Parton elastic 2  2 interactions  - P =  /3)

Information from non-equilibrium: Elliptic Flow Information from non-equilibrium: Elliptic Flow x y z pxpx pypy c 2 s =dP/d    v 2 /  measures the efficiency of the convertion of the anisotropy Coordinate from Coordinate to Momentum space Fourier expansion in p-space  |  viscosity  EoS Massless gas  =3P -> c 2 s =1/3 Bhalerao et al., PLB627(2005) More generally one can distinguish: -Short range: collisions -> viscosity -Long range: field interaction ->  ≠ 3P D. Molnar & M. Gyulassy, NPA 697 (02) 2v   time c 2 s = 0.6 c 2 s = 0.1 Measure of P gradients Hydrodynamics =0 c 2 s = 1/3 Parton Cascade

If v 2 is very large v 4 >0 require To balance the minimum v 4 >0 require v 4 ~ 4% if v 2 = 20% At RHIC a finite v 4 observed for the first time ! More harmonics needed to describe an elliptic deformation -> v 4 P. Kolb

Viscosity cannot be neglected it violates causality, but it violates causality, II 0 order expansion needed -> Israel-Stewart tensor based on entropy increase ∂  s   P. Romatschke, PRL99 (07) Relativistic Navier-Stokes     two parameters appears +  f ~ f eq reduce the p T validity range

Transport approach Collisions ->  ≠0 Field Interaction ->  ≠3P Free streaming C 23 better not to show… Discriminate short and long range interaction: Collisions ( ≠ 0) + Medium Interaction ( Ex. Chiral symmetry breaking )  decrease

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

a)collisions switched off for  <  c =0.7 GeV/fm 3 b)  /s increases in the cross-over region, faking the smooth transition between the QGP and the hadronic phase Two kinetic freeze-out scheme Finite lifetime for the QGP small  /s fluid! At 4  /s ~ 8 viscous hydrodynamics is not applicable! No f.o. This gives also automatically a kind of core-corona effect

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

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 v 2q ~0.08 needed by Coalescence scaling ? Is it compatible with a fluid  /s ~ ? I° Hot Quark

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

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

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

NJL Mean Field Two effects:  ≠ 3p no longer a massless free gas, c s <1/3  Chiral phase transition Associated Gap Equation free gasscalar field interaction Fodor, JETP(2006) NJL gas

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

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

200 AGeV for central collision, b=0 fm. Dynamical evolution with NJL

Does the NJL chiral phase transition affect the elliptic flow of a fluid at fixed  /s? S. Plumari et al., PLB689(2010) - NJL mean field reduce the v 2 : attractive field - If  /s is fixed effect of NJL compensated by cross section increase - v 2   /s not modified by NJL mean field dynamics!

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

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

Transport approach can pave the way for a consistency among known v 2,4 properties:  breaking of v 2 (p T )/  & persistence of v 2 (p T )/ scaling  v 2 (p T ), v 4 (p T ) at  /s~ can agree with what needed by coalescence (QNS) by coalescence (QNS)  NJL chiral phase transition do not modify v 2   /s  Signature of  /s(T): large v 4 /(v 2 ) 2 Summary for ligth QGP Next Steps for a quantitative estimate:  Include the effect of an EoS fitted to lQCD  Implement a Coalescence + Fragmentation mechanism

A Nearly Perfect Fluid T f ~ 120 MeV ~ 0.5 ~ 0.5 For the first time very close to ideal Hydrodynamics to ideal Hydrodynamics Finite viscosity is not negligible No microscopic description ( ->0)  Blue shift of dN/dp T hadron spectra  Large v 2 /   Mass ordering of v 2 (p T )

Jet Quenching Nuclear Modification Factor How much modification respect to pp?  Jet gluon radiation observed  Jet gluon radiation observed:  all hadrons R AA <<1 and flat in p T  photons not quenched -> suppression due to QCD away near Medium Jet triggered angular correl.

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

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

Hadronization Modified Baryon/Mesons Au+Au p+p PHENIX, PRL89(2003) Quark number scaling Dynamical quarks are visible Collective flows Enhancement of v 2 v 2q fitted from v 2  GKL Coalescence scaling

Heavy Quarks  m c,b >>  QCD  m c,b >>  QCD produced by pQCD processes (out of equilibrium)  eq >  QGP  eq >  QGP with standard pQCD cross section (and also with non standard pQCD) non standard pQCD) Hydrodynamics does not apply to heavy quark dynamics (f ≠f eq ) pQCD “D” QGP- RHIC Equilibration time npQCD

 v 4 more sensitive to both  /s and f.o.  v 4 (p T ) at 4  s  could also be consistent with coalescence  v 4 generated later than v 2 : more sensitive to properties at T  T c What about v 4 ? Relevance of time scale !

Effect of EOS on v 2 Decrease in v 2 of about 40% H. Song and U.Heinz

Very Large v 4 /(v 2 ) 2 ratio not very much depending on  /s Ratio v 4 /v 2 2 not very much depending on  /s and not on the initial eccentricity and not on particle species … see also M. Luzum, C. Gombeaud, O. Ollitrault, arxiv: Same Hydro with the good dN/dp T and v 2

 /s 1 1 T/T c QGP 2 2  V 2 develops earlier at higher  /s  V 4 develops later at lower  /s -> v 4 /(v 2 ) 2 larger -> v 4 /(v 2 ) 2 larger Effect of  /s(T) on the anisotropies Hydrodynamics Effect of finite  /s+f.o. Effect of  /s(T) + f.o. |y|<1 v 4 /(v 2 ) 2 ~ 0.8 signature of  / s close to phase transition!

If the system if very dense   one can derive and add the three-body collision that make the transition from the dilute to the dense system: See: Zhu and Greiner PRC71 (2004)