A FAST HADRON FREEZE-OUT GENERATOR

, R. Lednicky, T.A. Pocheptsov: Joint Institute for Nuclear Research, Dubna, Russia

I.P. Lokhtin, A.M. Snigirev , L.V. Malinina: Moscow State University, Institute of Nuclear Physics, Russia Iu.A. Karpenko, Yu.M. Sinyukov: Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

N.S. Amelin

FASTMC

a part of Universal Hydro Kinetic Model (UHKM)

(PRC 74 064901(2006))

Outline

1. Introduction- motivation.

2. Physical framework of the model .

3. Hadron generation procedure .

4. Model parameters.

5. Examples of calculations with generator and comparison with RHIC experimental data

- particle ratios- pseudorapidity- pt-spectra- correlation radii

6. Conclusions, perspectives, status

http://uhkm.jinr.ru

The creation of the UHKM Universal Hydro Kinetic Model was initiated by Nikolai Amelin.

- LHC very high hadron multiplicities fairly fast MC- generators for event simulation. (I. P. Lokhtin and A. M.Snigirev, Phys. Lett. B378, 247 (1996), Z. Phys. C76, 99 (1997).)

- FASTMC- fast Monte Carlo procedure of hadron generation

- Matter is thermally equilibrated. Particle multiplicities are determined by the temperature and chemical potentials. Statistical model. Chemical freeze-out.

- Particles can be generated on the chemical or thermal freeze-out hypersurface represented by a parameterization or a numerical solution of relativistic hydrodynamics.

- Standard parameterizations of the hadron freeze-out hypersurface and flow velocity profile under the assumption of a common chemical and thermal freeze-out.so-called Bjorken-like (BlastWave) and Hubble-like parametrizations. (F. Retiere and M. Lisa, Phys.Rev. C70 (2004) 044907; THERMINATOR: Thermal heavy-Ion generator A. Kisiel, T. Taluc, W. Broniowski, W. Florkowski, nucl-th/0504047)

FASTMC: Introduction-Motivation

Introduction-Motivation

-The C++ generator code is written under the ROOT framework.

- Besides standard space-like sectors associated with the volume decay, the hypersurface may also include non-space-like sectors related to the emission from the surface of expanding system (Kiev- Nantes model).

- Decays of hadronic resonances is included (presently u,d and s quarks only). Note that it is not pure “Blast-Wave” because the resonances are included

1. We consider the hadronic matter created in heavy-ion collisions as a hydrodynamically expanding fireball with the equation of state of an ideal hadron gas.

2. “concept of effective volume” T=const and µ=const the total yield of particle species is: , total co-moving volume, ρ-particle number density

3. in central Au+Au or Pb+Pb collisions at GeV

parametrizations of the T and µB:

4. All macroscopic characteristics of particle system (T,µi) are determined via a set of equilibrium distribution functions in the fluid element rest frame:

effiii VTN ),(

2002.2 NNs

NN

NNBBBBse

dscbaT

1)(;)( 4

),;(4))(),(;(),( 0*2*

0

***0*

0

*3i

eqii

eqii

eqi TpfpdpxxTpfpdT

1)/]exp([)2(

1),;(

0*30*

Tp

gTpf

i

ii

eqi

)()exp()(

2),( 2

1

12

2 T

kmK

T

k

kTm

gT i

k

ik

ii

ieqi

Physical framework of the model: Hadron multiplicities

We suppose that a hydrodynamic expansion of the fireball ends by a sudden system breakupat given T and chemical potentials. Momentum distribution of produced hadrons keeps the thermal character of the equilibrium distribution.

),);(()()(

33

30

i

x

eqi

i Txupfpxdpd

Ndp

Physical framework of the model: Hadron momentum distribution

),);(()(),(33

60

, ieq

ii

i Txupfpxnpdd

NdppxW

vnnnn

nvnunn

)()1(

),(00*1*

00*

),;()(),(),( 0******,, i

eqiii TpfpxnpxWpxW

}0,0,0,1{** un

Cooper-Frye formula:

vpppp

pvpp

)()1(

),(00*1*

*00

The invariant weight

It is convenient to transform the four-vectors intothe fluid element rest frame,

and calculate the weight in this frameIn the case when the normal four-vector coincides with the fluid element flow velocity

the weight is independent of x and isotropic in p. 100% efficiency

the four-momenta of the generated particles transformed back to the fireball rest frame using the velocity field v.

Physical framework of the model: Freeze-out surface parameterizations

At relativistic energies, due to dominant longitudinal motion, it is convenient to substitute the Cartesian coordinates t, z by the Bjorken ones

2/122 )( zt }sin,cos{},{ rryxr

zt

zt

ln2

1, ,

Similarly, it is convenient to parameterize the fluid flow four-velocity at a point x in terms of the longitudinal z and transverse r fluid flow rapidities:

)(1

)(1ln

2

1)(

xv

xvx

z

zu

)(cosh)(1

)(cosh)(1ln

2

1)(

xxv

xxvx

ur

uru

,

}sin,,cos{ rrrx

}sincosh,sinsinh,cossinh,cos{cosh)( uuuuuuuuxu Representing the freeze-out hypersurface by the equation the hypersurface element:

),,( r

drdddrdd

dxdxdxd 3

for the azimuthaly symmetric hypersurface ),( r )(r we further assume the longitudinal boost invariance

The local quantities (such as particle density) are then functions of τ and r only

For the simplest freeze-out hypersurface const drdd23

Physical framework of the model: Freeze-out surface parameterizations

Assuming and the linear transverse flow rapidity profile: u

maxuu R

r here R is the fireball transverse radius;

The total effective volume for particle production at const

)1coshsinh(2)()( maxmaxmax

2

max

2

00)(

3max

min

uuu

u

R

r

x

eff

RddrdrxuxdV

We refer this choice of the freeze-out hyper-surface and the flow 4-velocity profile as the Bjorken-like parametrization

We also consider so called Cracow model scenario corresponding to the Hubble-like freeze-out hypersurface and spherically-symmetric Hubble flow with four-velocity Hxxu /)(

The Bjorken model with hypersurface constzt 2/122 )(

constzyxt 2/12222 )(

2RV Heff

)()( xuxn

)()( xuxn

Hadron generation procedure

Initialization of the chosen model parameters

Calculation of Veff and particle number densities

effiii VTN ),( The mean multiplicities

Simulation of particle freeze-out 4-coordinates in the fireball rest frame :on each hypersurface segment accord. to the element by sampling uniformly distributed r,η, φ

Calculation ofthe corresponding collective flow four-velocities

The particle three-momenta in the fluid element rest framesaccording to the probabilityby sampling uniformly distributed cos(θ*p) and φ*p

von Neumann rejection/acceptance procedure to account for diff. between the true prob. and prob. corresponding to 3-5. Residual weight simulated x, p are accepted W> ξ- a test variable randomly simulated in [0,W_max], otherwise-> 3

***2**0 cos),;( ppieq

i dddppTpf

the hadron four-momentum isboosted to the fireball rest frame

The two-body, three-body and many-body decays are simulated with the branching ratios

calculated via ROOT utilities;--Boltzmann equation solver

Multiplicities by Poisson distr.

*0

33 dud

)1( *0*0

**

pnpn

W resi

It should be stressed that a high generation speed is achieved due to 100 % generation efficiency of the freeze-out four-coordinates and four-momenta in steps 3-5 as well as due to a weak non-uniformity of the residual weight Win the cases of practical interest.

For example, in the Bjorken-like model, the increase of the maximal transverse flow rapidity from zero to 0.65 leads only to a few percent decrease of the generation speed. Compared, e.g., to THERMINATOR our generator appears more than two order of magnitude faster.

Hadron generation procedure

Validation of the fast MC procedure

In the Boltzmann approximation for the equilibrium distribution function,the transverse momentum spectrum in the Bjorken-like model takesthe form:

Exact formulae (solid line), the corresponding MC results (black triangles),MC results obtained with a constant residual weight (black points).

65.0max u

2max utransverse flow rapidity

Validation of the MC procedure:

Model parameters:1. Number of events to generate.

2. Thermodynamic parameters at chemical freeze-out: T temperature

{µB, µS, µQ} chemical potentials per a unit charge

3. As an option, γS < 1 taking into account the strangeness suppression

4. Volume parameters: τ -the freeze-out proper time R- firebal transverse radius

5. -maximal transverse flow rapidity for Bjorken-like parametrization

6. ηmax -maximal space-time longitudinal rapidity which determines the rapidity interval [- ηmax, ηmax] in the collision center-of-mass system.

7. To account for the violation of the boost invariance, an option corresponding to the substitution of the uniform distribution of the space-time longitudinal rapidity by a Gaussian distribution.

8. Option to calculate T, µB using phenomenological dependence of

maxu

)(),( BNNB Ts

The parameters used to model central Au+Au collisions at = 200 GeV

parameter Bjorken-like Hubble-like

T, GeV 0.165 0.165 µB, GeV 0.028 0.028µS, GeV 0.007 0.007µQ, GeV -0.001 -0.001γS 1 (0.8) 1 (0.8) τ, fm/c 6.1 9.65 R, fm 10.0 8.2 ηmax 2 (3,5) 2 (3,5) ρu max 0.65 -

s

FASTMC PHENIX

π -/π + 0.98 0.984 ± 0.004K-/K+ 0.94 0.933 ±0.008K-/ π + 0.21 0.162 ± 0.001p-/p 0.71 0.731 ± 0.011

Particle number ratios near mid-rapidity in central Au + Au collisions s= 200 GeV

Examples of calculations and comparison with RHIC experimental data:

Particle ratios near mid-rapidity in central Au + Au collisions at = 130 GeV.

T = 0.168 GeV, µB = 0.041 GeV, µS = 0.010 GeV and µQ = -0.001 GeV.

s

PHOBOS data on pseudo-rapidity spectrum of charged hadrons in central Au+Au collisions at =200 GeV (black points)our MC results obtained within the Bjorken-like and Hubble-likemodels for different values of (lines).

Single freeze-out scenario allowsto fix the effective volume

s

max

2RVeff

Pseudo-rapidity spectrum of charged hadrons

. p_t spectraThe mid-rapidity PHENIX data on π, K and proton p_t spectra in Au+Au collisions at 200 GeV with our MC results obtained within the Bjorken-like and Hubble-like models.

The overestimated slope of the kaon and proton p_t spectracan also be related with the oversimplified assumption of a commonthermal and chemical freeze-out or insufficient number of theaccounted heavy resonance states.

For kaons, the discrepancy can be diminished with the help of the strangeness suppression parameter γs of 0.8

Due to the effects of QS and FSI,the momentum correlations of two or more particles at small relative momenta in their center-of-mass system are sensitive to the space-time characteristics of the productionprocess so serving as a correlation femtoscopy tool.

Momentum correlations

q = p1- p2 , x = x1- x2

out transverse pair velocity vt

side

long beam

The corresponding correlation widths are usually parameterized in terms of the Gaussian correlation radii R_i:

)2(1),( 2,

22222221 longoutlongoutlonglongsidesideoutout qqRqRqRqRppCF

We choose as the reference frame the longitudinal co-moving system (LCMS)

.

w=1+cos qx

T=165 MeV T=100 MeV T=130 MeV

Momentum correlations

STAR collaboration data (open points)

FASTMC Tth=Tch=165 MeV, R=10 fm, τ=6.1 fm/c (solid line)

FASTMC Tth=Tch=165 MeV, R=8 fm, τ=9 fm/c (dashed line)

FASTMC Tth=Tch=165 MeV, R=8 fm, τ=9 fm/c (dashed line)with negative r-t correlation coefficient (red dotted line)

2RVeff

The concept of a later thermal freeze-out occurring at Tth<Tch and with no multiplicity constraint on the thermal effective volumecan help to resolve this problem:

More realistic models as compared with the simpleBjorken-like and Hubble-like ones (particularly, consider a more complex formof the freeze-out hypersurface taking into account particle emissionfrom the surface of expanding system M.S.Borysova, Yu.M.Sinyukov, S.V.Akkelin,B.Erazmus and Iu.A.Karpenko, Phys. Rev. C 73, 024903 (2006)

and study the problem of particle rescattering and resonance excitation after thechemical and/or thermal freeze-out (only minor effect of elastic rescatterings on particle spectra and correlations is expected N.S.Amelin, R.Lednicky, L.V.Malinina, T.A.Pocheptsov and Y.M.Sinyukov, Phys. Rev. C 73, 044909 (2006)

For the latter, our earlier developed C++ kinetic code can be coupled to FASTMC.

F. Retiere and M. Lisa, Phys.Rev. C70 (2004) 044907

Momentum correlations: other approaches

Conclusions -We have developed a MC simulation procedure and the corresponding C++ code allowing for a fast but realistic description of multiple hadron production in central relativistic heavy ion collisions.

- A high generation speed and an easy control through input parameters make our MC generator code particularly useful for detector studies.

-As options, we have implemented two freeze-out scenarios with coinciding and with different chemical and thermal freeze-outs.

-Also implemented are various options of the freeze-out hypersurface parameterizations

-The generator code is quite flexible and allows the user to add other scenarios and freeze-out surface parameterizations as well as additional hadron species in a simple manner.

-We have compared the RHIC experimental data with our MC generation results obtained within the single freeze-out scenario and Bjorken-like and Hubble-like freeze-out surface parameterizations.

http://uhkm.jinr.ruThe fast MC procedure is extendedto describe non-central collisions. Scenarios with single f.o. and thermal f.o. are implemented in the model

We started to study the possibility to add the QGSM inelastic cascade part to UKM elastic cascade.

SPHES is tested and very soonwill be accessible for users in our site.

Generation from the external filecontaining SPHES output is tested and very soon will be accessible in our site.

FASTMChydro + high-pt part related to the partonic states: We are studying influence of the mini-jet production on CFs at RHIC energies. Jet-quenching model PYTHIA/PYthiaQUENched is implemented.

FASTMS is implemented in AliRoot

Status and Perspectives

Additional slides

Preliminary:

High-p_t part related to the partonic states created in ultra-relativistic heavyPYTHYA+PYQUEN (I.P.Lokhtin and A.M.Snigirev, Eur. Phys. J. C 45, 211 (2006).)

Jet-quenching model PYQUEN/PYTHIA is implemented.

The fitted correlation radii and strength parameter are compared with those measured by STAR collaboration

The Bjorken-like model describes the decrease of the correlation radii with increasing k_t but overestimates their values.The situation is even worth with the Hubble-like model which ismore constraint than the Bjorken-like one and yields the longitudinal radius by a factor two larger.

Momentum correlations