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J.M. Burward-Hoy Lawrence Livermore National Laboratory for the PHENIX Collaboration
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  1. Source Parameters from Identified Hadron Spectra and HBT Radii for Au-Au Collisions at sNN = 200 GeV in PHENIX J.M. Burward-Hoy Lawrence Livermore National Laboratory for the PHENIX Collaboration Motivation • The objective is to measure the characteristics of the particle emitting source from both spectra and HBT radii simultaneously. • An interpretation of the data assuming relativistic hydrodynamic expansion is presented. • The study uses PHENIX Preliminary data as presented by T. Chujo and A. Enokizono.

  2. Detecting , K, p in PHENIX HBT AnalysisPHENIX Electromagnetic Calorimeter Single Kp SpectraPHENIX Time of Flight momentum resolution p/p ~ 1%  1% p TOF resolution 120 ps EMCal resolution 450 ps QM2002: Particle Yields I

  3. z A Simple Model for the Source • Model by Wiedemann, Scotto, and Heinz , Phys. Rev. C 53, 918 (1996) E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev. C 48, 2462 (1993) • Fluid elements each in local thermal equilibrium move in space-time with hydrodynamic expansion. • No temperature gradients • Boost invariance along collision axis z. • Infinite extent along rapidity y. • Cylindrical symmetry with radius r. • Particle emission • hyperbola of constant proper time 0 • Short emission duration • t < 1 fm/c r t z QM2002: Particle Yields I

  4. Transverse Kinetic Energy Spectra 5% Au-Au at sNN = 200 GeV • Kinetic energy spectra broaden with increasing mass, from  to K to p (they are not parallel). QM2002: Particle Yields I

  5. <pT> (GeV/c) Open symbols: sNN = 130 GeV Mean Transverse Momentum • Mean pt increases with Npart and m0, indicative of radial expansion. • Relative increase from peripheral to central greater for (anti)p than for , K. • Systematic uncertainties:  10%, K 15%, and (anti-)p 14% Npart QM2002: Particle Yields I

  6. 1/mt dN/dmt A Tfo t() mt  Reproducing the Shape of the Single Particle Spectra Radial position on freeze-out surface  = r/R Particle density distribution f() is independent of  Shape of spectra important t f()  parameters normalization A freeze-out temperature Tfo surface velocity T 1 Linear flow profile () = T <T > = 2T/3 S. Esumi, S. Chapman, H. van Hecke, and N. Xu, Phys. Rev. C 55, R2163 (1997) Minimize contributions from hard processes (mt-m0) < 1 GeV Exclude  resonance region pT < 0.5 GeV/c QM2002: Particle Yields I

  7. Fitting the Transverse Momentum Spectra • Simultaneous fit in range (mt -m0 ) < 1 GeV is shown. • The top 5 centralities are scaled for visual clarity. • Similar fits for negative particles. QM2002: Particle Yields I

  8. Fitting the Transverse Momentum Spectra • Simultaneous fit in range (mt -m0 ) < 1 GeV is shown. • The top 5 centralities are scaled for visual clarity. • Similar fits for positive particles. QM2002: Particle Yields I

  9. For All Centralities2 Contours in Parameter Space Tfo and T PHENIXPreliminary In each centrality, the first 20 n- contour levels are shown. From the most peripheral to the most central data, the single particle spectra are fit simultaneously for all pions, kaons, and protons. PHENIXPreliminary QM2002: Particle Yields I

  10. A close-up …Most Central and Most Peripheral For the 5% spectra T = 0.7 ± 0.2 syst. Tfo = (110  23 syst.) MeV For the most peripheral spectra: T = 0.46 ± 0.02 stat. 0.2 syst. Tfo = 135  3 stat.  23 syst. MeV QM2002: Particle Yields I

  11. The Parameters Tfo and T vs. Npart • Expansion parameters in each centrality • Overall systematic uncertainty is shown. • A trend with increasing Npart is observed: • Tfo and T • Saturates at mid-central QM2002: Particle Yields I

  12. f = 0 f = 0.3 f = 0.6 f = 0.9 Use analytical forms for the radii from Wiedemann, Scotto, and Heinz Expansion from the kT Dependence of HBT Radii • Ref:PRC 53 (No. 2), Feb. 1996 T = 150 MeV, R = 3 fm, 0 = 3 fm/c Ro (fm) RL (fm) Rs (fm) PHENIX Preliminary ++ Ro (fm) RL (fm) parameters geometric radius R freeze-out temperature Tfo flow rapidity at surface T freeze-out proper time 0 Rs (fm) QM2002: Particle Yields I

  13. Fitting the kT Dependence of HBT Radii • 2 contours in parameter space Tfo and T • The contours are not closed • In this region of parameter space, the minimum 2 value is found • These contours are the n- values relative to this minimum QM2002: Particle Yields I

  14. HBT Radii and Single Particle Spectra PHENIXPreliminary • Spectra and RL are consistent within 2.5 . • Ro and Rs disagree with the spectra. • Rs prefers high flow and low temperatures • T > 1.0 and Tfo < 50 MeV • Ro prefers • T > 1.4 and Tfo > 100 MeV • (R-contours not closed) QM2002: Particle Yields I

  15. From the spectra (systematic errors): T = 0.7 ± 0.2 syst. Tfo = 110  23 syst. MeV Using spectra information to constrain HBT fits… PHENIX Preliminary Rs (fm) Ro (fm) RL (fm) • 10% central positive pion HBT radii (similar result for negative pion data). • Systematic uncertainty in the data is 8.2% for Rs, 16.1% for Ro, 8.3% for RL. ++ R = 9.6±0.2 fm 0 = 132 fm/c QM2002: Particle Yields I

  16. From the spectra (systematic errors): T = 0.7 ± 0.2 syst. Tfo = 110  23 syst. MeV Using spectra information to constrain HBT fits… PHENIX Preliminary Rs (fm) Ro (fm) RL (fm) • 10% central negative pion HBT radii. • Systematic uncertainty in the data is 8.2% for Rs, 16.1% for Ro, 8.3% for RL. -- R = 9.7±0.2 fm 0 = 132 fm/c QM2002: Particle Yields I

  17. Conclusions For the most peripheral spectra: T = 0.5 0.2 syst. (< T> = 0.3 ± 0.2 syst. ) Tfo = 135  23 MeV • Expansion measured from spectra depends on Npart. • Saturates to constant for most central. • Used simple profiles for the expansion and particle density distribution • Linear velocity profile • Flat particle density • Within this hydro model, no common source parameters could be found for spectra and all HBT radii simultaneously. … to the 5% spectra T = 0.7 ± 0.2 syst. (< T> = 0.5 ± 0.2 syst.) Tfo = 110  23 MeV Rs prefers T > 1.0 and Tfo < 50 MeV Ro prefers T > 1.4 and Tfo > 100 MeV QM2002: Particle Yields I

  18. positive negative + - dN/dy/0.5Npart K+ K- p p Open symbols: sNN = 130 GeV Npart Particle Yields per Npart pair vs. Npart PHENIX Preliminary • Yield per pair Npart shown on a log scale for visual clarity only. • Linear dependence on Npart. • Relative increase from peripheral to central greater for K than for , (anti-)p. • Systematic uncertainties shown as lines. QM2002: Particle Yields I

  19. Year-1 Mean transverse momentum 20+/- 5 % increase • Mean pt with Npart , m0 radial flow • Relative increase from peripheral to central same for , K, (anti)p • (Anti)proton significant  from pp collisions 20+/- 5 % increase Open symbols: pp collisions QM2002: Particle Yields I

  20. 130 GeV Year-1:Fitting the Single Particle Spectra Simultaneous fit(mt -m0 ) < 1 GeV (see arrows) PHENIX Preliminary PHENIX Preliminary Exclude  resonances by fitting pt > 0.5 GeV/c The resonance region decreases T by ~20 MeV. This is no surprise! Sollfrank and Heinz also observed this in their study of S+S collisions at CERN energies. NA44 also had a lower pt cut-off for pions in Pb+Pb collisions. PHENIX Preliminary QM2002: Particle Yields I

  21. Year-1: Single Particle Spectra 130 GeV PHENIX Preliminary QM2002: Particle Yields I

  22. Year-1: An example of a fit to ++. . . PHENIX Preliminary 130 GeV The 2 is better for the higher order fit. . . QM2002: Particle Yields I

  23. HBT Radii and Single Particle Spectra • HBT radii suggest a lower temperature and higher flow velocity • Use best fit of singles and convert  to  • Singles and HBT radii are within 2 130 GeV Tfo PHENIX Preliminary f QM2002: Particle Yields I

  24. t() 20 fm/c 2 fm/c  5 fm/c From Hydrodynamics:Radial Flow Velocity Profiles Velocity profile at  ~ 20 fm/c “freeze-out” hypersurface At each “snapshot” in time during the expansion, there is a distribution of velocities that vary with the radial position r Plot courtesy of P. Kolb QM2002: Particle Yields I

  25. parameters normalization A freeze-out temperature Tfo surface velocity t 1/mt dN/dmt A Tfo t() mt f()   1 Hydrodynamics-based parameterization 1/mt dN/dmt = A  f()  d mT K1( mT /Tfo cosh  ) I0( pT /Tfo sinh  ) t integration variable   radius r = r/R definite integral from 0 to 1 particle density distribution f() ~ const linear velocity profile t() = t surf. velocity t ave. velocity <t > = 2/3 t boost () = atanh( t() ) minimize contributions from hard processes fit mt-m0<1 GeV QM2002: Particle Yields I

  26. Two Approaches to Calculating HBT Radii. . . (After assuming something about the source function. . .) A numerical approach is to • numerically determine C(K,q) from S(K,q) • C(K,q) ~ 1 + exp[ -qs2Rs2(K) – q02Ro2(K) – ql2Rl2(K)-2qlqoRlo2(K)] • there is an “exact” calculation of these radii (full integrations) • there are lower-order and higher-order approximations (from series expansion of Bessel functions). • The lowest-order form for Rs was used in Phenix PRL. (A similar expression is used by NA49). • The higher-order approximation is very good when compared to the exact calculation for Rs and RL. What I’m doing QM2002: Particle Yields I

  27. Important Assumptions Used. . . As is also assumed in calculating particle spectra Integration over  is done exactly. • Boost invariance (vL = z/t). Space-time rapidity equals flow rapidity • Infinitely long in y. • In LCMS, y and L = 0. • Integrals expressed in terms of the modified Bessel functions: For HBT radii, approximations are used in integration over x and y. • Saddle point integration using “approximate” saddle point • Series expansion of Bessel functions • Assume mT/T>1 QM2002: Particle Yields I

  28. Calculating the HBT Radii Linear flow rapidity profile Defined weight function Fn f = 0 Rs f = 0.3 f = 0.6 f = 0.9 T = 150 MeV, R = 3 fm, 0 = 3 fm/c parameters geometric radius R freeze-out temperature T flow rapidity at surface f freeze-out proper time 0 Constants are determined up to order 3 from Bessel function expansion QM2002: Particle Yields I

  29. The Analytical Evaluation of the HBT Radii Linear flow rapidity profile Up to order 3 in Bessel function expansion QM2002: Particle Yields I