Pion correlations in hydro-inspired models with resonances

1. 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, Warsaw 2) Akademia Świętokrzyska, Kielce 3) Institute of Nuclear Physics, Polish Academy of Sciences, Cracow

2. 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-out

3. 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

4. 2. Freeze-out hypersurface and flow Cracow single-freeze-out model 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) 2 geometric parameters:τ, ρmax (generalized) blast-wave model all these forms describe well the transverse-momentum spectra !!! 3 geometric parameters:τ, a, ρmax

5. Cracow Blast-wave a=0.5 Blast-wave a=0.0 Blast-wave a=-0.5

6. 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. Rafelski Comput. Phys. Comm. 167 (2005) 229

7. the complete emission function is obtained as the sum over all possible decay channels splitting functions in momentum freeze-out hypersurface thermal distribution of primordial particles THERMINATOR generates events, sets of particles with the spacetime and momentum distributions described by the emission function S(x,p)

8. 4.1 Correlation Function – Basic Definitions one-particle and two-particle pion distributions the measured correlation function model assumptions relate the correlation function to the emission function squared wave function of a pair

9. 4.2 Monte-Carlo Method average momentum of the pair momentum difference by definition of the Monte-Carlo method, the integration is replaced by the summation over particles or pairs of particles in the numerical calculations Δ = 5 MeV

10. 4.3 Reference Frames 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 generalized momentum difference then one is able to calculate the wave function also in PRF ! the correlation function is a histogram of the squares of the wave function calculated for each pair in PRF but tabulated in LCMS !

11. 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 2) The Coulomb interaction is included

12. 4.5 Fitting procedure 1) if the simple wave function is used, the 3D correlation function is fitted with the standard gaussian formula 2) when the Coulomb wave function is used, the 3D correlation function is fitted with the Bowler-Sinyukov formula here KCoul is the squared Coulomb wave function integrated over a static gaussian source

13. 5. Results legend for the next plot: 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

14. 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)

15. All pions Primordial pions Points: projections of 3D CF Lines: projections of 3D fit |qx|<5 MeV |qx|<10 MeV |qx|<30 MeV 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

16. 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

17. all pions primordial pions separation distributions of pion pairs, blast-wave model with resonances, a=-0.5 the lines show the separation distributions which are the result of the fitting of the corresponding correlations function by a gaussian parameterization [CgaussSgauss  pair distr.] the effect of the resonances is visible in long-range tails

18. ω other ρ primordial resonance vivisection of the previous plot for all pions • the pions are divided into four groups: • those coming from the decays of ρ • those coming from the decays of ω • other, coming from the decays of other resonances than ρ or ω • primordial (primary) in all three directions we observe long-tails, „other” resonances give similar effects as the rho meson long tails in r give peaks for small values of q, this effect leads to lowering of the intercept

19. 6. Conclusions • 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 • our approach is as close as possible to the experimental treatment of the correlations (two-particle method, Coulomb included) • 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) • future: connection to the advanced hydro evolution, Chojnacki’s talk