Longitudinal fluid dynamics for relativistic heavyion collisions

1
Longitudinal fluid dynamics for relativistic
heavy-ion collisions

• Igor N. Mishustin

with L.M. Satarov, A.V. Merdeev, H. Stöcker
Frankfurt Institute for Advanced Studies,
J.W. Goethe Universität, Frankfurt am Main
Kurchatov Institute, Russian Research Center,
Moscow
recent publications Phys. Rev. C \bf 75,
024903 (2007) hep-ph/0606074 Phys. Atom. Nucl.
(in press) hep-ph/0611099
2
Contents

• Introduction
• - Landau- versus Bjorken-like initial
  conditions
• The model
• - fluid dynamics in light-cone coordinates
• - equation of state
• - initial conditions
• - particle spectra at freeze-out
• - feeding from resonance decays
• Results
• - dynamical evolution of matter
• - rapidity spectra of hadrons and BRAHMS
  data
• Conclusions

most central (0-5)
at 200 GeV
(PRL 2004,2005)
3
Earlier ideal hydro studies
11 dimensional models
Landau (1953), Melekhin (1958), Shuryak
(1972) Bjorken (1983) Mishustin Satarov
(1983) expansion of spherical fireball
Rischke Gyulassy (1995), Escola et al (1996)
21 dimensional models
Baym, Friman et al. (1983),

Blaizot Ollitrault (1987), Heinz et al (1999),
Bass Dumitru (2000), Pokrovsky et al (2000),
Teany et al (2001), , Huovinen (2001)
31 dimensional models
Amsden et al (1978), Stöcker, Maruhn, Greiner
(1980), Roshal Russkikh (1981), Rischke et al
(1995), Aguiar, Kodama et al. (2001), Hirano
et al (2001),
multi-fluid models
Amsden et al (1978), Clare Strottman (1986),
Csernai et al (1987), Mishustin, Russkikh,
Satarov (1988), Katscher et al (1995),
Brachmann et al (2000), Ivanov, Russkikh,
Toneev (2003)
4
Landau and Bjorken models
central collision of equal nuclei at
differ mostly by initial conditions
proper time
space-time rapidity
5
Equations of ideal hydrodynamics
Local conservation laws of baryon charge and
energy-momentum
Net baryon density
( in present study)
Collective (flow) 4-velocity
(units
equation of state (EOS)
Additional inputs
initial conditions
6
11 dimensional (longitudinal) flow
y
z
(beam axis)
R
x
Only longitudinal flow
Flow 4-velocity
Flow rapidity
Fluid-dynamical quantities
7
Transition to light-cone coordinates
cartesian coordinates
hyperbolic (light-cone) coordinates
proper time
space-time (pseudo)rapidity
light cone
inverse transformation
Such transition is motivated by asymptotic
behaviour at
for any initial conditions
8
Ideal hydrodynamics in t-? coordinates
after transition
Bellac (1996), Escola et al (1998)
Bjorken scaling solution for
solved numerically by flux-corrected transport
algorithm SHASTA
Boris Book (1973), Rischke et al (1995)
9
Equation of state
Bag-like parameterization (baryon-free matter)
Shuryak et al (2001)

- hadronic phase
- mixed phase
latent heat

• quark-gluon phase

Pressure
and temperature
- continuous at
Entropy density
Sound velocity
10
Parameters of equations of state
Critical temperature of deconfinement phase
transition (baryon-free matter)
Recent lattice calculations (hep-ph/0608013)
for
11
Comparison of different EOSs
EOS-I, II, III include
phase transition (PT) to quark-gluon plasma
pure hadronic EOS
soft EOS
hard EOS
12
Comparison with lattice data
Karsch Laermann (2003)
- number of
quark flavours
EOS-III ( ) strongly disagrees
with lattice data in hadronic region TltTc
13
Initial conditions
T. Hirano et al (2002)
flattened Gaussian
all calculations are made for initial proper time
experimental constraints on total energy of
produced particles are used to reduce the number
of model parameters
14
Integrals of motion
line in plane with
For any space-time hypersurface
(B0 for baryon-free case)
total baryon charge
in cm frame
total 4-momentum
total entropy
(ideal fluid)
For our initial conditions
BRAHMS data for central AuAu at
total energy of produced particles
numerical errors (E,S const) for
15
Total energy of secondary hadrons
Initial state set A
BRAHMS data (central AuAu)
well reproduced by the model
E1,3 are sensitive to initial conditions
16
Contours of equal e0
all contours correspond to the total energy
A,B,C
17
Initial energy density profiles
Finite size profiles in ? - space
deviations from Bjorken scaling even for
table-like profile
these profiles correspond to the same total
energy of produced hadrons E
18
Initial temperature profiles
All three phases of matter appear already in the
initial state
19
Best fit parameters of initial state
Fitting
rapidity distributions in central AuAu
collisions at
the results for EOS-I
Set A Gaussian-like initial profile of energy
density
20
Particle spectra at freeze-out
transition from equilibrated fluid to
collisionless expansion (freeze-out)
at some freeze-out hypersurface
we choose the freeze-out condition
Cooper Frye (1974)
freeze-out temperature (model parameter)
4-momentum of the particle
longitudinal rapidity
element of freeze-out hypersurface
we take for all particles
(chemical equilibrium in baryon-free case)
21
Feeding from resonance decay
For two-body decays
branching ratio
mass specrum of resonance R
rest-frame energy and momentum of particle i
zero-width approximation is used for all
resonances
Direct calculations for most dominant resonances
Other resonances included by enhancement factor
mean multiplicity of particle i in R-decays
22
Temperature profiles at different t
with PT
without PT
only forward parts (? gt 0) are shown
clear manifestation of mixed phase at
23
Flow rapidity evolution
with PT
without PT

• local minima of appear due to small
  acceleration in mixed phase

• for soft EOSs in hadronic phase

24
Entropy density evolution
with PT
without PT
Bjorken model
accuracy of 21 models ?

(drops by 15 at )
even for
small sensitivity of particle rapidity spectra
small sensitivity to PT
25
Isotherms in t-? plane
with PT
without PT
Initial state set A
2p-data (STAR) hadron emission times
10 fm/c
mixed phase
26
Rapidity distribution of pions
feeding
for best fits
- pion yield is larger for smaller TF (for
- contribution of resonance decays 35 at
midrapidity
(dotted curves)
- best-fit initial energy density is sensitive
to TC
27
Rapidity distribution of kaons
- sensitivity to freeze-out temperature is weak
for kaons
- best fits with
- contribution of resonance decays 45-50 at
midrapidity
28
Rapidity distribution of antiprotons
feeding
for best fits
- contribution of resonance decays (mainly )
55 at midrapidity !
- nonzero chemical potential should be added in a
more realistic model
thermal model ahalysis of RHIC data
29
Temperature profiles at different t (II)
Landau-like
Initial state
Bjorken-like
total energy
similar behaviour at large
30
Flow rapidity evolution (II)
Bjorken-like
Landau-like
31
Isotherms in t-? plane (II)
for table-like initial profile of energy density
(set D)
back-bended parts of isotherms appear at ?gt?0
32
Isotherms in t-? plane (III)
with PT
without PT
Landau-like initial state
33
Rapidity distributions of p and K mesons (II)
Landau-like initial state
34
Conclusions

• best agreement with BRAHMS data for central AuAu
  collisions

at 200 GeV is achieved for

• the only unsatisfactory prediction too long
  freeze-out times
• 50 fm/c for pions

possible solutions transverse expansion,
chemical nonequilibrium, explosive hadronization
from deconfined phase
35
Bjorken model for ideal gas
ideal gas EOS
Scaling hydro
Estimate of energy density from experimental data
(valid at freeze-out)
central AuAu at
(STAR, PHOBOS)
for
this estimate agrees with our results
36
BRAHMS data
PRL 93 (2004), 94 (2005)
Gaussian fits
close to Landau prediction
for
37
Entropy density
38
Equilibrium densities of pions
contribution of ? is dominant at all T
equilibrium density of pions hidden in
resonances exceeds density of direct pions at
39
Flow rapidity evolution - II
For hard EOS in hadronic phase
with PT
without PT
stronger acceleration of fluid
40
Isotherms of flow rapidity
with PT
without PT
Initial state set A
with from
Flow rapidity isotherm
41
Rapidity distributions of p and K mesons
sensitivity to phase transition
initial state
(unrealistic)
42
Rapidity distribution of pions (II)
43
Rapidity distribution of kaons (II)