Energy dependence of femtoscopy scales in A A collisions and predictions for LHC

1
Energy dependence of femtoscopy scales in AA
collisions and predictions for LHC
Yu. M. Sinyukov Bogolyubov Institute for
Theoretical Physics, Kiev
In collaboration with Yu. Karpenko
Workshop on Particle Correlations and Femtoscopy
WPCF-2010 Kiev, 14-18 September, 2010
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The evidences of space-time evolution of the
thermal matter in AA collisions

• Rough estimate of the fireball lifetime for AuAu
  Gev

In pp all femto-scales are
AA is not some kind of superposition of the of
order 1 fm !
individual collisions of nucleons of nuclei
The phenomenon of space-time evolution of the
strongly interacting matter in AA collisions
What is the nature of this matter at the early
collision stage? Whether does the matter becomes
thermal?
Particle number ratios are well reproduced in
ideal gas model with 2 parameters T,
for collision energies from AGS to RHIC
thermalchemical equilibrium
3
Collective expansion of the fireball.

• Observation of the longitudinal expansion
• It was conformed by NA35/NA49 Collaborations
  (CERN), 1995 !
  
• 
  
• Observation of transverse (radial) collective
  flows

Effective temperature for different particle
species (non-relativistic case)
radial
collective flow

• Observation of elliptic flows

HYDRODYNAMICS !
4
Expecting Stages of Evolution in
Ultrarelativistic AA collisions
t
Relatively small space-time scales (HBT puzzle)
8-20 fm/c
Early thermal freeze-out T_th Tch
150 MeV
7-8 fm/c
Elliptic flows
1-3 fm/c
Early thermalization at 0.5 fm/c
0.2?(LHC)
or strings

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Pre-thermal transverse flow
5
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Collective velocities developed between 0.3
and 1.0 fm/c
Central collisions
Collective velocity developed at pre-thermal
stage from proper time tau_0 0.3 fm/c by
supposed thermalization time tau_th 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 . Impact parameter
b0.
Yu.S. Acta Phys.Polon. B37 (2006) 3343 Gyulassy,
Yu.S., Karpenko, Nazarenko Braz.J.Phys. 37 (2007)
1031. Yu.S., Nazarenko, Karpenko Acta
Phys.Polon. B40 1109 (2009) .
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Collective velocities and their anisotropy
developed between 0.3 and 1.0 fm/c
Non-central collisions b6.3 fm
Collective velocity developed at pre-thermal
stage from proper time 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 . Impact parameter
b6.3 fm.
8

Basic ideas for the early stage developing of
pre-thermal flows
Yu.S. Acta Phys.Polon. B37 (2006) 3343 Gyulassy,
Yu.S., Karpenko, Nazarenko Braz.J.Phys. 37 (2007)
1031.
For nonrelativistic gas
For thermal and non-thermal expansion

at
Hydrodynamic expansion gradient pressure
acts
So, even if

and
Free streaming Gradient of density leads to
non-zero collective velocities
In the case of thermalization at later stage it
leads to spectra anisotropy
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Summary-1
Yu.S., Nazarenko, Karpenko Acta Phys.Polon. B40
1109 (2009)

• The initial transverse flow in thermal matter
  as well as its anisotropy are developed at
  pre-thermal - either partonic, string or
  classical field (glasma) - stage with even more
  efficiency than in the case of very early perfect
  hydrodynamics.
• Such radial and elliptic flows develop no
  matter whether a pressure already established.
  The general reason for them is an essential
  finiteness of the system in transverse direction.
• The anisotropy of the flows transforms into
  asymmetry of the transverse momentum spectra
  only of (partial) thermalization happens.
• So, the results, first published in 2006,
  show that whereas the assumption of (partial)
  thermalization in relativistic A A collisions
  is really crucial to explain soft physics
  observables, the hypotheses of early
  thermalization at times less than 1 fm/c is not
  necessary.

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Phenomenological model of pre-thermal evolution
Akkelin, Yu.S. PRC 81, 064901 (2010)
Matching of nonthermal initial conditions and
hydrodynamic stage

• If some model (effective QCD theory) gives us
  the energy-momentum tensor at time , one
  can estimate the flows and energy densities at
  expected time of thermalization , using
  hydrodynamic equation with (known) source terms.
• This phenomenological approach is motivated by
  Boltzmann equations, accounts for the energy and
  momentum conservation laws and contains two
  parameters supposed time of thermalization
  and initial relaxation time .

IC
Eqs
where
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HydroKinetic Model (HKM) of AA collisions I.
Matter evolution in chemically equilibrated
space-time zone
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Locally (thermally chemically) equilibrated
evolution and initial conditions (IC)
t
Tch
IC for central AuAu collisions
The effective" initial distribution is the one
which being used in the capacity of initial
condition bring the average hydrodynamic results
for fluctuating initial conditions
x
is Glauber-like profile
I.
Initial rapidity profiles
Yu. Karpenko talk at this Workshop
where
II.
is CGC-like profile
and are only fitting parameters
in HKM
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Equation of state in (almost) equilibrated zone
EoS from LattQCD (in form proposed by Laine
Schroder, Phys. Rev. D73, 2006).
MeV
Crossover transition, LattQCD is matched with an
ideal chemically equilibrated
multicomponent hadron resonance gas at Yu.
Karpenko talk at this Workshop
Particle number ratios
F. Karsch, PoS CPOD07026, 2007
are baryon number and strangeness susceptibilities
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HKM II. Evolution of the hadronic matter in
non-equlibrated zone.
t
Decay of the system and
spectra formation
Tch
x
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Soft Physics measurements
A
x
Landau, 1953
t
??K
Cooper-Frye prescription (1974)
A
p(p1 p2)/2 q p1- p2
QS correlation function Space-time
structure of the matter evolution
Long
p1
p
Out
p2
Side
BW
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Cooper-Frye prescription (CFp)
t
t
z
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 (hydrocascade) and the hydro
  method of continuous emission starts to develop.

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Hybrid models HYDRO UrQMD (Bass, Dumitru
(2000))

t
t
t
UrQMD
HYDRO
z
r
The initial conditions for hadronic cascade
models should be based on non-local equilibrium
distributions

• 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)
• The average energy density and pressure of input
  UrQMD gas should coincide with what the hadro gas
  has just before switching.
• At r-periphery of space-like hypsurf. the
  system is far from l.eq.

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Possible problems of matching hydro with
initially bumping IC
RIDGES?
The example of boost-invariant hydroevolution for
the bumping IC with ten narrow high energy
density tubes (r 1 fm) under smooth Gaussian
background (R5.4 fm)
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(No Transcript)
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Continuous Emission
The back reaction of the emission on the fluid
dynamics is not reduced just to energy-momen-tum
recoiling of emitted particles on the expan-ding
thermal medium, but also leads to a
re-arrangement of the medium, producing a
devia-tion of its state from the local
equilibrium, ac-companied by changing of the
local temperature, densities, and collective
velocity field. This complex effect is mainly a
consequence of the fact that the evolution of the
single finite system of hadrons cannot be split
into the two compo-nents expansion of the
interacting locally equi-librated medium and a
free stream of emitted particles, which the
system consists of. Such a splitting, accounting
only for the momentum-energy conservation law,
contradicts the unde-rlying dynamical equations
such as a Boltzmann one.
t
x
F. Grassi, Y. Hama, T. Kodama (1995)
Akkelin, Hama, Karpenko, Yu.S PRC 78 034906 (2008)
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Yu.S. , Akkelin, Hama PRL 89 , 052301 (2002)
Karpenko PRC 78, 034906
(2008).
Hydro-kinetic approach

• MODEL
• is based on relaxation time approximation for
  emission function of relativistic finite
• expanding system
• provides evaluation of escape probabilities and
  deviations (even strong)
• of distribution functions DF from local
  equilibrium
• 3. accounts for conservation laws at the particle
  emission
• Complete algorithm includes
• solution of equations of ideal hydro
• calculation of non-equilibrium DF and emission
  function in first approximation
• solution of equations for ideal hydro with
  non-zero left-hand-side that
• accounts for conservation laws for
  non-equilibrium process of the system
• which radiated free particles during
  expansion
• Calculation of exact DF and emission function
• Evaluation of spectra and correlations.

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Boltzmann equations and probabilities of
particle free propagation
Boltzmann eqs (differential form)
and
are G(ain), L(oss) terms for p. species
Probability of particle free propagation (for
each component )
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Spectra and Emission function
Boltzmann eqs (integral form)
Index is omitted everywhere
Spectrum
Relax. time approximation for emission function
(Yu.S. , Akkelin, Hama
PRL, 2002) For (quasi-) stable particles
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Kinetics and hydrodynamics below Tch 165 MeV
For hadronic resonances

where
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Equation of state in non-equilibrated zone
EoS
MeV

Pressure and energy density of multi-component
Boltzmann gas
At hypersurface the hadrons are in
chemical equilibrium with some barionic chemical
potential which are defind from particle
number ratio (conception of chemical freeze-out).
Below we account for the evolution of all
N densities of hadron species in hydro
calculation with decay resonances into expanding
fluid, and compute EoS dynamically for each
chemical composition of N sorts of hadrons in
every hydrodynamic cell in the system during the
evolution. Using this method, we do not limit
ourselves by chemically frozen or chemically
equilibrated evolution, keeping nevertheless
thermodynamically consistent scheme.
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EoS used in HKM calculations for the top RHIC
energy
The gray region consists of the set of the points
corresponding to the different hadron gas
compositions at each occurring during the
late nonequilibrium stage of the evolution.
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System's decoupling and spectra formation

• Emission function
• For pion emission

• is the total collision rate of the pion, carrying
  momentum p with all the hadrons h in the system
  in a vicinity of point x.

is the space-time density of pion production
caused by gradual decays during hydrodynamic
evolution of all the suitable resonances H
including cascade decays
The cross-sections in the hadronic gas are
calculated in accordance with UrQMD .


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The following factors reduces space-time scales
of the emission and Rout/Rside ratio

• Akkelin, Hama, Karpenko, Yu.S, PRC 78, 034906
  (2008)

• essentially non-flat initial energy density
  distributions (Gaussian, Glauber, CGC)
• more hard transition EoS, corresponding to
  cross-over (not first order phase transition!)
• fairly strong transverse flow at the late stage
  of the system evolution. It is caused by
• developing of flows at very early pre-thermal
  stage
• additional developing of transv. flow due to
  shear viscosity (Teaney, 2003)
• effective increase of transv. flow due to
  initially bumping structure (Grassy, Hama, Kodama
  2008)
  
• 
  
• correct description of evolution and decay of
  strongly interacting and chemically/thermally
  non-equilibrated system after hadronisation!
• Karpenko, Yu.S.
  PRC 81, 054903 (2010)

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Energy dependence of space-time scales
Iu. Karpenko, Yu.S. PLB 688, 50 (2010)
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Pion spectra at top SPS, RHIC and two LHC
energies in HKM
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Long- radii at top SPS, RHIC and two LHC energies
in HKM
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Side- radii at top SPS, RHIC and two LHC energies
in HKM
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Out- radii at top SPS, RHIC and two LHC energies
in HKM
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Out- to side- ratio at top SPS, RHIC and two LHC
energies in HKM
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Emission functions for top SPS, RHIC and LHC
energies
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The ratio as function on in-flow
and energy
At some p
2
1
1
2
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Conclusion

• The main mechanisms that lead to the paradoxical
  behavior of the interferometry scales, are
  exposed.
• In particular, decrease of
  ratio with growing energy and saturation of
  the ratio at large energies happens due to a
  magnification of positive
  correlations between space and time positions of
  emitted pions and a developing of pre-thermal
  collective transverse flows.
• The process of decoupling the fireballs created
  in Au Au collision at RHIC energies lasts from
  about 8 to 20 fm/c, more than half the fireballs
  total lifetime. The temperatures in the regions
  of the maximal emission are different at the
  different transverse momenta of emitting pions T
  75110 MeV for pT 0.2 GeV/c and T 130135
  MeV for pT 1.2GeV/c.
• At LHC energies the decay of the central part of
  the fireball is estimated as from 10 to 27 fm/c.

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BACK UP SLIDES
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Momentum transverse spectra of protons in HKM for
top RHIC energy and different types of profiles
(CGC and Glauber) of initial energy density
without and with including of the mean field
effect for protons (12 of the proton transverse
rapidity field off in the interval (0-1))
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Saddle point approximation
Spectrum
Emission density
where
Normalization condition
Eqs for saddle point
Physical conditions at
NPQCD-2009
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Dnepropetrovsk May 3 2009
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Cooper-Frye prescription
Spectrum in new variables
Emission density in saddle point representation
Temporal width of emission
Generalized Cooper-Frye f-la
NPQCD-2009
41
Dnepropetrovsk May 3 2009
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Generalized Cooper-Frye prescription
t
0
Escape probability
r
RANP08
42
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Yu.S. (1987)-particle flow conservation K.A.
Bugaev (1996) (current form)
Nov 3-6
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Momentum dependence of freeze-out
Here and further for PbPb collisions we
use initial energy density
Pt-integrated
EoS from Lattice QCD when Tlt 160 MeV, and EoS of
chemically frozen hadron gas with 359 particle
species at Tlt 160 MeV.
RANP08
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Nov 3-6
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The pion emission function for different pT in
hydro-kinetic model (HKM)The isotherms of 80 MeV
is superimposed.
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The pion emission function for different pT in
hydro-kinetic model (HKM). The isotherms of 135
MeV (bottom) is superimposed.
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Transverse momentum spectrum of pi- in HKM,
compared with the sudden freeze-out ones at
temperatures of 80 and 160 MeV with arbitrary
normalizations.
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Conditions for the utilization of the generalized
Cooper-Frye prescription

• 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.
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)
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.
NPQCD-2009
47
Dnepropetrovsk May 3 2009
48
Soft Physics measurements
A
x
Landau, 1953
t
??K
Cooper-Frye prescription (1974)
A
p(p1 p2)/2 q p1- p2
QS correlation function Space-time
structure of the matter evolution
Long
p1
p
Out
p2
Side
BW
48
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Conclusions
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.