Imaging in Heavy-Ion Collisions

1
Imaging in Heavy-Ion Collisions

• Talk Outline
• Background
• Nontrivial test problem
• Inverting a 3d correlation
• Implications for HBT Puzzle
• CorAL Status

• 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.
2
The Koonin-Pratt Formalism
We will invert to get S(r) directly
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

4
The Test Problem, part 1
Emission function is Gaussian with finite
lifetime, in lab frame

• T 50 MeV -- source temperature
• Rx Ry Rz 4 fm
• ? 20 fm/c, ballpark of ? lifetime (23 fm/c)

5
The Test Problem, part 2
Combine with ?? kernel to get correlation using
Scott Pratts CRAB code
6
Breaking Problem into 1d Problems
Expand in Ylms and Legendre polynomials
Where
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
• Rinv 5.38 /- 0.13 fm
• ?? 0.647 /- 0.029

• Same 30 undershoot for fits and images
• Gaussian fits cant 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!)
RS 3.42 /- 0.21 RO 7.02 /- 0.25 RL
5.04 /- 0.31 S(r0) 10 /- 1x10-5 ?
0.442 /- 0.067
11
Core Halo Model
Nickerson, Csörgo?, Kiang, Phys. Rev. C 57, 3251
(1998).
f is fraction of ps emitted directly from core,
effectively l f2 in source function
12
Core Halo Model, cont.

• From exploding core, with Gaussian shape
• Flow profile simple, but adjustable
• From decay of emitted resonances, w has most
  likely lifetime (23 fm/c)

13
Simple Model, Interesting Results
Set RxRyRz4 fm, tf/o10 fm/c, T175 MeV, f0.56
High velocity ps get bigger boost. Boost
finite t ? tail, but only modest core increase,
in L,O directions.
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Focus on Outwards Direction
Lifetime effects have dramatic affect on
non-Gaussian halo in outwards direction.
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Focus on Outwards Direction
Gaussian RO fits Baseline 5.63 Instant
f/o 4.78 No w 5.35 No lifetime 4.48 hmax0.35
6.51 radii in fm, fit w/ rlt15 fm, fit
tolerance 1
but only modest core changes.
16
Focus on Sidewards Direction
Minor changes deep in tail reflecting geometry
of w induced halo. All curves have core radii
4 fm.
17
Focus on Longitudinal Direction
Rapidity cut eliminates tail, otherwise similar
to outwards direction.
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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
• Dont 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, p-0, K-, L, 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)

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Extra Slides
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Low Relative Momentum Correlations
Background, pairs from different event, same
cuts (looks like signal, but no entanglement)
Signal, pairs from same event (these guys are
entangled)

• When C1, 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.
24
What do the terms mean?

• l 0 Angle averaged
• correlation, get access to Rinv
• l 1 Access to Lednicky offset,
• i.e. who emitted first (unlike only)
• l 2 Shape information
• access to RO, RS, RL
• C20 ? RL
• ?C00-(C20C22) ? RS ,RO
• 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 gt 0 terms giving Here the ?ln are the
zeros of the Spherical Bessel function.
26
Setting knots
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.
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
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Representing the Source Function
Radial dependence of each term in terms of Basis
Splines
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The inversion process
Koonin-Pratt eq. in matrix form
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Constraints for 3d problem(s)
Inversion can be stabilized with constraints.
Constraints we use
31
Outwards/Sidewards Ratio
Gaussian fits driven by r lt 10 fm where ratio
close to unity.
32
Simple Model, Interesting Results
Turn off resonance production (f1)
L,O tail reduced somewhat, S tail gone. Core
lifetime causes tail same size as resonances.
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Simple Model, Interesting Results
Instant freeze-out (tf/o0 fm/c)
Kills core-induced non-Gaussian tail, but
resonance induced tail of similar magnitude.
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Simple Model, Interesting Results
No resonances (f1), instant freeze-out (tf/o0
fm/c)
Essentially kills non-Gaussian tail. Obviously
need finite time effect for tail.
35
Simple Model, Interesting Results
Apply PHENIX acceptance cut of hmax0.35
Kills contribution from high pL pairs, so kills
cores 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
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Simple model, interesting results
No flow, small temperature
NO PLOT YET, HARD TO SAMPLE FROM SMALL
TEMPERATURES