Pion correlations in hydro-inspired models with resonances
description
Transcript of Pion correlations in hydro-inspired models with resonances
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
)()(,
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1)4(
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pc
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xm
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pd
xm
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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