Pion correlations in hydro-inspired models with resonances

A. Kisiel1, W. Florkowski2,3, W. Broniowski2,3, J. Pluta1

(based on nucl-th/0602039, to be published in PRC)

1) Warsaw University of Technology, Warsaw2) Akademia Świętokrzyska, Kielce3) Institute of Nuclear Physics, Polish Academy of Sciences, Cracow

1. Hydro-inspired models

the measured particle spectra and correlations reflect properties of matter at the stage when particles stop to interact, this moment is called the kinetic (thermal) freeze-out

hydro-inspired models use concepts borrowed from relativistic hydrodynamics but they do not include the complete time evolution of the system, they help us to verify the idea that matter, just before the kinetic freeze-out is locally thermalized and exhibits collective behavior, the observables are expressed in terms of thermal (Bose-Einstein, Fermi-Dirac) distributions convoluted with the collective expansion

freeze-outPC M & clust. hadronization

N FD

N FD & hadronic TM

PC M & hadronic TM

C YM & LG T

string & hadronic TM

we assume one universal freeze-out for all processes (inealstic and elastic processes cease at the same time, also emission of strange and ordinary hadrons happens at the same moment)

simplifying but very fruitful assumption, gives good description of particle yields, transverse-momentum spectra, pion invariant-mass distributions, balance functions, azimuthal asymmetry v2 series of papers by: W. Broniowski, WF, B. Hiller, P. Bożek, A. Baran,

D. Prorok talk tomorrow evening

consistent with sudden hadronization (explosion) scenario at RHIC, J.Rafelski and J.Letessier, PRL 85 (2000) 4695

in the single-freeze-out model the thermodynamic parameters, such as temperature T and baryon chemical potential μB, are obtained from the analysis of the hadron abundances (ratios of the multiplicities)

in this talk the results obtained with the Monte-Carlo version of the single-freeze-out model are presented

THERMINATOR (THERMal heavy-IoN generATOR), A. Kisiel, T. Tałuć, W. Broniowski, and WF Comp. Phys. Comm. 174 (2006) 669

Cracow single-freeze-out model

const,2222 zrt

zyx rtrr 2222 ~,

(generalized) blast-wave model

const,,222 aart z

2. Freeze-out hypersurface and flow

const,,~

sin,~

cos

v

t

r

tv

tvv z

for boost-invariant and cylindrically symmetric models the freeze-out hypersurface is defined by the freeze-out curve in Minkowski space t - ρ (rz = 0)

t

r

t

r

t

r

t

rv zyx ,,

all these forms describe well the transverse-momentum spectra !!!

2 geometric parameters: τ, ρmax

3 geometric parameters: τ, a, ρmax

Cracow Blast-wavea=0.5

Blast-wavea=0.0

Blast-wavea=-0.5

3. Emission function

in our calculations all well established resonances are taken into account,381 particle types with 1872 different decay modes are included

the Cracow and blast-wave models are treated on the same footing, the only important difference resides in the definition of the freeze-out hypersurface

THERMINATOR uses the same input as SHARE, G. Torrieri, S. Steinke, W. Broniowski, WF, J. Letessier, J. RafelskiComput. Phys. Comm. 167 (2005) 229

)()(,

,

...,),(

1)4(

1

3

21

111

)4(1

41121

1

13

12

222

)4(2

42212

23

11

11

22

2

NNNNN

NNNNNNNNN

N

N

NN

NNNNNNNN

N

N

pc

xupfxm

pxpxdedppB

E

pd

xm

pxxdedppB

E

pd

xm

pxxdedppB

E

pdpxS

NN

NN

thermal distribution of primordial particles

freeze-out hypersurfacesplitting functions inmomentum

the complete emission function is obtained as the sum over all possible decay channels

c

c pxSpxS ),(),(

THERMINATOR generates events, sets of particles with the spacetime and momentum distributions described by the emission function S(x,p)

4.1 Correlation Function – Basic Definitions

one-particle and two-particle pion distributions

2

31

321231 21,,

pdpd

dNEEppW

pd

dNEpW ppp

the measured correlation function 2111

21221

,,

pWpW

ppWppC

model assumptions relate the correlation function to the emission function

2224

1114

2**222

4111

4

,,

,,,,

pxSxdpxSxd

rkpxSxdpxSxdkqC

2**, rk squared wave function of a pair

4.2 Monte-Carlo Method

210 ,,21

ppEEqqq pp

210 ,2

1,

21ppEEkkk pp

average momentum of the pair

momentum difference

i jjiji

i ijjiji

ppkppq

rkppkppq

kqC

21

,21

,

2**

otherwise02

||,2

||,2

||if1 zyx pppp

by definition of the Monte-Carlo method, the integration is replacedby the summation over particles or pairs of particles

in the numerical calculations Δ = 5 MeV

4.3 Reference Frames

*pairaofframerestthetoboostfurther

long

longoutsideaxis)-z (around rotation direction)-z(in boost

21 0,

,,,, q

kk

qqqkqpp

for each pair the following transformations are made:

i) from the laboratory frame to the longitudinal co-moving system (LCMS), using the Bertsch-Pratt decomposition, and subsequently ii) from LCMS to the pair rest frame (PRF)

in the pair-rest frame we calculate the relative distance and the generalizedmomentum difference **, kr

PRFin ,0~,~ *2

kqk

kqkqq

then one is able to calculate the wave functionalso in PRF !

the correlation function is a histogram of the squares of the wave functioncalculated for each pair in PRF but tabulated in LCMS !

4.4 Wave Functions

we consider two options for the wave function:

1) The simplest wave function is taken into account which includes symmetrization over the two identical pions but neglects all dynamical interactions

**22cos1,

2

1 ****

rkee QrkirkiQ

2) The Coulomb interaction is included

iiFeiiFeAe rkirkic

iQC c ,1,,1,2

1)(

****

function trichypergeome radius,Bohr ,

factor Gamow shift, phase Coulomb

1*

****

Faak

rkrk

Acc

4.5 Fitting procedure

long2

long2

side2

side2

out2

out2 )()()(exp1 qkRqkRqkRC

1) if the simple wave function is used, the 3D correlation function is fittedwith the standard gaussian formula

2) when the Coulomb wave function is used, the 3D correlation function is fitted with the Bowler-Sinyukov formula

long2

long2

side2

side2

out2

out2

Coul )()()(exp1 qkRqkRqkRKC

here KCoul is the squared Coulomb wave function integrated over a static gaussian source

5. Results

resonances NOT included, only primordial pions, simple wave function, gaussian fit

resonances included, simple wave function, gaussian fit

resonances included, Coulomb wave function, Bowler-Sinyukov fit

STAR experimental data

pions from weak decays included

legend for the next plot:

Cracow a=0.5

a=0.0 a=-0.5

decays of resonances increase the radii by about 1 fm (no van der Waals corrections)

Allpions

Primordialpions

|qx|<5 MeV

|qx|<10 MeV

|qx|<30 MeV

Points:projectionsof 3D CF

Lines:projectionsof 3D fit

projections of the pion correlation function for the blast-wave model with resonances, a = - 0.5

simple wave function is used and the results are fitted with a standard gaussian formula

0.25 GeV < kT < 0.35 GeV

the projections of the correlation function (symbols) and the projections of the 3D fit (lines) are compared

deviations between the function and the fit reflect the fact that the underlying two-particle distributions are not gaussian

projections lower the intercept

again projections of the pion correlation function for the blast-wave model with resonances, a = - 0.5

but now the Coulomb wave function is used and the results are fitted with the Bowler-Sinyukov formula

0.25 GeV < kT < 0.35 GeV

the projections of the correlation function (symbols) and the projections of the 3D fit (lines) are compared

Coulomb interactions dig holes at low values of q, the Bowler-Sinyukov formula works very well!

concepts to extract the properties of the correlation function from its behavior at q=0 are useless

separation distributions of pion pairs, blast-wave model with resonances, a=-0.5

primordial pions all pions

the lines show the separation distributions which are the result of the fitting of the corresponding correlations function by a gaussian parameterization[CgaussSgauss pair distr.]

the effect of the resonances is visible in long-range tails

ρ

ω

primordial

other the pions are divided into four groups:

1) those coming from the decays of ρ2) those coming from the decays of ω3) other, coming from the decays of

other resonances than ρ or ω

4) primordial (primary)

in all three directions we observe long-tails, „other” resonances give similar effects as the rho meson

resonance vivisection of the previous plot for all pions

long tails in r give peaks for small values of q, this effect leads to lowering of the intercept

6. Conclusions

1) simulatanoues description of the transverse-momentum spectra and the correlation radii is possible in the hydro-inspired models – special choice of the freeze-out hypersurface must be made

2) our approach is as close as possible to the experimental treatment of the correlations (two-particle method, Coulomb included)

3) the role of the resonances is analyzed in detail, some earlier expectations were confirmed (decrease of intercept, the role of omega meson), some not (increase of the radii due to the strong decays of resonances)

4) future: connection to the advanced hydro evolution, Chojnacki’s talk