2. L.P. Csernai 1 Freeze-out and constituent quark formation in a space-time layer

3. L.P. Csernai 2 Constant pre FO temperature contour from hydro for the upper hemisphere, x>0

4. L.P. Csernai 3 Matching Conditions for pre/post FO change:  Conservation laws !!!  Nondecreasing entropy If the final state is out of Eq., the energy-momentum tensor and f(x,p) have to be evaluated, and the above eqs. solved!!! [L.P. Csernai, Sov. JETP, 65 (l987) 216.] [ Anderlik et al. Phys.Rev.C 59 (1999) 3309] [ Tamosiunas and Csernai, Eur. Phys. J. A20 (2004) 269]

5. L.P. Csernai 4 Rapid FO and recombination into Constituent Quarks A: Hydro history [file] > QGP (q,g)  BAMPS  CQ-s B: Hydro history [file] > QGP (q,g)  FO model in a layer  CQ-s

6. L.P. Csernai 5 The choices of post FO distributions are: Jüttner - distr. / Timelike FO Cut - Jüttner - distr. / Spacelike FO - No physical ground for these choices, although conservation laws can be enforced. - Post FO distr. must not be an eq. distr. - Physical, transport processes must create the post FO distribution  takes space and time!

7. L.P. Csernai 6 Sudden FO at a sharp 3-dimensional space-time hyper-surface Gradual FO in an extended 4-dimensional space-time volume [J.Knoll] For large systems (vs. mfp) this space-time volume is a “layer”,  there is a dominant direction (gradient) of this change of f(x,p). Let that be: “t” or “s” (It must not be time-like!) From kinetic theory:

8. L.P. Csernai 7 Mod. Bolz. Tr. Eq. [Csernai et al., Eur. Phys. J. A 25, 65-73 (2005)] Projected to the direction of dominant change this leads to: where [E. Molnar, et al., PHYSICAL REVIEW C 74, 024907 (2006) ]

9. L.P. Csernai 8 Freeze out in a finite layer The corresponding equations for both space-like and time-like freeze out /wo re-thermalization The solution : Space-like Time-like [ E. Molnar, et al., J.Phys.G34 (2007) 1901; Phys.Rev.C74 (2006) 024907; Acta Phys.Hung. A27 (2006) 359; V.K. Magas, et al., Acta Phys. Hung.A27 (2006) 351. ] This should be supplemented with a recombination process into hadrons / constituent quarks.

10. L.P. Csernai 9 The invariant “ Escape” probability Escape probability factors for different points on FO hypersurface, in the RFG. Momentum values are in units of [mc] AB C D EF t’ x’ [RFG]

11. L.P. Csernai 10 Freeze out distribution with rescattering from kinetic model across a layerV=0 [V. Magas, et al.,] Heavy Ion Phys.9:193-216,1999

12. L.P. Csernai 11 Analytic fit to Kinetic Model Solution :.. [ K. Tamosiunas and L.P. Csernai, Eur. Phys. J. A20 (2004) 269]

13. L.P. Csernai 12 Cancelling Juttner Distribution [Karolis Tamosiunas et al.]

14. L.P. Csernai 13 This was up to 2005 – 2006 New developments from 2006: v1 confirmed at RHIC Indication of Mach Cones around jets CNQ scaling of flow : FO & Hadronization

15. L.P. Csernai 14 Freeze Out Rapid and simultaneous FO and “hadronization” Improved Cooper-Frye FO: - Conservation Laws: - Post FO distribution: Hadronization ~ CQ-s - Pre FO: Current and, QGP - Post FO: Constituent and - are conserved in FO!!! Choice of F.O. hyper-surface / layer [L.P. Csernai, Sov. JETP, 65 (l987) 216.] [ Cancelling Juttner or Cut Juttner distributions.]

16. L.P. Csernai 15 Constituent quark number scaling of v 2 (KE T ) Collective flow of hadrons can be described in terms of constituent quarks. Observed n q – scaling  Flow develops in quark phase, there is no further flow development after hadronization R. A. Lacey (2006), nucl-ex/0608046. CNQ scaling

17. L.P. Csernai 16 Thermal smearing is influenced by the pre-FO parton distribution  strong BTE does not take this into account correctly: LOCAL molecular chaos fails Modified BTE with non-local Collision term is vital: [Modified Boltzmann Transport Equation, V.K. Magas, L.P. Csernai, E. Molnar, A. Nyiri and K. Tamosiunas, Nucl. Phys. A 749 (2005) 202-205. / hep-ph/0502185] [Modified Boltzmann Transport Equation and Freeze Out, L.P. Csernai, V.K. Magas, E. Molnar, A. Nyiri and K. Tamosiunas, Eur. Phys. J. A 25 (2005) 65 -73. / hep-ph/0505228] FO description should include, (i) partonic thermal smearing, (ii) conservation & entropy increase, (iii) Cooper-Frye type of evaluation of post FO distribution (iv) constituent quarks or Quarkyonic Matter Simultaneous FO & recombination

18. L.P. Csernai 17 b=70% b-max. Flow in hydro, before F.O. b=30% b-max. b= 0

19. L.P. Csernai 18 Flow in hydro, after appr.(*) F.O. b=30% b-max. (*) Thermal smoothing in z-direction only with T FO = 170 MeV and m FO = 139 MeV (both fixed). Transverse smoothing would further reduce the magnitude of v1 (and v2). Correct FO description is of Vital Importance ! Freeze Out

20. L.P. Csernai 19 Hadronization via recombination  /n q Momentum distribution of mesons in simple recombination model: Local f q (p µ u µ ) is centered at the local u, & meson Wigner function: momentum conservation comoving quark and antiquark: for the momentum distribution of mesons we get: for baryons, 2  3 flow moments: [Molnar D. -NPA774(06)257]

21. L.P. Csernai 20  Elliptic flow of mesons: For baryons: Scaling Variables of Flow: 1st step: Flow asymmetry: V 2 / n q  V 2 scales with n q i.e., flow develops in QGP phase, following the common flow velocity, u, of all q-s and g-s. Mass here does not show up (or nearly the same mass for all constituent quarks). Then flow asymmetry does not change any more. In a medium p T is not necessarily conserved, K E T = m T – m might be conserved  scaling in the variable K E T [J. Jia & C. Zhang, 2007]

22. L.P. Csernai 21  E T 2nd step: p t / n q  K E T / n q = m o (√(1+u 2 ) - 1) / n q  E T  u << 1 : m o u T 2 / 2  u >> 1 : m o u T Thus, scaling flow indicates dependence (equilibration) of transverse energy, i.e., not only the flow velocity but the constituent quark mass, m o, participates. Flow momentum changes while energy equilibrates in a finite system (Canonical Ensemble). The final stages of hadronization do not change the flow-asymmetry, but locally the constituent quarks complete their "dress up" in their local region by redistributing energy to reach equilibrium. Quarkyonic Matter : No gluons / Asymptotic freedom (weakly interacting) A new phase between high T QGP and Hadronic Phase, Especially at higher baryon densities (FAIR)

23. L.P. Csernai 22 QGP – Bag model EoS  Constituent quark gas

24. L.P. Csernai 23 From QGP To CQ matter or Quarkyonic matter A)CQ is in chemical equilibrium  Energy-mom. B)CQ has the same # of q and q-bar as QGP

25. L.P. Csernai 24 CQ matter in ch. Eq.

26. L.P. Csernai 25 CQ matter out of ch. Eq.

27. L.P. Csernai 26 Acceleration, non-relativitic limit Acceleration if P QGP > P hadr

28. L.P. Csernai 27 For n_B = 0

29. L.P. Csernai 28 In general the FO hyper-surface is not orthogonal to the flow velocities, so this acceleration (deceleration) is an essential consequence of the correct FO description! In early simplified approach [see mentioned in L.P. Csernai: Introduction to Relativistic Heavy Ion Collisions] it was argued that in a flow one can choose a ragged FO hyper-surface like this to the right: tt xx The simplified approach, violates momentum conservation [!] and decreases flow effects! Acceleration is stronger at the edge near to space-like FO, left side. Fully space-like FO leads to strong acceleration as only outgoing particles can FO! FAIR P dV

30. L.P. Csernai 29

31. L.P. Csernai 30

32. L.P. Csernai 31 OUTLOOK for F. O.   CNQ scaling indicates QGP, simplifies F.O. description to Const. Quarks. This requires, however, Modified BTE description   Space – Time volume or layer Freeze Out required   A rapid process should be quantitatively described. The kinetic approach does not provide a time or spatial scale for the Hadronization of QGP!   Larry McLerran [GSI, 9.2.2009] predicts an intermediate Quarkyonic phase   Igor Mishustin [CPOD, GSI, 9.7.2007] sets an estimate for “Explosive Hadronization”. See also earlier work: [Csernai and Mishustin, PRL 74 (95) 5005]   Choice of FO Surface or Layer  Hydro history

33. L.P. Csernai 32 The END

34. L.P. Csernai 33

35. L.P. Csernai 34