L.P. Csernai 1

Freeze-out and constituent Freeze-out and constituent quark formation in a quark formation in a

space-time layerspace-time layer

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Constant pre FO temperature contour from hydro for the upper hemisphere, x>0

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Matching Conditions for pre/post FO change: Conservation laws !!! Conservation laws !!!

Nondecreasing entropyNondecreasing 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]

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Rapid FO and recombination into Constituent QuarksRapid FO and recombination into Constituent Quarks

A: Hydro history [file] > QGP (q,g) A: Hydro history [file] > QGP (q,g) BAMPSBAMPS CQ-s CQ-s

B: B: Hydro history [file] > QGP (q,g) Hydro history [file] > QGP (q,g) FO model in a layerFO model in a layer CQ-s CQ-s

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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!

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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:

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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) ]

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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.

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The invariant “ Escape” probabilityThe invariant “ Escape” probability

Escape probability factors for different points on FO hypersurface, in the RFG. Momentum values are in units of [mc]

A B C

D E F

t’

x’

[RFG[RFG]]

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Freeze out distribution with rescattering from kinetic model across

a layerV=0V=0

[V. Magas, et al.,] Heavy Ion Phys.9:193-216,1999

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Analytic fit to Kinetic Model Solution:

.

.

[ K. Tamosiunas and L.P. Csernai, Eur. Phys. J. A20 (2004) 269]

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Cancelling Juttner Distribution [Karolis Tamosiunas et al.]

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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

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Freeze OutRapid 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

0,0

NT

0)()( pfp

q qqq NN and

q q

[L.P. Csernai, Sov. JETP, 65 (l987) 216.]

[Cancelling Juttner orCut Juttner distributions.]

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Constituent quark number scaling of vConstituent quark number scaling of v22 (KE (KETT ) )

Collective flow of hadrons can be described in terms of constituent quarks.

Observed nObserved nqq – scaling – scaling

Flow develops in quark phase, Flow develops in quark phase, there is no further flow there is no further flow development after hadronizationdevelopment after hadronization

R. A. Lacey (2006), nucl-ex/0608046.

CNQ CNQ scalingscaling

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• 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

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b=70% b-max.

Flow in hydro, before F.O.

b=30% b-max.

b= 0

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Flow in hydro, after appr.(*) F.O.b=30% b-max.

(*) Thermal smoothing in z-direction only with TFO = 170 MeV and mFO = 139 MeV (both fixed).

Transverse smoothing would further reduce the magnitude of v1 (and v2).

Correct FO description is of Correct FO description is of Vital Importance !Vital Importance !

Freeze Out

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Hadronization via recombination Hadronization via recombination /n /nqq

Momentum distribution of mesons in simple recombination model:

Local fq(pµuµ) 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 3flow moments:

[Molnar D. -NPA774(06)257]

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Elliptic flow of mesons:

For baryons:

Scaling Variables of Flow:

1st step: Flow asymmetry: V2 / n q V2 scales with nq i.e., flow developsin 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 pT is not necessarily conserved, K ET = mT – m might be conserved scaling in the variable K ET [J. Jia & C. Zhang, 2007]

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2nd step: pt / nq K ET / nq = mo (√(1+u2) - 1) / nq EETT u << 1 : mo uT

2 / 2 u >> 1 : mo uT

Thus, scaling flow indicates dependence (equilibration) of transverse energy, i.e., not only the flow velocity but the constituent quark mass, mo, 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)

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From QGPFrom QGP

To CQ matter or Quarkyonic matterTo CQ matter or Quarkyonic matter

A)A) CQ is in chemical equilibrium CQ is in chemical equilibrium Energy-mom. Energy-mom.

B)B) CQ has the same # of q and q-bar as QGPCQ has the same # of q and q-bar as QGP

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CQ matterCQ matterin ch. Eq.in ch. Eq.

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CQ matterCQ matterout of ch. Eq.out of ch. Eq.

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Acceleration, non-relativitic limitAcceleration, non-relativitic limit

Acceleration if PQGP > Phadr

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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:

t t

x x

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

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OUTLOOK for F. OUTLOOK for F. O.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

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The END

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