SPS????????????? Mean-Field effects at SPS energies @ ???? RCNP, 4 Nov. 2004

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SPS?????????????Mean-Field effects at SPS
energies _at_ ???? RCNP, 4 Nov. 2004

• ??? ?? ?? (M. Isse)
• ?????
• ?? ?? ?? (N. Otuka)
• IOP P.K.?? (P.K. Sahu)
• Frankfurt U. ?? ? (Y. Nara)
• ??? ?? ? (A. Ohnishi)

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• ????
• ??????(?????)
• ???????????????????
• ?????
• ????????JAM
• ????????????
• ??
• ?AGS E895, E877????ltpxgt,v2
• ?SPS NA49????v1,v2
• ?????????????
• ???

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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.

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Collective Flow Measurements in Heavy-Ion
Collisions
Accelerator Inc. Energy (AGeV) Particle Year
GSI-SIS(FOPI) LBNL-Bevalac(EOS) 0.12 Ni,Nb, Au,Pb 1986
BNL-AGS (E877) 11 Au 1993
CERN-SPS(NA49) 158,40, 20,30,60,80 Pb 1995
BNL-AGS (E895) 28 Au 1997
BNL-RHIC 6565,100100, 3131 Au 2000
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Collective Flows

• Anisotropic collective flows (ltpxgt, v1,v2)
  emerges in non-central collisions.
• Very sensitive to the EOS.

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Previous Works (1)
P.K.Sahu, W.Cassing, U.Mosel and A.Ohnishi, NPA
672(2000),376

• F is the slope of ltpxgt 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).

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Previous Works (2)

• Momentum dependent mean field is necessary to
  describe heavy ion collisions from SIS to AGS
  energies (0.111 A GeV)
• This work also used Boltzmann Equation based MF
  model
• P.Danielewicz, R.Racey, W.G.Lynch, Science
  298(2002),1592

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Previous Works (3)

• Momentum dependent soft mean field well describe
  azimuthal anisotropy in 0.4 A GeV AuAu
  collisions.
• Azimuthal dependence of mean kinetic energy
  can be fit via ltEkingtE0kin DEkincos2f .
• FOPI Collaboration and P. Danielewicz, PRL
  92(2004),072303

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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 (Einclt11 A GeV).
• Now anisotropic flow data in SPS energies are
  available.
• NA49 Collab.(C. Alt et al), PRC
  68(2003),034903

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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 gt 24 GeV), color strings
  are formed and they decay into hadrons according
  to Lund string model PYTHIA.
• At high energies(?s gt 10 GeV), multiple mini-jet
  production is included using eikonal formalism
  for pQCD.

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

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Including Mean Field

• 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.

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Density dependent potential

• First term is given as Skyrme type zero-range
  approximated interaction where ? f(r,p) dp
  ? (r).
• a,b,g and Cex(k)are parameter to give saturation
  property.
• Second term is a momentum dependent part.

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

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Including Mean Field

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

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Mean Field at AGS energies

• Einc2-11 AGeV

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Sideward Flowltpxgtvs y (Proton)
Comparison with AGS E895 data PRL 84(2000),5488

• Mean momentum of sideward emitted particles in
  mid-central collisions.
• Momentum dependent mean-field (MH,MS) well
  reproduces 2 to 8 AGeV data.

• E895

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Sideward Flowltpxgtvs y (Proton,Pion)
Comparison with AGS E877 data PRC 56(1997)3254

• Mean momentum of sideward emitted particles in
  mid-central collisions.
• Momentum dependent mean-field (MH,MS) well
  reproduces 2 to 8 AGeV data.

• E877

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Mean Field at SPS energiesComparison with SPS
NA49 data NA49 Collab.(C. Alt et al), PRC
68(2003),034903

• Einc40 and 158 AGeV
• Time step dt0.1
• Nucleons feel MF
• (resonance and other baryons, anti-baryons dose
  not feel MF)

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Directed Flow v1 vs y (Proton)
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?
ltpxgt will be calculated via integrating v1 with
pT multiplicity weight as
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Directed Flow v1 vs y (Pion)
We find that pions are emitted to escape nucleons
at mid-rapidity.
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Directed Flow v1 vs PT (Proton)

• We take ylt1.5 and averaged with the sign. NA49
  take 0ltylt2.1.
• The reason of narrow range is to omit counting
  nucleons in spectator.
• yproj2.234 (40AGeV)
• yproj2.912(158AGeV)

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Directed Flow v1 vs PT (Pion)
In 40 A GeV MS is good, although in 158 A GeV no
MF(CS) seems good.
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Elliptic Flow v2 vs y (Proton)
All MF well suppress the proton v2.
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Elliptic Flow v2 vs y (Pion)
All MF on nucleons also well suppress the pion v2.
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Elliptic Flow v2 vs PT (Proton)
We take also ylt1.5 and averaged with the sign
as did in v1analysis. NA49 take 0ltylt2.1. We find
MH and MS well suppress v2.
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Elliptic Flow v2 vs PT (Pion)
We find all MF give a bit over estimate at lower
pt region, but tendency is good.
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Incident energy dependence of v2

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

Proton
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Discussion

• Time scale to form flows
• Conditions of MF
• ?Which particle feels MF ?
• (only Nucleons / all Baryons)
• ? Timestep of the calculation
• MF on higher(RHIC) energies

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Time evolution

• Einc40 AGeV, PbPb
• 4ltblt8 fm collisions
• ylt0.8yproj ,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.
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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)

Pion
Proton
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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)

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v2 of ?s62 AGeV (Preliminary)
(Einc2050 AGeV)
Proton
Pion
We expect MF effects even lower RHIC energies.
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v2 vs pT
Proton
Pion
?s62

• For experimentalists.
• ?We want precise anisotropic flow study at lower
  RHIC energies as SPS-NA49 paper.

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Summary

• We investigate heavy-ion collisions from AGS to
  SPS energies (2158 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.