1. Imaging in Heavy-Ion Collisions D.A. Brown, A. Enokizono, M. Heffner, R. Soltz (LLNL) P. Danielewicz, S. Pratt (MSU) RHIC/AGS Users Meeting 21 June 2005 This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. Talk Outline: Background Nontrivial test problem Inverting a 3d correlation Implications for HBT Puzzle CorAL Status

2. The Koonin-Pratt Formalism We will invert to get S(r) directly The pair’s final state wave- function, we can compute this if we know the interaction. This is the kernel K(q,r). The source function, it gives the probability of producing a pair a distance r apart in the pair CM frame Source function related to emission function: We work in Bertsch-Pratt coords., in pCM

3. 3d HBT at RHIC Boost may have interesting observable consequence (Rischke, Gyulassy Nucl. Phys. A 608, 479 (1996)): If there is a phase-transition, hydro evolution will slow in mixed phase. Will lead to long-lived source Huge difference in Outward/Sideward radii R out /R side == 1! (in LCMS) Experiment analysis wrong? Coulomb correction? Theory screwed up? Interpretation confused? Instant freeze-out? Gaussian fits?

4. The Test Problem, part 1 T = 50 MeV -- source temperature R x = R y = R z = 4 fm  = 20 fm/c, ballpark of  lifetime (23 fm/c) Emission function is Gaussian with finite lifetime, in lab frame: (lab  pair CM boost) +  == non-Gaussian source!

5. The Test Problem, part 2 Combine with  +  + kernel to get correlation using Scott Pratt’s CRAB code

6. Breaking Problem into 1d Problems Where Expand in Ylm’s and Legendre polynomials:

7. Breaking Up Problem, cont.

8. Imaging Summary  Use CorAL version 0.3 * Source radial terms written in Basis Spline representation – knots set using Sampling Theorem from Fourier Theory – use 3rd degree splines * Use full Coulomb wavefunction, symmetrized * Use generalized least-square for inversion * Use constraints to stabilize inversion * Cut off at finite l, q in input correlation

9. Comparison to 1d Source Fits Fits & imaging extract same core parameters: R inv = 5.38 +/- 0.13 fm = 0.647 +/- 0.029 Same 30% undershoot for fits and images Gaussian fits can’t get halo CorAL Gaussian fit == “best” Coulomb correction

10. 3d Results Nail radii Height low by 30% Nail integral of source Resolve halo in L & O directions Measured Gaussian parameters (from source!): R S : 3.42 +/- 0.21 R O : 7.02 +/- 0.25 R L : 5.04 +/- 0.31 S(r=0): 10 +/- 1x10-5 : 0.442 +/- 0.067 Can extract the important physics!

11. Core Halo Model Nickerson, Csörgo˝, Kiang, Phys. Rev. C 57, 3251 (1998).      f is fraction of  ’s emitted directly from core, effectively  = f 2 in source function

12. Core Halo Model, cont. From exploding core, with Gaussian shape: Flow profile simple, but adjustable: From decay of emitted resonances,  has most likely lifetime (23 fm/c)

13. Simple Model, Interesting Results Set R x =R y =R z =4 fm,  f/o =10 fm/c, T=175 MeV, f=0.56 High velocity  ’s get bigger boost. Boost + finite   tail, but only modest core increase, in L,O directions.

14. Focus on Outwards Direction Lifetime effects have dramatic affect on non-Gaussian halo in outwards direction.

15. Focus on Outwards Direction … but only modest core changes. Gaussian R O fits: Baseline:5.63 Instant f/o:4.78 No  :5.35 No lifetime:4.48  max =0.35:6.51 radii in fm, fit w/ r<15 fm, fit tolerance 1%

16. Focus on Sidewards Direction Minor changes deep in tail reflecting geometry of  induced halo. All curves have core radii ~ 4 fm.

17. Focus on Longitudinal Direction Rapidity cut eliminates tail, otherwise similar to outwards direction.

18. Take Home Messages from Model * Need emission duration + particle motion to get lifetime effect * Finite lifetime can create tail, w/o changing core radii substantially * RHIC HBT puzzle could be Gaussian fit missing tail from long lifetime * Resonance effects detectable, but similar to freeze-out duration effects * Don’t be fooled by acceptance effects Since source tails hide in Coulomb hole of correlation, imaging RHIC data should shed light on puzzle

19. CorAL Features * Variety of kernels: – Coulomb, NN interactions, asymptotic forms – Any combination of p, n,  , K +-, , plus some exotic pairs * Fit a 1d or 3d correlation w/ variety of Gaussian sources * Directly image a correlation, in 1d or 3d * Build model correlations/sources from – OSCAR formatted output, – Blast Wave, – variety of simple models * Build model correlations/sources in Spherical or Cartesian harmonics

20. CorAL Status * Merging of three related projects: – original CorAL by M. Heffner, fitting correlations w/ source convoluted w/ full kernel – HBTprogs in 3d by D. Brown, P. Danielewicz, imaging sources from correlation data w/ full kernels – CRAB (++) by S. Pratt, use models to generate correlations, sources w/ full kernels * Rewritten in C++, open source, nearly stand alone (depends on GSL only) * Developed on MacOS X, linux, cygwin * Time scale for 1.0 release: end of summerish (sorry, probably not in time for QM2005)

21. Extra Slides

22. Low Relative Momentum Correlations Signal, pairs from same event (these guys are entangled) Background, pairs from different event, same cuts (looks like signal, but no entanglement) When C=1, there is no correlation Removes features in spectra not caused by entangled particles A versatile observable: can use nearly any type of pair

23. 3d HBT at RHIC Work in Bertsch-Pratt coordinates in pair CM frame: Boost from lab  pair CM means lifetime effects transformed into Outwards/Longitudinal direction. Long (beam) Out Side

24. What do the terms mean? l = 0 : Angle averaged correlation, get access to R inv l = 1 : Access to Lednicky offset, i.e. who emitted first (unlike only) l = 2 : Shape information: access to R O, R S, R L: C 20  R L  C 00 -(C 20 ±C 22 )  R S,R O l = 3 : Boomerang/triaxial deformation (unlike only) l = 4 : Squares off shape

25. Sampling Theorem If we can ignore FSI, the 1-dimensional kernels are: The l = 0 term is just a Fourier Transform. Sampling Theorem says for a given binning in q-space, source uniquely determined at certain r points (termed collocation points) spaced by: Sampling Theorem may be generalized to the l > 0 terms giving Here the  l n are the zeros of the Spherical Bessel function.

26. To reconstruct the source at these collocation points, choose knots: In general the spin-zero like pair meson kernel is much more complicated than this: and these knots are only an approximation to the best knots. Setting knots

27. Representing terms in source We use basis splines to represent the radial dependence of each source term: Basis splines are piece-wise continuous polynomials: Setting knots for the splines is crucial for representing source well 0 th -order 1 st -order 2 nd -order …

28. Representing the Source Function Radial dependence of each term in terms of Basis Splines: Generalized Splines: – N b =0 is histogram, – N b =1 is linear interpolation, – N b =3 is equivalent to cubic interpolation. Recursion relations == fast to evaluate, differentiate, integrate Resolution controlled by knot placement R Form basis for R 2, but not orthonormal

29. The inversion process Koonin-Pratt eq. in matrix form: kernel not square kernel possibly singular noisy data error propagation this is an ill-posed problem Practical solution to linear inverse problem, minimize : Most probable source is: With covariance matrix:

30. Constraints for 3d problem(s) Inversion can be stabilized with constraints. Constraints we use:

31. Outwards/Sidewards Ratio Gaussian fits driven by r < 10 fm where ratio close to unity.

32. Simple Model, Interesting Results Turn off resonance production (f=1) L,O tail reduced somewhat, S tail gone. Core lifetime causes tail same size as resonances.

33. Simple Model, Interesting Results Instant freeze-out (  f/o =0 fm/c) Kills core-induced non-Gaussian tail, but resonance induced tail of similar magnitude.

34. Simple Model, Interesting Results No resonances (f=1), instant freeze-out (  f/o =0 fm/c) Essentially kills non-Gaussian tail. Obviously need finite time effect for tail.

35. Simple Model, Interesting Results Apply PHENIX acceptance cut of  max =0.35 Kills contribution from high p L pairs, so kills core’s tail in L direction

36. Simple model, interesting results No flow Lowers source, but not much else

37. Simple model, interesting results Small temperature NO PLOT YET, HARD TO SAMPLE FROM SMALL TEMPERATURES

38. Simple model, interesting results No flow, small temperature NO PLOT YET, HARD TO SAMPLE FROM SMALL TEMPERATURES