Chiho Nonaka

Hydro + Cascade Model at RHICHydro + Cascade Model at RHICHydro + Cascade Model at RHICHydro + Cascade Model at RHIC

In Collaboration with

Steffen A. Bass (Duke University & RIKEN)

October 30, 2004@DNP, Chicago

Duke University, Chiho Nonaka

• Introduction• 3-d Hydrodynamic Model• Hydro + Cascade Model• Results (hadron spectra, elliptic flow)• Summary

Contents

Chiho Nonaka

Introduction • Hydrodynamic Model at RHIC

–HBT

Possible solution?

Morita, Muroya, CN and Hirano,PRC66:054904,2002

–Elliptic flow

Huovinen et.al, PLB503

–Elliptic flow

Hirano and Tsuda, PRC66

Morita et al., PRC66

–Single particle spectraSuccess Failure?

Chiho Nonaka

Freeze-out, Final Interactions • Freeze-out

• Tf &

Thermal model T = 177 MeV = 29 MeV

UrQMD

• Final interactions

Markert @QM2004

• Freeze-out is not universal for all hadron spectra. • Rescattering and regeneration is important.

• universal Tf for all hadrons in Hydro

Thermal + radial flow fit

STAR, nucl-ex/0307024

Chiho Nonaka

Hydro + Cascade ModelBass and Dumitru, PRC61,064909(2000)Teaney et al, nucl-th/0110037

Full 3-d Hydrodynamics•EoS 1st order phase transition•QGP + excluded volume model

( Improved) Cooper-Fryeformula (Reco)

UrQMD

t fm/c

Final interactions

Monte Carlo

Hadronization

Hirano and Tsuda, PRC66(2002)054905

Treatment of freeze-out in transport model is determined by mean free path.

• Hydro + hadron transport model

Key:– Freeze-out condition ex. Chemical and kinetic freeze-out

– Final interactions

Chiho Nonaka

3-d Hydrodynamic Model• Hydrodynamic equation

– Baryon number density conservation

• Coordinates

• Lagrangian hydrodynamics– Tracing the adiabatic path of each volume element – Effects of phase transition of observables

• Algorithm – Focusing on conservation law

∂νT = 0 )( : ννννν ε uugpuuTT −−=

velocity local : pressure : density energy : ε up

∂ { ( , }n T uB ) = 0

,0},({ = ∂ uTs ) ∂

{ ( , }n T uB ) = 0

Flux of fluid

€

(τ ,x,y,η ) :

€

τ = t 2 − z2 , η = tanh−1 z

t

⎛

⎝ ⎜

⎞

⎠ ⎟

Chiho Nonaka

Trajectories on the Phase Diagram

• Lagrangian hydrodynamics

　　　　

effect of phase transition

temperature and chemical potential of volume element of fluid

C.N et al., Eur. Phys.J C17,663(2000)

Chiho Nonaka

Parameters• Initial Conditions

– Energy density

– Baryon number density

– Parameters

– Flow longitudinal: Bjorken’s Solution

• Equation of State– 1st order phase transition– QGP phase (Bag model), mixed phase, hadron phase (up to 2GeV) (excluded volume model)– Bag constant:

• Hydro UrQMD

€

ε(x,y,η ) =εmaxW (x,y;b)H(η )

€

W (x,y;b) :

wounded nuclear model

€

nB (x,y,η ) = nBmaxW (x,y;b)H(η )€

H(η ) = exp −η −η 0( )

2

2σ η2 θ η −η 0( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

€

εmax ,nBmax ,σ η ,η 0

€

B1

4 = 233 [MeV]

€

PH = PQGP

€

at Tc =160 MeV, μ c = 0 MeV

€

Tu =140 MeV

Chiho Nonaka

Hadron Spectra (I) • Pure Hydro

– Central collision– Parameters

• Hydro works well up to PT ~ 2 GeV

€

ε0 = 6.0 (GeV/fm3), nB 0 = 0.14 fm-3, η 0 =1.5, σ η =1.4,

Tf =140 (MeV)

Chiho Nonaka

Hadron Spectra (II)• Hydro + UrQMD

• Many pions are produced in UrQMD. • Low PT resonances • High PT interactions•Transition temperature is too low.

• PT slope becomes flatter. Extra radial flow in UrQMD

The initial condition for Hydro + UrQMD Is different from that for pure hydro.

Chiho Nonaka

Elliptic Flow• Pure Hydro

•Centrality 5-10 % Hydro works well.

Centrality dependence

Chiho Nonaka

Elliptic Flow (II) • Hydro + UrQMD preliminary

•In UrQMD elliptic flow becomes small ?•Shape of elliptic flow as a function of

Chiho Nonaka

Summary• Hydro + Cascade Model

– Hadron Spectra, elliptic flow – Effect of resonances, final interactions in experimental data

• Work in progress– Parameter – Initial Conditions – EoS (QCD critical point) ,CN and Asakawa nucl-th/0410078

– Parton Cascade Model – Hadronization mechanism

Recombination + Fragmentation model, Duke Group

　　

BACK UP

Chiho Nonaka

( )∑=

Δ+−Δ+∂+Δ∂+=Δ+3

1

),(),()()(),()(n

nnnn

tt

mm ittXittXivtivitvttv

Numerical Calculation • Step 1.

• Step 2.

• Step 3.

titu

ituitXittX

t

mmm Δ+=Δ+

),(),(

),(),(

ATns

Tsn

itTittT BB

ns B ⎭⎬⎫

⎩⎨⎧

∂∂

−∂∂

Δ+=Δ+ ),(),(

1),(),(

,

ATnT

sTs

T

nititt B

B

ns B⎭⎬⎫

⎩⎨⎧

∂∂

−∂∂

Δ+=Δ+ ),(),(

1),(),(

,

⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂∂

∂−

∂∂

∂∂

=ΔT

TnTsTn

T

Ts BBns B

),(),(),(),(,

μ

μ

μ

μ

μμ

1),(),(

),(),(−

Δ+Δ+=

ittdittuitditu

A tt

tt

σσ

Coordinates move in parallel with baryon number current and entropy density current.

local velocity: )( ttvm Δ+

from hydro eq.

temperature and chemical potential

0},({ = ∂ uTs )

∂ { ( , }n T uB ) = 0

CPU time is almost proportional of # of lattice points.