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Transport and Hydrodynamic Model for Ultra-relativistic Heavy Ion Collisions

Yu-Liang Yan

China Institute of Atomic Energy

Collaborators: Yun Cheng (CCNU, China) Dai-Mei Zhou (CCNU, China) Bao-Guo Dong (CIAE, China)

Xu Cai (CCNU, China) Ben-Hao Sa (CIAE, China)

L. P. Csernai (UiB, Norway)

May. 7 - 13, 2014, CCNU, Wuhan, China

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Outline

• Introduction

• PACIAE model

• Hydrodynamic model

• Transport and Hydrodynamic hybrid Model (PACIAE+HYDRO+ PACIAE)

• Results and Summary

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Stages of heavy ion collisions

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– The PACIAE model is a parton and hadron cascade model , which is based on PYTHIA.

– PYTHIA is a famous model for relativistic hadron-hadron collisions.

– The PACIAE model is composed of

(1) Parton initialization

(2) Parton evolution

(3) Hadronization

(4) Hadron evolution

PACIAE model

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Particle list

particle order position and four momentum

particle 1particle 2particle 3……

x, y, z, px, py, pz, E of particle 1x, y, z, px, py, pz, E of particle 2x, y, z, px, py, pz, E of particle 3……

Collision list

collision order collision pair

123

……

collision time

Part. i and j Part. l and mPart. l and n

……

col. time for ij col. time for lmcol. time for ln

……

nn collision happens if their least approaching distance min

totd

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Remnant

Remnant Initial state radiation

Final state radiation

Hadronization

kf

lf

Rescattering ?

Decay

)(xf ai

)(xf aj

h

a

b

…

ˆd

dt

• Sketch for pp simulation in PYTHIA

Parton Distribution Function

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1) Parton Initializationa) Nucleus-nucleus collision is decomposed into

nucleon-nucleon (NN) collisions.

b) NN collision is described by the PYTHIA model, and the string fragmentation is switched-off.

c) The diquarks (anti-diquarks) are broken into quarks (anti-quarks), so the consequence of this NN collision is a partonic final state (quarks, anti-quarks, and gluons, beside a few remnants).

PACIAE model

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3) Hadronization

4) Hadron Evolution (Hadron Rescattering)

String Fragmentation (SF) model in PYTHIA, and Coalescence model. The coalescence model is used in the hybrid model.

Only and their antiparticles arecondsidered, and the usual two-body collision model is used.

/),(,,,,,,, Jknp

PACIAE model

2) Parton Evolution (Parton Recattering)Only 2→2 process are considered, 2 → 2 Leading-Order (LO-) pQCD differential cross sections. (Replaced by the Hydro evolution in the hybrid model.)

Reference ： Ben-Hao Sa, et al, Comp. Phys. Comm. 183 (2012) 333-346.

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Hydrodynamical model

Relativistic fluid dynamics (FD) is based on the assumption of local equilibrium and the conservation laws.

The MIT bag model is used for the equation of state (EoS).

Algorithms for solving the hydrodynamic equations: PIC, SHASTA, and RHHLE etc. Particle in Cell (PIC) method is used in hydro code.

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1) The baryon charge related marker particles are introduced to represent the motion of the fluid.

Particle in Cell (PIC) method

a) The momentum and energy are distributed evenly among the marker particles.

b) Each marker particle is given an effective velocity.

c) In a time step t, if the marker particle crosses to a new cell, then it transfers the corresponding amount of baryon charge, momentum, and energy from the donor cell to the acceptor cell.

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2) Update the baryon charge, momentum, and energy values. Use the updated values to solve hydrodynamic equations. 3) Proceed the next time step of the calculations.

Particle in Cell (PIC) method

Reference ：Phys. Rev. C 17 (1978) 2080Phys. Lett. B 261 (1991) 352

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Particle in Cell(PIC) method

Initial marker particle position

Time step is 100 cycles.

Time step is 200 cycles.

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Initial state for Hydrodynamic model

There are many initial state models, such as: • perturbative QCD (hard scatterings) plus

saturation model,• Glauber-based parametrization method, • color glass condensate inspired model, • effective string rope model, • transport model, etc.

In the HYDRO-PACIAE model, the effective string rope model is used to generate initial state.

We choose the transport model PACIAE to generate initial state in the new hybrid model.

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Quantities of cells in Hydro

Quantities of particles in

PACIAE model1), Particle number: N

2), Particle type: KF code

3), Momentum and energy of each particle: E, px, py, pz

4), Coordinates and time of each particle: vx, vy, vz, T

1), Energy density: Ecf

2), Baryon density: n

3), Pressure: p

4), Velocity: vx, vy, vz

5), Temperature and Entropy: T and sq

Physical Quantities in Hydro and PACIAE model

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From PACIAE model to Hydro

• PACIAE model is Monte Carlo transport model.

• Event-by-event fluctuations of the initial state can naturally included.

• The energy, momentum, and baryon number density are described by three-dimensional Gaussian distribution.

}2

)]([)()(exp{

),,(

2

222

pzpp

cf

zzyyxxN

zyxE

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Case 1: Isochronous transition, t = Constant. Case 2: Transition takes place on the hyperboloid, τ=Constant.Case 3: Realistic and complex condition.

From Hydro to PACIAE model

)0,0,0,1(d

),0,0,1( vd

celleachindifferentd

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dppxfpd

dNE i ),(

3Cooper-Frye formula

Solutions of Boundary Conditions

Fermi distribution for Quark and anti-quark:

1)2(

1)( /)(3 T

iii ie

gf

Bose distribution for Gluon:

1)2(

1)( /)(3 T

iii ie

gf

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1. Initial stage for hydrodynamic

(effective string rope model (ESRM)

and/or PACIAE model).

2. Hydrodynamic evolution (Hydro code).

3. Translate hydro into parton

(Cooper-Frye formula).

4. Hadronization (PACIAE model).

5. Hadron rescattering (PACIAE model).

Structure of the hybrid model

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Charged Hadron Pseudorapiditydistribution:

Results of the Hybid model

Reference ： Yu-Liang Yan, et al, J. Phys. G: Nucl. Part. Phys. 40 (2013) 025102

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Transverse momentum distribution from Hybrid and PACIAE model

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

10-1

100

101

102

103

(1/2p

T)d

N2

/dp

Td

[(G

eV/c

)-2]

pT(GeV/c)

Pb+Pb 2.76TeV

PACIAE Hybrid ALICE

Total charged particles

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0.0 0.5 1.0 1.5 2.0 2.5 3.010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

40-50%

-0.5<<0.5

Pb+Pb 2.76TeV

20-30%

10-20%

5-10%

30-40%

0-5%

(1

/2p

T)d

N2

/dp T

d[(

GeV

/c)-2

]

pT(GeV/c)

Transverse momentum distribution of pion

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0.5 1.0 1.5 2.0 2.5 3.0

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

40-50%

30-40%

-0.5<<0.5

(1/2p

T)d

N2

/dp

Td[

(Ge

V/c

)-2]

pT(GeV/c)

Pb+Pb 2.76TeV

K

20-30%

10-20%

5-10%

0-5%

Transverse momentum distribution of kaon

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0.5 1.0 1.5 2.0 2.5 3.0

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

40-50%

Pb+Pb 2.76TeV

30-40%

20-30%

10-20%

5-10%

pT(GeV/c)

(1/2p

T)d

N2

/dp

Td[

(Ge

V/c

)-2]

p0-5%

-0.5<<0.5

Transverse momentum distribution of proton

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Elliptic flow from Hybrid and PACIAE model

Deformation parameter in PACIAE:

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222

,4

,xy

xyrp

rprrp C

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Elliptic flow for pion, kaon, and proton

0.5 1.0 1.5 2.0 2.50.00

0.05

0.10

0.15

0.20

0.25

0.30

pT(GeV/c)

Pb+Pb 2.76TeV

pion kion proton

v 2

40-50%

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Summary

• Based on the PACIAE and Hydrodynamic model, a hybrid model, ESRM +HYDRO+PACIAE and PACIAE+HYDRO+PACIAE, is established.

• The ESRM (or PACIAE)+HYDRO+PACIAE model is a partonic based integrated hybrid model, which can reproduce the experimental data well.

• The PACIAE+HYDRO+PACIAE hybrid model can be used to study the partonic degrees of freedom, quark number scaling, and the initial state fluctuations, etc.

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Thanks for yourattention!