Nantes June 15, 2009 GDRE - 2009 Spectra and space-time scales in relativistic heavy ion collisions:...
If you can't read please download the document
Nantes June 15, 2009 GDRE - 2009 Spectra and space-time scales in relativistic heavy ion collisions: the results based on HydroKinetic Model (HKM) Based
Nantes June 15, 2009 GDRE - 2009 Spectra and space-time scales
in relativistic heavy ion collisions: the results based on
HydroKinetic Model (HKM) Based on: Yu.S., I. Karpenko, A. Nazarenko
J. Phys. G: Nucl. Part. Phys. 35 104071 (2008); S.V. Akkelin, Y.
Hama, Iu. Karpenko, Yu.S., PR C 78, 034906 (2008 ) Yu. Sinyukov,
BITP, Kiev Yu.S. Two lectures in Acta Phys Polon. 40 (2009)
Slide 2
Nantes June 15, 2009 2 Heavy Ion Experiments E_lab/A (GeV)
Slide 3
3 Thermodynamic QCD diagram of the matter states The
thermodynamic arias occupied by different forms of the matter
Theoretical expectations vs the experimental estimates
Slide 4
4 UrQMD Simulation of a U+U collision at 23 AGeV
Slide 5
Dnepropetrovsk, 3 May 2009 5 Soft Physics measurements x t A A
KK p=(p 1 + p 2 )/2 q= p 1 - p 2 (QS) Correlation function
Space-time structure of the matter evolution, e.g., T ch and ch
soon after hadronization (chemical f.o.) Radial flow Landau/Cooper-
Frye prescription
Slide 6
NPQCD-2007 6 Collective transverse flows P T Initial spatial
anisotropy different pressure gradients momentum anisotropy v
2
Slide 7
7 Energy dependence of the interferometry radii Energy- and
kt-dependence of the radii Rlong, Rside, and Rout for central Pb+Pb
(Au+Au) collisions from AGS to RHIC experiments measured near
midrapidity. S. Kniege et al. (The NA49 Collaboration), J. Phys.
G30, S1073 (2004).
Slide 8
8 Expecting Stages of Evolution in Ultrarelativistic A+A
collisions Early thermalization at 0.5 fm/c 0.2?(LHC) Elliptic
flows t Relatively small space-time scales (HBT puzzle) Early
thermal freeze-out: T_th Tch 150 MeV 10-15 fm/c 7-8 fm/c 1-3 fm/c
or strings
Slide 9
9 HBT PUZZLE & FLOWS Possible increase of the
interferometry volume with due to geometrical volume grows is
mitigated by more intensive transverse flows at higher energies:,
is inverse of temperature Why does the intensity of flow grow? More
more initial energy density more (max) pressure p max BUT the
initial acceleration is the same HBT puzzle Intensity of collective
flows grow Time of system expansion grows Initial flows ( < 1-2
fm/c) develop
Dnepropetrvsk May 2009 NPQCD-2009 11 Duration of particle
emission is taken into account by means of enclosed freeze-out
hypersurface: v i =0.35 fm/c volume emission surface emission
Slide 12
Dnepropetrovsk May 3 2009 NPQCD-2009 12 Ro/Rs ratio and initial
flows
Slide 13
Dnepropetrovsk May 3 2009 NPQCD -2009 13 Hydrodynamic
expansion: gradient pressure acts Free streaming: Gradient of
density leads to non-zero collective velocities For nonrelativistic
gas So, even if and : Yu.S. Acta Phys.Polon. B37 (2006) 3343;
Gyulassy, Yu.S., Karpenko, Nazarenko Braz.J.Phys. 37 (2007) 1031.
:at For thermal and non-thermal expansion In the case of
thermalization at later stage it leads to spectra anisotropy Basic
ideas for the early stage: developing of pre-thermal flows
Slide 14
Dnepropetrovsk May 2009 NPQCD-2009 14 Comparision of flows at
free streaming and hydro evolution
Slide 15
15 Boost-invariant distribution function at initial
hypersurface McLerran-Venugopalan CGC effective FT (for
transversally homogeneous system) Transversally inhomogeneous
system: of the gluon distribution proportional to the ellipsoidal
Gaussian defined from the best fit to the density of number of
participants in the collisions with the impact parameter b.
A.Krasnitz, R.Venugopalan PRL 84 (2000) 4309; A. Krasnitz, Y. Nara,
R. Venugopalan: Nucl. Phys. A 717 (2003) 268, A727 (2003) 427;T.
Lappi: PRC 67 (2003) 054903, QM 2008 (J.Phys. G, 2008) is the
variance of a Gaussian weight over the color charges of partons If
one uses the prescription of smearing of the -function as, then. As
the result the initial local boost-invariant phase-space density
takes the form
Slide 16
16 Developing of collective velocities in partonic matter at
pre-thermal stage ( Yu.S. Acta Phys. Polon. B37, 2006 ) Equation
for partonic free streaming in hyperbolic coordinates between
Solution where
Slide 17
Collective velocity developed at pre-thermal stage from proper
time tau_0 =0.3 fm/c by supposed thermalization time tau_i = 1 fm/c
for scenarios of partonic free streaming and free expansion of
classical field. The results are compared with the hydrodynamic
evolution of perfect fluid with hard equation of state p = 1/3
epsilon started at tau_0. Impact parameter b=0. Yu.S., Nazarenko,
Karpenko: Acta Phys.Polon. B40 1109 (2009)
Slide 18
Collective velocity developed at pre-thermal stage from proper
time tau_0 =0.3 fm/c by supposed thermalization time tau_i = 1 fm/c
for scenarios of partonic free streaming. The results are compared
with the hydrodynamic evolution of perfect fluid with hard equation
of state p = 1/3 epsilon started at tau_0. Impact parameter b=6.3
fm. Yu.S., Nazarenko, Karpenko: Acta Phys.Polon. B40 1109
(2009)
Slide 19
is a fitting parameter Initial parameters becomes to be
isotropic at =1 fm/c: The "effective" initial distribution - the
one which being used in the capacity of initial condition bring the
average hydrodynamical results for fluctuating initial conditions:
where is a fitting parameter For central Au+Au (Pb+Pb)
collisions
Slide 20
20 Hydro-Evolution: Equation of State EoS from LattQCD (in form
proposed by Laine & Schroder, Phys. Rev. D73, 2006) MeV The EoS
accounts for gradual decays of the resonances into expanding
hadronic gas consistiong of 359 particle species with masses below
2.6 GeV. The EoS in this non chemically equilibrated system depends
now on particle number densities of all the 359 particle species.
Since the energy densities in expanding system do not directly
correlate with resonance decays, all the variables in the EoS
depends on space-time points and so an evaluation of the EoS is
incorporated in the hydrody- namic code. We calculate the EoS below
in the approximation of ideal multi- component hadronic gas.
MeV
Slide 21
Hydro-evolution MeV Chemically equilibrated evolution of the
quark-gluon and hadron phases. For : MeV The system evolves as non
chemically equilibra- ted hadronic gas. The conception of the
chemical freeze-out imply that mainly only elastic collisi- ons and
the resonance decays among non-elastic reactions takes place
because of relatively small densities allied with a fast rate of
expansion at the last stage. Thus, in addition to (1) the equa-
tions accounting for the particle number conser- vation and
resonance decays are added: where denote the average number of i-th
particles coming from arbitrary decay of j-th resonance, is
branching ratio, is a number of i-th particles produced in decay
channel.
Slide 22
Cooper-Frye prescription (CFp) t z t r CFp gets serious
problems: Freeze-out hypersurface contains non-space-like sectors
artificial discontinuities appears across Sinyukov (1989), Bugaev
(1996), Andrelik et al (1999); cascade models show that particles
escape from the system about whole time of its evolution. Hybrid
models (hydro+cascade) and the hydro method of continuous emission
starts to develop.
Slide 23
Hybrid models: HYDRO + UrQMD ( Bass, Dumitru (2000)) t z t r
The problems: the system just after hadronization is not so dilute
to apply hadronic cascade models; hadronization hypersurface
contains non-space-like sectors (causality problem: Bugaev, PRL 90,
252301, 2003); hadronization happens in fairly wide 4D-region, not
just at hypersurface, especially in crossover scenario. t HYDRO
UrQMD The initial conditions for hadronic cascade models should be
based on non-local equilibrium distributions 20 May 2009 23
Slide 24
24 t x F. Grassi,Y. Hama, T. Kodama Continuous emission
Hydro-kinetic approach is based on combination of Boltsmann
equation and hydro for finite expanding system; provides evaluation
of escape probabili- ties and deviations (even strong) of distri-
bution functions from local equilibrium; accounts for conservation
laws at the particle emission; PROVIDE earlier (as compare to
CF-prescription) emission of hadrons, because escape probability
accounts for whole particle trajectory in rapidly expanding
surrounding (no mean-free pass criterion for freeze-out) Yu.S.,
Akkelin, Hama: PRL. 89, 052301 (2002); + Karpenko: PRC 78 034906
(2008). Basic ideas for the late stage
Slide 25
25 * Is related to local Hydro-kinetic approach MODEL provides
evaluation of escape probabilities and deviations (even strong) of
distribution functions [DF] from local equilibrium; is based on
relaxation time approximation for relativistic finite expanding
system; (Shear viscosity ). accounts for conservation laws at the
particle emission; Complete algorithm includes: solution of
equations of ideal hydro [THANKS to T. Hirano for possibility to
use code in 2006] ; calculation of non-equilibrium DF and emission
function in first approximation; [Corresponding hydro-kinetic code:
Tytarenko, Karpenko,Yu.S.(to be publ.)] Solution of equations for
ideal hydro with non-zero left-hand-side that accounts for
conservation laws for non-equlibrated process of the system which
radiated free particles during expansion; Calculation of exact DF
and emission function; Evaluation of spectra and correlations.
Slide 26
26 and are G(ain), L(oss) terms for p. species Boltzmann eqs
(differential form) Escape probability (for each component )
Boltzmann equations and Escape probabilities
Slide 27
27 Boltzmann eqs (integral form) Spectra and Emission function
Index is omitted everywhere Spectrum Method of solution (Yu.S. et
al, PRL, 2002)
Slide 28
28 Saddle point approximation Emission density Spectrum where
Normalization condition Eqs for saddle point : Physical conditions
at
Slide 29
29 Cooper-Frye prescription Spectrum in new variables Emission
density in saddle point representation Temporal width of emission
Generalized Cooper-Frye f-la
Slide 30
30 Generalized Cooper-Frye prescription: 30 r t 0 Escape
probability Yu.S. (1987)-particle flow conservation; K.A. Bugaev
(1996) (current form)
Slide 31
Nov 3-6 RANP08 31 Momentum dependence of freeze-out Here and
further for Pb+Pb collisions we use: initial energy density EoS
from Lattice QCD when T< 160 MeV, and EoS of chemically frozen
hadron gas with 359 particle species at T< 160 MeV.
Pt-integrated
Slide 32
The pion emission function for different pT in hydro-kinetic
model (HKM) The isotherms of 80 MeV is superimposed.
Slide 33
The pion emission function for different pT in hydro-kinetic
model (HKM). The isotherms of 135 MeV (bottom) is
superimposed.
Slide 34
Transverse momentum spectrum of pi in HKM, compared with the
sudden freeze-out ones at temperatures of 80 and 160 MeV with
arbitrary normalizations.
Slide 35
35 Conditions for the utilization of the generalized
Cooper-Frye prescription i)For each momentum p, there is a region
of r where the emission function has a sharp maximum with temporal
width. ii) The width of the maximum, which is just the relaxation
time ( inverse of collision rate), should be smaller than the
corresponding temporal homogeneitylength of the distribution
function: 1% accuracy!!! iii) The contribution to the spectra from
the residual region of r where the saddle point method is violated
does not affect essentially the particle momentum spectrum. Then
the momentum spectra can be presented in Cooper-Frye form despite
it is, in fact, not sadden freeze-out and the decaying region has a
finite temporal width. Also, what is very important, such a
generalized Cooper-Frye representation is related to freeze-out
hypersurface that depends on momentum p and does not necessarily
encloses the initially dense matter. iiii) The escape probabilities
for particles to be liberated just from the initial hyper-surface
t0 are small almost in the whole spacial region (except peripheral
points)
Slide 36
36 Inititial conditions in Pb+Pb and Au+Au colliisions for SPS,
RHIC and LHC energies We use 0.2 (the average transverse flow =
0.17) for SPS energies and 0.35 ( = 0.26) for all the RHIC and LHC
energies. Note that this parameter include also a correction of
underestimated transverse flow since we did not account for the
viscosity effects. For the top SPS energy GeV/fm 3 = 4.7 GeV/fm 3
), for the top RHIC energy GeV/fm 3 = 8.0 GeV/fm 3 ) We also
demonstrate results at = 40 GeV/fm 3 and = 50 GeV/fm 3 that
probably can correspond to the LHC energies 4 and 5.6 A TeV. As for
latter, according to Lappy, at the top LHC energy recalculated to
the time 1 fm/c is 0.07*700=49 GeV/fm 3. The chosen values of
approximately correspond to the logarithmic dependence of the
energy density on the collision energy.
Slide 37
Energy dependence of pion spectra
Slide 38
Energy dependence of the interferometry scales, R_long
Slide 39
Energy dependence of the interferometry scales, R_side
Slide 40
Energy dependence of the interferometry scales, R_out
Slide 41
Energy dependence of ratio
Slide 42
42 Pion emission density at different energies in HKM
Slide 43
The ratio as function on in-flow and energy If the extreme
internal fluid element has initial transverse velocity and undergo
mean acceleration, then. Suppose, that the fluid elements, that
finally form almost homogeneous in time emission with duration time
are initially situated within radial interval shifted to the
periphery of the system.
Slide 44
Pion and kaon interferometry Long- radii (preliminary)
Dnepropetrovsk May 2009 NPQCD-2009 44
Slide 45
Pion and kaon interferometry Side- radii (preliminary)
Dnepropetrovsk May 3 2009 NPQCD-2009 45
Slide 46
Pion and kaon interferometry Out- radii (preliminary)
Dnepropetrovsk May 3 2009 NPQCD-2009 46
Slide 47
Hydrodynamics with initial tube-like fluctuations (formation of
ridges, initially 10 small tubes)
Slide 48
Hydrodynamics with initial tube-like fluctuations (formation of
ridges, initially 1 small tube)
Slide 49
49 Conclusions-1 The following factors reduces space-time
scales of the emission and Rout/Rside ratio. developing of initial
flows at early pre-thermal stage; more hard transition EoS,
corresponding to cross-over; non-flat initial (energy) density
distributions, similar to Gaussian; early (as compare to standard
CF-prescription) emission of hadrons, because escape probability
account for whole particle trajectory in rapidly expanding
surrounding (no mean-free pass criterion for freeze-out) Viscosity
[Heinz, Pratt] The hydrokinetic approach to A+A collisions is
proposed. It allows one to describe the continuous particle
emission from a hot and dense finite system, expanding
hydrodynamically into vacuum, in the way which is consistent with
Boltzmann equations and conservation laws, and accounts also for
the opacity effects. 49
Slide 50
50 Conclusions-2 The CFp might be applied only in a generalized
form, accounting for the direct momentum dependence of the
freeze-out hypersurface corresponding to the maximum of the
emission function at fixed momentum p in an appropriate region of
r.
Slide 51
51 Conclusions-3 A reasonable description of the pion spectra
and HBT (except some an overestimate for ) in cental Au+Au
collisions at the RHIC energies is reached with the value of the
fitting parameter or the average energy density at the initial time
The initial time fm/c and transverse width 5.3 fm (in the Gaussian
approximation) of the energy density distribution are obtained from
the CGC estimates. The EoS at the temperatures corresponds to the
lattice QCD calculations at The used temperature of the chemical
freeze-out MeV is taken from the latest results of particle number
ratios analysis (F. Becattini, J.Phys. G, 2008). The anisotropy of
pre-thermal transverse flows in non-central collisions, bring us a
hope for a successful description of the elliptic flows with
thermalization reached at a relatively late time:1-2 fm/c.
Slide 52
CONCLUSION FINAL
Slide 53
Nov Dnepropetrovsk May 3 2009 NPQCD-2009 53 Hybrid models:
HYDRO + UrQMD ( Bass, Dumitru (2000) ) t z t r The problems: the
system just after hadronization is not so dilute to apply hadronic
cascade models; hadronization hypersurface contains non-space-like
sectors (causality problem: Bugaev, PRL 90, 252301, 2003);
hadronization happens in fairly wide 4D-region, not just at
hypersurface, especially in crossover scenario. t HYDRO UrQMD The
initial conditions for hadronic cascade models should be based on
non-local equilibrium distributions