Fits to Single Particle Spectra and HBT Radii

1. Fits to Single Particle Spectra and HBT Radii • J. Burward-Hoy • Lawrence Livermore National Laboratory • for the PHENIX Collaboration • Outline of Topics • Modeling the Source • Hydrodynamics-based parameterizations used • Calculating the Single Particle Spectra • Calculating the HBT radii • Results • Conclusions RHIC/INT Winter Workshop 2002

2. Modeling the Source • Interaction region Assembly of classical boson emitting sources in space-time region • The source S(x,p) is the probability boson with p is emitted from x Determines single-particle momentum spectrum E d3N/dp3 =  d4x S(x,p) Determines the HBT two-particle correlation function C(K,q) C(K,q) ~ 1 + |  d4x S(x,K) exp(iq·x) | 2/|  d4x S(x,K) |2 where K = ½(p1 + p2) = (KT, KL), q = p1 – p2 The LCMS frame is used (KL = 0) • In the hydrodynamics-based parameterizations: assume something about the source S(x,p) Gaussian particle density distribution Linear flow (rapidity or velocity) profile Instantaneous freeze-out at constant proper time (“sharp”) RHIC/INT Winter Workshop 2002

3. 1/mt dN/dmt A Tfo t() mt  Calculating the Single Particle Spectra 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() ~ exp(- 2/2) f() parameters normalization A freeze-out temperature Tfo surface velocity t  1 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 Ref: E. Schnedermann, J. Sollfrank, and U. Heinz, Phys. Rev. C 48, 2462 (1993) Ref: S. Esumi, S. Chapman, H. van Hecke, and N. Xu, Phys. Rev. C 55, R2163 (1997) RHIC/INT Winter Workshop 2002

4. Fitting the Single Particle Spectra Simultaneous fit(mt -m0 ) < 1 GeV (see arrows) PHENIX Preliminary 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. RHIC/INT Winter Workshop 2002

5. PHENIX Preliminary 118-126 MeV 0.71-0.73 5% Central Single Particle Spectra T = 1224 MeV t = 0.72 0.01 2/dof = 30.0/40.0 Note: For the 5-15% centrality (in S.C. Johnson’s talk), Tfo = 125 MeV and t = 0.69 RHIC/INT Winter Workshop 2002

6. Two Approaches to Calculating HBT Radii. . . Ref:PRC 53 (No. 2), Feb. 1996 (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)] An analytical way is to determine exact forms for the radii. • 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 is used in S.C. Johnson’s talk for Rs. (A similar one 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 RHIC/INT Winter Workshop 2002

7. 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 RHIC/INT Winter Workshop 2002

8. 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 RHIC/INT Winter Workshop 2002

9. An example of a fit to ++. . . PHENIX Preliminary The 2 is better for the higher order fit. . . RHIC/INT Winter Workshop 2002

10. 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 Tfo PHENIX Preliminary f RHIC/INT Winter Workshop 2002

11. Conclusions • Fully normalized, centrality selected , K, p/pbar spectra in 130 GeV Au-Au collisions are measured in PHENIX (Summer 2000 runs) • The data suggest radial flow at RHIC • A simple model that assumes hydrodynamic behavior is fit simultaneously to the 5% central data. • Good 2 fits for a finite range of anti-correlated parameters • Closed contours result • PHENIX HBT analysis results are within 2. • Using analytical calculation of radii to fit to data (higher order expansion which results in better 2 fits than the lowest order) • Contours are not closed in Tfo and f To Do: • Try to understand why the contours are not closed. • Repeat using numerical integration of HBT radii as a comparison to the analytical results. • Fit to Ro data as well . . . RHIC/INT Winter Workshop 2002

12. r z  ~ q/p y r 0 x Detecting , K, p in PHENIX DC main bend plane DC resolution p/p ~ 0.6%  3.6% p pt = p sin(0) TOF resolution 115 ps PC1 and Event vertex polar angle RHIC/INT Winter Workshop 2002

13. The Analytical Evaluation of the HBT Radii Linear flow rapidity profile Up to order 3 in Bessel function expansion RHIC/INT Winter Workshop 2002

14. f = 0 f = 0.3 f = 0.6 f = 0.9 An example: the Calculated Radii An example using: T = 150 MeV R = 3 fm 0 = 3 fm/c with different expansion velocities, f Rs RL RHIC/INT Winter Workshop 2002