SPS エネルギーでの平均場の効果Mean-Field effects at SPS energies

@ 大阪大学 RCNP, 4 Nov. 2004

北大理　 一瀬 昌嗣 (M. Isse)☆ 共同研究　　　　　　　　　　原研　 大塚 直彦 (N. Otuka)IOP Ｐ．Ｋ．サフ (P.K. Sahu)Frankfurt U. 奈良 寧 (Y. Nara)北大理 大西 明 (A. Ohnishi)

今日の話の流れ イントロ 実験の状況 ( フロー計測 ) 平均場入り輸送模型でのこれまでの結果 模型の説明 カスケード模型 JAM 運動量依存平均場の導入 結果 AGS E895, E877 との比較 <px>,v2 SPS NA49 との比較 v1,v2 平均場の効き方について考察 まとめ

Introduction Heavy-ion collisions provides information about nuclear

equation of state (EOS). EOS gives the some static thermal property of nuclei

(B.E., radius, …) We have to rely mostly on theoretical estimates to know the

high density and/or high temperature EOS. Other transport models shows the collective flows are very

sensitive to the EOS. Strong collective flows are measured in 1984 at Bevalac.

Followed Experiments also show radial or sideward expansions.

Momentum dependence on collective flows are studied from around 1990. In order to distinguish momentum and density dependence, we have to investigate wide incident energy range.

Collective Flow Measurements in Heavy-Ion Collisions

Accelerator Inc. Energy

(AGeV)Particle Year

GSI-SIS(FOPI)

LBNL-Bevalac （ EOS)

0.1~2Ni,Nb, Au,Pb

1986~

BNL-AGS (E877) 11 Au 1993~

CERN-SPS(NA49)158,40,

20,30,60,80Pb 1995~

BNL-AGS (E895) 2~8 Au 1997~

BNL-RHIC65+65,100+1

00, 31+31 Au 2000~

Collective Flows Anisotropic collective

flows (<px>, v1,v2) emerges in non-central collisions.

Very sensitive to the EOS.

22

2

1 ,

T

y

T

x

T

x

p

p

p

pv

p

pv

Previous Works (1)

F is the slope of <px> and normalized y at mid-rapidity.

F decreases above 2 AGeV as a function of incident energy.

Small F means small pressure to sideward direction, namely the created matter is soft.

Boltzmann equation based model (RBUU) well reproduce the data below 11 AGeV (SIS to AGS energies).

[P.K.Sahu, W.Cassing, U.Mosel and A.Ohnishi, NPA 672(2000),376]

Previous Works (2) Momentum dependent mean field is necessary to describe

heavy ion collisions from SIS to AGS energies (0.1~11 A GeV) This work also used Boltzmann Equation based MF model [P.Danielewicz, R.Racey, W.G.Lynch, Science 298(2002),1592]

Previous Works (3) Momentum dependent soft mean field well describe azimuthal anisotropy in 0.4 A

GeV Au+Au collisions.

Azimuthal dependence of mean kinetic energy can be fit via <Ekin>=E0kin –

Ekincos2

[FOPI Collaboration and P. Danielewicz, PRL 92(2004),072303]

Motivation Many previous succeeded works used Boltzmann

equation based model to describe mean field(MF). We would like to take other approach.

Cascade model + QMD type MF MF effects in heavy-ion collisions are well studied up

to AGS energies (Einc<11 A GeV). Now anisotropic flow data in SPS energies are

available. [NA49 Collab.(C. Alt et al), PRC 68(2003),034903]

Ref.[Y.Nara et al.PRC61(2000),024901]

Hadron-String Cascade JAM JAM describes heavy-ion collision by multiplying

hadron-hadron collision in the energy range of Einc = 1-160 AGeV and over.

All established hadronic states with masses up to around 2 GeV with isospin and antiparticles.

Inelastic hadron-hadron collisions produce resonance at lower energies.

At higher energies(s > 2~4 GeV), color strings are formed and they decay into hadrons according to Lund string model PYTHIA.

At high energies(s > 10 GeV), multiple mini-jet production is included using eikonal formalism for pQCD.

Including Mean Field(MF) To improve description of

hadron-hadron binary collisions. MF works in evolution stage after collisions.

We adopt a framework of constraint Hmiltonian dynamics RQMD/S [T.Maruyama et al. PTP 96(1996),263] into JAM.

N-body Hamiltonian with MF and their time derivatives are analytically given.

We include density dependent potential with (without) momentum dependent potential [MH,MS (H,S)]. They are parameterized to give saturation at =0 and two type EOS. The curvature represents incompressibility.

Including Mean Field

Density dependent potential

HS MH

MS

First term is given as Skyrme type zero-range approximated interaction where f(r,p) dp = (r).

and Cex(k)are

parameter to give saturation property.

Second term is a momentum dependent part.

Momentum dependent potential Lorentzian type

momentum dependent mean-field which simulates the exchange term of Yukawa potential.

The Schrödinger Equivalent Potential is a functional derivative of potential energy U=V/f.

This parameterization is chosen to reproduce real part of optical potential taken by Hama et al. of nucleon-nucleus collision experiments.

Including Mean Field

In the actual simulation we use these equations for each i-th particles.

Mean Field at AGS energies

Einc=2-11 AGeV

Sideward Flow<px>vs y (Proton)

Mean momentum of sideward emitted particles in mid-central collisions.

Momentum dependent mean-field (MH,MS) well reproduces 2 to 8 AGeV data.

• E895

Comparison with AGS E895 data [PRL 84(2000),5488]

Sideward Flow<px>vs y (Proton,Pion)

Mean momentum of sideward emitted particles in mid-central collisions.

Momentum dependent mean-field (MH,MS) well reproduces 2 to 8 AGeV data.

• E877

Comparison with AGS E877 data [PRC 56(1997)3254]

Mean Field at SPS energiesComparison with SPS NA49 data

[NA49 Collab.(C. Alt et al), PRC 68(2003),034903]

Einc=40 and 158 AGeV

Time step : dt=0.1 Nucleons feel MF (resonance and other baryons, anti-baryons dose not feel MF)

Directed Flow v1 vs y (Proton)

<px> will be calculated via integrating v1 with pT multiplicity weight as:

Momentum dependent MF MH,MS also well reproduces 40 AGeV data.In 158 AGeV MH, MS show negative slope at mid-rapidity, while density dependent MF H,S show positive.The observed ‘wiggle’ can be explained in momentum dependent MF?

Directed Flow v1 vs y (Pion)

We find that pions are emitted to escape nucleons at mid-rapidity.

Directed Flow v1 vs PT (Proton)

We take |y|<1.5 and averaged with the sign. NA49 take 0<y<2.1.The reason of narrow range is to omit counting nucleons in spectator.yproj=2.234 (40AGeV)yproj=2.912(158A

GeV)

Directed Flow v1 vs PT (Pion)

In 40 A GeV MS is good, although in 158 A GeV no MF(CS) seems good.

Elliptic Flow v2 vs y (Proton)

All MF well suppress the proton v2.

Elliptic Flow v2 vs y (Pion)

All MF on nucleons also well suppress the pion v2.

Elliptic Flow v2 vs PT (Proton)

We take also |y|<1.5 and averaged with the sign as did in v1analysis. NA49 take 0<y<2.1.

We find MH and MS well suppress v2.

Elliptic Flow v2 vs PT (Pion)

We find all MF give a bit over estimate at lower pt region, but tendency is good.

Incident energy dependence of v2

Proton

Momentum dependent mean-field well describe the integrated negative proton elliptic flow at lower incident energies.

Discussion

Time scale to form flowsConditions of MF Which particle feels MF ? (only Nucleons / all Baryons)

Timestep of the calculationMF on higher(RHIC) energies

Time evolution

Einc=40 AGeV, Pb+Pb4<b<8 fm collisionsy<0.8yproj ,hadrons

V2 are formed gradually in a large time scale, v1 is formed in a very short time.V2 can grow without MF, v1 cannot.

MF for only Nucleons(N) or All Baryons(B) (1)

We compare different conditions in MS type MF on sideward flow.

Small difference between two time steps (N,0.1 and N,0.5)

Large difference between MF included species (N,0.5 and B,0.5)

PionProton

MF for only Nucleons(N) or All Baryons(B) (2)

We compare different conditions in MS type MF on sideward flow.

Visible difference between species (N,0.5 and B,0.5) (N,0.1 and N,0.5)

v2 of s=62 AGeV (Preliminary)

We expect MF effects even lower RHIC energies.

(Einc=2050 AGeV)

ProtonPion

s=62

For experimentalists….We want precise anisotropic flow study at

lower RHIC energies as SPS-NA49 paper.

v2 vs pT

Proton Pion

Summary We investigate heavy-ion collisions from AGS to SPS

energies (2~158 AGeV) by using hadron-string cascade JAM with covariant mean-field model RQMD/S.

We adopt two-type of mean-field potentials which are momentum dependent and independent. The momentum dependent interaction improves the description well at SPS energies(40 and 158 AGeV).

Mean Field between nucleons affects also pion distributions. It improve description.

Our results suggests momentum dependent interaction have essential role to form corrective flows. The nuclear incompressibility dose not vary resulting corrective flows so much.