heavy ion collisions with 3 - - dim dim ultra

Collective flow analysis of Collective flow analysis of heavy ion collisions with 3

heavy ion collisions with 3 - - dim dim ultra

ultra - - relativistic hydro relativistic hydro

Together with:

Yun Cheng

Szabolcs Horvát Volodymyr Magas

Etele Molnár

Dan Strottman

Miklós Zétényi

Multi Module Modeling

M 1

– Initial state -- pre eq., Yang-Mills flux tube model M 2

– Fluid dynamics -- (near) Thermal equilibrium

M 3

– Final Freeze-out -- simultaneous Hadronization & FO (recomb.)

Collective dynamics  Flow observables

• V_1 & V_2 observed and analyzed

• CQN scaling  Flow develops in QGP

Goal:

How these 3 stages and transport processes influence the observables

We have been working on related problems

We divide the reaction to into stages: (i) initial, (ii) FD stage in local equilibrium with an EoS, at high, RHIC or LHC we have QGP, (iii) sudden hadronization and freeze out.

- Accordingly our reaction model is built up as a Multi Module Model (MMM).

- The initial stage/module at RHIC/LHC has no shocks, but YM field theory - The CFD module is updated considerably.

- Flow shows constituent quark number scaling (CNQ), this is implemented in the FO and hadronization description after the hydro. [FAIR, CERN]

- Interesting new observation, that quark and anti-quark numbers remain

unchanged during sudden hadronization, but the effective quark degeneracy must increase in CNQ.

- The final Hadronization and freeze out module is updated considerably.

- The early results arising from conservation laws across time-like FO hyper-

Global Flow ^{Directed Transverse}

flow

Elliptic flow Elliptic flow

3

flow component (anti - flow)

Squeeze out

Initial state _– Landau, complete stopping

Works well at low energies

Bjorken initial state – complete transparency

Initial state is boost invariant – all quantities depent only on t, not on y give rise to 2+1D simple hydro models

Very popular at

ultra-relativistic energies

Does not conserve

energy and momentum!!!

How to conserve momentum?

At low energies – fire streak picture

[Myers, Gosset, Kapusta, Westfall]

Tilted initial state

String rope --- Flux tube --- Coherent YM field

Initial state

3

flow component

Initial state – reaching equilibrium

Initial state by V. Magas, L.P. Csernai and D. Strottman Phys. Rev. C64 (2001) 014901

Nucl. Phys. A 712 (2002) 167–204

Flow is a diagnostic tool Flow is a

Flow is a diagnostic diagnostic tool tool

Impact Impact par., par., b b

Transparency Transparency – – string tension, string tension, A A

Equilibration Equilibration time,

time, Tf Tf

Consequence:

Consequence:

v v

(y), v (y), v

(y), (y), … …

Why should we measure v_1 ???

Why should we measure v_1 ???

3 3 - - Dim Hydro for RHIC (PIC) Dim Hydro for RHIC (PIC)

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Hydro

The relativistic Euler equations used are:

Here and in the following work, N is the particle number, M is the momentum, E is the energy and P is the pressure, all defined in the calculational frame.

They are related to the rest frame quantities by the relations:

All quantities are given in the program (i.e., dimensionless) units. In the notation of Harlow et. al (PIC code)

Similarly to Harlow et. al we introduce the notation:

Then further:

Solution of equations between the rest and calculation frame

General Equations

In Harlow et. al one writes P = b e + h; in their case, h( x ) depends on the density and represents the compression pressure. In our case h = −4/3 B

.

Solving for e in eq. (8):

Then inserting into eq. (9) yields

or

Inserting into this equation:

which is the form found in Amsden et. al setting b = 2/3 :

In the QGP case one has:

Pressure independent of density

Things simplify a bit if the pressure depends only on the energy e rather than

explicitly on density through the function h above. This is the case in the QCD

plasma wherein h = −4/3 B

. In this case we have

which upon insertion into the equation for the momentum leads to

Hence,

Solving for x where So,

One must take the minus sign to ensure

when

. From this we can obtain e , eq. (11), P, v and n.

Limits

The approach outlined above will not always work; occasionally, x will be greater

than 1. These equations can be used to put restrictions on allowed values of B,C,

....

Then:

If B > B

or

Harlow et. al claim that C > B

leads to x > 1. The equation above indicates the

situation is a bit more complicated than this in that the compression pressure

would need to be included in their case.

Zero B

In this case

Thus

with the last expression for b = 1/3 . Solving for x

We need to take the − in the ± so that x  0 when y  0. Note that for x  1, y  1.

If P = 0,

consequently

with the result that

This is attractive because for edge cells for which we don’t know the proper density

(i.e., how much of a cell is filled), this definition of x does not depend on the volume

filled.

Viscosity vs. “numerical viscosity”

-Viscosity is important (phase tr., initial state, stability, etc.) - Several numerical solution methods, finite resolution

- E.g. Lax method:

- Discretized in 1D, using the notation:

where

- Doing the same for the Euler equation yields

- A similar study for the FCT method results in num. kinetic viscosity:

Theoretical [D. Molnar, U. Heinz, et al., ] Theoretical [D. Molnar, U. Heinz, et al., ] η η = 50 – = 50 – 500 MeV/fm 500 MeV/fm

c, Re º c, Re º 10 – 10 – 100 100 For Δ For Δx=1fm, x=1fm, Δ Δt=0.9fm/c, t=0.9fm/c, ρ ρ =300MeV  =300MeV 

η η

= 167 MeV/fm = 167 MeV/fm

c c

Numerical “viscosity”

is not negligible !!!

Viscosity vs. “numerical viscosity” contd.

Particle in Cell method.

Particle in Cell method.

Better resolution than the Better resolution than the cell- cell - size would allow! size would allow!

“Marker particles “ Marker particles” ” = = Lagrangian

Lagrangian fluid cells. Large fluid cells. Large number of these.

number of these.

Randomly placed to avoid Randomly placed to avoid

“ “ ringing instabilities ringing instabilities ” ” and and

other grid related instabilities!

other grid related instabilities!

Runs very stable up to very Runs very stable up to very high energies, much beyond high energies, much beyond the principle applicability of the principle applicability of CFD approach.

CFD approach.

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Figure: In the PIC method Lagrangian fluid elements, called Markers, move in a decartian coordinate grid. At very high energies, to avoid instabilities arising from the computational grid, marker particles are randomized in our approach. The figure shows Marker particle positions in the central plane of an explosion (z is the beam direction), assuming an initial Landau state [15] with an energy density of 40 GeV/fm3. A total of 1.5 million marker particles are used to describe the three-dimensional nucleus [unpublished].

M2

Figure: Test of maximum baryon number density in the explosive final stage of expanding Quark-gluon Plasma after an ultra-relativistic heavy ion reaction, where the initial collision energy was 65 times the rest mass of the colliding nuclei. The result weakly depends on the ratio of the grid size in the direction of the collision and the length of the time-step which is 0.5 fm/c. Implementation of implicit methods and Newton-Krylov solvers for the relativistic hydrodynamics will significantly decrease the fluctuations and increase the accuracy. (Unpublished.)

Figure: Time evolution of the energy density in the central plane assuming an initial Landau state [15], which can be formed in a central (b=0) collision of two nuclei. The expansion is dominantly in the beam-, z-direction. The dynamics were described by a relativistic three-dimensional hydrodynamic model [unpublished].

Au+Au

Au+Au at 60+60 A GEV, b= 0.25 (R_pat 60+60 A GEV, b= 0.25 (R_p + R_t+ R_t) at ) at

t=0 t=0

Plotted: e, energy density, [GeV/fm

Plotted: e, energy density, [GeV/fm³³], in the rest frame of the cell. [tnc10 –], in the rest frame of the cell. [tnc10 – low res.]low res.]

P/T side reversed!

 / 

Au+Au

Au+Au at 60+60 A GEV, b= 0.25 (R_pat 60+60 A GEV, b= 0.25 (R_p + R_t+ R_t) at ) at

t=0 t=0

Plotted: e, energy density, [GeV/fm

Plotted: e, energy density, [GeV/fm³³], in the rest frame of the cell. [tnc16 –], in the rest frame of the cell. [tnc16 – high res.]high res.]

Au+Au

Au+Au at 60+60 A GEV, b= 0.5 (R_pat 60+60 A GEV, b= 0.5 (R_p + R_t+ R_t) at ) at

t= 1.902 fm/c t= 1.902 fm/c

Plotted: E, energy density, [GeV/fm

Plotted: E, energy density, [GeV/fm³³], in the calculational (CM) frame. Contour ], in the calculational (CM) frame. Contour lines are at 5, 2.5, 5, 8 [GeV/fm

lines are at 5, 2.5, 5, 8 [GeV/fm³³] and E_{max] and E_{max} = 9.19 GeV/fm} = 9.19 GeV/fm³³ ..

Au+Au E

=65 GeV/nucl. b=0.5 b

A

=0.08 => σ~10 GeV/fm

e [ GeV / fm ³ ] T [ MeV]

t=0.0 fm/c, T

= 420 MeV, e

= 20.0 GeV/fm

L

= 1.45 fm, L

=0.145 fm

. .

EoS: p= e/3 - 4B/3

Au+Au E

=65 GeV/nucl. b=0.5 b

A

=0.08 => σ~10 GeV/fm

e [ GeV / fm ³ ] T [ MeV]

t=9.1 fm/c, T

= 417 MeV, e

= 19.6 GeV/fm

L

= 1.45 fm, L

=0.145 fm

. .

Au+Au

Au+Au at 65+65 A GEV, b= 0.3 (R_pat 65+65 A GEV, b= 0.3 (R_p + R_t+ R_t) (String tension A=0.08, Tf) (String tension A=0.08, Tf=4.5 fm/c).=4.5 fm/c).

Au+Au

Au+Au at 65+65 A GEV, b= 0.4 (R_pat 65+65 A GEV, b= 0.4 (R_p + R_t+ R_t) (String tension A=0.065, Tf) (String tension A=0.065, Tf=6 fm/c).=6 fm/c).

Au+Au

Au+Au at 65+65 A GEV, b= 0.5 (R_pat 65+65 A GEV, b= 0.5 (R_p + R_t+ R_t) (String tension A=0.065, Tf) (String tension A=0.065, Tf= 5 fm/c).= 5 fm/c).

b=70% b-max.

Flow in hydro, before F.O.

b=30% b-max.

b= 0

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).

Correct FO description is of Correct FO description is of

Vital Importance ! Vital Importance !

Freeze Out

„3 ^rd flow” component

Hydro

[Csernai, HIPAGS ’93]

[Phys.Lett. B458 (99) 454]

Csernai & Röhrich

Rapidity distribution of v

Rapidity distribution of v

, v , v

, , nq nq

b = 70% b = 70%

nn_qq scalingscaling

pp_TT = p= p_TT / / nn_qq

v ₁ ( η ): system-size dependence

G. Wang / STAR

QM 2006 :

Best case: 200 GeV Au + Au

G. Wang / STAR

QM 2006 :

[G. Wang / STAR –

Nucl. Phys. A 774 (2006) 515–518]

[J. Chen / STAR – J. Physics 35 (2008) 044072]

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. hypersurface









  N

T

0 ) ( )

(  

 p

f p

q _q

q

q

N

N and

q q

M3

“Detonations across time-like fronts in relativistic systems” - ‘87

Burning and

radiating outside shell

M3

Preventing turbulence

The instability of deflagration- (flame-) front is not desirable at supersonic fronts.

With increasing temperature the radiation becomes

dominant and stabilizes the

flame front.

The radiative transfer also modifies the dissipative transport. This is of vital importance, because radiative transport propagates with the speed of light, and able to stabilize processes which cannot be stabilized by mechanical pressure. This is actually the reason of the failure of different rocket engines, and the success of the space shuttle rockets as well as of the implosion devices in the nuclear bombs. One should just look at the extremely stable, blue-ultraviolet flame fronts (15 000 oK) of the Space-Shuttle's liquid fuel rockets stabilized by radiative energy-momentum transfer, in contrast to the hardly stable, turbulent red flames at ignitions with lower temperatures. [Picture, NASA]

Interestingly the space-time picture of hadronization and freeze out of expanding and cooling QGP is very similar to time-like detonations [1].

Recognized also in

[LV. Bravina et al., PL 354B (95)192.]

Thus, if the process is rapid, i.e. sudden hadronization and freeze out, then it can and must be described by the same formalism.

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

Matching Conditions for core/crust boundary

  Conservation laws Conservation laws

  Nondecreasing entropy Nondecreasing entropy

If the final state is out of Eq., the energy-momentum tensor has to be evaluated, and the above eqs. solved!!!

[ Anderlik et al. Phys.Rev.C 59 (99) 3309]

M3

M3

M3

M3

Entropy;

bulk visc.

FAIR Recom- bination:

N

reduced

in FO !!!

• Recombination, reduces N, makes the FO even more rapid and sudden!

• 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 of (iv) constituent quarks (for flow observables).

Parton Cascade (MD !) and recombination model is a good alternative!

Simultaneous FO & recombination

FO hypersurface

T

=139 MeV

[B. Schlei, LANL 2005]

Freeze out:

Freeze out:

V.K. Magas, V.K. Magas, E. Molnar.

E. Molnar.

Improved calculation of FO hypersurface

Rapid and simultaneous FO and “hadronization” can and must be assumed based on experiments as well as studies of phase transition dynamics.

Experiments indicate small source size and large strangeness abundance, as well as CNQ scaling. This means flow and strangeness develop in QGP phase and no time is left for reestablishing chemical balance among light and heavy strange hadrons, or to change the flow via interactions among hadrons.

Conservation Laws across hypersurface

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Space-like hypersurface

Observed

Observed n n

– – scaling scaling

there is no further flow

development after hadronization development after hadronization

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

M3 CNQ scaling

CNQ scaling

Let us consider sudden freeze out and hadronization from QGP:

• Start with 2 flavours (u,d)  end with 3 flavours (u,d,s)

• Start with massless quarks and B_bag  end with massive constituent quarks (CQs)

• Start with and in QGP  end with either

(a) keeping all quarks post FO, i.e. both (very fast FO) (b) keeping only , & re-equilibrating CQs (fast)

Although, these processes happen gradually, during the reaction, the rate of quark equilibration increases exponentially due to increasing quark degeneracy, so we simplify our treatment assuming that these processes happen in the FO layer.

For a time-like FO surface, in RFF, with v₀ = v = 0  n_B = n_B0 & e = e₀ and T:

q q

B

n n

n   n ~  n

 n

n

n n

& ~

C q C

q

C

n n

n ~  

M3

For small, finite incoming velocities the velocity change (due to pressure change), can be obtained from the momentum conservation:

Fig. The ratio of post and pre FO velocity as function of ε and n for B_bag = 397GeV/ fm³. The freeze out may accelerate or decelerate the flow, depending on the initial state.

Fig. The ratio of post and pre freeze out velocity , δ = (vx – v0)/v0 [%]. Contour lines of δ are shown at values -10%, 0%, 10% as function of ε and n for B = 0.397GeV / fm3.

“

The difference between these two choices is illustrated in Fig. 1. The lower panel of Fig. 1 shows a schematic structure of Milekhin’s hypersurface. In practical calculations, the fragments of Milekhin’s hypersurface

are so tiny that the whole hypersurface looks like that in the upper panel of Fig. 1, however, with the normal vector to

each tiny fragment coinciding with the four- velocity.

Therefore, Milekhin’s method in fact conserves energy, but to see it one should consider it on a

discontinuous hypersurface. … “ [V. N. Russkikh & Yu. B. Ivanov,

Phys. Rev. 76, 054907 (2007)]:

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

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P dV

Measurable, v

, calculated at FO from pre- & post- FO flow pattern

At earlier FO, ~ 7.1 fm/c, the FO accelerates expansion and increases v₂ due to higher QGP pressure. Later, at ~ 9.2 fm/c, QGP pressure becomes low or negative, which decreases v₂ at

SUMMARY

• Initial state is decisive and can be tested by v1 & v2

• v1 dominates in semi-central collisions

• v2 dominates in more peripheral collisions

• position of v1 peak depends on b, σ, Tf.

• Viscosity is important both in hydro and in the initial dynamics

• Numerical viscosity should be taken in correction

• F.O. : entropy condition  space like FO is weak at RHIC / LHC &

• important at FAIR

•  bulk viscosity limits space like F.O. >> FAIR

• CNQ scaling indicates QGP, simplifies F.O. description to Const. Quarks.

This requires, however, Modified BTE description

The END

Reserve

Stability, Reynolds number

- kinematic viscosity

- viscosity - density

- length - velocity

In an ideal fluid any small perturbation increases and leads to turbulent flow. For stability

sufficiently large viscosity and/or heat conductivity are needed!

Re < 1000 - 2000

(Calculations are also stabilized by numerical viscosity.)

Interesting and important: in RFD detonation fronts are stabilized by radiation and heat conductivity . E.g. :

- Rocket propulsion

Re – studies in HICs

Theoretical [D. Molnar, U. Heinz, et al., ] Theoretical [D. Molnar, U. Heinz, et al., ] η η = 50 = 50 – – 500 MeV/fm 500 MeV/fm

c Re c Re º º 10 10 – – 100 100 Exp.: 50

Exp.: 50 – – 800 Mev 800 Mev/nucleon energies 80 /nucleon energies 80’ ’s s [Bonasera [ Bonasera , Schurmann , Schurmann , Csernai] , Csernai]

scaling analysis of flow parameters.

scaling analysis of flow parameters. Re Re º º 7 7 – – 8 ! 8 ! (more dilute, more viscous matter)

(more dilute, more viscous matter)

In both cases

In both cases η η/s /s ª ª 1 (0.5 – 1 (0.5 – 5) , 5) ,

This is a value large enough to keep the This is a value large enough to keep the flow laminar in Heavy Ion Collisions !!!

flow laminar in Heavy Ion Collisions !!!

In superstring theory, „based on analogy between black hole physics and equilibrium thermodynamics, ... there exist solutions called black branes, which are black holes with translationally invariant horizons. ... these

solutions can be extended to hydrodynamics, ... and black branes possess hydrodynamic characteristics of ... fluids: viscosity, diffusion constants, etc.”

In this model the authors concluded that η / s = 1 / 4π

And then they „speculate” that in general η / s > 1 / 4π or η / s > 1.

They argue that this is a lower limit especially for such strongly interacting

systems where up to now there is no reliable estimate for viscosity, like the

QGP. According to the authors: the viscosity of QGP must be lower than

that of classical fluids.

(Kovtun, et al., PRL 2005)

With With Kapusta Kapusta and and McLerran McLerran we we have studied these results and have studied these results and assumptions and found that : assumptions and found that :

-η vs. T has a typical decreasing and then increasing behaviour, due to classical reasons (Enskog’21) - η/s has a minimum exactly at the critical point in systems, which

have a liquid-gas type of transition - η vs. T shows a characteristic shows a characteristic behaviour

behaviour in all in all systems near the systems near the

critical point (not only in the case of

critical point (not only in the case of

He). He).

Viscosity – Momentum transfer

Via VOIDS

Via VOIDS Via PARTICLESVia PARTICLES

[ Enskog, 1921 ]

Helium (NIST)

Water (NIST) QGP (Arnold, Moore, Yaffe)

This phenomenon can help us This phenomenon can help us to detect experimentally the to detect experimentally the critical point:

critical point:

η can be determined from (i) fluctuation of flow parameters and from (ii) scaling properties

[Prakash, Venugopalan, .]

Stability of the core/crust HS.

• Landau-Lifsitz: mechanical stability is limited  V2,

rocket engine- gas-turbine- accidents,

• Fusion device instabilities

• Solved by Bethe /Los

Alamos publ. - Zeldovich, Raiser: High Temp.

phenomena

RT – instabilities in Tokamak

• The figure

above shows three-

dimensional isosurfaces of the pressure as the instability develops along ridges

dominantly

aligned along

the ambient

magnetic field.

Sun-surface - plasma

The picture was made The picture was made using the Swedish Solar using the Swedish Solar Telescope on the Canary Telescope on the Canary Island of La Palma. The Island of La Palma. The filaments' newly revealed filaments' newly revealed dark cores are seen to be dark cores are seen to be thousands of kilometers thousands of kilometers long but only about 100 long but only about 100 kilometers wide.

kilometers wide.

Resolving features 100 Resolving features 100 kilometers wide or less At kilometers wide or less At optical wavelengths,

optical wavelengths,

these images are sharper these images are sharper than even current space than even current space- - based solar observatories based solar observatories can produce. Recorded can produce. Recorded on 15 July 2002

on 15 July 2002

Viscosity/Scaling are important to extrapolate to other scales !

Viscosity/Scaling are important to extrapolate to other scales !